MATHEMATICAL MODEL OF THE DYNAMICS OF ...

MATHEMATICAL MODEL OF THE DYNAMICS OF PSYCHOTHERAPY by Michael Douglas Norman

A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Florida Atlantic University Boca Raton, FL December 2012

Copyright by Michael Douglas Norman 2012

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VITA Michael Douglas Norman was born on February 9th, 1982 in Bethesda, Maryland, the son of Janice Barbara Norman and Douglas Owen Norman. After graduating high school at Westford Academy in Westford, Massachusetts (2000), he attended the University of Massachusetts at Amherst, where he earned his Bachelor of Science in Computer Systems Engineering (2005). He then began his career as an engineer for the Bose Corporation. Strongly influenced by his father’s military-based research into complexity, he applied for and was awarded a scholarship to attend the New England Complex Systems Institute’s Summer School (2006). This led him to pursue a higher degree in the science of complexity; in the Fall of 2009, he entered the Complex Systems and Brain Sciences Ph.D. program at Florida Atlantic University. While there, he taught Complex Systems, Astronomy, Chemistry and Psychology. He has been independently supported by multiple prestigious fellowships from federal, state and private sources. In his free time, Michael enjoys snowboarding, motor sports, music production, science fiction and taking his dogs places.

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ACKNOWLEDGEMENTS This work would not have been possible without the support of Dr. Janet Blanks. I would like to acknowledge Dr. Larry Liebovitch, Dr. Paul Peluso, Dr. John Gottman, Dr. Viktor Jirsa, Dr. Urszula Strawinska-Zanko, Dr. Silke Dodel and Florida Atlantic University’s Center for Complex Systems and Brain Sciences, all of whom played critical roles in this work. This material is based upon work supported by the National Science Foundation under Grant No. 0638662.

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ABSTRACT Author:

Michael Douglas Norman

Title:

Mathematical Model of the Dynamics of Psychotherapy

Institution:

Florida Atlantic University

Dissertation Advisor: Dr. Janet C. Blanks Degree:

Doctor of Philosophy

Year:

2012 This is a novel attempt to produce a rigorous mathematical model of a complex

system. The complex system under study is the relationship between therapists and their clients. The success of psychotherapy depends on the nature of the relationship between a therapist and a client. We use dynamical systems theory to model the dynamics of the emotional interaction between a therapist and client. We determine how the therapeutic endpoint and the dynamics of getting there depend on the parameters of the model. Previously Gottman et al. [26] used a very similar approach (physical-sciences paradigm) for modeling and making predictions about husband-wife relationships. They modeled interactions using difference equations and then compared the behavior of those equations to the experimental affect data coded from video of married couples in a 15-minute discussion. The parameters they determined in this way had high predictive value of whether the marriages were stable and also gave new insights into the dynamics of how couples interact. Since that novel approach shed light on the

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dyadic interaction between couples we thought that it also had the possibility to give us new insights into the relationship between therapist and client. We describe the emotional state of both therapist and client with coupled, first order, nonlinear ordinary differential equations (ODE’s). The rate of change of the emotional state of the therapist and client is proportional to their previous state, their uninfluenced state when alone, and an influence function which depends on the state of the other person. We formulated influence functions based on the research literature on psychotherapy and the therapeutic alliance. We then determined the critical points from the intersection of the nullclines and used a numerical ODE solver (Matlab ODE113) to compute the trajectories from different initial conditions. To empirically validate this approach, 73 unique therapy sessions were videorecorded. Four of these interactions (chosen by our psychotherapy expert) were selected to be modeled and were coded using Gottman’s Specific Affect Coding System. The results validate this prototypical approach to psychotherapy; we have shown that human interaction (in the context of psychotherapy) can be quantified and modeled using differential equations.

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DEDICATION To my loving family for all their support: Ally, Mom, Dad, Joe, Nibbler, Chewy and Yogi – I could not have done this without you.

MATHEMATICAL MODEL OF THE DYNAMICS OF PSYCHOTHERAPY

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

Psychotherapy’s Mixed Success . . . . . . . . . . . . . . . . . . . . .

1

1.2

The Importance of Relationships . . . . . . . . . . . . . . . . . . . .

1

1.2.1

The Therapeutic Relationship . . . . . . . . . . . . . . . . . .

2

1.2.2

The Real Relationship . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Relationships as Complex Systems . . . . . . . . . . . . . . . . . . .

5

2 Complex Systems and Modeling . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1

The Importance of Perspective . . . . . . . . . . . . . . . . . . . . . .

7

2.2

The Pitfalls of Reductionism . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4

2.3.1

Nonlinearities in Social Systems . . . . . . . . . . . . . . . . .

11

2.3.2

Nonlinearities in Psychotherapy . . . . . . . . . . . . . . . . .

12

2.3.3

Self-Organization and Humans . . . . . . . . . . . . . . . . . .

13

2.3.4

Agency and Consciousness . . . . . . . . . . . . . . . . . . . .

14

2.3.5

Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

The Power of Interdisciplinary Science . . . . . . . . . . . . . . . . .

15

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2.4.1

Mathematical Modeling of Relationships . . . . . . . . . . . .

16

Psychotherapy and Recovery as a Nonlinear Dynamic Process . . . .

16

2.5.1

Emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.5.2

Extending Gottman’s Approach . . . . . . . . . . . . . . . . .

17

Dynamical Systems Theory and Two-Dimensional Analysis of Nonlinear Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.6.1

Dynamical Systems Theory . . . . . . . . . . . . . . . . . . .

18

2.6.2

Modeling with Differential Equations . . . . . . . . . . . . . .

21

The Concepts of Influence and Inertia . . . . . . . . . . . . . . . . . .

22

2.7.1

Influence and Inertia in a Dynamic Physical System . . . . . .

22

2.7.2

Influence and Inertia in a Dynamic Social System . . . . . . .

23

Modeling Dyadic Interaction and Dynamics . . . . . . . . . . . . . .

23

2.8.1

The Marriage Model . . . . . . . . . . . . . . . . . . . . . . .

23

2.8.2

The Competition-Cooperation Model . . . . . . . . . . . . . .

24

The Psychotherapy Model . . . . . . . . . . . . . . . . . . . . . . . .

26

2.9.1

Parameters and Equations of the Model . . . . . . . . . . . .

26

2.9.2

Influence Functions . . . . . . . . . . . . . . . . . . . . . . . .

27

2.9.3

Analysis and Solution

. . . . . . . . . . . . . . . . . . . . . .

30

2.9.4

Phase Portraits . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.5

2.6

2.7

2.8

2.9

3.1

3.2

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.1.1

Experimental Overview . . . . . . . . . . . . . . . . . . . . . .

39

3.1.2

Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.1.3

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2.1

41

Specific Affect Coding System (SPAFF) . . . . . . . . . . . .

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3.2.2

Selecting the Data to Model . . . . . . . . . . . . . . . . . . .

42

3.2.3

The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4 Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.3

4.1

Data Transformation: Discretizing, Weighting and Summing the Affect 47

4.2

Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.2.1

Threshold Autoregressive Bilinear Model . . . . . . . . . . . .

48

4.2.2

Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2.3

Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2.4

Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.2.5

Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

5 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.1

5.2

5.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.1.1

Nullclines and Phase Portraits . . . . . . . . . . . . . . . . . .

67

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.2.1

Empirical Phase Portraits . . . . . . . . . . . . . . . . . . . .

70

5.2.2

Variant Phase Portraits . . . . . . . . . . . . . . . . . . . . .

73

5.2.3

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.3.1

Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.3.2

Master Practitioners . . . . . . . . . . . . . . . . . . . . . . .

94

5.3.3

Trilinear Modeling . . . . . . . . . . . . . . . . . . . . . . . .

94

5.3.4

Instrumentation Leveraging . . . . . . . . . . . . . . . . . . .

95

5.3.5

Inertial Functions . . . . . . . . . . . . . . . . . . . . . . . . .

95

5.3.6

Question the Assumptions . . . . . . . . . . . . . . . . . . . .

95

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5.3.7 5.4

Other Dyads . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

The Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

A Appendix: Psychotherapy Research Publication . . . . . . . . . . . . . . . 107 B Appendix: Cognitive Neurodynamics Publication . . . . . . . . . . . . . . 124 C Appendix: Dynamical Systems Theory . . . . . . . . . . . . . . . . . . . . 136 C.1 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 C.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 D Appendix: Instructions to Participants . . . . . . . . . . . . . . . . . . . . 141 E Appendix: RRI and WAI Surveys . . . . . . . . . . . . . . . . . . . . . . . 142 F Appendix: Facial Action Coding System . . . . . . . . . . . . . . . . . . . 148 G Appendix: Permissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Bibliography

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

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LIST OF TABLES

3.1

SPAFF Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.2

RRI and WAI Summary Scores . . . . . . . . . . . . . . . . . . . . .

45

4.1

Discretized Noldus Data: Case 1 . . . . . . . . . . . . . . . . . . . . .

53

4.2

Weighted and Summed SPAFF: Case 1 . . . . . . . . . . . . . . . . .

54

5.1

Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.2

Case 1 Model Variants . . . . . . . . . . . . . . . . . . . . . . . . . .

76

F.1 FACS Code Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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LIST OF FIGURES

2.1

Trajectory of a Pendulum . . . . . . . . . . . . . . . . . . . . . . . .

32

2.2

IHW (Ht ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.3

Cooperation-Competition Influence Functions . . . . . . . . . . . . .

34

2.4

FC (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.5

FT (T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.6

FC (C) and FT (T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.7

Theoretical Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . .

38

3.1

Noldus Therapist Data . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.2

Noldus Client Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.1

Time Series: Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.2

Time Series: Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.3

Time Series: Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.4

Time Series: Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.5

FC (C): Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.6

FT (T ): Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.7

FC (C): Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.8

FT (T ): Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.9

FC (C): Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.10 FT (T ): Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.11 FC (C): Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

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4.12 FT (T ): Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

5.1

Nullclines: Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.2

Nullclines: Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.3

Nullclines: Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.4

Nullclines: Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.5

Phase Portrait: Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.6

Vector Field: Case 1, Attractor 1 . . . . . . . . . . . . . . . . . . . .

82

5.7

Vector Field: Case 1, Attractor 2 . . . . . . . . . . . . . . . . . . . .

83

5.8


84

5.9


85

5.10 Phase Portrait: Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.11 Phase Diagram: Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . .

87


88


89


90

5.15 Phase Portrait: Variant 1

. . . . . . . . . . . . . . . . . . . . . . . .

97


. . . . . . . . . . . . . . . . . . . . . . . .

98

5.17 Phase Portrait: Variant 2 with Overlay . . . . . . . . . . . . . . . . .

99


. . . . . . . . . . . . . . . . . . . . . . . . 100


. . . . . . . . . . . . . . . . . . . . . . . . 101


. . . . . . . . . . . . . . . . . . . . . . . . 102


. . . . . . . . . . . . . . . . . . . . . . . . 103


. . . . . . . . . . . . . . . . . . . . . . . . 104


. . . . . . . . . . . . . . . . . . . . . . . . 105


. . . . . . . . . . . . . . . . . . . . . . . . 106

A.1 Psychotherapy Research p. 1 . . . . . . . . . . . . . . . . . . . . . . . 108 xiv

A.2 Psychotherapy Research p. 2 . . . . . . . . . . . . . . . . . . . . . . . 109 A.3 Psychotherapy Research p. 3 . . . . . . . . . . . . . . . . . . . . . . . 110 A.4 Psychotherapy Research p. 4 . . . . . . . . . . . . . . . . . . . . . . . 111 A.5 Psychotherapy Research p. 5 . . . . . . . . . . . . . . . . . . . . . . . 112 A.6 Psychotherapy Research p. 6 . . . . . . . . . . . . . . . . . . . . . . . 113 A.7 Psychotherapy Research p. 7 . . . . . . . . . . . . . . . . . . . . . . . 114 A.8 Psychotherapy Research p. 8 . . . . . . . . . . . . . . . . . . . . . . . 115 A.9 Psychotherapy Research p. 9 . . . . . . . . . . . . . . . . . . . . . . . 116 A.10 Psychotherapy Research p. 10 . . . . . . . . . . . . . . . . . . . . . . 117 A.11 Psychotherapy Research p. 11 . . . . . . . . . . . . . . . . . . . . . . 118 A.12 Psychotherapy Research p. 12 . . . . . . . . . . . . . . . . . . . . . . 119 A.13 Psychotherapy Research p. 13 . . . . . . . . . . . . . . . . . . . . . . 120 A.14 Psychotherapy Research p. 14 . . . . . . . . . . . . . . . . . . . . . . 121 A.15 Psychotherapy Research p. 15 . . . . . . . . . . . . . . . . . . . . . . 122 A.16 Psychotherapy Research p. 16 . . . . . . . . . . . . . . . . . . . . . . 123 B.1 Cognitive Neurodynamics p. 1 . . . . . . . . . . . . . . . . . . . . . . 125 B.2 Cognitive Neurodynamics p. 2 . . . . . . . . . . . . . . . . . . . . . . 126 B.3 Cognitive Neurodynamics p. 3 . . . . . . . . . . . . . . . . . . . . . . 127 B.4 Cognitive Neurodynamics p. 4 . . . . . . . . . . . . . . . . . . . . . . 128 B.5 Cognitive Neurodynamics p. 5 . . . . . . . . . . . . . . . . . . . . . . 129 B.6 Cognitive Neurodynamics p. 6 . . . . . . . . . . . . . . . . . . . . . . 130 B.7 Cognitive Neurodynamics p. 7 . . . . . . . . . . . . . . . . . . . . . . 131 B.8 Cognitive Neurodynamics p. 8 . . . . . . . . . . . . . . . . . . . . . . 132 B.9 Cognitive Neurodynamics p. 9 . . . . . . . . . . . . . . . . . . . . . . 133 B.10 Cognitive Neurodynamics p. 10 . . . . . . . . . . . . . . . . . . . . . 134 B.11 Cognitive Neurodynamics p. 11 . . . . . . . . . . . . . . . . . . . . . 135

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E.1 RRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 E.2 WAI Client Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 E.3 WAI Client Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 E.4 WAI Therapist Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 E.5 WAI Therapist Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 G.1 Elsevier Permission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 G.2 Taylor & Francis Permission . . . . . . . . . . . . . . . . . . . . . . . 152

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CHAPTER 1 INTRODUCTION 1.1

PSYCHOTHERAPY’S MIXED SUCCESS

One quarter of all adults in the United States have a diagnosable mental disorder [58] for which psychotherapy is a proven method of treatment. Only one quarter of those with a disorder seek therapy, and half of those who seek therapy drop out after the first session [38, 41, 51, 52]. Psychotherapy works for some, but not all.

1.2

THE IMPORTANCE OF RELATIONSHIPS

It has been shown that one of the most important factors in predicting the success (or failure) of therapy is the therapeutic relationship (the relationship between a therapist and client). Horvath et al. have shown that significant correlations between therapeutic outcome and the strength of the relationship, as measured with the Working Alliance Inventory [32, 33], exist. These correlations are only moderate, however (from 0.22 to 0.29 [32]). There is no universal agreement as to what the most critical elements of this relationship are, but some studies have shown that it is the personal (or real) relationship between the therapist and client, as opposed to the theoretical framework being utilized by the therapist [60, 38, 41, 51, 52]. This implies that the manner in which a therapist interacts with a client (i.e., to construct the therapeutic relationship) is more important than the properties of the therapeutic relationship itself.

1

1.2.1

The Therapeutic Relationship

The effectiveness of psychotherapy is 40% dependent on the characteristics of the client outside of therapy [44, 61, 41]). This tells us that if the client has the “wrong” characteristics (which will be a subject of this investigation), then therapy can be, at best, only 60% effective. The remaining 60% is determined by: elements common to all schools of therapy (30%), expectancy (15%), and specific techniques or theoretical approaches (15%) [44, 61, 41]. It is noteworthy that the elements common to all schools of therapy are essentially a decomposition of the interpersonal relationship that develops between client and therapist. In fact, this relationship is twice as important as the specifics of the applied therapeutic approach. Investigations into psychotherapy commonly refer to this relationship (or aspects of it) as the ’therapeutic relationship’ [44, 61, 19, 57]. The American Psychological Association’s Task Force on Empirically Supported Therapy Relationships has defined the psychotherapy relationship as: “the feelings and attitudes that therapist and client have toward one another, and the manner in which these are expressed ” ([56], p. 7). The therapeutic relationship is the most important component of a successful therapeutic encounter [44, 61], but how to define what constitutes a high-quality therapeutic relationship it unclear at present [44, 61, 38, 19]. The therapeutic relationship is a human encounter disguised as a professional engagement. The client experiences the personality of the therapist, and the therapist experiences the personality of the client. The client is evaluating and reacting to the personality of the therapist, no matter what treatment, theory, or technique is employed [44, 61, 60]. In a review of over 2,000 process-outcome studies since 1950 [59] several therapist

2

variables were identified that consistently produced positive results: “Therapist credibility, skill, empathic understanding, and affirmation of the patient, along with the ability to engage with the patient, to focus on the patient’s problems, and to direct the patient’s attention to the patient’s affective experience, we’re highly related to successful treatment” ([41], p. 22). Investigations into the role of the therapeutic relationship in successful therapy have been done by looking at compliance (i.e., attending scheduled appointments) vs. non-compliance (i.e., appointment cancellations and therapy drop-outs). There are three major factors that contribute to treatment compliance: “(1) the perceived warmth and friendliness of the therapist; (2) talking to the client about something that was of importance to the client; and (3) talking to the client in a structured manner. These three relational factors were shown to be more important in determining outcome than client diagnosis and demographics” ([61], p. 41). A further review of the impact that the therapeutic relationship has on successful therapeutic outcomes can be found in Appendix A.

1.2.2

The Real Relationship

When specifying a subset of an entire interaction (i.e., the therapeutic relationship is a subset of the entire relationship), one may lose dynamics that are crucial predictors of the success or failure of the overall relationship. In order to recover the elements of the entire relationship that may be lost, the concept of the ’real relationship’ was introduced by Greenson [28] and reviewed by Gelso [19]. While originally meant to be complementary to the concept of the therapeutic relationship, it becomes apparent that the two definitions overlap heavily. Represented here is the cutting-edge embodiment of the most current trend in psychotherapy research, which is toward unbiased analysis of the entire relationship 3

and drawing conclusions from empirical evidence. One of the most vital residual elements of a relationship that correlates strongly with relationship quality is emotional valence (or state) [26, 61, 51]. Emotional valence also happens to be the only element which can be empirically observed via behavior; behavioral display of emotion is known as affect (see Appendix A for more details). The emotional dynamics of psychotherapy have never been observed or modeled in a direct fashion before, but it seems that certain features of this relationship can be captured and predicted by a model that describes how a dyad reacts to themselves and each other.

1.3

MODELING

In order to model a system (e.g., a human relationship), we must theorize which variables are to be identified and how they can be delineated from the rest of the system. Measurements are taken and used to inform the model. This empiricallyinduced, data-driven model is then used to look for insights and make predictions; John Gottman has revolutionized the study of marriage by utilizing this strategy [26] and has inspired this model. It is important to note that attempting to differentiate and model a subset of the entire interaction between a therapist and client (e.g., a model based on the Working Alliance) would intrinsically hinder the investigation, and is not what is being done. In other words, it would be difficult to dissect the interaction and try to categorize which affective exchanges are part of the Working Alliance versus which ones are not. The holistic approach of modeling the entire interaction, without prejudice, is a much more appropriate method of investigation.

4

1.4

RELATIONSHIPS AS COMPLEX SYSTEMS

A relationship is a system. A linear mathematical relationship (such as y = x + 1) allows one to easily determine the structure of the system; for any value of x, you can determine y. The system, in its entirety, is reducible to one equation that describes the relationship between x and y. The structure of the system is a line, and the behavior can be described as ’x is one less than y’, or ’y is one more than x’. The structure is produced by computing the behavior from many initial conditions. The structure also gives a complete description of the behavior; given a line with no other information, one could reduce it to an equation. This binding of behavior and structure is the beauty of mathematics. Some systems are not so simple. Some systems are complex; their structure can only be observed once it has evolved from the dynamics between the subsystems (i.e., the ’structure’ of the relationship is based on the behavior of the participants). A human relationship is a complex system because the dynamical properties of the system are not deducible from properties of the components/subsystems alone (i.e., we cannot predict the dynamics of a relationship based on genetic information or from demographical data, e.g., income, relationship status, etc., or from phenotypical data, e.g., height, weight, etc. of the biological systems participating in the relationship); it is emergent. There is no linear mathematical structure underlying the dynamics. We do not purport to be able to reduce the entire complexity and depth of human interaction to a set of equations, although it has been shown that by rigorous and careful application of objective measurement and mathematical modeling, some great insights may be gained [26]. It is of importance to note that, unlike the subsystems x and y from the system y = x + 1, the subsystems of interest here, the therapist and client, are able to act

5

on their own behalf; a concept known as agency. In reality, a human relationship is emergent; the model should allow for emergence as well. Modeling the real relationship (i.e., the entire interaction) as an emergent product of the patterns present in the dynamics of the agents’ interaction is a complexity-based approach.

6

CHAPTER 2 COMPLEX SYSTEMS AND MODELING 2.1

THE IMPORTANCE OF PERSPECTIVE

In order for one to begin to study a system, a perspective must be chosen. When attempting to model a system, a researcher must focus on either the structure of the system or its behavior. Many times, this perspective is assumed (or taken for granted) without any explicit thought. In a structure-focused investigation, the natural progression is a whole/part decomposition analysis of the system where the behavior is left implied, or something to be deduced from the articulation of the various components of the system. It is critical to note that this perspective is recursive. This is due to the intuitive notion that the whole/part decomposition analysis can be repeated on the components of the first system. This type of analytical perspective is typically mediated by available technology. The need for higher resolution, greater magnification, higher sampling rates and the like are ever-present. Critical information (i.e., behavior, dynamics) is lost when modeling by continual dissection of the system in this recursive manner. The underlying assumption of those that adopt this perspective is that the information can be restored; this tends to come down to a matter of faith that further dissection by science (i.e., some time in the future) will provide the missing links as technological limitations are overcome. In a behavior-focused investigation, one naturally begins asking questions which revolve around what the system under study does, and how it accomplishes it. Real

7

world systems (natural or artificial) are not static; they do things. The relative success or failure of a system’s behavior has a direct effect on it’s adaptations and the evolution of it’s structure. Key to understanding behavior is understanding relationships. The backbone of most relationships is communication; communication between and among components help define the relationships in a system. Even seemingly static artificial systems, such as a bridge, are actually designed so that the dynamic forces (i.e., behavior) are safely distributed. The design is such that the forces are communicated (via the relationships) to load-bearing members in a manner that prevents resonance (a communication loop) of forces. The argument here is that one should focus on the relationships, not the dissection of components. One should attribute parameters to components that bear weight on (i.e., are relevant to describing) the relationships and communication between them, allowing the dynamics of the relationship to be an emergent product of the components’ definitions. A complex systems perspective allows the relationships (at the scale of interest) to dictate the properties of the system’s components, rather than the other way around.

2.2

THE PITFALLS OF REDUCTIONISM

The source of the shared intuitive notion that one must disassemble systems to better understand them (which, as it happens, immediately destroys any trace of the dynamics) comes from the philosophy of reductionism [18, 37]. Reductionism has produced remarkable achievements, to be sure. But, as a child disassembles a radio to discover how it functions only to have a non-working radio upon reassembly, scientists (especially those brave enough to embark on cross-disciplinary ventures) are

8

finding that the pieces do not fit and their collective ’radio’ is non-functional. Reductionism is a tool and, sometimes, a philosophy. Reductionism is not a law of physics, but many of our physical laws are predicated on it. Quantum mechanics is wonderful example of this; when reality is dissected to this degree (or scale, if you prefer), observations cannot be made without causing an effect to the state/dynamics of the system being observed [67]. This scale of description tells us very little about the dynamics of the system at larger scales. This runs counter to the reductionist viewpoint that has thus far been the dominant theme in the majority of scientific endeavors. Reductionism is a wonderful tool; it has led us to breakthroughs such as the sequencing of the human genome, allowing us to predict which humans are predisposed to certain genetic diseases before they emerge [65, 46]. The field of epigenetics has showed us that although the static sequence of genes plays a large role in the development of an organism and its resultant phenotype, the dynamics between an organism and its environment and between the systems of its own biology play a crucial role as well [64]. The development of the organism cannot be reduced to the sequence of genes alone. Phenotypes produced from genetic knock-outs do not show what the knocked out gene ’controls’. Rather, they show what processes are disrupted when it is removed. A tube amplifier with it’s vacuum tube removed will squeal. The errant conclusion of a reductionist would be that the vacuum tube is a squeal suppressor.

2.3

NONLINEAR DYNAMICS

In his book, “Nonlinear Dynamics and Chaos” [70] , Steven Strogatz writes about dealing with the simple nonlinearity that exists when describing the temporal behavior of a pendulum using ordinary differential equations (p. 7):

9

dx1 = x2 dt dx2 g = − sinx1 dt L

(2.1) (2.2)

“Nonlinearity makes the pendulum equation very difficult to solve analytically. The usual way around this is to fudge, by invoking the small angle approximation sinx ≈ x for x 0, and in virtually all cases c = 1. The system is coupled via the influence functions FC (C) and FT (T ). When translating the model system from discrete (Gottman’s difference equation representation, Equations 2.3 and 2.4) to continuous (our new differential equation representation, Equations 2.7 and 2.8), the parameters r and a must be converted to m and b, respectively. Converting a to b requires no extra steps, as b = a [43]. As shown by Liebovitch et al. [43]:

m=r−1 2.9.2

(2.9)

Influence Functions

The theoretical influence functions described in this section were constructed to be more flexible than the bilinear model; the difference being that these influence functions are formed from three line segments. They are quite useful as a starting point for developing an empirically-based bilinear model [44, 61]. See Appendix A for a more detailed description of the influence functions.

27

   0.5C + 0.5    FC (C) = C + 0.5      −0.5C + 2    5T − 0.1    FT (T ) = 0.5T − 0.1      −3T + 13.9

C≤0 01 T ≤0 04

Now we will discuss the influence functions and their empirical basis. The client’s influence on the therapist (FC (C)) is shown in Figure 2.4. When the client’s affect is negative (C < 0), the therapist will exhibit affect that is more positive. Although, in the case of extreme negativity, the therapist may begin to exhibit neutral and even negative affect [7, 19, 20, 31, 57, 66]. When the client if affectively neutral, the therapist would generally utilize strategies to elicit more positive emotions, either by encouraging the client or having the client focus on their strengths and abilities, in the hope that this may change the client’s affect [7]. If the client seems stuck in neutral, the therapist may try to elicit any affect on the part of the client (which could even be negative). Although if not part of a broader strategy, the therapist “going negative” is generally a bad tactical move and may undermine the therapeutic relationship [41, 19, 31, 57, 66]. When the client’s affect moves from neutral to positive, the therapist will also exhibit more positive affect. There comes a point, however, when the client’s affect becomes positive enough that the therapist may begin to exhibit more neutral affect and the client’s positivity will sustain itself without active encouragement from the therapist [41, 7, 19, 20, 31, 57, 66]. Figure 2.5 shows how the client’s emotional state depends on the therapist’s

28

(FT (T )). The affectively negative therapist is likely to cause a client to be even more negative. The client may interpret the therapist’s negative valence as a sign of the therapist’s disappointment in the client. The client may even feel that the therapist is being judgmental. The therapist’s affect in this case may be the result of frustration with the client’s stubbornness or lack of progress in treatment, but it could also be the therapist’s fears that their own performance is not up to par [7, 1]. This is to be expected of a novice therapist or one about to burnout. The therapist may not be aware that they are displaying negative affect that is driving the client (who is aware of it) to an even more negative state. This is an indicator of the therapeutic rupture, which is a predictor of premature termination of therapy[52, 57]. It should be kept in mind that there are times when the display of negative emotion may be beneficial to the therapeutic relationship. Specifically, appropriate confrontation or expressions of disappointment may be the optimal course of action in some situations. The therapist strategically initiating a therapeutic rupture can sometimes have a long term benefit for the client. The success of this sort of strategy is highly dependent on the skill of the therapist and the strength of the therapeutic relationship [41, 19, 20, 31, 66, 1, 56]. When the therapist’s affect is neutral, the client is likely to be either slightly negative or neutral. This is particularly true during the early stages of the therapeutic process. Some clients may not be influenced by a therapist displaying neutral affect. Of course, there is always the possibility that the client will interpret (i.e. project) the therapist’s neutral affect as a signal of disinterest, possibly causing a negative reaction on the part of the client [41, 7, 19, 20, 31, 57, 66]. When the therapist’s affect changes from neutral to positive, the client may stay neutral (or slightly negative) initially [41, 7, 19, 20, 31, 66, 56]. As the therapist’s affect moves to slightly higher levels of positivity, the client may react by displaying more 29

neutral affect (a move in the positive direction) [66, 69]. This could be a indicator that the client is beginning to emotionally invest themselves in the therapist’s message or is at least beginning to experience positive results due to the therapeutic intervention. There is the possibility that a positive steady state will emerge here, where therapeutic gains can theoretically be maximized [57]. As the therapist’s affect moves into the more extreme ranges of positivity, the client might turn negative. The client becoming “turned off” by this extremely positive affect can be the result of perceiving the therapist as being disingenuous or too “pollyannish”.

2.9.3

Analysis and Solution

For a complete analysis and solutions to the the ODEs, see Appendix B. The real power of the model does not come out of the analytic solution per se, but rather from computational simulation of the system from many different initial conditions. If we plot all of these integrations on a shared graph, we produce what is called a phase portrait, which are introduced in Section 2.9.4.

2.9.4

Phase Portraits

Matlab’s ODE113 function was used to integrate the system. The result of the computations is a landscape where two critical points emerge (this landscape can be seen in Figure 2.7); one is at (C, T )1 = −1.6, −0.3, and the other is at (C, T )2 = 0.3, 0.8. Examination of the flow reveals that (C, T )1 is a saddle point and (C, T )2 is a stable attractor. Linear stability analysis was used to confirm this, with the saddle point having one positive and one negative eigenvalue, and the stable point having two negative eigenvalues. See Appendix B for more details. The evolution of the system to the stable point would represent a successful therapeutic endpoint. For example, if either the client or therapist are extremely positive, 30

they both end up mildly positive, which is the desired outcome. The existence of the saddle point brings about trajectories that are pushed into negative space permanently. This would be the termination of therapy; a “black hole” from which the therapeutic relationship dies. The relationship depicted in Figure 2.7 will most probably end up at the stable point, assuming the therapist initially displays positive affect. If the client starts therapy with a very negative affect (C = −5 or −4), then the therapist must be positive enough to overcome the tendency of the trajectory to end up moving towards the saddle point and further into negative space. It is also interesting to note that if the client begins mildly negative (C = −1) or neutral, the therapist is able to match the negative emotion and still manage to draw the relationship towards the positive stable point. If the client starts therapy with extremely positive affect, the therapist can display some negative or neutral emotion and move the relationship towards the stable point. This method of “going negative” can be a useful strategy to bring the client to a positive valence from a mildly negative one. It is also useful for “tamping down” a client’s unrealistically positive affect. In this scenario, the client and therapist have equal influence on each other, which means that if the therapist starts with extremely high positive affect, the client will draw them down to the stable point. The most important factor for this relationship to succeed is the therapist’s starting affect; in most cases it must not be negative. It should be noted that if the client starts therapy extremely negative, the relationship will end up being pushed into more negative space by the saddle point, regardless of the therapist’s actions. For more discussion of this theoretical landscape, see Appendix B.

31

x2

(x1(t), x2(t))

x1

(x1(0), x2(0))

Figure 2.1: Trajectory of a Pendulum in Phase Space

32

Influence on wife at t

Husband at t-‐1

Figure 2.2: Bilinear Influence Function: IHW (Ht )

33

Cooperation

Competition

Figure 2.3: Cooperation-Competition Influence Functions Fy (y) and Fx (x). Reprinted by permission of Elsevier [43]. See Appendix G for permission statement.

34

5

4

3

Therapist

2

1

0

−1

−2

−3

−4

−5 −5

−4

−3

−2

−1

0

1

2

3

Client

Figure 2.4: FC (C): Client Influence on the Therapist

35

4

5

5

4

3

2

Client

1

0

−1

−2

−3

−4

−5 −5

−4

−3

−2

−1

0

1

2

3

Therapist

Figure 2.5: FT (T ): Therapist Influence on the Client

36

4

5

5 Fc(C) Ft(T) 4

3

Therapist

2

1

0

−1

−2

−3

−4

−5 −5

−4

−3

−2

−1

0

1

Client

Figure 2.6: Influence functions

37

2

3

4

5

5

4

3

Therapist

2

1

0

−1

−2

−3

−4

−5 −5

−4

−3

−2

−1

0

1

2

3

4

Client

Figure 2.7: Phase portrait of the theoretical model integrated with parameter values m1 ,2 = −1, b1 ,2 = 0 and c1 ,2 = 1

38

5

CHAPTER 3 THE EXPERIMENT 3.1 3.1.1

METHOD Experimental Overview

Thirty-five graduate students in the Department of Counselor Education at Florida Atlantic University were invited and agreed to participate in this IRB-approved study. Participants were selected at random to take on the role of either the therapist or the client and be video-recorded doing a 15-minute simulated initial clinical interview. 73 unique interviews were recorded; no exact combinations of client/therapist were ever repeated.

3.1.2

Participants

The participant pool consisted of 35 graduate students in Counselor Education. Four of the participants were males. All participants were over 18.

Instructions to Participants Participants were instructed to either conduct themselves as a therapist or a client. If a student was chosen to be a therapist, they were simply told to be as authentic and natural as possible. If a student was chosen to be a client, they were told to draw on personal experience and talk about a real problem that they did not mind sharing. See Appendix D for specifics.

39

3.1.3

Procedure

The data was collected in the Education building of the Boca Raton campus of Florida Atlantic University under the supervision of Counselor Education associate professor and advisor of this dissertation, Paul Peluso. All data collection events took place during the Spring semester of 2011 (January - May). The sessions were conducted in closed, private therapy rooms with the only equipment present being the cameras and tripods. The participants were monitored (via the computers capturing the live feeds) from across the hall in a separate room where the rest of the hardware was stationed.

Technical Procedure Two virtually identical audio/video data collection apparatuses (each hereby referred to as a ’rig’) were employed in this experiment. Each rig consisted of two digital video cameras, each on a tripod. Two chairs were positioned facing each other, with a videocamera facing each chair with the entire upper body of the participant framed on either the right or left hand side of the viewfinder. The analog output of these cameras was fed into a split-screen video integrator, which takes the right half of one picture and the left half of the other and combines them into a single, undistorted, analog signal containing the streaming video data from both cameras. The output of this device was then fed into an analog-to-digital converter whose output was captured by standard distributions of iMovie (running on OS X 10.4) via firewire. The analogto-digital converter also received the audio input from one of the cameras, which was sufficient to capture the audio for both participants as they were seated within 5’ of each other for the duration of the experiment.

40

Surveys (RRI and WAI) Scores on the two measures, the Real Relationship Inventory [39], and the Working Alliance Inventory, Short Form [33, 74] were collected to serve as a measure of the quality of the therapeutic interaction (although they were not directly used in the data analysis). Each instrument is attached in Appendix E.

3.2

DATA ANALYSIS

All experimental participants were fully compliant with the procedure. Each fifteen minute interview was video-recorded and then the emotional dynamics between the client and therapist coded using the Specific Affect (SPAFF) coding system [26]. Under the direction of Paul Peluso, in consultation with Gottman’s Relationship Research Institute, an expert SPAFF observer analyzed the videos to produce a time series of emotional state for each person based on their displayed affect.

3.2.1

Specific Affect Coding System (SPAFF)

Developed by John Gottman and James Murray, the Specific Affect Coding System (SPAFF) provides a way to leverage the universality of emotional expression discovered by Paul Ekman (see Appendix F) to create a model of the dynamics between two people [26].

The Codes In Gottman et al.’s extensive, ongoing investigation into a mathematical model of marriage, twenty-one types of dynamic affect were discovered. They are shown in Table 3.1. For an extensive description of these codes, see the SPAFF Manual [27].

41

Coding Procedure Details on SPAFF coding can be found in the SPAFF manual [27]. All coding was done on a Noldus Observer console[55].

3.2.2

Selecting the Data to Model

Of the 73 videos recorded, four were selected to be modeled (see Table 3.2). This paring down of the data was due to multiple factors including: the time it takes to model each interaction, video quality, therapy quality (both measured and expert-evaluated) and the cost involved in having the data coded. The project’s psychotherapy expert, Paul Peluso, selected the four to be modeled. Although the psychotherapeutic instruments played no direct role, they will undoubtedly be of use for future work. The RRI and WAI data for these four interaction cases can be found in Table 3.2. Shown are the summary scores for the WAI-Therapist instrument (WAI-T), WAI-Client instrument (WAI-C), and the RRI. Participants 4 and 6 are male; participants 1, 2, 3, 5, 7 and 8 are female.

3.2.3

The Data

In Figure 3.1, a snippet of the Noldus output for the therapist is shown (and in Figure 3.2, the client). The extremely long length of the four data sets prevents them from being practically included, but these two figures should give the reader a sense of the data.

3.3

HYPOTHESIS

It is hypothesized that identifiable patterns will emerge from the modeling of psychotherapy, and that these patterns and the parameters of the model will provide

42

insight into the true nature of the psychotherapeutic relationship.

43

Table 3.1: SPAFF Codes

Index

Affect

Weight

1

Disgust

-3

2

Contempt

-4

3

Belligerence

-2

4

Low Domineering

-1

5

High Domineering

-1

6

Criticism

-2

7

Anger

-1

8

Tension

0

9

Tense Humor

2

10

Defensive

-2

11

Whining

-1

12

Sadness

-1

13

Stonewalling

-2

14

Neutral

1/10

15

Interest

2

16

Low Validation

4

17

High Validation

4

18

Affection

4

19

Humor

4

20

Surprise/Joy

4

21

Physical Affection

4

44

Table 3.2: RRI and WAI Summary Scores

Case

Therapist

Client

WAI-T

WAI-C

RRI

1

Participant 1

Participant 2

67

68

116

2

Participant 3

Participant 4

67

60

118

3

Participant 5

Participant 6

39

70

110

4

Participant 7

Participant 8

72

28

56

Figure 3.1: Noldus Data: Therapist

45

Figure 3.2: Noldus Data: Client

46

CHAPTER 4 DATA FITTING 4.1

DATA TRANSFORMATION: DISCRETIZING, WEIGHTING AND SUMMING THE AFFECT

The raw, coded data is a sequence of affect codes and their duration. This data is transformed into a second-by-second account of the 15-minute interaction. Whichever code holds for the majority of a given second becomes the discretized value for that second. See Table 4.1 for an example. For the corresponding codes for each code number, see Table 3.1. In order to quantify the emotional gain carried by each type of qualified affect, a weighting system was developed by Gottman et. al. [26, 47]. This system attributes a weighting to each emotional affect. Positive affect has a positive weight, and negative affect has a negative weight. The weights of each type of affect are shown in Table 3.1. The final step in treating the data for modeling is to sum the affect codes over six-second intervals as this is the average length of a turn-of-speech in a conversation [26, 47]. A 15-minute interaction with 900 (seconds) data points breaks down into 150 data points (each representing a six-second window) per person. An example can be seen in Table 4.2.

Windows with 0.6 or less are rounded down to zero, as this indicates that neutral

47

was the dominant code in that window. In order to get more numerical variability in windows where neutral is mixed in with other codes, the weighting of neutral is 1/10. Setting windows of 0.6 or less to zero restores the neutrality of the neutral code, while still preserving the dynamics it takes part in [26, 47]. The dominance of neutral in this sample is to be expected, as on the scale of whole (i.e., 15-minute) interactions, the majority of interaction is found to be neutral [26]. Uninfluenced parameters are determined from these neutral points, so they are very important. The time series’ for the four cases are presented in Figures 4.1 - 4.4.

4.2

MODEL FITTING

In Gottman’s original bilinear model [26], the threshold between the different linear regimes of both influence functions was fixed at zero, and the best fit line calculated for each regime (positive or negative) separately by computing the least squares from the weighted and summed SPAFF data in that regime (see Figure 2.2 for an example). We use this as a starting point for our modeling scheme, but we then proceed to slide the threshold up and down the entire regime to computationally determine the least squares estimation of the parameters by comparing all possible fits of these lines [30].

4.2.1

Threshold Autoregressive Bilinear Model

Although Gottman et al. used an iterative process for parameter estimation [26], Hamaker et al. [30] have shown that this system is actually a case of the more general threshold autoregressive model (TAR), initially investigated by Tong and Lim [73] and used widely in economic theory. The specific case is known as a closed loop TAR system (TARSC, [73]). In this type of system, two variables serve as each other’s threshold (i.e., the client’s affect is the threshold for the therapist’s dynamic

48

regime change, and vice-versa). By abstracting the system to a more general representation (that of a TAR), one is able to utilize the tools already available for analyzing TARs. The ability to simultaneously (rather than iteratively) estimate the model’s parameters from the data set is one of the benefits. This means that our inertia (r or m) and uninfluenced parameters (a or b) are computed at the same time as the influence functions using the TARSC method described by Hamaker et al. [30]. All parameters and influence functions are obtained by least squares estimation. Gottman’s previous modeling method fixes the thresholds between the two linear regimes to zero. This method [26] has since been improved upon by allowing the threshold of influence to be adjusted for the best fit. Previously, the marriage modeling had been done by comparing how the negative affect of one dyad member influenced the negative affect of the other (X,Y thresholds of influence functions were fixed at zero). Hamaker et al. [30] has recently shown that by adjusting the threshold (as a free parameter) between the positive and negative regimes of affect when defining the influence function (rather than assuming it be zero), a better fit to the data can be achieved. Note that the threshold’s definition of being the point of zero influence is not violated, as the influence is still zero between regimes. What follows are the best fit (least sum of squared residuals) bilinear models for the four cases, as computed in R. The influence functions were fitted using the standard R distribution (R for Mac OS X GUI 1.40-devel Leopard build 32-bit) running on an Intel Core 2 Due with OS X Lion. The ’dyad’ package (open-source and freely available on CRAN [30]) was used to derive the least squares bilinear influence, inertia and uninfluenced parameters.

49

4.2.2

Case 1

For Case 1, the influence function describing the client’s influence on the therapist, FC (C), is   −1.441703C + 6.4876635 C ≤ 4.5 FC (C) =  −0.2278052C + 1.0251234 C > 4.5

(4.1)

which can be seen graphically with data points overlaid (as generated by R) in Figure 4.5. For Case 1, the influence function describing the therapist’s influence on the client, FT (T ), is

FT (T ) =

 

−0.6376065T + 5.2283733

T ≤ 8.2

(4.2)

 −0.01525227T + 0.125068614 T > 8.2

which can be seen graphically in Figure 4.6.

4.2.3

Case 2

For Case 2, the influence function describing the client’s influence on the therapist, FC (C), is

FC (C) =

 

−1.069827C + 2.4606021

C ≤ 2.3

 0.00579687C − 0.013332801 C > 2.3

50

(4.3)

which can be seen graphically in Figure 4.7. For Case 2, the influence function describing the therapist’s influence on the client, FT (T ), is   1.169578T − 2.6900294 T ≤ 2.3 FT (T ) =  0.0859379T − 0.19765717 T > 2.3

(4.4)


4.2.4

Case 3

For Case 3, the influence function describing the client’s influence on the therapist, FC (C), is   −0.17866C + 0.78903972 C ≤ 4.2 FC (C) =  −0.3653419C + 1.53443598 C > 4.2

(4.5)

which can be seen graphically in Figure 4.9. For Case 3, the influence function describing the therapist’s influence on the client, FT (T ), is   −0.1427034T + 0.91330176 T ≤ 6.4 FT (T ) =  0.04416631T − 0.282664384 T > 6.4


51

(4.6)

4.2.5

Case 4

For Case 4, the influence function describing the client’s influence on the therapist, FC (C), is   −0.01153827C − 0.005769135 C ≤ −0.5 FC (C) =  0.132025C + 0.0660125 C > −0.5

(4.7)

which can be seen graphically in Figure 4.11. For Case 4, the influence function describing the therapist’s influence on the client, FT (T ), is   −0.03981842T + 0.63709472 T ≤ 16 FT (T ) =  0.1014746T − 1.6235936 T > 16


52

(4.8)

Table 4.1: Discretized Noldus Data: Case 1

Second

Therapist

Client

1

14

14

2

14

14

3

14

14

4

14

8

5

14

8

6

14

8

7

8

8

8

17

8

9

17

16

10

14

16

11

14

16

12

14

16

13

14

16

14

14

8

15

14

8

16

14

14

17

14

14

18

14

14

19

8

14

20

8

8

53

Table 4.2: Weighted and Summed SPAFF: Case 1

Second

Therapist

Client

1

0.6

0.3

2

8.3

16

3

0.6

4.3

4

0.1

0.4

5

8.2

0.6

6

8.4

0.5

7

12.3

0.1

8

8

0.1

9

0

0.6

10

0.4

0.6

11

16

0.6

12

4.1

0.6

13

4

0.6

14

4

0.6

15

4

0.4

16

12

0.3

17

24

0.1

18

24

16.2

19

22

4.5

20

12

8.4

54

Figure 4.1: Time Series: Case 1

55


56


57


58

15 10 5 Influence

0 -5 -10 -15 0

5

10

15 Score

Figure 4.5: FC (C): Case 1

59

20

15 10 Influence

5 0 -5

-5

0

5

10 Score

Figure 4.6: FT (T ): Case 1

60

15

20

25

10 5 Influence

0 -5 -10

0

5 Score


61

10

10 5 -5

0

Influence

-5

0

5

10 Score


62

15

20

15 10 5 Influence

0 -5 -10 -15

-10

-5

0 Score


63

5

10

10 5 Influence

0 -5 -10

-5

0

5

10

15

Score


64

20

25

15 10 5 Influence

0 -5 -10 -15

-10

-5

0

5 Score


65

10

15

15 10 5 -10

-5

0

Influence

0

5

10

15 Score


66

20

CHAPTER 5 RESULTS & DISCUSSION 5.1

RESULTS

5.1.1

Nullclines and Phase Portraits

The most interesting analysis of a dynamical system is done computationally. By integrating the system from many different initial conditions, and plotting the values of the key variables against one another, a visualization of the systems dynamics through all possible states is produced (see Liebovitch et al.’s work [44] for a discussion of limit cycles and a walkthrough of the integration process). The influence functions were fitted using the standard R distribution (R for Mac OS X GUI 1.40-devel Leopard build 32-bit) running on an Intel Core 2 Duo with OS X Lion. The ’dyad’ package (open-source and freely available on CRAN [30]) was used to derive the best fit bilinear influence functions to the data. For a thorough discussion of the threshold autoregressive process see Section 4.2 and Hamaker et al.’s 2009 manuscript [30]. The ODE113 package on the 64-bit version of Matlab 7.12.0.635 (R2011a) running on a Intel Core 2 Duo with Mac OS X Lion (10.7.3) was used to compute the nullclines and derive the phase portraits.

67

Nullclines The model is formed by Equations 2.7 and 2.8. In order to analyze the system’s dynamics (derived from the empirical cases discussed in Chapter 3) we need to incorporate the other derived parameters, m and b, with the influence functions, FC (C) and FT (T ). Recall that m is the inertia and the constant b is the uninfluenced, or alone, state of the person. Table 5.1 shows the model’s parameter values for each case, which have been estimated from the data as discussed in Chapter 4. It should be noted that for all of these cases, c1 and c2 have been fixed at 1. The resulting nullclines are depicted in Figures 5.1 - 5.4. Table 5.1: Model Parameters

Case

m1

m2

b1

b2

1

-0.53184

-0.89315

3.5672

3.6423

2

-1.06873

-0.49868

7.61278

-0.17219

3

-0.69884

-0.66196

5.7614

-0.00054703

4

-0.56664

-0.5832

3.8934

0.20766

68

Phase Portraits As discussed in Chapter 2 and elaborated on in Appendix C, the intersection points of the nullclines create the critical points that dictate the flow of system that can be visualized in the phase portraits. The phase portraits for the four cases are shown in Figures 5.5 - 5.10. To illustrate how the model has extracted patterns from the data in the form of these phase portraits, one can compare them to the phase diagrams of the interactions (Figures 5.11 - 5.14). These phase diagrams have the same dimensionality as the portraits (i.e., client state by therapist state) but present the raw data in place of the data derived to fit the model. In Figure 5.5, the phase portrait for Case 1 is shown. By visual inspection, it is clear that there are two attractors present in this system, both in positive-positive space. In Figure 5.8, the phase portrait for Case 2 is shown. By visual inspection, it is clear that there is an attractor in positive-therapist space, and just slightly above neutral for the client. In Figure 5.9, the phase portrait for Case 3 is shown. By visual inspection, it is clear that there is an attractor in positive-therapist space, and just slightly above neutral for the client. In Figure 5.10, the phase portrait for Case 4 is shown. By visual inspection, it is clear that there is an attractor in positive-therapist space, and just slightly above neutral for the client.

69

5.2

DISCUSSION

5.2.1

Empirical Phase Portraits

The sheer existence of stable attractor states in this system is a conclusion of great importance. Arbitrary parameter values and influence functions can produce systems with unstable or saddle points, and attractors at extreme numerical ranges; in all of these cases, trajectories are thrown off to plus or minus infinity. The presence of a stable attractor in positive or neutral space for both client and therapist for all of the four empirical cases is a clear indication that this novel method of modeling the dynamics of psychotherapy has merit. In comparing the values shown in Table 5.1 to the resulting phase space portraits shown in Figures 5.5 - 5.10, quite an apparent pattern can be seen to emerge: the client with a higher (3.6423 vs. < 1) uninfluenced (i.e., alone, b2 ) state, as is seen in Case 1, ends up with two attractors in the system; both of which show a positive outcome for the client. In Cases 2, 3 and 4, the b2 value is less than 1, and the trajectories for these clients all end up with attractors close to neutral (in the client dimension). This seems to support the old adage that “clients have to want to change” [51]. In other words, if the client is positive when uninfluenced (regardless of the initial conditions of the simulation), therapy will be at least moderately successful (based on the literature, we suppose that a successful outcome would be in positive emotional space for the client, however these conclusions are tentative and will be subject to further empirical validation). The reverse situation also holds true: the therapist’s b value is the most significant determiner of therapist emotional state at the stable attractor. This is in agreement with Gottman et al.’s findings [26] that the greatest predictor of future marriage stability is the uninfluenced states. If both the husband and wife have positive uninfluenced parameters, the marriage is much more likely to

70

succeed. Looking at Case 1 (Figure 5.5) phase portrait, we find some interesting dynamics. This is a two attractor landscape, and both of the attractors are quite close together and positive for the client. This is an indication of what we believe is a good therapistclient fit. No matter what the initial conditions are, the system will be directed toward a successful outcome (although one is more successful than the other). As was just shown, the uninfluenced parameters play a large role in determining these dynamics, so it is of no surprise that both these participants have positive uninfluenced states. The b1 value is 3.5672 and the b2 value here is 3.6423. The upper-right quadrant contains both of the attractors. If the client or therapist start out maximally positive (C, T = 24, 24), the net effect will be a damping down to a more balanced emotional state. In the upper-left quadrant, virtually all trajectories flow to the less-positive client stable point. This makes sense, as a client who is negative dealing with a therapist who is overly positive cannot be expected to have the most efficient outcome [51, 44]. It is interesting to note that the therapist’s affect becomes even more positive as the client’s affect becomes less negative. This could be considered the therapist’s attempt to persuade the client out his or her negativity. In the lower left quadrant we have an interesting dynamic. If the client’s negative affect is not met by equal or greater negativity on the part of the therapist, therapy will end up at the less positive client attractor. This is in agreement with psychotherapeutic literature on paradoxical intervention [51]. In the lower-right quadrant, the therapist starts out negative and the client positive. As the system plays out, the client has a surge of increasing affect (ironically, perhaps an attempt on the part of the client to persuade the therapist out of their negative state). All trajectories in this quadrant will flow to the more-positive client space. To further illustrate the dynamics around the critical points in Case 1, the eigen71

vectors at various initial conditions (i.e., the vector field) can be seen for the first attractor (located at C, T = 5.4272, 6.3101) in Figure 5.6 and for the second attractor (located at C, T = 4.0843, 7.8342) in Figure 5.7. The open source MATLAB package pplane was used to compute these vector fields. The pplane package could not handle the piecewise linear influence functions as inputs, so the fields were computed separately for each critical point. Looking at Case 2 (Figure 5.8), the dynamics are similar to Case 1. This is a single attractor landscape, with the attractor virtually neutral for the client and positive for the therapist. The b value for the client is much less than in the first case (b2 = −0.17219 here), and this seems to play a significant role in the resulting dynamics in this system. The therapist’s uninfluenced state is the most positive of any of our participants (b1 = 7.61278), but this seems to fail to have the desired effect on the client (namely, trying to evoke positive affect). All trajectories flow to the single attractor state. Looking at Case 3 (Figure 5.9), we see very similar dynamics to Case 2. This is also a single attractor landscape with a very low (slightly negative, in fact) uninfluenced parameter (b2 = −0.00054703). Once again, the result of engaging in therapy with a client that has a neutral uninfluenced parameter is unsuccessful. The therapist, having a positive uninfluenced parameter (b1 = 5.7614), ends up fairly positive at the conclusion of the encounter. Looking at Case 4 (Figure 5.10), we see a landscape nearly identical to Cases 2 and 3. Once again, a single attractor is present, and all of the trajectories flow to it. This stable attractor is virtually neutral in the client dimension. Once again, we have a client whose uninfluenced parameter is close to zero (b2 = 0.20766), and a therapist with a positive uninfluenced parameter (b1 = 3.8934). It is interesting to note that our most successful therapy session (Case 1), has 72

the lowest b1 parameter value of the four. This means that our least positive-whenuninfluenced therapist was our most effective.

5.2.2

Variant Phase Portraits

Through many simulations, varying m (inertia) did not seem to have a significant effect on the outcome of therapy (see Figure 5.15), but changing the b values did. What is the effect of a client being more positive when uninfluenced? What is the effect of a therapist being more positive when uninfluenced? The pattern that emerges in all four cases, that the uninfluenced parameter (b) has a significant impact on the resulting therapeutic endpoint, can be explored and predicted by using this nonlinear dynamical systems model. The ability to alter the parameters of the empirically derived cases of this model, to see what would have happened had circumstances in therapy been different, is powerful indeed. By varying the inertia (m) and uninfluenced state (b) of our model, we can find out what would happen if the inertias are set to the same parameters as in the theoretical model (e.g., m1 ,2 = −1) as well as the impact of altering the uninfluenced state of the agents. The set of parameter variations are shown in Table 5.2. These are all variations to Case 1. We see what happens to the system as the uninfluenced parameter for the therapist (b1 ) is held at zero and the uninfluenced parameter of the client (b2 ) is taken through incremental steps from 1 to 4 (Figures 5.16 to 5.20). We then see the effect that holding b2 at zero and increasing b1 incrementally from 1 through 4 has on the system (Figures 5.21 to 5.24). Looking at the most minimalist case of uninfluenced client positivity (Variant 2), where b1 = 0 and b2 = 1, we can see that there is a two attractor landscape (Figure 5.16). One attractor is neutral for the client, and the other is positive. If the client and therapist each start out neutral, the system will evolve to the more-positive client 73

attractor, where the therapist is slightly negative. In the upper right quadrant, most trajectories evolve to the more positive client attractor, meaning that if the client and therapist are both initially positive, the stable state will be in positive-client space unless the therapist is overly positive (above the dashed black line shown in Figure 5.17). In the lower left quadrant, it becomes clear that if the therapist wants the client to end up at the positive client attractor, the therapist must match or exceed the client’s negative affect (by staying to the right of the red line in Figure 5.17). This echoes the empirical findings of Dreher et al. [13] that the therapist’s exhibition of contempt seems to catalyze an increase in positive affect in the client in subsequent therapy sessions; the therapist in that study also displays disgust in 9 of the 11 sessions. In the upper left quadrant, improvement in the client’s affect is guaranteed as the trajectories all flow from negative client space to the neutral attractor. Although not the optimum emotional result for therapy, bringing the client from negative to neutral is far better than no gains at all. The therapist’s affect is rapidly increased as the client’s affect becomes less negative; this may be the therapist maneuvering the client out of their negative space by using lots of positive affect. This may be the preferred outcome if the client’s negative emotion is grief or anger, as neutral would represent “acceptance” of the situation. At the very least, the client will feel “less bad” and probably be able to make more rational decisions (as oppose to ones made out of negative emotion which could be irrational or in bad judgement). There is a cautionary point for therapists not too treat client negativity with too much positivity if they want the relationship to evolve to the more-positive client attractor. This might be the best argument for paradoxical interventions, and the difference between master clinicians and novice ones. Novice clinicians are more likely to respond to initial client negativity with pos74

itivity (taking the “safe route”); the likely result of this would be the client in a neutral state and the therapist in a positive one [51]. A more experienced therapist would be more likely to take greater risks (with the possibility of greater rewards) by using the approach of “going negative” (a paradoxical intervention) and working with the client “where they are” emotionally. This allows the therapist to work through the emotion with the client and with the goal of guiding the client to a more positive emotional state [51]. In the lower right quadrant, we see that any initial conditions will bring the client to the maximally positive space: the more positive stable attractor. There is an indication of over-enthusiasm (or perhaps unrealistic positivity) on the part of the client, as the client’s affect increases (except for maximally positive initial conditions) above the value of the more positive attractor before reaching it. This oscillation of client affect could be an indication of the client’s ambivalence to change. Variants 3 to 5 demonstrate that the more positive the uninfluenced client is, the more positive the stable state of therapy will be. The less-positive client attractor disappears once b2 ≥ 2 and we are left with single attractor landscape that always flows to the more-positive client attractor. Variants 6 to 9 show very clearly that a similar pattern exists when the b values are reversed.

75

Table 5.2: Case 1 Model Variants

∗

Variant

m1

m2

b1

b2

1

-1

-1

∗

∗

2

∗

∗

0

1

3

∗

∗

0

2

4

∗

∗

0

3

5

∗

∗

0

4

6

∗

∗

1

0

7

∗

∗

2

0

8

∗

∗

3

0

9

∗

∗

4

0

Parameter unchanged

76

dT/dt dC/dt

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

Client

Figure 5.1: Nullclines: Case 1

77

10

15

20

dT/dt dC/dt

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

Client


78

10

15

20

dT/dt dC/dt

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

Client


79

10

15

20

dT/dt dC/dt

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

Client


80

10

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client

Figure 5.5: Phase Portrait: Case 1

81

15

20

Figure 5.6: Vector Field: Case 1, Attractor 1

82

Figure 5.7: Vector Field: Case 1, Attractor 2

83

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


84

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


85

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


86

15

20

Figure 5.11: Phase Diagram: Case 1

87


88


89


90

5.2.3

Conclusions

Psychotherapy has been shown to be an effective method for treating mental illness [38, 41, 51]. Mental illness is the leading cause of disability (in the U.S.); mental illness afflicts one in four adults in the United States [58]. The dismal frequency of psychotherapy dropout[52, 5] shows there is a need to invest in new tools and techniques to bring psychotherapy into the realm of hard science, so that the intuitions of master practitioners may be abstracted and used to generate mathematical laws of human dynamics. A prototypical experiment has been done here; a dynamical systems model of psychotherapy has never been attempted before. We have taken a major step and shown that empirical psychotherapy dynamics can be quantified and modeled. Identifiable patterns have emerged from this modeling, and these patterns and the parameters of the model have provided objective insight into the true nature of the psychotherapeutic relationship.

The Client What the model has clearly shown is that the client has to be at least slightly positive for therapy to be successful. This is evidenced by the fact that the client evolves to a more positive stable point during therapy than their uninfluenced (i.e., state when alone) parameter if and only if they are positive when uninfluenced. This is evidenced in all of the empirical cases (Figures 5.5 - 5.10; discussion of the differences in b2 between Cases 1 and 2-4, and the corresponding results of therapy can be found in Section 5.2.1). To test this finding, we can use the model to see what would have happened in successful therapy (Case 1), if the client is affectively neutral when uninfluenced.

91

As can be seen in Figures 5.21 - 5.24, the client is virtually neutral at each of the attractors in each system when b2 = 0, despite a therapist who is very positive when uninfluenced, as in Figure 5.24. As is seen when comparing the b2 values and resulting stable states in Figures 5.16 through 5.20, the result of therapy is positive for the clientonly if the client is positive to begin with, as is demonstrated by comparing these to Figures 5.21 through 5.24. These findings show that the uninfluenced values (b1 and b2 ) are the most important factor in therapeutic success. A client must be positive; he or she must want to participate in the relationship. If the client is affectively neutral (or negative) when uninfluenced, therapy will not be successful. It is not necessarily a negative result that psychotherapy is only successful for people who are positive about participating in the experience (e.g., people who are “happy” to go to therapy). This conclusion changes the discourse of questions one wants to ask. It is evident that there is a heterogenous population and that therapy, like medicine, may not work for everyone. Asking, “what are the characteristics of someone who will be helped by therapy?” is a different question than, “what type of psychotherapy works for everyone?” or, “what is the best type of psychotherapy?” and then averaging or looking for a net effect. There is an effort to practice individualized medicine, based on a patient’s genetic makeup, to understand which medicines work for which people; perhaps this “individualized” paradigm needs to be applied to psychotherapy.

The Therapist Interesting conclusions can be drawn about how a therapist’s parameters contribute to their overall success with clients by inspecting the system’s trajectories in phase space. 92

As can be seen in Case 1 (Figure 5.5), and made more prominent by Variation 2 (Figure 5.16): given that two critical points exist (both attractors in positivepositive space), and one is more positive for the client than the other, it would be advantageous for the therapist to maneuver the client to the point of greater positivity (for the client). The model suggests in what ways the therapist might do that. If the client’s initial condition is negative and the therapist is at least as negative as the client (as is the case with Figure 5.16), the client’s trajectory flows to the attractor with a higher emotional state for the client (i.e., what we suppose is the desired attractor of the interaction). The paradoxical intervention of the therapist “going negative” [51] is shown to be successful when the client has a positive uninfluenced state (b2 ). Here, the model is telling us that, for a client with positive b, purposefully “going negative” will not have the desired effect unless the therapist goes “all the way” with it and is more negative than the client. This is most likely because the client will view the therapist’s attempt at intervention as inauthentic if the client’s negative intensity is not met or exceeded [51]. For the client to end up at the positive attractor, the therapist can be in any state except for being much more positive than the client (e.g., more than twice a client’s score is not good for the client). If the therapist is overly positive, the client will end up at the neutral attractor. This tells us that if the therapist is overly positive, the client may not be convinced and therapy will not evolve to positive-client space.

5.3

FUTURE WORK

The model’s ability to quantify dyadic interaction, specifically in the context of psychotherapy, provides a tremendous amount of opportunity for future work and further exploration.

93

5.3.1

Training

By altering the parameters of an empirically derived implementation of this model, a practitioner can see what would have happened had a session been handled differently. This can be used to predict how to best treat a client in the future. It would also enable the therapist to investigate (and think) about the relationships they forge with their clients from a completely new and objectively measurable perspective. Even in the simplest of applications, it is believed that by getting therapists to think from a dynamical systems perspective, therapeutic gains may be improved [51, 44, 61].

5.3.2

Master Practitioners

With the proof-of-concept complete, full psychotherapy sessions with highly experienced practitioners are currently being video-recorded. Questions such as, ’what might a phase portrait derived from the dynamics between a client and a master practitioner look like and how would it differ from these novice cases?’ and, ’how do the dynamics between psychotherapists from differing schools of thought compare?’ are in need of answering. This will hopefully lead to therapists being able to make predictions about how to best treat clients, and may aid in training novice therapists to that end.

5.3.3

Trilinear Modeling

It remains to be seen what insights might be gained from fitting a trilinear model, such as the one theorized by Liebovitch, Peluso, Norman and Gottman [44, 61] to empirical psychotherapy data.

94

5.3.4

Instrumentation Leveraging

Perhaps the subjectively measured data (from the RRI and WAI; see Appendix E for details) could be used to inform parameters of the model; for example, to fine-tune the inertia parameter.

5.3.5

Inertial Functions

In this prototypical model, the inertial parameters are derived from the whole interaction, and are global across all the different linear regimes. It makes intuitive sense that perhaps a person’s inertia at any given moment is dependent on the regime they are in at that moment. It might be that a person’s inertia is higher in their negative emotional space than it is in their positive. Inertial functions could be used to replace the inertia constant and will be a topic of future investigation.

5.3.6

Question the Assumptions

It may prove useful to revisit prior assumptions such as interaction window size (which is currently six-seconds). Many procedural decisions were aligned with Gottman’s marriage studies [26] in order to minimize unforeseen conditional constraints on the subsequent analysis.

5.3.7

Other Dyads

Many other dyads may lend themselves to this modeling approach. Teacher-student, coach-player, salesperson-customer, doctor-patient, etc. are all different dyadic relationships that could be informed by this modeling approach.

95

5.4

THE BEGINNING

The overwhelming importance of the psychotherapeutic relationship is just starting to be recognized. In this experiment we have explored, in detail, a method for capturing and quantifying this relationship. The tools that have been developed over the course of this project are opening up the field of psychotherapy by enabling new hypotheses to be formed that would never before have been possible. This is due to the novel application of rigorous, mathematical structure to a nebulous discipline that previously had none.

96

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client

Figure 5.15: Phase Portrait: Variant 1

97

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


98

15

20

Figure 5.17: Phase Portrait: Variant 2 with Overlay

99

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


100

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


101

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


102

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


103

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


104

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


105

15

20

20

15

Therapist

10

5

0

−5

−10

−15

−20

−20

−15

−10

−5

0

5

10

Client


106

15

20

APPENDIX A PSYCHOTHERAPY RESEARCH PUBLICATION Reprinted by permission of Taylor & Francis [61]. See Appendix G for permission statement.

107

Psychotherapy Research, January 2012; 22(1): 4055

A mathematical model of psychotherapy: An investigation using dynamic non-linear equations to model the therapeutic relationship

PAUL R. PELUSO1, LARRY S. LIEBOVITCH2, JOHN M. GOTTMAN3, MICHAEL D. NORMAN4, & JESSICA SU4 1 Florida Atlantic University, Counselor Education, Boca Raton, USA; 2Division of Mathematics and Natural Sciences, Queens College, Queens, USA; 3Relationship Research Institute, Seattle, USA & 4Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA

Downloaded by [Florida Atlantic University] at 13:28 08 February 2012

(Received 3 January 2011; revised 17 August 2011; accepted 2 September 2011)

Abstract Mathematical models, such as the one developed by Gottman et al. (1998, 2000, 2002) to understand the interaction between husbands and wives, can provide novel insights into the dynamics of the therapeutic relationship. A set of nonlinear equations were used to model the changing emotional state of a therapist and client. The results suggest: (1) The person that is most responsive to the other achieves the most positive state, (2) the emotional state of the client oscillates before reaching its final state, (3) therapy is least successful when the therapist starts from a negative state, and (4) there is an inverse relationship between models that change only the influence parameter and models that change only the inertia parameter, creating a series of four basic models to work with. These theoretical models require further, empirical investigation to test the derived parameters. If validated, or revised based on observations of therapist-client relationships in development, they could provide specific direction in creating successful therapeutic relationships for training clinicians and those already in practice.

Keywords: dynamical systems; alliance; emotion in therapy; process research; psychotherapist training/supervision/ development; statistical methodology; technology in psychotherapy research and training

The rates of psychological disorders are staggering. According to the National Institute of Mental Health (NIMH), one in four adults in the United States suffers with a diagnosable mental illness (NIMH, 2010). These disorders exact a physical, emotional and economic toll on the individuals afflicted with them, their families, their communities, and the nation as a whole. They are the leading cause of disabilities in the United States. Psychotherapy has been shown to be an effective method for treating a wide array of mental illness (Kazdin, 2008; Lambert & Barley, 2002; Mozdzierz, Peluso, & Lisecki, 2009). However, only 25% of individuals who do have a diagnosable mental illness will come to therapy; and of those, approximately half will drop out after the first session (Muran, Gorman, Eubanks-Carter, et al., 2009). This rate of attrition has remained consistent over the last 50 years (Barrett, Chua, Crits-Christoph, Gibbons, & Thompson, 2008). The major question that has confronted clinicians and researchers alike continues to be: if therapy is successful and effective, then why don’t clients stay?

Lambert and Barley (2002) summarized the available research and concluded that certain variables contributed significantly different percentages to successful therapeutic outcomes. They found that extratherapeutic change (defined as client characteristics outside the therapeutic interaction) contributes 40% towards the effectiveness of therapy; ‘‘common factors’’ (that is, elements common to all therapies) contribute 30%; expectancy (hope, ‘‘placebo effects,’’ etc.) contributes 15%; and techniques or theoretical approaches contribute 15%. Lambert and Barley carefully point out that the data accumulated from psychotherapy research demonstrate that common, interpersonal factors contribute more to therapeutic outcome than specific techniques. The literature has traditionally tried to describe the salient qualities of the ‘‘common factors’’ in therapy by referring to therapeutic relationship (Gelso, 2009; Norcross, 2010; Norcross & Wampold, 2011). Indeed, without a positive therapeutic relationship between client and counselor, nothing significant would be accomplished in therapy. Norcross (2002),

Correspondence concerning this article should be addressed to Paul R. Peluso Ph.D., Florida Atlantic University, Counselor Education, 777 Glades Rd., Bldg 47, Rm. 270, Boca Raton, 33431 USA. Email: [email protected] ISSN 1050-3307 print/ISSN 1468-4381 online # 2012 Society for Psychotherapy Research http://dx.doi.org/10.1080/10503307.2011.622314

Figure A.1: Psychotherapy Research p. 1

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A mathematical model of psychotherapy


reporting on Division 29 of the American Psychological Association’s Task Force on Empirically Supported Therapy Relationships, defined the psychotherapy relationship as: ‘‘the feelings and attitudes that therapist and client have toward one another, and the manner in which these are expressed’’ (p. 7; emphasis added). The therapeutic relationship is an essential element in counseling therapy clients. The quality of the therapeutic relationship is one of the most potent and lasting indicators of successful treatment. However, the qualities of this relationship and the interaction between the therapist and client that are most effective to help change negative behaviors or habits are not yet well understood (Kazdin, 2008; Gelso, 2009; Norcross & Wampold, 2011).

The Therapeutic Relationship Even though it is a professional engagement between a client and a practitioner, at its core the therapeutic relationship is essentially a human encounter. It is the person of the therapist that the client experiences, evaluates, and reacts to in treatment seemingly no matter what sort of treatment, theory, or technique the therapist espouses to practice (Orlinsky & Howard, 1977). Orlinsky, Grave, and Parks (1994) reviewed more than 2000 process-outcome studies since 1950, and identified several therapist variables that were consistently shown to have a positive impact on treatment outcome: ‘‘Therapist credibility, skill, empathic understanding, and affirmation of the patient, along with the ability to engage with the patient, to focus on the patient’s problems, and to direct the patient’s attention to the patient’s affective experience, were highly related to successful treatment’’ (cited in Lambert & Barley, 2002, p. 22). Centorrino et al. (2001) demonstrated how important the therapeutic relationship is in determining successful treatment outcomes. They investigated factors associated with outpatient mental health treatment compliance (i.e., keeping scheduled clinic appointments) versus noncompliance (i.e., failure to keep appointments and treatment drop-outs). Only three factors contributed to treatment compliance: (1) The perceived warmth and friendliness of the therapist; (2) talking to the client about something that was of importance to the client; and (3) talking to the client in a structured manner. These three relational factors were shown to be more important in determining outcome than client diagnosis or demographics. Lastly, clients who felt that they were going to be listened to by a therapist, rather than merely treated by a medical professional with medications, were more likely to be compliant (a prerequisite for eventual success in therapy).

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In their meta-analysis of 79 studies of psychotherapy outcome, Martin, Garske, and Davis (2000) determined that the connection between the therapeutic relationship (and a specific element of the relationship, the ‘‘working alliance’’) and outcomes is consistent. In addition, Martin et al. suggest that there is a therapeutic/healing effect in the relationship itself, and that if an appropriate alliance is established, a client will experience the relationship as therapeutic regardless of other psychological interventions. Martin et al. found that other variables do not seem to influence the strength/quality of relationship, and that the finding was consistent regardless of the type of measure that is used; when the alliance assessment is taken (early or late in therapy); and type of treatment provided. Horvath and Bedi (2002) summarized 90 outcome studies conducted between 1976 and 2000 and concluded that clients who were aligned with their therapist had positive outcomes. More recently, Muran et al. (2009) and Gelso (2009) concluded that client and therapist ratings of the strength of the therapeutic alliance consistently predicted success in treatment. Lastly, Horvath, Del Re, Fluckiger, and Symonds (2011) followed up their previous research with an analysis that included twice the number of studies as before, and found a similar (though slightly higher) effect size (r .275) between therapeutic outcome and the strength of the alliance.

The Real Relationship It is evident from the empirical evidence that a strong affective bond resulting in a consistent therapeutic alliance is an important vehicle in determining successful treatment outcomes (Mozdzierz, Peluso & Lisecki, 2009). However, given the high number of rates of premature dropout in therapy, there is an element that seems to be missing. Over 30 years ago, Greenson (1965, 1967), as cited in Gelso, 2009) delineated between the professional elements of the therapeutic encounter (therapeutic relationship, the working alliance, and the issues of transference and countertransferrence) and the personal encounter. He believed that this personal encounter, called the real relationship, was fundamental to the success of the therapeutic encounter, and that these overlapping elements are often difficult to distinguish from one another (as cited in Gelso, 2009). While they may seem indistinguishable, according to Gelso (2009), the working alliance and the real relationship are different in that ‘‘(w)hereas the working alliance is an artifact of the treatment in the sense that its only reason for existence is to get the work done, the real relationship exists to one degree or another any time two or more people relate to one another’’ (p. 257). Indeed, Greenson, Gelso


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and others define the real relationship as one that contains genuineness, realism, and (positive) emotional valence. It is an emergent property of any two people coming together for mutual benefit. As a result, both client and therapist contribute to creating this element of the therapeutic encounter. A key element to the real relationship is valence, or how positive or negative the individuals’ attitudes and feelings are toward one another. It is a critical element of the development of the real relationship (and thus, the working alliance). According to Gelso, discussing the importance of the emotional valance of the therapeutic relationship is not traditionally held in as high esteem as the concepts of therapist neutrality and distance. Nonetheless, the therapeutic relationship, by virtue of the real relationship, is not unlike any other human relationship (e.g., marriage, friendship, etc.) between two people, and is highly dependent on the valence of relationship between client and therapist. Clearly, one of the most powerful mechanisms of change between a therapist and a client is the therapeutic relationship. However, the essentials of how to create this relationship*the interaction between client and therapist*have not been modeled or observed in a direct fashion. Kazdin (2008) advocated that researchers should, ‘‘attempt to understand more about the many change processes and how they can be triggered, activated, exploited, and trained’’ (p. 157). According to Kazdin, merely researching the strength of the alliance is not enough. Instead, intervening process must be studied that can show the mechanisms of therapy at work. These studies must include a demonstration of the timeline (e.g., showing cause and effect of the therapeutic process on a client), and an explanation of how the therapeutic process works to improve client functioning and lead to positive treatment outcomes (Kazdin, 2008). Norcross and Wampold (2011) highlight the concept of therapist responsiveness to a client’s needs as both a particular strength of the therapeutic alliance, and a difficult construct to quantify and measure: Effective psychotherapists are responsive to the different needs of their clients in different cases, and within same case, at different moments. Successful responsiveness can confound attempts to find naturalistically observed linear relations of outcomes with therapist behaviors (e.g., cohesion, positive regard). Because of such problems, the statistical relations between the relationship and outcome cannot always be trusted. By being clinically attuned and flexible, psychotherapists make it more difficult in research studies to discern what works (p. 100). We believe that a dynamical systems approach that uses non-linear differential equations can provide a method for modeling some of these complex rela-

tional dynamics, and provide information on the nature and composition of the working alliance, and the impact that the emotional valence within the relationship (viz. the ‘‘real relationship’’) has on the successful progress of therapy (or the lack therof) that has remained elusive.

Mathematical Models: Using Dynamic Nonlinear Equations to Model Human Relationships The most common paradigms used in social science to analyze experimental data are to determine the functional relationships between dependent and independent variables, the correlations between different variables, or the most meaningful factors that account for the data as determined by their statistical properties (e.g., by ANOVA). By contrast, in a recent issue of American Psychologist, Vallacher, Coleman, Nowak, and Bui-Wrzosinska (2010) advocated the usefulness of dynamical systems for the modeling and study of complex phenomena using simple linear formulae. Dynamic nonlinear mathematical equations allow for investigators to understand complex systems that are apt to change (as with the ‘‘responsiveness’’ that Norcross and Wampold described above). The ultimate goal in creating and using these equations is to define the dynamics of a system and evaluate how the system changes over time (Gottman, Murray, Swanson, Tyson, & Swanson, 2002). This approach is common in the physical sciences, but rarely used in the social sciences (Gottman & Notarius, 2000, 2002), though this is beginning to change (see Vallacher et al., 2010). A dynamical systems approach does not to seek correlations between variables, but: (1) uses experience, data, and intuition as the starting point to develop a mathematical model, (2) uses mathematics to ‘‘solve’’ the model, that its to determine its mathematical properties, and (3) then uses those properties to make predictions which can be empirically tested. Over the last 15 years, dynamical systems approaches have begun to be used in social, developmental and clinical psychology to describe human systems. Granic and her colleagues (Granic & Hollenstein, 2003; Granic & Lamey, 2002) have used dynamical systems to successfully model parent-child interactions and discover attractors in the system and other relational patterns that could guide intervention. Gardener, Burr and Wiedower (2006) have even suggested that dynamical systems may be able to provide some traditional schools of family therapy that were based on systems theory with a mathematical tool to explain various theoretical components (e.g., homeostasis) as well as some


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A mathematical model of psychotherapy quintessential family therapy techniques (e.g., paradoxical interventions, circular questioning, etc.).


Gottman et al.’s Work Gottman and his associates (Cook et al., 1995; Gottman et al., 2002; Gottman, Coan, Carrere, & Swanson, 1998) have used dynamic nonlinear equations to model the interaction between husbands and wives in relation to each other. Specifically, they utilized this quantitative mathematical approach to be able to understand the changes in the marital dyad between the members of the couple, and the relative influence that each has on the other (Gottman et al., 1998). These equations (one for the wife, one for the husband) allowed the researchers to model both the positive and negative interactions of the couple. They can be used to determine the stable ‘‘steady states’’ or points of homeostasis within the system (i.e., the relationship). These steady states function as an anchor point that brings the system back to homeostasis if the system is perturbed, or moved by a force away from homeostasis. In regulated systems, these steady states can act as attractors that, if the system is perturbed, can pull the system towards that steady state. Gottman et al. (2002) utilized a metaphor of a rubber band to describe the properties of attractors; if the band is stretched, it snaps back into its original state when it is released. Systems with stable steady states work in a similar fashion. Gottman et al. (1998) used equations that yielded scores based on two components: (1) The interpersonal influence from one spouse to the other, and (2) the individual’s own dynamics (or the uninfluenced behavior). The uninfluenced behavior may consist of their present emotional state, any past interactions that the couple have had (particularly with regard to conflict), any prior experiences (e.g., family-of-origin experiences), or any personal characteristics and dispositions (e.g., dysphoria or optimism). The influence component was defined as function of one person’s ability to move or change the other person’s emotional state (positively or negatively) from the uninfluenced steady state. Further, Gottman and his associates were able to model the process for repairing and dampening the other person’s emotional states, which created more steady states. Ideally, the more and better positive steady states, and the fewer negative steady states that a relationship has, the more satisfied and regulated the marital system is. This modeling has allowed for predictions about changes in the relationship to be made, and how to best intervene with the couple.1 This work has not been without critique, however (see Stanley, Bradbury, & Markman, 2000, and Gottman, Carre`re, Swanson, & Coan, 2000, for an

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in-depth critique and response). In particular, Heyman and Slep (2001) have cautioned against the overgeneralization of prediction without additional cross-validation (a point that Gottman, Swanson, & Swanson, 2002 and Coan & Gottman, 2007, subsequently addressed). Vallacher et al. (2010) specifically affirmed Gottman and his colleagues’ approach as a viable application of dynamical systems to realworld relationships (or ‘‘conflicts’’).

Application to Therapeutic Treatment: Formulation of the Mathematical Model While the original target of Gottman’s work was married couples, his use of dynamical systems and the equations used to model the interaction can be customized to reflect the unique characteristics of the therapist-client relationship. Of particular interest is the emotional influence of the therapist and client on each other in the creation of an effective (or ineffective) therapeutic relationship. Like couples, we would expect that the theoretical mathematical model could be used to describe and predict successful and unsuccessful (therapeutic) relationships depending on the parameters or conditions of the relationship. It is important to note that we are not reproducing the dynamics of marital relationships and applying them to the therapeutic relationship. Instead, we are using a dynamical systems approach (which has been used to model various other dyadic interactions, such as couples’ relationships) and applying the unique properties of the therapist and client relationship. Thus, modeling the therapeutic relationship between therapists and clients should allow researchers to be able to evaluate the quality of the relationship and the effectiveness of specific interventions that might create some significant therapeutic gains with a predictability that has not yet been seen in the clinical research literature. The information would allow researchers to study this relationship in detail, to be able to ascertain the parameters of the relationship, and to be able to see how specific intervention strategies can predict changes in clients as well as see how specific intervention strategies actually produce changes in clients. The starting point for the present project is the dynamic nonlinear mathematical equations that Gottman et al. (2002) used to model the emotional valence of marital dyads. We alter these equations to uniquely model the elements of the therapist-client relationship. We will also demonstrate*through computer simulations of the solutions of these equations*how varying some of the parameters creates new dynamics within the therapeutic relationship, and how this might affect the outcome of


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therapy. Lastly, we will discuss some insights for clinicians, and future directions for further assessing the therapeutic relationship and validating the various models found in the simulations.


The Equations of the Therapeutic Relationship In the model, we use the differential equations that Liebovitch et al. (2008) derived from Gottman et al.’s (2002) original marital difference equations (see Appendix for details). However, the major difference is the creation of influence functions for both the therapist and the client. These influence functions are essentially the mathematical ‘‘blueprint’’ for how each ‘‘actor’’ in the system will behave in reaction to the other ‘‘actor’’ (i.e., how the therapist responds to the client or how the client reacts to the therapist). The influence functions were created by considering what would be a ‘‘typical’’ therapist’s emotional reaction to a client’s display of negative, neutral, or positive emotion. We likewise considered what a client’s reaction would be to a

therapist’s display of negative, neutral, or positive emotion (see below for details). These influence functions were based on both our own clinical experience as well as from the literature on the therapeutic alliance (Bohart & Tallman, 2010; Duncan, Miller, Wampold, & Hubble, 2010; Gelso, 2009; Gelso & Hayes, 2002; Horvath & Bedi, 2002; Horvath et al., 2011; Lambert & Barley, 2002; Norcross, 2002, Norcross, 2010; Norcross & Wampold, 2011; Safran, Muran, & Eubanks-Carter, 2011; Safran, Muran, Samstang, & Stevens, 2002) and the literature on the behavior of expert therapists (Skovholt & Jennings, 2004). The result is two distinct influence functions that create a picture of what each person in the therapist-client dyad would be doing in relation to the other person at different levels of affect (from extremely negative to extremely positive). It is important to note that, at this point in the project, these influence functions are speculative in nature and provide a starting point for this exploratory project. They are presented visually in Figure 1(ac).

Figure 1. The influence functions for the model where thetherapist and client respond with equal intensity to each other. How the client influences the therapist is shown in (a) and how the therapist influences the client is shown in (b). The intersection of these functions, shown in (c), identifies two critical points (in the upper right and lower left).


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Description of client’s influence function. Figure 1(a) shows the client’s influence function on the therapist’s behavior. Negative affect. When the client’s affect is negative, the therapist will most likely exhibit more positive affect to try to draw out or encourage the client. However, they may, under prolonged ‘‘exposure’’ to the client’s negative affect, begin to exhibit neutral and even negative affect in the face of extreme negative behavior. This would create a steady state in the negative-negative space, which would effectively be the death of therapy*a ‘‘black hole’’ from which the therapeutic relationship dies (Bohart & Tallman, 2010; Gelso, 2009; Gelso & Hayes, 2002; Horvath & Bedi, 2002; Horvath et al., 2011; Norcross, 2010; Norcross & Wampold, 2011; Safran et al., 2002, Safran et al., 2011). Neutral affect. When the client is affectively neutral, therapists will generally utilize strategies to elicit more positive emotions. They will attempt to encourage clients, or try to get the client to focus on their strengths and abilities, in the hopes that this change of focus will change the client’s affect (Bohart & Tallman, 2010). At the same time, therapists may try to provoke any affect from the client (which may sometimes be negative). However, unless tied to a broader strategy, this is generally born out of frustration and may undermine the therapeutic alliance (Gelso, 2009; Horvath & Bedi, 2002; Lambert & Barley, 2002; Norcross, 2010; Safran, et al., 2002, Safran, et al., 2011). We will discuss this below. Positive affect. As the client’s affect moves from neutral to positive, initially, the therapist will also exhibit more positive affect. However, there is a point where, as the client’s affect becomes more positive, the therapist may begin to take a more neutral affective stance, as the therapist no longer needs to actively ‘‘encourage’’ the client, but the positive affect (presumably from some positive behavior change or symptom relief) sustains itself (Bohart & Tallman, 2010; Gelso, 2009; Gelso & Hayes, 2002; Horvath & Bedi, 2002; Lambert & Barley, 2002; Norcross, 2002; Norcross & Wampold, 2011; Safran et al., 2002). Description of therapist’s influence function. Figure 1(b) shows the therapist’s influence function on the client’s behavior. Negative affect. When a therapist exhibits negative affect, the client is likely to react even more negatively. The client may experience therapist negative emotion as judgmental, or a signal of

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some disappointment in the client. This may be the result of therapist frustration with either the pace of treatment or the client’s unwillingness to change, or fears about the therapist’s own performance in conducting therapy (Anderson, Lunnen, & Ogles, 2010; Bohart & Tallman, 2010). It is reasonable to suspect that this would be a part of novice therapists’ practice, but could also be reflective of therapists who may be on the brink of burnout. The therapist may not even acknowledge the frustration, but it may get picked up by the client (Safran et al., 2011), and move the therapy towards the more negative end of the graph. This is an indicator of a therapeutic rupture, which in turn is a predictor of premature termination form therapy (Muran et al., 2009; Norcross, 2010). At the same time, there may be circumstances when a display of negative emotion may be beneficial to the therapeutic relationship. In particular, appropriate confrontation or expressions of disappointment may be necessary feedback to the client (Safran et al., 2011). Again, the immediate result may be a therapeutic rupture, but if it is done purposefully or strategically, it may have a long-term benefit for the client. The success of this strategy depends a lot on the skill of the therapist and the strength of the therapeutic relationship (Anderson et al., 2010; Gelso, 2009; Gelso & Hayes, 2002; Horvath & Bedi, 2002; Lambert & Barley, 2002; Norcross, 2002; Norcross & Wampold, 2011; Safran et al., 2002, 2011). Neutral affect. When the therapist is affectively neutral, most clients are likely to be either slightly negative or neutral (particularly early in the therapeutic process). Some clients may not be influenced one way or another to a therapist’s neutral affect, unless they find (i.e., project) it to be a signal of therapist disinterest (e.g., the ‘‘tabula rasa’’ of psychoanalysis), at which point clients may react negatively, feeling that the therapist is too detached or ‘‘clinical’’ (Bohart & Tallman, 2010; Gelso, 2009; Gelso & Hayes, 2002; Horvath & Bedi, 2002; Lambert & Barley, 2002; Norcross, 2010; Safran et al., 2002). Positive affect. As the therapist’s affect moves from neutral to positive, initially, the client may remain neutral, or slightly negative (Bohart & Tallman, 2010; Gelso, 2009; Gelso & Hayes, 2002; Horvath & Bedi, 2002; Lambert & Barley, 2002; Norcross, 2002; Safran et al., 2002). However, as the therapist’s affect becomes more positive, the client may respond positively by exhibiting more neutral affect (Skovholt & Jennings, 2004; Safran et al., 2002). This could be a sign of the client either ‘‘buying into’’ the therapist’s message, or a sign that the client is beginning to experience some positive


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results from the therapeutic intervention. A positive steady state may emerge at this point, where therapeutic gains may be maximized (Norcross, 2010; Norcross & Wampold, 2011). However, as the extreme expressions of positive affect on the part of the therapist, the client might turn ‘‘negative’’ (i.e., get ‘‘turned off’’, especially if they perceive that it is disingenuous or too ‘‘pollyannish’’) (Horvath et al. 2011; Safran et al., 2011).

trenched in a fruitless struggle, develop a therapeutic rupture, and eventually lead to a termination of the relationship (Muran et al., 2009; Safran et al., 2011). At this point, the therapist, despite previous efforts at using positive or neutral affect to elicit some positive behavior change for a client, may*out of desperation or frustration*resort to expressing negative emotion to provoke a reaction. However, this is often done without much reflection and is frequently ineffective (Anderson et al., 2010).


Analysis of the Mathematical Models We now analyze the dynamics of the mathematical model by: (1) First identifying the values of the behavior state of the therapist and client that define the "critical points’’2 which can represent the final steady-state values that they reach at the conclusion of therapy, (2) then determining which values of the initial behavior state of the therapist and client will reach those final steady states, and (3) finally investigating*through computer simulation*how the dynamics of the relationship between therapist and client depends on the parameters of the model, and how the relationship changes based on the initial parameters of the models (i.e., different therapistclient combinations). First, it was important that we plot both the client and therapist influence functions on the same graph. The points where they intersect tell us the critical points that could represent their steady states or attractors for the system. This is illustrated in Figure 1(c) where the two influence functions are plotted on a graph of the behavior state of the therapist versus the behavior state of the client. On that plot we can see two intersections that identify two critical points. The stable critical point in the upper right of Figure 1(c) is a good outcome, because both the therapist and client have positive emotional states for that stable steady state. It is at a point where the therapist has more positive affect than the client. This seems*for this simulated system*to be the most beneficial spot for a therapeutic relationship to be in. One interpretation is that, at this point, the client is at least in the positive space, but may be trying to consolidate the new behaviors or way of thinking with their present life. If they have been initially successful, then the therapist will have a good working relationship with the client to allow him or her to generalize to other areas in the client’s life (Norcross, 2010; Norcross & Wanpold, 2011). The unstable critical point in the lower right of Figure 1(c) is in a negative-negative space. We will see (below) that the dynamics near this critical point drags the client into ever more negative behavior states. Therapeutic relationships that move in this direction are in serious jeopardy of becoming en-

Dynamics of the Therapeutic Relationship We now continue the analysis of the model presented in Figure 1 by determining the dynamics, how the expression of emotions between the therapist and client evolve together in time. We do this by using a plot called a ‘‘phase-space.’’3 Each point in this phase space corresponds to one value T for the behavior state of the therapist on the vertical axis and one value C for the behavior state of the client on the horizontal axis. The therapist and client each start with an initial behavior state called their ’’initial condition.‘‘ Their initial condition, having one value of T for the therapist and one value for C for the client, is represented by one point in the plot. The dynamics of the therapeutic relationship can then be visualized by following the path of this point, called its ’’trajectory," as it moves through the plot (much like a swimmer caught in a current) towards or way from an attractor. Figure 2 shows the phase-space plot of the trajectories of the therapist and client where the therapist and client have equal influence (meaning that the client and the therapist react emotionally to each other in an equal way) with each other. This corresponds to the same parameters and influence functions shown in Figure 1(c)4. The lines (trajectories) in Figure 2 clearly show that many initial conditions are attracted to a stable endpoint at the critical point in the upper right corner or they will be attracted to the critical point in the lower left of the figure. For example, if a therapist starts the therapeutic interaction at a 5 and the client starts at a 3, the point were the two intersect is the initial condition. The line that passes through that point represents the trajectory that the relationship is projected to take toward the stable point. This does not mean that the relationship is predetermined to go to that point, but that, according to the system dynamics, it indicates that*over time*unless there is an effort made to move away from the attractor point, the relationship will end up at the homeostasis point emotionally. In this instance, where client and therapist have equal influence with each other, the relationship will


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cross & Wampold, 2011). The only exception is if the client is initially very negative, in which case, the relationship will invariably be pulled towards negative emotional states.

Figure 2. Phase-space plot of the dynamics of the therapeutic relationship corresponding to equations (1a) and (1b) and the influence functions in Figure 1. In this model the therapist and client respond with equal intensity to each other. Each point in the plot reflects the behavior state of the therapist (vertical axis) and the client (horizontal axis). At the stable steady-state attractor both the therapist and client are in mildly positive states.

likely end up in at this positive attractor, as long as the therapist begins with positive emotion. However, if the client starts therapy in a very negative emotional state (where ‘‘C’’ starts at 5 or 4) then the therapist must be more positive in order to overcome the movement towards the negative state (‘‘T’’ starting at 2 or higher). Furthermore, in this model, if client begins therapy with a mild negative state (1) or is neutral, the therapist can also match the negative emotion and still attract the relationship towards the positive stable steady state (approximately 1 for therapist and client). In addition, if the client starts therapy with very positive affect (2, to 5) the therapist can also display some negative or neutral emotion and still draw the relationship to the positive steady state. ‘‘Going negative’’ can be a strategy for the therapist to either bring a client who is mildly negative or neutral ( 1 or 0) about therapy into a positive space. It may also be a strategic method for ‘‘tamping down’’ a client who is displaying highly (and possibly unrealistically) positive emotions. Since, in this scenario, both the client and the therapist are equally influential of the other, one can look at the other side of the coin. Specifically, if the therapist initially is highly positive (5 to2) and the client is either negative or positive, the therapist will be drawn down toward the positive stable steady state. The key, it seems, for this relationship is that (in most instances) the therapist must avoid beginning with a negative affect (Nor-

Producing different models by changing the parameters of the therapeutic relationship. The dynamics of the therapeutic relationship shown in Figure 2 represents our first model using our basic set of parameters and influence functions. The power of a mathematical model is that we can now explore, through computer simulations, what happens when we change those parameters. In particular, we varied the parameters of the therapist’s or client’s reactions to each other’s influence, and the strength of therapist’s or client’s response to their previous emotional state (or level of inertia). In other words, how reactive one person is to the other person’s emotional display determines the level of influence. In addition, how strong a person’s response is to their previous emotional state is a reflection of the level of inertia or entrenchment that one person has to how they have been feeling all along, as well as how quickly they may be willing to abandon or consciously change it. In each of the figures that follow, one or more of the parameters of the equations (1a) and (1b) has been changed from the base model in Figure 2 to simulate different aspects of the relationship between therapists and clients. We discovered that there were three distinct ‘‘types’’ of relationships that emerged (in addition to the initial model detailed above), which will describe each below. Model 1: The highly responsive client (or influential therapist) yields positive outcome. Figure 3 shows the phase-space plots of the trajectories of the therapist and client if the client responds very strongly to the therapist (sT 0C 10 rather than 1). This may be in indication of a very influential or skilled therapist. Just as in Figure 2, there are two critical points, one of which is a stable steady-state attractor where the client is very positive and the therapist is moderately positive, and the other is an unstable critical point. There are some noteworthy issues. First is that the therapeutic relationship is attracted toward the positive steady state. Second, the emotional state of the relationship spirals, or oscillates up and down in time before reaching a final steady state. This seems to be in line with clients’ frequent oscillations (i.e., ambivalence) regarding change. Third is that the client winds up more positive than the therapist, and that the relationship is attracted to the steady state rapidly (as indicated by the ‘‘tight’’ spiraling). This seems to be a very good outcome for therapy. Fourth is a cautionary finding that the client will likely respond very strongly to any


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Figure 3. Phase-space plot of the dynamics of the therapeutic relationship when the client responds very strongly to the therapist. There is one stable steady-state attractor (middle right) where the client is in a positive state and the therapist is in a mildly positive state. The other trajectories are drawn towards the increasingly negative states (lower left).

negative input from the therapist. As shown in Figure 3, any time that the therapist starts with a negative ‘‘score’’ on the therapist axis, the relationship will be directed towards the unstable critical point and from there to increasingly negative values. On the other hand when the therapist starts with positive scores, the relationship will be directed towards the positive steady state. The only exception to this is when the client scores start very negative ( 5 to 3) and the initial therapist scores are only neutral or slightly positive therapist scores (0 to 1). Thus, just as in Figure 2, as long as the therapist begins with positive emotion, the client will reach this positive outcome, unless the client starts very negative about treatment. Fifth, this model may be an ‘‘ideal’’ scenario for a brief therapy, where change is swift (as indicated by the close lines) and the client is satisfied. Second, just as in Figure 2, as long as the therapist begins with positive emotion, the client will reach this positive outcome, unless the client starts very negative about treatment. An additional simulation (not illustrated) was run that modeled a relationship if the client responds more weakly to their own previous state (aC 0.1 rather than 1.0), but with equal influence (sT 0C and sC0T 1). This may be indicative of a client who is very motivated to change (i.e., responds weakly to their pervious state) because they are not so ‘‘entrenched.’’ The client may not be so rigid in his or her own point of view, or they are so ‘‘sick and tired’’ of how bad things have gotten that they are

not clinging to their old ideas or behaviors. As a result, they are open to accepting influence from their therapist. This is a very positive or optimistic therapeutic relationship. As with Figure 3, there are two critical points, one stable and one unstable, both in the same relative places as in Figure 3 (as a result, we have omitted these phase-space plots), though there are several important points worth discussing. First, if the client begins with very negative emotion, the therapeutic relationship is unlikely to move towards the positive attractor, no matter how positive the therapist is initially. Clearly, in this scenario, if the client is very negative (e.g., toward therapy*a ‘‘precontemplator’’), then the therapist taking a neutral stance is probably not going to effect much change. Second, the movement here towards the steady states is slower, therefore the therapist needs to be patient with the pace of therapy, otherwise the client may pick up on the therapist’s frustrations and the relationship become attracted to the negative steady state. This is important feedback to therapists in training, or therapists who may be struggling with clients who are ready for change, but might not have much hope that they can be helped (i.e., feeling very negative and ‘‘sick and tired’’ of the status quo). Model 2: Being too responsive to clients (and not very influential) produces mediocre outcomes. Figure 4 shows the phase-space plot of the trajectories of the therapist and client if the therapist responds very strongly to the client

Figure 4. Phase-space plot of the dynamics of the therapeutic relationship when the therapist responds very strongly to the client. There is one stable steady-state attractor (middle top) where the therapist is in a positive state and the client is in a neutral state. The other trajectories are drawn towards the increasingly negative states (lower left).


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A mathematical model of psychotherapy (sC 0T 10 rather than 1). Again, there are two critical points, one stable and one unstable. However, in this case, the stable attractor is located where the therapist is very positive and the client is neutral. Again, there are oscillations between positive and negative feelings before settling on the steady state. Looking at Figure 4, and following along, the client is very influential and persuasive, or the therapist is very responsive to the client. In this model, if the client starts negatively (from 5 to 3), no matter how positively the therapist begins the therapy, the relationship will be drawn towards the unstable critical point and towards ever more negative states (a very poor outcome). If the client starts mildly negative (1 to 2), and the therapist starts neutral or mildly positive, the relationship will also be drawn towards these negative outcomes. If the therapist is more positive when the client starts slightly negative, then the relationship will be attracted towards the other steady state where the therapist is positive and the client is neutral. However, if the client starts positive (1 to 5), the therapist can be either positive or negative, and the relationship will be attracted to the steady state. For most clients, this will mean moving from a positive emotional state to a neutral state (and probably not a great outcome). However, there may be some scenarios where this is a ‘‘positive result’’ therapeutically. For example, if a client is unrealistically positive (e.g., euphoric or manic), then employing strategies (even using/displaying negative emotion) to bring them towards a more realistic view (e.g., neutral) might be beneficial. In addition, clients who are mildly negative initially, in this scenario, may reach a neutral state (a better outcome emotionally than at the beginning). In most cases, however, this configuration is not optimal for successful therapy. Another simulation (not illustrated) was run with the parameters of the influence functions set for the therapist and client if the therapist responds more weakly to their previous state (aT 0.1 rather than 1.0, and sT 0C and sC0T 1). Again, because of the similarity of the plot, it is not re-printed here. In this phase-space plot, clients who begin therapy with negative emotion can very quickly move to a positive feeling state and then to neutral, as long as the therapist is neutral or positive, emotionally. The only appreciable difference to Figure 4 is that, since the therapist responds more slowly to changes, it takes more time for the therapeutic relationship to approach the steady state (again, because the lines are spread further apart). This may represent a therapist who is more patient with the client, especially clients who are initially negative about therapy, but who are ‘‘won over’’ (at least in part). The problem is that, in this model, there is a strong

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pull towards the negative attractor that may be tough to overcome. Another explanation for why this may be happening is that novice therapists in this situation may seem ‘‘wishy-washy’’ or unsure about their strategy or approach to therapy. Therefore, they may seem to be less directive, which can become frustrating to the client who is seeking direction or more active help from the therapist. The therapist may feel good about the therapy they are delivering, and think that they are being attentive to the client, but the client probably doesn’t feel very satisfied. The client may feel like the one who is ‘‘leading’’ the therapy and that it is not ‘‘going’’ anywhere (or moving forward), or that therapist is not doing much. The key for the best outcome in this model is a positive emotional stance by the therapist. Model 3: Client and therapist exert strong influence and produces complex system. Figure 5 shows the phase-space plot of the trajectories of the therapist and client if both the therapist and the client respond very strongly to one another (sC 0T sT 0C 10 rather than 1). There are a total of four critical points. Two are stable attractors and two are unstable critical points. The stable attractors that are present roughly correspond to the attractors in Figure 3 (where the client is more positive than therapist) and Figure 4 (where the client is neutral and the therapist is positive). In addition, there is unstable critical point where the therapist is neutral and the client is slightly negative and another unstable critical point where both the therapist and

Figure 5. Phase-space plot of the dynamics of the therapeutic relationship when the therapist and client both respond very strongly to each other. There are two stable steady-state attractors, similar to those in Figures 3 and 4.


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client are positive. This is one of the more active and dynamic systems for therapy, as there are three potential outcomes. First, if the client is initially extremely negative (5 to 2), and the therapist is initially neutral, or not positive enough (1 or 2), the relationship will be attracted towards the very negative region. Second, if the therapist is initially extremely positive, and the client begins slightly negative, neutral or slightly positive, then they will end up at the neutral attractor (for the client). Third, however, if the therapist is initially positive and the client is initially negative, or if the therapist is initially positive to neutral while the client is initially positive, then the relationship will end up at the attractor which is positive for both. In other words, therapist neutrality (unless in the face of very positive emotion from the client) is not always predictive of good outcome for the relationship. Additionally, the client will either be ‘‘won over’’ to the far positive steady state, or will stay entrenched in a neutral affect (and probably terminate). Therapists who react very strongly to their clients may seem to be really ‘‘in tune’’ with their client. They may be able to connect with their clients quickly, and be able to intervene effectively with their clients. However, they may also be more sensitive and either over-react or react negatively to the client. At the same time, clients who react strongly are apt to be influenced by their therapists. These clients may not necessarily be motivated for change initially (i.e., if they initially begin therapy in a negative state), but may ‘‘feed off’’ the therapist’s enthusiasm and influence. This may be indicative of clients who are emotional labile and prone to shifts in mood. As with Figures 3 and 4, an additional simulation was run (not illustrated) where the parameters for the influence functions were changes so that both the therapist and the client respond very weakly to their previous state (aC aT if, 0.1 rather than if, 1.0). Again, the result is very similar to Figure 5 (with two stable attractors and two unstable critical points) except that more time passes before the relationship arrives at a stable steady state (because the spirals are not a close as in the other figures). This type of therapeutic relationship would have therapists and clients alike who may be more present-focused, and more process-oriented. They may be more reflective and introspective, but may also be more likely to adopt a solution-focused approach where the therapist and client collaborate on solutions rather than problems (see Duncan, Miller, Wampold & Hubble, 2010). Thus we would expect many of the same dynamics to result as the model in Figure 5, but taking longer to achieve.

Conclusion The purpose of this paper was to show that dynamical systems could be used to model the emotional exchange and strength of the therapeutic relationship in a way that would provide useful information for clinicians. The simulations were run on the basis of the influence functions, and yielded some interesting theoretical models of what happens in effective and ineffective therapy (i.e., symptom reduction, problem resolution, and client satisfaction). From the simulations, four different models emerged so that, after investigating the different phase-space plots, we can begin to argue or hypothesize that some strategies (or approaches) will be more effective given certain conditions or parameters. While it can be said that this is already being done in the literature (e.g., Gelso, 2009; Muran et al., 2009; Safran et al., 2011), these simulations give us a starting point for both explaining and predicting why this happens in therapy. Gottman et al. (2002) seemed to understand the potential of this approach beyond the marital dyad: The model (that was developed) has given birth to a theoretical language about the mechanism of change . . . The model provides the language . . . suggests variables that can be targeted for change using interventions. In short, the model leads somewhere. It helps us raise questions, helps us wonder what the parameters may be related to and why. It raises questions of etiology . . . Thus, it is likely that the major contribution of the model will be the theoretical language and mathematical tools it provides. It will give us a way of thinking . . . that we never had before. (2002, p. 172, parentheses added) With these simulations, clinical researchers can begin to predict the strength or quality of the therapeutic relationship (and subsequently outcome in therapy) based on the value of certain initial values at the beginning of therapy. Using dynamical systems we can make predictions about how negative outcomes result from some of these beginning values, and (more importantly) how to switch strategies (or approaches) in therapy to get a better outcome.

Implications for Practice There were several interesting findings that the simulations revealed, which are worth pursuing. First, an overall analysis of the models shows that the person (therapist or client) who is most responsive to the other winds up being the most positive, that is, in the positive steady state. This seems, on its


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A mathematical model of psychotherapy face, to support Gelso’s (2009) assertion that therapist neutrality is not always the best strategy, and may even be detrimental to the therapeutic relationship (as indicated by Norcross & Wampold, 2011). Next, in most, but not all, of these examples, the steady state is reached through a spiral trajectory. That is, the client (and therapist) will go through up and down emotional swings before reaching their final steady states. This seems to be validated by clinical observations dating back to Freud and his concept of resistance, regarding clients vacillating and displaying ambivalence about making lasting changes. This is also reflective of the work of Miller and Rollnick (2002) and the Motivational Interviewing approach on resolving ambivalence rather than merely ‘‘confronting’’ it (specifically in light of the finding that the therapeutic relationship suffers when therapists are negative). Third, for most simulations, the client winds up in a very negative emotional state if the therapist starts in a negative emotional state. This was an overwhelming finding, and*though it might seem self-evident*it should be reinforced to practitioners and trainees everywhere. Fourth, for most simulations, even if the client starts in a negative emotional state, if the therapist is initially neutral or positive, the client will wind up in a positive emotional state. The impact of the valance of the relationship (positive or negative) is crucial in the ultimate outcome of therapy: The client may genuinely and realistically not especially like the person of the therapist, or the client may realistically perceive or ‘‘subceive’’ negative reactions on the therapist’s part of which the therapist is perhaps unaware. There can indeed be negative reactions within the context of a real relationship. At the same time, negative valence would reflect a weaker real relationship. (Gelso, 2009, p. 255) By and large, this is an optimistic finding for the practice of psychotherapy, and also seems to support over 50 years of findings that psychotherapy is generally effective (Lambert & Barley, 2002; Norcross, 2010; Norcross & Wampold, 2011). Also, we found that increasing the influence of the other person yields the same type of phase-space trajectories as responding more weakly to the previous emotional state. The difference is that responding more weakly to the previous emotional state*i.e., increased resistance to change*means that it will take a longer time to effect change. Thus, the two key variables that therapists must be mindful of is the amount of influence they and the client have over each other in a given therapeutic relationship, and how strong the response is to their own previous

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emotional state. On the therapist’s part, this ‘‘resistance’’ might be indicative of long-held beliefs about therapy (theoretical approaches, stance of the therapist, etc.), and for the client, it might be the result of their own long-held beliefs about life or whether they are ready for (or feel that they need to) change. However, in light of Gelso’s (2009) finding about the role of the ‘‘real relationship’’ in the therapeutic endeavor, understanding how these variables affect the emotional valence of the relationship can have a direct effect on the strength of the therapeutic relationship and working alliance. Indeed, as Hill and Knox (2009) have detailed, monitoring and addressing these relationship issues are important in enhancing outcomes. The present findings can provide a roadmap for clinicians to monitor themselves, their clients, and the therapeutic relationship, relative to the emotional valence of both the client and the therapist. In addition, therapists can determine which of the four models their particular therapist-client relationship is following, and whether they are heading towards a positive or negative steady state. At the same time, these findings must be evaluated in a real-world context with actual therapeutic relationships. This is the next step in this work.

Future Research While the results of this investigation are both exciting and promising, there are several questions that remain to be answered. Thus, the next logical step in this project is to subject the models that emerged from the simulations to empirical testing. Interestingly, Aharonovich, Amrhein, Bisaga, Nunes, E. & Hasin (2008) recommend to researchers that the ‘‘dynamic aspect of the (therapeutic) process can be studied empirically through analysis of codes assigned to patient statements in recorded therapy sessions’’ (p. 557, parentheses added). As Gelso (2009) noted, the therapeutic relationship is both a ‘‘real’’ and a professional relationship where the therapist is explicitly trying to influence the client, and the client will likely be resisting this influence (actively or passively). Therefore, we would expect that the dynamics of the therapeutic dyad would reflect a more direct and active process of influence by the therapist as well as active influence (resistance or reactance) by the client. Following from the simulations of this relationship, we can create some testable hypotheses about how to tailor therapeutic interactions based on the parameters of these models. What we don’t know is how long each of the models identified takes to get to the steady state. We also don’t know what exactly makes up very negative, negative, neutral, positive, and very positive affect.


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Specifically, therapists and clients could be videotaped and their therapeutic interactions could be coded to see if they fit one of the four models discovered through the simulations in the present paper (or if new models may need to be created to better fit with the observed data). This is one of the strengths of a dynamical systems approach (Liebovitch et al., 2008; Vallacher et al., 2010). The initial session could be videotaped and coded, and then a follow-up session could be recorded to see the fully developed therapeutic relationship. In addition, research with real therapist-client dyads would be able to designate those relationships that make up successful therapy, and those that do not, based on clinical outcome measures, client satisfaction measures, and self-report measures of the therapeutic relationship. It would be possible to sample both expert and novice therapists to see if there are any systematic differences between them regarding the models that they fit into. Lastly, it might be possible to discover how to intervene strategically with a client or with a therapist when the relationship is pulled toward a negative attractor. Another powerful aspect of this kind of research would be identifying how the affective elements of the therapeutic relationship impact the mechanisms of change over time (either positively or negatively). At this point, specifically identifying the amount of time as a parameter is one of a number of empirical questions that we are interested in pursuing, but that we cannot definitively answer at this time (see Liebovitch, Peluso, Norman, Su, & Gottman, in press, for more details). A frequent problem that researchers using dynamical systems have faced is estimating the parameters from the data and then applying the best fit to the data. Models can be created, but if there is no way to evaluate their goodness of fit to real data, then the models are illustrative and speculative at best, but may not hold up to the most basic principle of science: falsifiability. The parameters of the model determine how the emotional state of the therapist and client depend on time. This predicted time series can be compared to an experimentally measured time series developed by a suitable coding of the emotional states measured from video recordings. The parameters can then be found, through numerical methods, that minimize the difference in the sum of the squares of the errors between the predicted and measured time series. This was the method used by Gottman et al. (2002). Recently, Hamaker and her colleagues (Hamaker, Zhang & Van der Maas, 2009) also devised a method for estimating the parameters for a model and compute a Bayesian Information Criterion (BIC) which can then be used to compare different influence func-

tions and determine the best fit for the given interaction. In fact Madhyastha, Hamaker, and Gottman (2011) have recently used this to test Gottman et al.’s (2002) original influence functions with interesting results. Thus, employing this same approach, it might be able to compare the different influence functions for therapists and clients and determine the best-fitting dynamical systems model, which may solve the problem of determining the effect of therapist ‘‘responsiveness’’ (differing levels of affective involvement that emerges within the context of client needs) on client outcomes that Norcross and Wampold (2011) raised. Summary The purpose of this paper was to demonstrate the usefulness of modeling the affective nature of the therapist-client relationship using dynamic systems. Based on the influence functions created to describe the client’s and therapist’s reactions, we have developed four archetypal therapeutic relationships based on this computer simulation of the parameters (see Appendix). These theoretical models provide a starting point for a larger discussion for both clinicians and researchers regarding the dynamics of the therapeutic relationship. In addition, there may be more ‘‘types’’, based on changes to the influence functions. Research using actual therapistclient dyads and coding the affective exchange is the next step in empirically validating, or altering, these models. While it may not be realistic to expect clinicians to routinely videotape and code the interactions with clients, we hope that future research might be able to better define the ranges of negative, neutral and positive affect, and provide clinicians with some clear exemplars of each within a therapeutic relationship. The ultimate goal is to detail the elements of the therapeutic relationship that can reduce the rates of premature dropout, and increase the effectiveness of clinicians. By looking at the mechanisms of the therapeutic relationship, in depth and detail, we aspire to shed some light into this important area. Notes 1

Once the equations were completed, Gottman and his associates sought to test their models on couples. Couples were videotaped, and the quality of their interaction was coded using the Specific Affect Coding System (SPAFF; Gottman et al, 1996). The SPAFF codes specific emotional behavior of a husband and wife in real time, and can be used in any conversation. According to Gottman et al., the SPAFF focuses solely on the affects expressed by the participants, drawing on facial expressions, vocal tones, and speech content to characterize the emotions that are displayed. SPAFF coders ‘‘categorized the affects displayed using five positive codes (interest validation,


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affection, humor and joy), 10 negative affect codes (disgust, contempt, belligerence, domineering, anger, fear/tension, defensiveness, whining, sadness, stonewalling), and a neutral affect code’’ (2002, p. 179). The final weighted scale ranged from 24 to 24, giving equal weighting to the five positive codes and the 10 negative codes. Couples were asked to have a 15-minute discussion about an area of ongoing conflict, which was videotaped for coding using the SPAFF. Each videotape was coded in its entirety by two independent observers, and then used to determine the parameters of the previously formulated nonlinear equations of the mathematical model. The results of their analyses yielded support for the predictive ability of the parameters of the model to discriminate between three separate criterion groups (happy stable couples, unhappy stable couples, and divorced couples), as well as discover the importance of each of the four parameters (husband and wife’s influence and uninfluenced steady states) and the ability to intervene therapeutically (for a full discussion of these results, please see Gottman et al., 2002). Critical points can be ‘‘stable’’ or ‘‘unstable.’’ If small changes in T and C bring both the therapist and client back to their values at the critical point, then that critical point is a stable steady state. This is a point of equilibrium that the relationship would gravitate towards (or be attracted to). It is therefore called an ‘‘attractor’’ of the relationship. However, if small changes in T and C always push the therapist and client further away from the critical point, then it is unstable and does not represent a final steady state. We will see in the computer simulations below that the critical point at the intersection of the influence functions in the upper right of Figure 1(c) is a stable steady state, while the critical point at the intersection of the influence functions in the lower left of Figure 1(c) is unstable. This has important (and potentially useful) implications for the therapeutic relationship. We used computer software (MATLAB ODE113) to calculate how the values of T and C, for the behavior state of the therapist and client respectively, in equations (1a) and (1b), change in time when they start from many different initial conditions. We then plot these trajectories on the phase-space. These simulations create a picture of how the therapeutic relationship might change given different initial conditions of the client and the therapist. These plots allow for a visual representation of the dynamics within the system, and form the basis for predictions within the model. In fact, if you overlay both Figure 1(c) and Figure 2, the intersection of the client and therapist influence function exactly correlates with the steady states in the phase space of Figure 2.

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Norcross, J.C. (2010). The therapeutic relationship. In B.L. Duncan, S.D. Miller, B.E. Wampold, & M.A. Hubble (Eds.), The heart and soul of change: delivering what works in therapy (2nd ed, pp. 113142). Washington DC: American Psychological Association. Norcross, J.C., & Wampold, B.E. (2011) Evidence-based therapy relationship: Research conclusions and clinical practices. Psychotherapy, 48(1), 98102. doi: 10.1037/a0022161. Orlinsky, D.E., Grave, K., & Parks, B.K. (1994). Process and outcome in psychotherapy. In A.E. Bergin & S.L. Garfield (Eds.), Handbook of psychotherapy and behavior change (pp. 257 310). New York, NY: Wiley. Orlinsky, D.E., & Howard, K.E. (1977). The thrapists’s experience of psychotherapy. In A.S. Gurman & A.M. Razin (Eds.), Effective psychotherapy: a handbook of research (pp. 566589). New York, NY: Pergamon. Prochaska, J.O., & DiClemente, C.C. (2003). The transtheoretical approach. In J.C. Norcross & M.R. Goldfried (Eds.), Handbook of psychotherapy integration (2nd ed., pp. 147171). New York, NY: Oxford University Press. Safran, J.D., Muran, J.C., & Eubanks-Carter, C. (2011) Repairing alliance ruptures. Psychotherapy, 48(1), 8087. doi: 10.1037/a0022140. Safran, J.D., Muran, J.C., Samstang, L.W., & Stevens, C. (2002). Repairing alliance ruptures. In J.C. Norcross (Ed.), Psychotherapy relationships that work: Therapist contributions and responsiveness to patient needs (pp. 235254). New York: Oxford University Press. Skovholt, T.M., & Jennings, L. (2004). Master therapists: exploring expertise in therapy and counseling. Boston: Allyn & Bacon. Stanley, S.M., Bradbury, T.N., & Markman, H.J. (2000). Structural flaws in the bridge from basic research on marriage interventions for couples. Journal of Marriage and the Family, 62, 256264. Vallacher, R.R., Coleman, P.T., Nowak, A., and Bui-Wrzosinska, L. (2010) Rethinking intractable conflict: The perspective of dynamical systems. American Psychologist, 65(4), 262278. doi: 10.137/a0019290. Wampold, B.E. (2010). The research evidence for common factors models: A historically situated perspective. In B.L. Duncan, S.D. Miller, B.E. Wampold, & M.A. Hubble (Eds.), The heart and soul of change: delivering what works in therapy (2nd ed, pp. 4981). Washington DC: American Psychological Association.

Appendix The variables T(t) and C(t) denote the therapist’s and client’s behavior scores respectively at time t. The parameters aT and aC denote the ‘‘inertia’’ to change, that is, the degree of dependence of the therapist’s and client’s behavior scores on their previous values respectively, where aT, aC B0. The smaller the absolute value of these parameters, the less is the dependency on previous behavior scores. The parameters bT and bC denote the therapist’s and client’s behavior scores respectively when alone. The last parts of equations are the influence functions, which model the effect that the therapist or client has on the other person, that is, they are a function of either the client’s or the therapist’s behavioral score.


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A mathematical model of psychotherapy The functions fC† T(C) and fT† C(T) denote the form of the influence of the client on the therapist and the influence of the therapist on the client respectively and sC† T and sT† C denote the strength of those influences. The complete therapy equations are then:


dT =dt ¼ aT T ¼ bT þ sC! TfC! TðCÞ dC=dt ¼ aC C þ bC þ sT ! CfT! C ðTÞ

(1a) (1b)

Translating these mathematical symbols into words, these equations state that the rate of change of the behavior state of the therapist and client (dT/dt and dC/dt) at time t is proportional to the sum of their ‘‘inertia’’ to change (aTT and aCC), their uninfluenced behavior state (bT and bC), and the influence from each other (sC 0T fC 0T(C) and sT 0C fT 0C(T)). In other words, both the therapist and the client begin each session with an initial behavior score, and then these scores change over time based on the influence (or interaction) of the other person over time. Steady states in the therapeutic relationship. The behavior state of both the therapist (T) and client (C) start from some initial values and both

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evolve in time. We are very interested in determining where they wind up, as this represents the outcome of the therapy for both the therapist and client. Such steady states, if they exist, must occur where the values of T and C are no longer changing in time, namely where the derivatives dT/dt0 and dC/ dt 0. The special values of T and C where this occurs are called the ‘‘critical points.’’ The equations for dT/dt 0 and dC/dt 0 are each called the ‘‘nullclines.’’ Since the critical points satisfy the equations of both nullclines, they lie at the intersections of the two nullclines. Therefore, we can easily see the values of these critical points graphically as the intersections of the nullclines on a plot where the behavior state of the therapist is on the vertical axis and the behavior state of the client is on the horizontal axis. For the first case that we study here, the client and therapist each have an equal strength of influence over each other, although the shape of their influence functions is each different (as described above). Since, for this case, the ‘‘inertia’’ parameters aT aC 1, the nullclines are the same as the influence functions.


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APPENDIX B COGNITIVE NEURODYNAMICS PUBLICATION Open Access: This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Cogn Neurodyn (2011) 5:265–275 DOI 10.1007/s11571-011-9157-x

RESEARCH ARTICLE

Mathematical model of the dynamics of psychotherapy Larry S. Liebovitch • Paul R. Peluso • Michael D. Norman • Jessica Su • John M. Gottman

Received: 23 March 2011 / Revised: 3 May 2011 / Accepted: 9 May 2011 / Published online: 22 May 2011 Ó The Author(s) 2011. This article is published with open access at Springerlink.com

Abstract The success of psychotherapy depends on the nature of the therapeutic relationship between a therapist and a client. We use dynamical systems theory to model the dynamics of the emotional interaction between a therapist and client. We determine how the therapeutic endpoint and the dynamics of getting there depend on the parameters of the model. Previously Gottman et al. used a very similar approach (physical-sciences paradigm) for modeling and making predictions about husband–wife relationships. Given that this novel approach shed light on the dyadic interaction between couples, we have applied it L. S. Liebovitch Charles E. Schmidt College of Science, Center for Complex Systems and Brain Sciences, Center for Molecular Biology and Biotechnology, Department of Psychology, Florida Atlantic University, Boca Raton, FL 33431, USA L. S. Liebovitch Division of Mathematics and Natural Sciences, Department of Physics, Queens College, City University of New York, Flushing, NY 11367, USA P. R. Peluso College of Education, Department of Counselor Education, Florida Atlantic University, Boca Raton, FL 33431, USA M. D. Norman (&) Charles E. Schmidt College of Science, Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA e-mail: [email protected] J. Su Charles E. Schmidt College of Science, Florida Atlantic University, Boca Raton, FL 33431, USA J. M. Gottman The Gottman Institute, University of Washington, Seattle, WA 98115, USA

to the study of the relationship between therapist and client. The results of our computations provide a new perspective on the therapeutic relationship and a number of useful insights. Our goal is to create a model that is capable of making solid predictions about the dynamics of psychotherapy with the ultimate intention of using it to better train therapists. Keywords Nonlinear phenomena Dynamical systems Dynamical systems theory Ordinary differential equations Biological systems Psychotherapy

Introduction One in four adults in the United States suffers with a diagnosable mental disorder. These disorders are the leading cause of disabilities and extract a physical and emotional toll on these individuals, their families, and their communities. Psychotherapy has been shown to be an effective method for treating these disorders (Lambert and Barley 2002; Kazdin 2008; Mozdzierz et al. 2009). Yet, only one quarter of those with these disorders seek psychotherapy and one half drop out after the first session (Muran et al. 2009). A therapist in possession of a better understanding of psychotherapy would be able to improve the success of therapy, reduce the client drop out rate, and yield better ways to train novice therapists. The success of psychotherapy depends on the nature of the therapeutic relationship between a therapist and a client. Studies have sought to identify the most essential elements of this relationship. Although, those elements are not fully understood, previous psychotherapy studies have reported that the essential element is the personal relationship between the therapist and client, rather than an abstract

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Figure B.1: Cognitive Neurodynamics p. 1

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theoretical framework used by the therapist (Lambert and Barley 2002; Kazdin 2008; Mozdzierz et al. 2009; Muran et al. 2009; Orlinsky and Howard 1977; Martin et al. 2000). This suggests that some features of this relationship can be represented by a model that describes how two people, a dyad, react to themselves and to each other. Previous studies of this relationship dyad have used the social science paradigm of determining the functional correlations between dependent and independent variables. Here we use a physical science paradigm to investigate the nature of this relationship. Just such a physical science paradigm approach, based on rigorous mathematical modeling, was pioneered by Gottman et al. (2002) to study the interactions between husbands and wives and it proved useful in understanding the stability (or instability) of their marriages. They showed that this approach gives insights into the dynamics of a marriage and has the power to make specific successful predictions about whether the marriage is stable or ends in divorce. We now modify and extend their approach of husband–wife dyads to analyze and understand the therapist–client dyad. Our work is the first mathematically rigorous model used in the study psychotherapy (with the exception of Gottman’s work, mentioned above). Psychotherapy studies have been done on a ‘case study’ basis or have used an intuitive approach with no mathematical backbone. In our approach, we formulate a rigorous mathematical model of the therapist–client relationship based on published empirical data and our own experience, determine the dynamical properties of that model, and then compare those properties to the known properties of the therapist–client relationship. We will show that this approach yields new insights into the therapeutic relationship. This model cannot, and is not intended to, represent the full nature of the complex human interaction in psychotherapy. However, the fact that it does reveal important insights suggests that some simple dynamical features may underlie the more complex behaviors that emerge in the therapeutic relationship. The long term goal is for such a theoretical mathematical model to be used to describe and predict successful and unsuccessful therapeutic relationships depending on the parameters or conditions of the relationship. Thus, modeling the therapeutic relationship between therapists and clients may allow researchers to be able to evaluate the quality of the relationship and the effectiveness of specific interventions that might create some significant therapeutic gains with a predictability that has not yet been seen in the clinical research literature. The information may allow researchers to see how specific intervention strategies can predict changes in clients as well as see how specific intervention strategies actually produce changes in clients. One of the fundamentally novel aspects of our type of approach (beyond the presence of math) is the perspective

from which the model is developed. Rather than dissecting the individual components of a system (the client and therapist) in order to study them independently (i.e. compiling an exhaustive survey of possible attributes), we focus on reproducing the emergent dynamics of the relationship that exists between the components. We look for the set of (relevant) properties of the components that play a dominant role in these relationships. It is the relationships that inform the descriptions and properties of the components, not the other way around. Mathematically, our model is based on coupled, ordinary, nonlinear differential equations. Differential equations have previously been used to model interaction at many scales, from human relationships [e.g. a love affair (Strogatz 1988, 1994)] to functional neurodynamics [e.g. neuronal populations (Ghosh et al. 2008)]. Our perspective shares some commonalities with agent-based modeling, where a system of agents, each following a set of (relatively simple) rules, can give rise to emergent dynamics. Agent-based modeling has been successful in reproducing emergent behaviors in large biological systems [e.g. bird flocking (Reynolds 1987), the spread of epidemics/ dynamics of populations (Chowell et al. 2003)], social systems [e.g. conflict (Cederman 2003), ethnic violence (Lim et al. 2007)], and learning [e.g. the impact of emotion on the strength of beliefs (Memon and Treur 2010)]. The advantage of our dynamical systems approach using ordinary differential equations is that we can analyze many of the properties of our system analytically. Our model is therefore much less computationally intensive than most agent-based simulations.

Model Our mathematical model is a system of two-dimensional, ordinary differential equations (ODEs) representing the emotional valance of a therapist and client. Our ODEs were based on Gottman et al.’s difference equations (Gottman et al. 2002) as reformulated by Larry Liebovitch into differential equations (Liebovitch et al. 2008). We have used a similar approach in models of conflict (Liebovitch et al. 2008) and gene regulatory networks (Liebovitch et al. dC 2009). The equations for dT dt and dt are dT ¼ m1 T þ b1 þ c1 FC ðCÞ dt dC ¼ m2 C þ b2 þ c2 FT ðTÞ: dt

ð1Þ ð2Þ

These equations correspond to the dynamics of the therapist, T (Eq. 1), and client, C (Eq. 2), respectively, where these variables are the emotional valence, or affect, of the

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and the rationale for their choice is explored in ‘‘Influence functions’’. The dynamics of the system’s behavior can be analyzed by identifying the critical points of the model, which can represent the final steady state values that the dyad reach at the conclusion of therapy. We then investigate for which initial conditions of T and C the client and therapist will reach the stable states. Finally, we will see how the dynamics of their behavior depend on the parameters of the model. Influence functions We now describe the influence functions and the empirical basis for their functional form. How the therapist’s valence depends on the client’s valence, FC(C), is shown in Fig. 1. When the client’s affect is negative, the therapist will exhibit more positive affect, though they may, under prolonged exposure to clients negative affect, begin to exhibit neutral and even negative affect in the face of extreme

5 4 3 2

Therapist Valence (T)

therapist and client. For example, a positive value of T would indicate the therapist is in a positive state, and a negative value of T indicates the therapist is in a negative state and similarly for the positive and negative values of C for the client. The variables m1 and m2 represent the therapist’s and client’s (respectively) inertia to change and b1 and b2 represent their emotional values when alone. The parameters c1 and c2 are the coupling/reactivity strengths, or scaling factors, of the influence functions. We assume c1,2 [ 0. The system is coupled via the influence functions FC(C) and FT(T). One of the major contributions of this model is the formulation of these influence functions. These are the blueprints for one actor’s emotional affect in response to the other actor’s. Specifically, the influence that the client has on the therapist FC(C) and the influence that the therapist has on the client FT(T). These functions are piecewise linear segments in the differential equations reflecting the dynamics of the therapist–client interaction. In other words, they dictate how the two members of the dyad will influence each other. Although these influence functions are based on published empirical data and our own experience, they are somewhat speculative in nature but nonetheless can still provide a starting point for this exploratory project. These functions are 8 C0 < 0:5C þ 0:5 FC ðCÞ ¼ C þ 0:5 0\C 1 ð3Þ : 0:5C þ 2 C[1 8 T 0 < 5T 0:1 FT ðTÞ ¼ 0:5T 0:1 0\T 4 ð4Þ : 3T þ 13:9 T [4

1 0 −1 −2 −3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

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Client Valence (C)

Fig. 1 FC(C), Client’s influence function on the therapist

negative behavior. This may even create a steady state in the negative–negative space, which would effectively be the death of therapy—a ‘‘black hole’’ from which the therapeutic relationship dies (Bohart and Tallman 2010; Gelso 2009; Gelso and Hayes 2002; Horvath and Bedi 2002; Norcross 2010; Safran et al. 2002). When the client is affectively neutral, therapists will generally utilize strategies to elicit more positive emotions. They will attempt to encourage clients, or try to get the client to focus on their strengths and abilities, in the hopes that this change of focus will change the clients affect (Bohart and Tallman 2010). At the same time, therapists may try to elicit any affect on the part of the client (which may sometimes be negative). However, unless tied to a broader strategy, this is generally born out of frustration and may undermine the therapeutic alliance (Lambert and Barley 2002; Gelso 2009; Horvath and Bedi 2002; Norcross 2010; Safran et al. 2002). As the client’s affect moves from neutral to positive, initially, the therapist will also exhibit more positive affect. However, there is a point where, as the client’s affect becomes more positive, the therapist may begin to exhibit more neutral affect, as the therapist no longer needs to actively encourage the client, but the positive affect sustains itself (Lambert and Barley 2002; Bohart and Tallman 2010; Gelso 2009; Gelso and Hayes 2002; Horvath and Bedi 2002; Norcross 2010; Safran et al. 2002). How the client’s valence depends on the therapist’s valence, FT(T), is shown in Fig. 2. When a therapist exhibits negative affect, the client is likely to react even more negatively. The client may experience therapist negative emotion as judgmental, or a signal of some disappointment in the client. This may be the result of

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Client Valence (C)

2 1 0 −1 −2 −3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

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5


Fig. 2 FT(T), Therapist’s influence function on the client

therapist frustration with either the pace of treatment, the client’s unwillingness to change, or fears about the therapist’s own performance in conducting therapy (Bohart and Tallman 2010; Anderson et al. 2010). It is reasonable to suspect that this would be a part of a novice therapist’s practice, but could also be reflective of therapists who may be on the brink of burnout. This frustration may not even be acknowledged by the therapist, but it may get picked up by the client, and move the therapy towards the more negative end of the graph. This is an indicator of a therapeutic rupture, which in turn is a predictor of premature termination from therapy (Muran et al. 2009; Norcross 2010). At the same time, there may be circumstances when a display of negative emotion may be beneficial to the therapeutic relationship. In particular, appropriate confrontation or expressions of disappointment may be necessary feedback to the client. Again, the immediate result may be a therapeutic rupture, but if it is done purposefully or strategically, it may have a long term benefit for the client. The success of this strategy depends a lot on the skill of the therapist and the strength of the therapeutic relationship (Lambert and Barley 2002; Gelso 2009; Gelso and Hayes 2002; Horvath and Bedi 2002; Safran et al. 2002; Anderson et al. 2010; Norcross 2002). When the therapist is affectively neutral, most clients are likely to be either slightly negative or neutral (particularly early in the therapeutic process). Some clients may not be influenced one way or another to a therapist’s neutral affect, unless they find (i.e., project) it to be a signal of therapist disinterest (e.g., the tabula rasa of psychoanalysis), at which point clients may react negatively (Lambert and Barley 2002; Bohart and Tallman 2010; Gelso 2009; Gelso and Hayes 2002; Horvath and Bedi 2002; Norcross 2010; Safran et al. 2002).

As the therapist’s affect moves from neutral to positive, initially, the client may remain neutral, or slightly negative (Lambert and Barley 2002; Bohart and Tallman 2010; Gelso 2009; Gelso and Hayes 2002; Horvath and Bedi 2002; Safran et al. 2002; Norcross 2002). However, as the therapist’s affect becomes more positive, the client may respond positively by exhibiting more neutral affect (Safran et al. 2002; Skovholt and Jennings 2004). This could be a sign of the client either buying into the therapists message, or a sign that the client is beginning to experience some positive results from the therapeutic intervention. A positive steady state may emerge at this point, where therapeutic gains may be maximized (Norcross 2010). However, as a consequence of extreme expressions of positive affect on the part of the therapist, the client might turn negative (i.e., get turned off, especially if they perceive that it is disingenuous or too pollyannish).

Analysis and solution To truly appreciate the insights that the model provides, one should look to the phase portraits that emerge from numerically integrating the system from various initial conditions. A phase portrait shows the directions and paths of emotional change for the dyad. These changes are a function of the dyad’s previous emotional state and the parameters of the model. Phase portraits produced by the model are explored in ‘‘Phase portraits’’. For the purposes of simplifying the model as much as possible while still preserving the dynamics, let us assume for this numerical analysis a system with parameters m1, m2 = -1, b1, b2 = 0, and c1, c2 = 1. It’s important to note that these particular parameter choices are evenly matched. This would be the sign of novice therapist, as an expert therapist might evoke more reactivity from a client than a client evokes in the therapist (e.g. c2 [ c1). It is also important at this point to understand the significance of a nullcline. The nullclines exist where the rate of change of the emotional valence of the client dC dt or therapist dT dt equals zero. In order to explore the dynamics of a system, one must start by finding the critical points, or states, in that system. These critical points are found where the nullclines intersect. In other words, the nullclines define the points in the system where the rate of change of the client and therapist both equal zero. With the values of the parameters mentioned above, the nullclines dT dt ¼ 0 and dC dt ¼ 0 become T ¼ FC ðCÞ and

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ð5Þ


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C ¼ FT ðTÞ

ð6Þ

which are shown in Fig. 3. What this means is that for these parameter values, the nullclines equate to the influence functions. The critical points at the intersection of these nullclines are ðC; TÞ1 ¼ ð1:6; 0:3Þ and ðC; TÞ2 ¼ ð0:3; 0:8Þ: Linear stability analysis is used in order to analyze the dynamics of this system. The stability of each critical point of the model (critical points are defined by the intersection of the nullclines; where the rate of change of both the client’s and therapist’s emotional state equals zero), can be analyzed using it’s corresponding Jacobian matrix. For this system, with these parameters, stability analysis reveals that the first critical point is a saddle (having one positive and one negative eigenvalue) and that the second is an attractor (having two negative eigenvalues). This attractor is the stationary state in the system which defines the values that the variables T and C reach once sufficient time has passed (as long as they are not captured by the unstable force of the saddle point which would result in C and T going to 1). The location of this stable state depends on the parameters in the system. The following equation can be used to generalize the process of stability analysis in the region immediately about a specific critical point. m1 I1 T T_ ¼ ð7Þ _ I2 m2 C C The critical points emerge from the system as a result of the parameter (and influence function) choices. They are

best visualized with a phase portrait, which shows the dynamics of the system from various initial conditions. We shall discuss phase portraits in ‘‘Phase portraits’’. If we generalize our influence function segments to the form FC(C) = M1C ? B1 and FT(T) = M2T ? B2, then the segments existing in the region of our critical point we are analyzing can be used to define the slopes I1 = c1M1 and I2 = c2M2. In other words, I1 and I2 are the influence function slopes (M1 and M2) multiplied by the reactivity scaling factors (c1 and c2). Once again, this is useful for understanding the dynamics of the relationship in the region immediately surrounding the critical point being analyzed. The influence functions are defined by fixed numbers (with fixed ranges for the individual segments); therefore it is beyond the scope of this model to have a purely analytical form of these functions (which are treated computationally). In order to approximate a general solution for any given set of intersecting line segments, we must incorporate the parameters that we set to zero in our numerical analysis (b1 and b2) and the ones we disregarded when doing our stability analysis (B1 and B2). Let us define b1 = c1B1 ? b1 and b2 = c2B2 ? b2. We can then express our system (for any given pair of lines) as T_ ¼ m1 T þ b1 þ I1 C C_ ¼ m2 C þ b2 þ I2 T The system has particular solutions T¼

ð8Þ

b1 I2 b2 m1 m1 m2 I1 I2

ð9Þ

and C¼

5

and general solution

4

I1 b2 m2 b1 m1 m2 I1 I2 b I 2 b2 m 1 C ¼ c3 eut þ c4 eut þ 1 m1 m2 I1 I2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ 12 ½tr þ tr 2 4det; u ¼ 12 ½tr tr 2 4det and tr = m1 ? m2 (the trace of the matrix from Eq. 7), det = m1m2 - I1I2 (the determinant of the matrix from Eq. 7). T ¼ c1 eut þ c2 eut þ

3 2


b2 I1 b1 m2 m1 m2 I1 I2

1 0 −1 −2 −3

Phase portraits Fc(C)

−4

F (T) T

−5 −5

−4

−3

−2

−1

0

1

2

3

4

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Client Valence (C)

Fig. 3 Influence functions, and therefore the nullclines (for the parameter values m1,2 = -1, b1,2 = 0 and c1,2 = 1)

The numerical integration function in Matlab, ODE 113, was used to numerically integrate our system of equations from different initial conditions. The results of the integration for the parameters defined in ‘‘Analysis and solution’’ are shown in Fig. 4. The

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(T = 5 to 2) and the client is either negative or positive, the therapist will be drawn down toward the positive stable steady state. The key, it seems, for this relationship to be successful, is that (in most instances) the therapist must avoid beginning with a negative affect. The only exception is if the client is initially very negative, in which case, the relationship will be pulled towards negative emotional states, no matter what the therapist does. Now, let’s explore what happens as we vary a parameter of our model. We will say that the client is more reactive than the therapist, meaning the client responds strongly to the therapist (c1 = 1, c2 = 10), which could be the mark of a skilled practitioner. Figure 5 shows the phase-space plots of the trajectories of the therapist and client if the client responds very strongly to the therapist (c2 = 10 rather than 1). This may be an indication of a very influential or skilled therapist. Just as in the first phase portrait (Fig. 4), there are two critical points, one of which is a stable steady state attractor where the client is very positive and the therapist is moderately positive, and the other is a saddle point. There are some noteworthy results. The therapeutic relationship is attracted toward the positive critical point (steady state attractor) and the emotional state of the relationship spirals, or oscillates up and down in time, before reaching this final steady state. This seems to be in line with clients’ frequent oscillations (i.e. ambivalence) regarding change. The client winds up more positive than the therapist, and the relationship is attracted to the steady state rapidly (as indicated by the tight spiraling). This seems to be a very good outcome for therapy. There is also a cautionary finding

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2

2



trajectories go to the stable attractor at (C, T)2 = (0.3, 0.8) or are pulled towards and then pushed away from the saddle point at (C, T)1 = (-1.6, -0.3). The evolution of the system to the stable point would represent a successful therapeutic endpoint. For example, if the client or therapist are initially maximally positive (we’ve all encountered people in this state from time to time), they both end up mildly positive, which is the desired outcome. In Fig. 4, where client and therapist have equal influence with each other, the relationship will likely end up at the positive attractor, as long as the therapist begins with positive emotion. However, if the client starts therapy in a very negative emotional state (C = -5 or -4) then the therapist must be more positive in order to overcome the movement towards the negative saddle point and into the ‘‘black hole’’. Furthermore, with these parameters of the model, if the client begins therapy with a mild negative state (C = -1) or is neutral, the therapist can also match the negative emotion and still attract the relationship towards the positive stable steady state (approximately T = 1 and C = 1). In addition, if the client starts therapy with very positive affect (C = 2 to 5) the therapist can also display some negative or neutral emotion and still draw the relationship to the positive steady state. Going negative can be a strategy for the therapist to either bring a client who is mildly negative or neutral (C = -1 or 0) about therapy into a positive space. It may also be a strategic method for tamping down a client who is displaying highly (and possibly unrealistically) positive emotions. Since, in this scenario, both the client and the therapist are equally influential of the other, one can look at the other side of the coin. Specifically, if the therapist initially is highly positive

1 0 −1

1 0 −1

−2

−2

−3

−3

−4

−4

−5 −5

−4

−3

−2

−1

0

1

2

3

4

−5 −5

5

−4

−3

−2

−1

0

1

2

3

4

5

Client Valence (C)

Client Valence (C)

Fig. 4 Phase portrait of the system integrated with parameter values m1,2 = -1, b1,2 = 0 and c1,2 = 1

Fig. 5 Phase portrait of the system integrated with parameter values m1,2 = -1, b1,2 = 0, c1 = 1 and c2 = 10

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here that the client will likely respond very strongly to any negative input from the therapist. As shown in Fig. 5, any time that the therapist starts with a negative affect on the therapist axis, the relationship will be directed towards the saddle point and from there to increasingly negative values. When the therapist starts with positive affect, the relationship will be directed towards the positive steady state. The only exception to this is when the client affect starts very negative (T = -5 to -3) and the initial therapist affect is only neutral or slightly positive (T = 0 to 1). Thus, just as in Fig. 4, as long as the therapist begins with positive emotion, the client will reach this positive outcome, unless the client starts very negative about treatment. This model may be an ideal scenario for a brief therapy, where change is swift (as indicated by the tight spiraling) and the client is satisfied. Just as in Fig. 4, as long as the therapist begins with positive emotion, the client will reach this positive outcome, unless the client starts very negative about treatment.

Discussion Relevant parameters The dynamics of our system at any given critical point can be determined by Eq. 7. If m1m2 is positive (as is the case if m1 = m2 = -1), and if I1 and I2 have opposite signs, the determinant is positive. If I1 and I2 have the same sign, the sign of the determinant depends on whether the product I1I2 is large enough to outweigh the positive influence of the inertial terms. The point here is if I1 and I2 have the same sign, and are large enough, then det \ 0 and we have a saddle. As stated, the slope I of a nullcline at the intersection point with another nullcline plays a large role in determining the dynamics at that critical point. The inertia of the actor m also has a significant role to play in the dynamics as well. We will now explore the importance of the determinant (det), trace (tr) and discriminant (tr2 - 4det) of the coefficient matrix of the system to analyzing the system’s dynamics at each of the critical points. If the det \ 0, we get a saddle point. If the det [ 0 and tr2 4det [ 0, the critical point is a stable point. If tr2 4det \ 0, then the critical point is a stable spiral. Recall that the coefficient matrix is determined by the parameters of the system and the influence functions (Table 1). Limit cycles We now show that for this system, with these influence functions, there is no limit cycle present. The reason a limit cycle is not desirable in this model is simple: a limit cycle

Table 1 Relevant parameters Signs of I1, I2

Magnitude

Expected behavior

Same

Large

Saddle

Same

Small

Attractor

Opposite

Small

Attractor

Opposite

Large

Spiral

would represent a never-ending oscillation of client and therapist emotional state, which would not be a realistic therapeutic outcome. Utilizing the concept of a trapping domain from Poincare-Bendixson’s theorem, we show that the flow of trajectories along the edges of a closed, appropriately defined region (what is important is the general shape of this region, rather than the specific equations defining it’s boundaries) such as the dashed line shown in Fig. 6 are inwards, towards the stable attractor point. If the flow never leaves the trapping domain, and we have shown that only a stable fixed point exists within this domain, then no limit cycle can exist. This does not preclude the existence of stable spiral behavior within the trapping domain, however. Once a trajectory of the system flows into the trapping domain, it will not escape. The trapping domain is depicted in Fig. 6. For the attractor point (C, T)2, which exists at the intersection of the nullclines (Eqs. 5 and 6) derived from Eqs. 3 and 4 (where 0 \ C B 1 and 0 \ T B 4) at ðC; TÞ2 ¼ ð0:3; 0:8Þ; dT becomes dt ¼ m1 T þ b1 þ c1 FC ðCÞ dT and for our standard dt ¼ m1 T þ b1 þ c1 ðC þ 0:5Þ; parameter values m1 = - 1, b1 = 0 and c1 = 1, this becomes dT ¼ T þ C þ 0:5 dt

ð10Þ

dC Likewise, dC dt ¼ m2 C þ b2 þ c2 FT ðTÞ becomes dt ¼ m2 C þ b2 þ c2 ð0:5T 0:1Þ and with the standard parameter values m2 = -1, b2 = 0 and c2 = 1, this becomes

dC ¼ C þ 0:5T 0:1 dt

ð11Þ

What follows is one of a range of possible domain sizes, but the concepts remain the same. First, let us define, as generally as possible (without being redundant), our trapping domain’s left boundary. For C = -4 and T ¼ 1 y 4; dC dt will always be positive and flow rightwards. To test this, we substitute these values into Eq. 11. The rate of change dC dt is positive and the flow is rightwards or inwards toward the stable point. Next, let us define our trapping domain’s upper boundary. For C = -4 B x B 4 and T ¼ 4; dT dt will always be negative and flow downwards. To test this, we substitute

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Bifurcation diagrams

5 4

We now present bifurcation diagrams showing how the critical points of the system change when independently changing the inertia, m2, of the client and the relative strength of the influences between the therapist and client, a = c2/c1.

2 1 0

Varying m2

−1 −2 −3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Client Valence (C)

Fig. 6 Visualization of the trapping domain (dashed line)

these values into Eq. 10. The rate of change dT dt is negative and the flow is downwards or inwards toward the stable point. Now, let us define our trapping domain’s right boundary. For C = 4 and T ¼ 1 y 4; dC dt will always be negative and flow leftwards. To test this, we substitute these values into Eq. 11. The rate of change dC dt is negative and the flow is leftwards or inwards toward the stable point. The trapping domain’s lower boundary is created by the saddle point in the negative–negative space (C, T)1 = -1.6, -0.3. We shall define this lower boundary by the lower dashed line seen in Fig. 6. It begins at C, T = -4, 1 and ends at C, T = 4, -1. The equation for this line segment is T = -0.25C, where -4 B C B 4. Recall that a saddle point is stable along one direction (attractive) and unstable along the other (repulsive). The unstable region of this saddle point is explicitly defined (as shown in ‘‘Analysis and solution’’) by our stability analysis and the integration of the system as seen in the phase portrait shown in Fig. 6. Any trajectory entering the trapping domain from this lower boundary will have been pushed in by the unstable force of the saddle point and will be pulled in by the stable attractor. On the other side of this saddle point (the lower left region of Fig. 6), trajectories are pushed into the therapeutic ‘‘black hole’’ from which there is no return. No limit cycle can exist in this system because any trajectory entering the trapping domain will never escape, and within the trapping domain there only exists a single critical point, which is a stable attractor. Any trajectory not eventually caught by the attractive force of the stable point that resides within the trapping domain will be subject to the unstable force of the saddle point and descend into further negativity.

Shown in Fig. 7 are the therapist values for the fixed points as m2 is varied from -3 to 3 in steps of 0.1. The other parameters of the model are m1 = -1, c1,2 = 1, b1,2 = 0. For m2 B -2.5, our saddle point ceases to exist. There is also a ‘‘hump’’ created by the shifting stable point from m2 = -0.75 to m2 = -0.25, with a peak at m2 = -0.5. The stable point and saddle exist at the same level of emotional valence for the therapist at m2 = 0 and for 0 \ m2 \ 1 the stable point begins to descend into negative emotional space for the therapist. At m2 = 1, the stable point ceases to exist and only the saddle remains for increasing values of m2. Shown in Fig. 8 are the client values for the fixed points as m2 is varied from -3 to 3 in steps of 0.1. The other parameters of the model are m1 = -1, c1,2 = 1, b1,2 = 0. This bifurcation diagram shows characteristics similar to our therapist plot. Specifically, the critical saddle point ceasing to exist at about m2 = -2.5. The stable point also shows signs of the hump at m2 = -0.5, but then continues in an upward trend right until the stable point ceases to exist at m2 = 1. The saddle point continues to exist for values of m2 [ -2.5, just as in Fig. 7.

5

0



3

−5

−10

−15 −3

−2

−1

0

1

2

3

Client Inertia (m2)

Fig. 7 Values of the attractor (.) and saddle (?) points of the therapist as a function of m2

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10

5 4

5

3


Client Valence (C)

0

−5

−10

−15

−20

2 1 0 −1 −2 −3

−25

−4 −30 −3

−2

−1

0

1

2

−5 −50

3

−40

Client Inertia (m2)

−30

−20

−10

0

10

20

30

40

50

Client Reactivity ( )

Fig. 8 Values of the attractor (.) and saddle (?) points of the client as a function of m2

Fig. 9 Values of the attractor (.) and saddle (?) points of the therapist as a function of a = c2/c1

It should be kept in mind that the inertia of an actor is thought of as a dampening force on the dynamics of the system (i.e. m2 \ 0 typically), so m2 values greater than zero will be unusual. Another way of interpreting a non negative m2 value would be that the therapist is exhibiting such a strong influence on the client, that the client has negative inertia. It appears that the actor with the least inertia (highest value of m2) will have the most positive emotional outcome, assuming the initial conditions are such that the trajectory goes to the stable point.

end state (the stable point) becomes more positive as well. This illustrates the first of our conclusions (below) that the person who is the most responsive ends up being the most positive.

Varying a Shown in Fig. 9 are the therapist values for the fixed points as a = c2/c1 is varied from -50 to 50 in steps of 1, by increasing the client’s coupling strength, c2. The parameter c1 = 1 and is held constant throughout. The other parameters of the model are m1,2 = -1, b1,2 = 0. For a \ 0, which would indicate a negative coupling strength on the client side (this is counterintuitive for a human dyad, much as positive values of m are counterintuitive), the stable point remains in slightly positive space for the therapist. A peak in the therapist’s emotional state is seen at around a = 2, which is exactly where the client’s emotional dynamic changes from a steep increase to a more gradual one. Shown in Fig. 10 are the client values for the fixed points as a is varied from -50 to 50 in steps of 1. The parameter c1 = 1 and is held constant throughout. The other parameters of the model are m1,2 = -1, b1,2 = 0. When a \ 0, we see that the stable attractor exists in the negative space for the client, while the saddle is in positive emotional space. When a [ 0 and the client’s reactivity scaling factor is raised, we see that the client’s emotional

Conclusions Through determining how the endpoints, stability, and dynamics of the system depends on various parameters, we have drawn a number of conclusions from this theoretical framework. The therapist or client who is the most responsive to the other ends up being the most positive If the client is more responsive to the affect of the therapist, the client reaches a more positive affect than the therapist. If the therapist is more responsive to the affect of the client, the therapist reaches a more positive affect than the client. This means that to achieve the most positive state for the client, which defines successful therapy, the therapist’s responses to the client should be moderated. This important conclusion of the model may provide a dynamical basis for understanding the intuitive empirical finding, over the last century of psychotherapy, of the advantages to be gained from the therapist presenting a low reactivity face to the client. In terms of the mathematics of the model, increasing the value of the influence function scaling factor c1,2 will result in an increase in the slope I1,2. This raises the respective coordinate along its axis, thus improving the emotional state of the person being influenced. Recall that our fixed point is defined for T by Eq. 8 and for C by Eq. 9. In order for this conclusion to hold true, the

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Client Valence (C)

10

5

0

−5 −50

−40

−30

−20

−10

0

10

20

30

40

50

Client Reactivity ( )

Fig. 10 Values of the attractor (.) and saddle (?) points of the client as a function of a = c2/c1

person’s uninfluenced emotional state (their emotional state when alone) must not be negative (b1,2 C 0). The final stable state of the client may be approached through emotional ups and downs For many of the scenarios we’ve studied, the steady state is reached through a spiral trajectory. This translates to the client and therapist each going through up and down emotional swings before reaching their final steady states. These emotional swings are not necessarily backsliding on the part of the client. Rather they are a direct result of the dynamics driven by the therapist and client influence functions, and therefore must be expected in these therapeutic relationships. A client who is less influenced by their own previous state takes longer to reach their final stable state A person who is less influenced by their own previous state has a slowed approach to a final steady state attractor. The smaller a person’s inertial term m, the more likely they are to oscillate before reaching their steady state. Complementarily, the lower the magnitude of the trace tr of the system matrix, the slower the spiral decays. A client who is less influenced by their own previous state follows a similar trajectory to one that is more responsive to their therapist Responding more weakly to one’s own previous emotional state yields a similar pattern of dynamics as responding more strongly to the other person. We define types of phase

portraits by the number and type of critical points that exist. Suppose a point is close to being a saddle point. The determining factor for that is whether det = m1m2 I1I2 \ 0. If we’re on the tipping point between being a saddle and an attractor, we can assume that I1 and I2 have the same sign, so the only question is whether they are big enough. In this case, increasing I1I2 will have the same effect as decreasing m1m2, that is to say, increasing the influences has the same effect as weakening your response to your previous emotional state. Now suppose a point is on the brink between being a stable point and a spiral. The determining factor is whether (m1 - m2)2 ? 4I1I2 \ 0. If we’re on the tipping point between being an attractor and a spiral, we can assume that I1 and I2 have opposite signs. In this case, increasing the influence functions has the same effect as bringing m1 and m2 closer together.

Summary As noted above, this model cannot, and is not intended to, represent the full nature of the complex human interaction in psychotherapy. However, the fact that it does reveal important insights about therapist neutrality, client emotional swings, and the reciprocal roles of inertia and influence between therapist and client, are consistent with therapists’ empirical experiences. This may suggest that some simple dynamical features may underlie the more complex behaviors that emerge in the therapeutic relationship.

Future work The present exploratory work is already quite a significant conceptual leap in trying to develop a new approach that may shed light on the dynamics of psychotherapy. We hope that it will serve as a firm starting point to further develop new theoretical and empirical studies. Theoretically, Gottman et al. (2002) found that changes in the influence functions with time were essential features of how relationships in marriages were improved (or worsened). We want to explore how changing these influence functions, during the course of therapy, can improve the therapeutic outcome. Experimentally, again as Gottman et al. (2002) did, we want to video record psychotherapy sessions, code the time dependent valence of the therapist and client, use that data to determine the best fit parameters of the model system, and then determine which of those parameters best correlates with independent measures of the success of therapy. We are especially interested in learning how the

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influence functions and dynamics differ between inexperienced and experienced therapists. The long term goal is to understand enough about the dynamics of psychotherapy to suggest which approaches are likely to be the most beneficial for the client and how those approaches can be empirically tested. This may also lead to new ways to train therapists to utilize those best approaches. Acknowledgments We would like to thank Dr. Viktor Jirsa (Director of Research, CNRS, Marseille; Associate Professor, Center for Complex Systems and Brain Sciences, Florida Atlantic University) for his support on this project. This material is based upon work supported by the National Science Foundation under Grant No. 0638662. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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and responsiveness to patient needs. Oxford University Press, New York Kazdin AE (2008) Evidence-based treatment and practice: new opportunities to bridge clinical research and practice, enhance the knowledge base, and improve patient care. Am Psychol 63:146–159 Lambert MJ, Barley DE (2002) Research summary on the therapeutic relationship and psychotherapy outcomes. In: Norcross JC (eds) Psychotherapy relationships that work: therapist contributions and responsiveness to patient needs. Oxford University Press, New York Liebovitch LS, Naudot V, Vallacher R, Nowak A, Bui-Wrzosinska L, Coleman P (2008) Dynamics of two-actor cooperation-competition conflict models. Physica A 387:6360–6378 Liebovitch LS, Shehadeh LA, Jirsa VK, Hu¨tt M-T, Marr C (2009) Determining the properties of gene regulatory networks from expression data. In: Das S, Caragea D, Welch S, Hsu WH (eds) Handbook of research on computational methodologies in gene regulatory networks. IGI Global, Hershey Lim M, Metzler R, Bar-Yam Y (2007) Global pattern formation and ethnic/cultural violence. Science 317(5844):1540–1544 Martin DJ, Garske MP, Davis MK (2000) Relation of the therapeutic alliance with outcome and other variables: a meta-analytic review. J Consult Clin Psychol 68:438–450 Memon ZA, Treur J (2010) On the reciprocal interaction between believing and feeling: an adaptive agent modelling perspective. Cogn Neurodyn 4:377–394 Mozdzierz G, Peluso PR, Lisiecki J (2009) Principles of counseling and psychotherapy: learning the essential domains and nonlinear thinking of master practitioners. Routledge, New York Muran JC, Gorman BS, Eubanks-Carter C et al (2009) The relationship of early alliance ruptures and their resolution to process and outcome in three time-limited psychotherapies for personality disorders. Psychother Theory Res 46(2):233–248 Norcross JC (2002) Psychotherapy relationships that work: therapist contributions and responsiveness to patient needs. Oxford University Press, New York Norcross JC (2010) The therapeutic relationship. In: Duncan BL, Miller SD, Wampold BE, Hubble MA (eds) The heart and soul of change: delivering what works in therapy, 2nd edn. American Psychological Association, Washington, D.C. Orlinsky DE, Howard KE (1977) The therapist’s experience of psychotherapy. In: Gurman AS (eds) Effective psychotherapy: a handbook of research. Pergamon, New York Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. Comput Graph 21(4):25–34 Safran JD, Muran JC, Samstang LW, Stevens C (2002) Repairing alliance ruptures. In: Norcross JC (eds) Psychotherapy relationships that work: therapist contributions and responsiveness to patient needs. Oxford University Press, New York Skovholt TM, Jennings L (2004) Master therapists: exploring expertise in therapy and counseling. Allyn & Bacon, Boston Strogatz SH (1988) Love affairs and differential equations. Math Mag 61(1):35 Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. AddisonWesley, Reading

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APPENDIX C DYNAMICAL SYSTEMS THEORY What follows is a computational example derived from the theoretical psychotherapy model [44].

C.1

CRITICAL POINTS

Critical points of the model, some of which represent final steady states, are determined by looking at where the derivatives

dT dt

and

dC dt

both equal zero. This is where

the the values of T and C are no longer changing in time. The equations for and

dC dt

dT dt

= 0 are called nullclines. Critical points must satisfy the equations for both

nullclines, so they must lie at the intersection of the two. dT = m1 T + b1 + c1 FC (C) = 0 dt dC = m2 C + b2 + c2 FT (T ) = 0 dt

(C.1) (C.2)

Transforming this, we get:

T =

−b1 − c1 FC (C) m1

(C.3)

C=

−b2 − c2 FT (T ) . m2

(C.4)

Influence functions FC (C) and FT (T ) are used to model the interaction of the system’s components (i.e., the client and therapist). For the purposes of simplifying the model as much as possible while still preserving the dynamics, let us assume 136

for this numerical analysis a system with parameters m1 , m2 = −1, b1 , b2 = 0, and c1 , c2 = 1. This yields the following nullclines (which are identical to the influence functions due to our choice of parameters): dT = −T + FC (C) = 0 dt dC = −C + FT (T ) = 0 dt

(C.5) (C.6)

which can be rearranged as:

T = FC (C)

(C.7)

C = FT (T )

(C.8)

By inspection (of Figure 2.6), it becomes clear that there are two critical points. One lies at the intersection of the first segments of each nullcline and the second at the intersection of the second segments of each of the nullclines. Analytically, these points are solved to be [C, T ]1 = [−1.6, −0.3] and [C, T ]2 = [0.3, 0.8]. Rearranging C = 5T − T =

1 C 2

1 , 10

we get T =

C+.1 . 5

Then by substituting this into

+ 21 , we get the client component of our first fixed point, which is -1.6.

Similarly, we can rearrange T = 21 C +

1 2

to be C = 2T − 1. Substituting that in to

1 C = 5T − 10 , we can solve for the therapist component of our first fixed point, which,

as previously stated, is -0.3. We use the same method to solve for the second critical point, using the influence functions defined by the second segments: T = C + C = 12 T −

1 . 10

137

1 2

and

C.2

STABILITY ANALYSIS

Now that we know where our critical points are, we must determine how they create the flow of the system. In particular, we’d like to know if they are going to be attractive, repellent or somewhere in between. Attractor critical points are also known as stable points, and these are the critical points which represent final, steady states. These attractor points can be seen as a place the therapeutic relationship gravitates towards. Small perturbations of the therapist or client’s affect from these points will always cause the relationship to be brought back to them. Saddle points are stable along one manifold (negative eigenvalue) but unstable along another (positive eigenvalue). The stable manifolds act to pull the flow of the system towards themselves, whereas the unstable manifolds push the flow away. These points do not constitute final steady states, but they affect the evolution of the therapeutic relationship towards (or away) from any final steady states. We do not find any unstable critical points (repellent with two positive eigenvalues) with the parameters used in this model. In order to determine what type of system we’re dealing with, we need to do a linear stability analysis in the regions directly around the critical points. In order to do this, we must first convert the system’s equations into matrix form for each of the critical points. For our first critical point, we can transform the system so the critical point is at the origin (by disregarding the constants associated with the influence functions). It is also helpful to recall at this point that our resulting nullclines do not have any constants either, as b1 = b2 = 0. We then write our system as:

138

     1 ˙ T T −1 2  =   5 −1 C C˙

(C.9)

For any matrix A, the condition for eigenvalues A~x = λ~x is equivalent to the condition (A − Iλ)~x = 0, an equation that has nonzero solutions if and only if det(A − Iλ) = 0. So in order to find the eigenvalues λ1 ,2 and perform a stability analysis, we must find the zeroes of det(A − Iλ), or in our case, solve the equation: 1 −1 − λ 2 =0 5 −1 − λ → λ2 + 2λ − 1.5 = 0 √ −2 ± 10 → λ1 ,2 = 2 √ 10 → λ1 ,2 = −1 ± 2

(C.10) (C.11) (C.12) (C.13)

Since our first eigenvalue, λ1 = 0.581, is positive and our second eigenvalue, λ2 = −2.581, is negative, we have a saddle for our first critical point. We will now do the stability analysis of our second critical point using the same method. First we write our system in matrix form:      −1 1 T˙ T  =   1 C˙ −1 C 2

(C.14)

Then, we find the eigenvalues of the second critical point by solving the equation: −1 − λ 1 =0 1 −1 − λ 2 → λ2 + 2λ +

139

1 =0 2

(C.15)

(C.16)

−2 ± → λ1 ,2 = 2 → λ1 ,2 = −1 ±

√

2

(C.17)

√

2 2

(C.18)

Since our first eigenvalue, λ1 = −0.2929, is negative and our second eigenvalue, λ2 = −1.7071, is negative, we have a stable attractor for our second critical point. For a thorough discussion of limit cycles in this model, see Liebovitch et al. [43, 44] or Appendix B.

140

APPENDIX D INSTRUCTIONS TO PARTICIPANTS You are to act as either a client or therapist, as directed by the experimenters. If asked to be a therapist, please render the best therapy you can. If asked to be a client, please discuss an issue that is not overly personal and you don’t mind sharing (no personally identifiable data will be kept). You will be videotaped and these videos will be analyzed by the research team. At the conclusion of the 15-minute session, a researcher will inform you that time is up. At this time the client will fill out the Real Relationship Inventory form and the Working Alliance Inventory-Client form. The therapist will fill out the Working Alliance Inventory-Therapist form.

141

APPENDIX E RRI AND WAI SURVEYS This page intentionally left blank.

142

Relationship Inventory.pdf ID#_____________ Real Relationship Inventory (Kelley, Gelso, Fuertes, Marmarosh, & Lanier, 2010) Directions: Please complete the items below in terms of your relationship with your therapist. Use the following 1–5 Likert scale in rating each item. Scale: (5) Strongly Agree; (4) Agree; (3) Neutral; (2) Disagree; (1) Strongly Disagree. Answer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

I was able to be myself with my therapist. My therapist and I had a realistic perception of our relationship. I was holding back significant parts of myself. I appreciated being able to express my feelings in therapy. My therapist liked the real me. It was difficult to accept who my therapist really is. I was open and honest with my therapist. My therapist’s perceptions of me seem colored by his or her own issues. The relationship between my therapist and me was strengthened by our understanding of one another. My therapist seemed genuinely connected to me. I was able to communicate my moment-to-moment inner experience to my therapist. My therapist was holding back his/her genuine self. I appreciated my therapist’s limitations and strengths. We do not really know each other realistically. My therapist and I were able to be authentic in our relationship. I was able to see myself realistically in therapy. My therapist and I had an honest relationship. I was able to separate out my realistic perceptions of my therapist from my unrealistic perceptions. My therapist and I expressed a deep and genuine caring for one another. I had a realistic understanding of my therapist as a person. My therapist did not see me as I really am. I felt there was a significant holding back in our relationship. My therapist’s perceptions of me were accurate. It was difficult for me to express what I truly felt about my therapist.

Figure E.1: Real Relationship Inventory

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144 Figure E.2: Working Alliance Inventory Client Form p. 1

© A. O. Horvath, 1981, 1982; Revision Tracey & Kokotowitc 1989.

Thank you for your cooperation.

Work fast, your first impressions are the ones we would like to see. (PLEASE DON'T FORGET TO RESPOND TO EVERY ITEM.)

This questionnaire is CONFIDENTIAL; neither your therapist nor the agency will see your answers.

W A !(S)

If the statement describes the way you always feel (or think) circle the number 7; if it never applies to you circle the number 1. Use the numbers in between to describe the variations between these extremes.

_____________________________________________________________________________________________________________ 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always _____________________________________________________________________________________________________________

Below each statement inside there is a seven point scale:

On the following pages there are sentences that describe some of the different ways a person might think or feel about his or her therapist (counsellor). As you read the sentences mentally insert the name of your therapist (counsellor) in place of _____________in the text.

Instructions

Short Form (C)

Working Alliance Inventory

145 Figure E.3: Working Alliance Inventory Client Form p. 2

W A !(S)

________________________________________________________________________________________________________________________________________________________________ 1. _______________ and I agree about the things I will need to do in therapy to help improve my situation. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 2. What I am doing in therapy gives me new ways of looking at my problem. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 3. I believe _______________ likes me. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 4. _______________ does not understand what I am trying to accomplish in therapy. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 5. I am confident in _______________ 's ability to help me. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 6. _______________ and I are working towards mutually agreed upon goals. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 7. I feel that _______________ appreciates me. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 8. We agree on what is important for me to work on. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 9. _______________ and I trust one another. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 10. _______________ and I have different ideas on what my problems are. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 11. We have established a good understanding of the kind of changes that would be good for me. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________ 12. I believe the way we are working with my problem is correct. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ________________________________________________________________________________________________________________________________________________________________

146

Figure E.4: Working Alliance Inventory Therapist Form p. 1

© A. O. Horvath, 1981, 1984, 1991; based on revision by Tracey & Kokotowitc 1989.

Thank you for your cooperation.

Work fast, your first impressions are the ones we would like to see. (PLEASE DON'T FORGET TO RESPOND TO E V E R Y ITEM.)

This questionnaire is C O N F I D E N T I A L; neither your therapist nor the agency will see your answers.

If the statement describes the way you always feel (or think) circle the number 7; if it n e v e r applies to you circle the number 1. Use the numbers in between to describe the variations between these extremes.

1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ___________________________________________________________________________________________________

___________________________________________________________________________________________________

Below each statement inside there is a seven point scale:

On the following pages there are sentences that describe some of the different ways a person might think or feel about his or her client. As you read the sentences mentally insert the name of your client in place of _____________in the text.

Instructions

Short Form T

W orking Alliance I nventory

147

Figure E.5: Working Alliance Inventory Therapist Form p. 2

WAI(T s) p.2

____________________________________________________________________________________________________________________________________________________________________________________ 1. _______________ and I agree about the steps to be taken to improve his/her situation. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 2. My client and I both feel confident about the usefullness of our current activity in therapy. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 3. I believe _______________ likes me. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 4. I have doubts about what we are trying to accomplish in therapy. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 5. I am confident in my ability to help _______________. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 6. We are working towards mutually agreed upon goals. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 7. I appreciate _______________ as a person. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 8 We agree on what is important for _______________ to work on. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 9. _______________ and I have built a mutual trust. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 10. _______________ and I have different ideas on what his/her real problems are. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 11. We have established a good understanding between us of the kind of changes that would be good for _______________. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________ 12. _______________ believes the way we are working with her/his problem is correct. 1 2 3 4 5 6 7 Never Rarely Occasionally Sometimes Often Very Often Always ____________________________________________________________________________________________________________________________________________________________________________________

APPENDIX F FACIAL ACTION CODING SYSTEM The first scientifically rigorous, empirical studies on emotional affect were done by Paul Ekman; a pioneer in the exploration of the relationship between emotion and facial expression, and is considered one of the 100 most eminent psychologists of the twentieth century [29]. He created the Facial Action Coding System (FACS) [15]. His life’s work has been portrayed in the FOX television series, “Lie to Me” [14, 6]. Paul Ekman showed that facial expressions are not culturally determined, an idea that had been popularized by Margaret Mead. He was able to show that facial expressions are universal, and thus must have a biological basis [16]. There are seven universal emotional expressions. This universality of expression lends itself to an objective measure of emotion. It was out of this that Ekman developed FACS [15]. The Facial Action Coding System (FACS) [17] involves analyzing the exact facial muscles being used to make an expression. Units of muscles that tend to fire together are grouped into Action Units. The seven universal expressions and examples of corresponding Action Units/FACS codes are shown in table F.1. The rigorous objectivity that FACS brought to bear on the study of facial expression allowed John Gottman to take the next step; he created a mathematical model of relationships, based on the universality of emotional expression that Ekman had shown. For more details on FACS, see the Facial Action Coding System Manual [17].

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Table F.1: FACS Code Examples

Affect

Action Codes

Happiness

6+12

Sadness

1+4+15

Surprise

1+2+5B+26

Fear

1+2+4+5+20+26

Anger

4+5+7+23

Disgust

9+15+16

Contempt

R12A+R14A

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APPENDIX G PERMISSIONS This page intentionally left blank.

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Figure G.1: Elsevier Permission

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Figure G.2: Taylor & Francis Permission

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