Constraints on synchronizing oscillator networks

Constraints on synchronizing oscillator networks David E. Cairnsy, Roland J. Baddeley and Leslie S. Smith Centre for Cognitive and Computational Neuroscience University of Stirling, Stirling Scotland, FK9 4LA y

Email [email protected] March 18, 1996

Abstract This paper investigates the constraints placed on some synchronized oscillator models by their underlying dynamics. Phase response graphs are used to determine the phase locking behaviours of three oscillator models. These results are compared with idealized phase response graphs for single phase and multiple phase systems. We nd that all three oscillators studied are best suited to operate in a single phase system and that the requirements placed on oscillatory models for operation in a multiple phase system are not compatible with the underlying dynamics of oscillatory behaviour for these types of oscillator models.

1 Introduction Following observations of oscillations and synchronization behaviour in cat visual cortex (Eckhorn et al., 1989; Gray et al., 1989a) a number of interpretations have been put forward to explain these results (Gray et al., 1989b; Eckhorn et al., 1988; Grossberg & Somers, 1991; Shastri, 1989; Sompolinsky et al., 1990). It has been suggested that a possible interpretation of the observed synchronization behaviour is that the brain could be using synchronized oscillations as a method of solving the binding problem (Von der Malsburg & Schneider, 1986). If a cluster of nodes which share a common property are synchronized, they are thus labelled as belonging to one group. Other synchronized nodes which are in a dierent phase of an oscillatory cycle are eectively labelled as a separate group. By using this method, it can be seen that a number of dierent entities may be stored simultaneously, each represented by a dierent phase in an oscillatory cycle. A fundamental requirement behind these theories is that groups of nodes should be able to move into and remain in separate synchronized phases. A simple but eective architecture which enables synchronization to take place is lateral coupling. Lateral connections between node pairs transfer a measure of the activation state of one node to the other. This causes a change in the period of the receiving node and therefore a change in its phase. We investigate the response of three generic oscillator models to this type of eect and determine whether or not they are capable of supporting multiple phases as required by the above theories in order to perform binding.

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2 Method Three studies were performed, one for each of the oscillator models. To provide a small but general cross section, we chose one simple oscillator model and two biological models. For the simple oscillator, a leaky integrator model was chosen to illustrate the most basic phase response one can obtain from a non-linear system [App. A.1]. As an example of models of cellular oscillations or potential pacemaker cells, a reduced version of the HodgkinHuxley cell membrane model (the Morris-Lecar oscillator) was chosen [App. A.2 (Rinzel & Ermentrout, 1989)]. At the multi-cellular level, an oscillator based upon an original model of excitatory / inhibitory cell cluster interactions by Wilson and Cowan (Wilson & Cowan, 1972) was used. [App. A.3 (Wang et al., 1990)] The technique for obtaining the phase response graphs was obtained from Rinzel and Ermentrout's original study of the dynamics of the Morris-Lecar model (Rinzel & Ermentrout, 1989). Each oscillator was driven by a constant input until the period of the oscillation had stabilized. This gave a base period b . Driving the oscillator by the constant input and starting from a point just after peak activation, trials were made for points across the phase of the oscillation. For each successive trial, the instant of delivery of an entraining signal to the oscillator was increased. Each entraining input was of a constant size and was delivered for a constant proportion (0.025) of the period of the oscillator. The entraining input caused a change in the period of the oscillator. A measure of the relative phase shift caused by the entraining input was calculated according to equation 1. The cumulative results of the trials allowed the production of phase response graphs for each oscillator.
 = b ? n

(1)

b


 Phase shift b Normal period n New period

3 Discussion The following discussion relates how the results of the study (shown in gure 1) compare with an idealized phase response behaviour that one would like in a single phase and a multiple phase system. For a single phase system where all nodes move toward a globally synchronized activation, the ideal phase response behaviour can be represented by the graph in gure 2.a. An entraining signal causes the phase of a receiving node to move in the direction of the phase of the node producing the signal. The degree of phase shift is proportional to the dierence between the two nodes and thus causes a steady convergence with minimal possibility of overshoot. The direction of phase shift is determined by the dierence in phase, the phase shift being in the direction of the shortest `route' to synchrony. A node with this form of behaviour will always attempt to synchronize with the originator of any signal and will only remain unperturbed when in synchrony. This represents an idealized behaviour for a single phase system, however any system that has zero phase shift at 0 and 1 with a monotonic decrease in phase in the region 0 - 0.5 and a monotonic increase in phase in the region 0.5 - 1.0 (with a discontinuity at 0.5) will cause synchronization to occur (Niebur et al., 1991; Sompolinsky et al., 1990). An example of the phase response for an oscillator in a multiple phase system is shown in gure 2.b. The oscillator maintains the requirements for phase locking with entraining signals arriving close to the phase of a receiving node. However, if the entraining signals arrive further out of phase then no phase shift occurs. This `dead zone' allows for the co-existence of multiple phase groups where inputs arriving from each out-of-phase group do not perturb the receiving group. The above phase response behaviour is atypical of most oscillatory dynamics. The frequency of a node is usually increased or decreased as

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a result of extra input. Only in cases where the `activation' of the node is saturated (for example when it has reached its peak or is in a refractory period), will little phase shift occur. The extended region of low response required for multiple phases is unlikely to be present in the basic dynamics of most oscillator models. Comparing these requirements with the phase graphs of the oscillators under study, it can be seen that they are best suited to single phase / synchronized activation systems. All three models exhibit an almost linear positive phase convergence in the latter half of their phase. In the case of the two neuro-physiologically based systems some negative movement is also observed in the rst half of the phase plane. Although none of the phase responses are ideal, they are sucient to allow all of the models to exhibit eective synchronization behaviour. Conversely, the oscillators studied do not show the type of behaviour necessary for a multiple phase system. They do not possess signi cant regions of low response to entraining input in their mid-phase region. Consequently these oscillators do not allow for separate phases to co-exist stablely in a system. They will always be attempting to cause global synchronization. This would favour a network where one population is synchronized against a background of incoherent activity (Sompolinsky et al., 1990; Niebur et al., 1991; Koch & Schuster, 1992) thus allowing gure ground seperation but not labelling of multiple objects by phase (Shastri, 1989).

4 Conclusion This paper indicates there is a limit to the number of stable phases one can expect a system of interacting oscillators to maintain and that this limit is low. The results give support to models which use similar oscillators to achieve low level synchronization for the purposes of coherent activation. For models which use synchronized oscillations and multiple phases as a method to solve the binding problem, they show that the number of phases available is likely to be signi cantly less than the minimum required to perform useful computation.

Acknowledgements The authors would like to thank the members of CCCN for useful discussions in the preparation of this paper, in particular Peter Hancock and Mike Roberts for their helpful comments on the draft versions. Roland Baddeley and David Cairns are both funded by SERC and Leslie Smith is a member of sta in the Department of Computing Science and Mathematics at the University of Stirling.

A Oscillator Models

A.1 Leaky integrator  x (t + 1 ) =

T + E + x(t)k x(t) <  x(t)  

0

(2)

T Tonic input (1.0) k = 0.95 E Entraining input (0.5,0.25)  = 19.93

A.2 Morris - Lecar

dv = ?i (v ; w ) + T + E ion dt

(3)

3

[w (v ) ? w ] dw =  1 dt w (v ) iion = gCa m1 (v )(v ? 1 ) + gK w (v ? vK ) + gL (v ? vL ) m1 (v ) = 0 :5  [1 + tanh f(v ? v1 )=v2 g]

(5) (6)

w1 (v ) = 0 :5  [1 + tanh f(v ? v3 )=v4 g]

(7)

w (v ) = 1 =cosh f(v ? v3 )=(2  v4 )g

v w T E

(4)

(8)

Voltage gCa = 1:1 Fraction of K + channels open gK = 2:0 Tonic input (0.28) gL = 0:5 Entraining input (0.14,0.07)

v1 = ?0:01  = 0:2 v2 = 0:15 vK = ?0:7 v3 = 0:0 vL = ?0:5 v4 = 0:3

A.3 Wang et al

dxi = ? xi + G [T xi ? T F ( yi ) + S + I ? H ] x xx xy i i i dt x x y dyi = ? yi + G (?T yi + T xi ) y yy yx dt y y x

Hi = 

Zt

Gr (v ) =

0

xi ( )exp [? (t ?  )]d 

1 1 + exp [?(v ? r )=r ]

F (x ) = (1 ? )x + x 2 x = 0:9 y = 1:0 x = 0:2 y = 0:2  = 0:4

x = 0:4 y = 0:6 x = 0:05 y = 0:05  = 0:2

(9) (10) (11)

r 2 fx ; y g

(0    1 )

(12) (13)

Txx = 1:0 Ii : Tonic input (0.3) Txy = 1:9 Si : Entraining input (0.15,0.075) Tyx = 1:3 Tyy = 1:2 = 0:14

References Eckhorn, R., Bauer, R., Jordan, W., Brosch, M., Kruse, W., Munk, M., & Reitboeck, H.J. 1988. Coherent Oscillations : A Mechanism of Feature Linking in the Visual Cortex? Biological Cybernetics, 60, pp121-130. Eckhorn, R., Reitboeck, H.J., Arndt, M., & Dicke, P. 1989. Feature Linking via StimulusEvoked Oscillations: Experimental Results from Cat Visual Cortex and Functional Implications from a Network Model. Proc. of the International Joint Conference on Neural Networks (Washington). Gray, C.M., Konig, P., Engel, A.K., & Singer, W. 1989a. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronisation which re ects global stimulus properties. Nature, 338, pp1698-1702. Gray, C.M., Konig, P., Engel, A.K., & Singer, W. 1989b. Synchronisation of oscillatory responses in visual cortex: A plausible mechanism for scene segmentation. Proceedings of the International Symposium on Synergetics of Cognition.

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Grossberg, S., & Somers, D. 1991. Synchronised Oscillations During Cooperative Feature Linking In A Cortical Model of Visual Perception. Neural Networks, Vol. 4, pp453466. Koch, C., & Schuster, H. 1992. A simple network showing burst synchronization without frequency locking. Neural Computation, Vol.4, 2. Niebur, E., Schuster, H.G., Kammen, D.M., & Koch, C. 1991. Oscillator-phase coupling for dierent two-dimensional network connectivities. Physical Review A, Vol.44, pp6895-6904. Rinzel, J., & Ermentrout, G.B. 1989. Analysis of Neural Excitability and Oscillations : In Methods in Neuronal Modeling - From Synapses To Networks Ed. Koch and Segen. M.I.T. Press. Shastri, L. 1989. From simple associations to systematic reasoning : A connectionist representation of rules,variables and dynamic bindings. Technical Report. University of Pennsylvania. Sompolinsky, H., Golomb, D., & Kleinfeld, D. 1990. Global processing of visual stimuli in a neural network of coupled oscillators. Proc. Natl. Acad. Sci. USA, Vol. 87, pp7200-7204. Von der Malsburg, C., & Schneider, W. 1986. A neural cocktail-party processor. Biological Cybernetics, 54, pp29-40. Wang, D., Buhmann, J., & von der Malsburg, C. 1990. Pattern Segmentation in Associative Memory. Neural Computation, 2, pp94-106. Wilson, H.P., & Cowan, J.D. 1972. Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, Vol. 12.

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Phase Shift

Phase Shift 0.1 0.08 0.06 0.04 0.02

-0.02 -0.04 0.2

(a)

(b)

Phase

0.4

0.6

0.8

Phase

Phase Shift 0.4 0.3

Input 0.50 Input 0.25

0.2 0.1

-0.1 -0.2 0.2

0.4 0.6 Phase

(c)

0.8

1

Figure 1: Phase response graphs (a) Leaky integrator model (b) Morris-Lecar cell membrane model (c) Wang et al cell cluster model. Each graph shows the change in phase which occurs when an oscillator is perturbed at a given point in its phase. The x axis gives the phase of the oscillator at the point it is perturbed and the y axis the degree of perturbation in terms of phase shift. The amount of entraining input I by which the oscillator is stimulated by is given as a fraction of the driving input of the oscillator.

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1

Phase Shift

Phase Shift

0.0

0.0

(a)

0.0

0.5

1.0

0.0

(b)

Phase

0.5

Phase

Figure 2: Idealized phase response graphs (a) Single phase system (b) Multiple phase system

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1.0