Connecting Cortical and Behavioral Dynamics ... - Semantic Scholar

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Communicated by Jack Cowan

Connecting Cortical and Behavioral Dynamics: Bimanual Coordination V. K. Jirsa A. Fuchs J. A. S. Kelso Program in Complex Systems and Brain Sciences, Center for Complex Systems, Florida Atlantic University, Boca Raton, Florida 33431, U.S.A.

For the paradigmatic case of bimanual coordination, we review levels of organization of behavioral dynamics and present a description in terms of modes of behavior. We briefly review a recently developed model of spatiotemporal brain activity that is based on short- and long-range connectivity of neural ensembles. This model is specified for the case of motor and sensorimotor units embedded in the neural sheet. Focusing on the cortical left-right symmetry, we derive a bimodal description of the brain activity that is connected to behavioral dynamics. We make predictions of global features of brain dynamics during coordination tasks and test these against experimental magnetoencephalogram (MEG) results. A key feature of our approach is that phenomenological laws at the behavioral level can be connected to a field-theoretical description of cortical dynamics. 1 Introduction In coordinated movements typically several states related to different behavioral patterns can be found—for example, different gaits of horses (Collins & Stewart, 1993; Schoner, ¨ Jiang, & Kelso, 1990) or different configurations among the joints for trajectory formation tasks (Buchanan, Kelso, & de Guzman, 1997; Kelso, Buchanan, & Wallace, 1991). These states have different stabilities dependent on external or internal control parameters. When such control parameters are manipulated, coordination states may become unstable, and the system exhibits a transition from one state to another. These phenomena have been investigated intensively experimentally and theoretically and mathematical models have been set up reproducing the experimentally observed coordination behavior as well as predicting new effects (see Haken, 1996; Kelso, 1995, for reviews). On the other hand recent magnetoencephalogram (MEG) and electroencephalogram (EEG) experiments (Kelso et al., 1992; Wallenstein, Kelso, & Bressler, 1995) have investigated the spatiotemporal brain dynamics during coordination of finger movements with external periodic stimuli. To accommodate these results, a c 1998 Massachusetts Institute of Technology Neural Computation 10, 2019–2045 (1998) °

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V. K. Jirsa, A. Fuchs, and J. A. S. Kelso

mathematical phenomenological model was developed describing ongoing brain activity (Jirsa, Friedrich, Haken, & Kelso, 1994). In Jirsa and Haken (1996a, 1996b), a more biologically motivated field theory of the spatiotemporal brain dynamics was elaborated, which combined properties of neural ensembles, including their short- and long-range connections in the cortex, in addition to describing the interaction of functional units embedded into the neural sheet. This approach was applied to a brain-coordination experiment (Kelso et al., 1992), where the subject’s task was to coordinate rhythmic behavior of one finger with an external acoustic stimulus. During this experiment, the MEG of the subject was recorded. Complex systems, such as the brain, have the general property that they perform low-dimensional behavior during transitions from one macroscopic state to another (Cross & Hohenberg, 1993; Haken, 1983, 1987). This behavior has also been found in the analyses (Fuchs, Kelso, & Haken, 1992; Jirsa, Friedrich, & Haken, 1995) of brain data from the coordination experiment in Kelso et al. (1992). On the basis of such analyses, the phenomenological model in Jirsa et al. (1994) describing the brain activity was derived qualitatively from the biologically motivated theory in Jirsa and Haken (1996a, 1997). In this article, we treat rhythmic coordination behavior between the subjects’ two index fingers, that is, two equivalent internal oscillators. This experimental condition fundamentally differs from the situation described in Kelso et al. (1992), where an internal oscillator (finger) interacts with an external oscillator (metronome). The goal of this article is to show how it may be possible to traverse levels of organization from the behavioral level to the brain level. For this purpose, we choose a bimanual coordination experiment (Kelso, 1981, 1984) in which a transition in coordinated behavior is observed between finger movements when a control parameter is changed. Our line of thought is as follows: • On the purely behavioral level we distinguish two levels of organization: First is the collective level, represented by the relative phase between the fingers. This collective variable characterizes the dynamic state in which the system is found. Second, the component level is represented by the positions of the individual fingers and their velocities. These component variables perform an oscillatory behavior and interact nonlinearily. When both levels of description are consistent, the description on the collective level can be derived from the component level. This traverse of scales from the collective to the component level has been performed by Haken, Kelso, and Bunz (1985) and is known in the literature as the HKB model. In section 2 we give a brief review of their results. The same strategy has been applied to a number of other systems (see Kelso, 1995). • Our next step (also in section 2) will be to transform the HKB model from the component level to the mode level. These modes describe the same dynamics as the HKB model (because it is still the same model),

Connecting Cortical and Behavioral Dynamics

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but represent the behavioral coordination states. From Friedrich, Fuchs, and Haken (1991), Kelso et al. (1992), and Fuchs et al. (1992), we know that low-dimensional behavioral dynamics can be reflected in the brain dynamics. Since we want to focus here on a bimanual coordination experiment where a phase transition is observed on the behavioral level, we also expect a low-dimensional brain dynamics in the EEG/MEG, which can be described in terms of modes as in Jirsa et al. (1994). Thus, we will take the HKB model in the representation of the behavioral modes as a guideline for the development of a model of the brain dynamics. • On the brain level we start our discussion with an introduction of a recently developed field theory of the brain (Jirsa and Haken, 1996a, 1997) in section 3.1. We specify this model for the bimanual coordination case by defining sensorimotor and motor areas as functional units. By using anatomical symmetry arguments and the working hypothesis of low-dimensional brain dynamics, we can reduce the field-theoretical description to a set of equations governing the temporal dynamics of two spatial brain modes. These equations correspond to the equations of the behavioral modes of the HKB model and thus allow us to connect the behavioral and the brain levels. From this treatment, we make predictions about the macroscopic spatiotemporal brain dynamics potentially observable in brain experiments using EEG/MEG. We test these predictions in a preliminary fashion against experimental MEG results from a bimanual coordination experiment. 2 The Level of Behavior Experimental studies by one of us (Kelso, 1981, 1984), as well as others (see Carson, Byblow, & Goodman, 1994, for a review) have shown that abrupt phase transitions occur in human finger movements under the influence of scalar changes in cycling frequency. Below a critical cycling frequency, two dynamical patterns or states are possible: an in-phase state where the finger movements are symmetric and an antiphase state where the finger movements are antisymmetric. Starting the finger movements in the antiphase state and increasing the cycling frequency, a spontaneous transition from antiphase to in-phase is performed at the critical frequency. Beyond this frequency, only the in-phase state is stable. Further, it is experimentally observed that the amplitude of the finger movements decreases when the cycling frequency is increased. These phenomena were theoretically modeled by Haken et al. (1985) by formulating a model system for the dynamics of the collective variable represented by the relative phase 8 between the fingers. This model system was then used as a guide to establish a model for the dynamics of the component variables represented by finger positions x1 and x2 . Here we give

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a brief summary of the results, distinguishing the collective and component level. 2.1 Collective Level. In complex systems, collective variables are not known in advance; they have to be identified. Transitions offer a useful entry point. In bimanual coordination, the collective level may be represented by the relative phase 8 between the fingers. This collective variable characterizes the coordinative state in which the system is found, and its dynamics is governed by ˙ = −a sin 8 − 2b sin 28, 8

(2.1)

where a and b are constant parameters but dependent on a control parameter, the cycling frequency Ä. The dynamics of 8 shows bistability at 8 = 0, π if b>

|a| 4

(2.2)

and monostability at 8 = 0 otherwise. Starting at 8 = π and then changing the parameters a and b such that b = |a|/4, the system undergoes a transition from 8 = π to 8 = 0 and remains in this state. 2.2 Component Level. The component level is represented by the positions x1 and x2 of the individual fingers and their velocities. These component variables perform an oscillatory behavior and interact nonlinearily. Two ordinary differential equations, again based on detailed experimental results, describe the dynamics of the individual fingers with the amplitudes x1 and x2 . This model system reads x¨ 1 + (Ax21 + Bx˙ 21 − γ )x˙ 1 + Ä2 x1 = (x˙ 1 − x˙ 2 ) · (α + β(x1 − x2 )2 ),

(2.3)

x¨ 2 + (Ax22 + Bx˙ 22 − γ )x˙ 2 + Ä2 x2 = (x˙ 2 − x˙ 1 ) · (α + β(x1 − x2 )2 ).

(2.4)

The left-hand sides of equations 2.3 and 2.4 describe the motion of the individual fingers, while the right-hand sides describe the coupling. These equations can be solved for small amplitudes by the following ansatz: xi = ri ei8i eiÄt + ri e−i8i e−iÄt

with

i = 1, 2,

(2.5)

where ri is a real-time dependent amplitude and 8i a real-time dependent phase. Mathematical analysis (Haken et al., 1985) reveals the following properties of the model system. In the steady state, the nontrivial amplitude of the oscillators is r2i = r2 =

γ A + 3BÄ2

(2.6)


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and depends on the cycling frequency Ä. Here ri is independent of the index i. The relative phase 8 = 81 − 82 obeys the dynamics in equation 2.1, and the parameters are given by a = −(α + 2βr2 ) and b = 1/2βr2 . An increase in the cycling frequency causes a decrease in the amplitude of the oscillators. The transition from antiphase 8 = π to in-phase 8 = 0 occurs at a critical amplitude, which is given by r2c =

γ −α . = A + 3BÄ2c 4β

(2.7)

The model system (see equations 2.3 and 2.4) reproduces the main experimental phenomena observed in the bimanual coordination experiment and predicts other properties, such as relaxation times, critical fluctuations, and switching times that have been checked quantitatively in experiments (Kelso, Scholz, & Schoner, ¨ 1986; Scholz, Kelso, & Schoner, ¨ 1987). 2.3 Mode Level. With the goal of connecting the behavioral coordination laws (see equations 2.1, 2.3 and 2.4) to brain dynamics, we introduce an alternative description of these phenomena in terms of symmetric and antisymmetric modes. These modes directly correspond to the behavioral states of the system, for which we seek correspondence at the brain level. We define the following variables: ψ˜ + = x1 + x2

ψ˜ − = x1 − x2 .

(2.8)

These variables represent modes of behavior where ψ˜ + corresponds to the symmetric (in-phase) mode and ψ˜ − to the antisymmetric (antiphase) mode. The backtransformation onto the amplitudes of finger movement reads x1 =

1 (ψ˜ + + ψ˜ − ) 2

x2 =

´ 1³ ψ˜ + − ψ˜ − . 2

(2.9)

In order to obtain the equations governing the dynamics of the new variables ψ˜ + , ψ˜ − we subtract and sum equations 2.3 and 2.4, respectively, and obtain ´ A ∂ ³ 3 ψ˜ + + 3ψ˜ −2 ψ˜ + ψ¨˜ + − γ ψ˙˜ + + Ä2 ψ˜ + + 12¶∂t µ 2 B ˙3 ψ˜ + + 3ψ˙˜ − ψ˙˜ + = 0 + 4 ´ A ∂ ³ 3 ψ˜ − + 3ψ˜ +2 ψ˜ − ψ¨˜ − − γ ψ˙˜ − + Ä2 ψ˜ − + 12¶∂t µ 2 B ˙3 ˙ ˙ ψ˜ − + 3ψ˜ + ψ˜ − = 2ψ˙˜ − · (α + β ψ˜ −2 ). + 4

(2.10)

The left-hand sides of equation 2.10 represent fully symmetric (with respect to the exchange of the indices + and −), nonlinearly coupled equations. The

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former coupling terms with α and β in the variables x1 , x2 now appear only in one equation solely in terms of the antisymmetric mode ψ˜ − . In order to treat the system in 2.10 analytically, we make the following ansatz: ψ˜ + = R+ ei8+ eiÄt + R+ e−i8+ e−iÄt

(2.11)

ψ˜ − = R− ei8− eiÄt + R− e−i8− e−iÄt ,

(2.12)

where R+ , R− denote real-time dependent amplitudes and 8+ , 8− the corresponding time-dependent phases. Inserting this ansatz into equations 2.10 and performing two approximations well known in nonlinear oscillator theory (rotating wave approximation and slowly varying amplitude approximation; see, e.g., Haken, 1983), we obtain the following equations for the amplitudes, 1 R˙ + = γ R+ − a(Ä) · (R2+ + 2R2− + R2− cos 2(8− − 8+ )) · R+ 2 1 ˙ R− = γ R− − a(Ä) · (R2− + 2R2+ + R2+ cos 2(8+ − 8− )) · R− 2 + αR− + βR3− ,

(2.13)

and for the phases ˙ + = −a(Ä)R2− sin 2(8− − 8+ ) 8

(2.14)

˙ − = −a(Ä)R2+ sin 2(8+ − 8− ), 8

(2.15)

where a(Ä) = 1/8(A + 3BÄ2 ). Defining the new variable φ = 8+ − 8− we can rewrite equations 2.14 and 2.15 as φ˙ = −a(Ä)(R2+ + R2− ) sin 2φ,

(2.16)

which has the only stable solutions φ = π2 , 3π 2 for nontrivial amplitudes R+ , R− . Thus, equation 2.13 can be reduced to 1 ∂V R˙ + = γ R+ − a(Ä) · (R2+ + R2− ) · R+ = − 2 ∂R+ 1 R˙ − = γ R− − a(Ä) · (R2+ + R2− ) · R− + αR− + βR3− 2 ∂V =− , ∂R−

(2.17)


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Figure 1: The potential V is plotted (top row) in dependence of R+ and R− as the frequency increases from left to right. The axis of R− points out of the page. The scale on the axes is in arbitrary units. The isoclines of the potential V are plotted in dependence of the oscillator amplitudes R+ , R− . The plus sign marks a local maximum, the minus sign a local minimum.

where the dynamics of R+ , R− can be expressed in terms of a gradient dynamics with the potential 1 1 1 1 V = − γ (R2+ + R2− ) + a(Ä) · (R2+ + R2− )2 − αR2− − βR4− . 4 4 2 4

(2.18)

Note that the equations 2.17 correspond to the standard equations of pattern recognition used in the synergetic computer (Fuchs & Haken, 1988; Haken, 1991). A linear stability analysis of equation 2.17 yields the same results as in Haken et al. (1985) and can be graphically presented in terms of the potential V in equation 2.18. In Figure 1 (upper row) the potential V is plotted in dependence of R+ and R− as the control parameter Ä increases from left to right. Here the R− -axis points out of the page. The corresponding isoclines of V are plotted below in arbitrary units of R+ , R− . Below the critical frequency bistability is present (left two pictures); either the symmetric or the antisymmetric mode is present. At the critical frequency Äc (third picture) the antisymmetric mode becomes unstable, and only the symmetric mode remains for higher frequencies (right-most picture). In the framework of this article we will use the behavioral mode system presented in equation 2.10 as a guideline to traverse scales of organization from the behavioral level to the brain level.

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3 The Level of Brain First, we briefly review a field-theoretical description of neural activity recently developed by Jirsa and Haken (1996a, 1996b, 1997). Then we specify the neural field equation with respect to the bimanual coordination experiment and discuss it in detail. 3.1 Field-Theoretical Description of Neural Activity. Let us consider an (n − 1)-dimensional closed surface 0 representing the neocortex in an n-dimensional space. This medium 0 shall consist of neural ensembles to which we assign two state variables describing their activity: dendritic currents generated by active synapses cause waves of extracellular fields, which can be measured by the EEG (Freeman, 1992), and intracellular fields measurable by the MEG (Williamson & Kaufman, 1987). Action potentials generated at the somas of neurons correspond to pulses. We call the magnitude of the neural ensemble average of the waves the wave activity ψj (x, t) with j = e, i where the indices distinguish excitatory and inhibitory activity and the magnitude of the neural ensemble average of the pulses the pulse activity ψj (x, t) with j = E, I distinguishing excitatory and inhibitory pulses. The location in 0 is denoted by x, the time point by t. These scalar quantities are related to each other via conversion operations (Freeman, 1992), which we define as follows: Z dX fj (x, X)Hj (x, X, t) , (3.1) ψj (x, t) = 0

where j = e, i, E, I. Here the function Hj (x, X, t) represents the output of a conversion operation and fj (x, X) the corresponding distribution function depending on the spatial connectivity. From the experimental results of Freeman (1992) it is known that the conversion from wave to pulse is sigmoidal within a neural ensemble; however, the conversion from pulse to wave is also sigmoidal, but constrained to a linear small-signal range. Assuming that excitatory neurons have only excitatory synapses and inhibitory neurons only inhibitory synapses (which is generally true; Abeles, 1991) we obtain the following relations between conversion output and pulses: ¶¸ · µ |x − X| He (x, X, t) = S ψE X, t − v ¶ µ |x − X| (3.2) ≈ ae ψE X, t − v ¶¸ · µ |x − X| Hi (x, X, t) = S ψI X, t − v ¶ µ |x − X| , ≈ ai ψI X, t − v

(3.3)


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and between conversion output and waves: ¶ ¶ · µ µ |x − X| |x − X| − ψi X, t − HE (x, X, t) = Se ψe X, t − v v ¶¸ µ |x − X| + pe X, t − v

¶ ¶ · µ µ |x − X| |x − X| − ψi X, t − HI (x, X, t) = Si ψe X, t − v v ¶¸ µ |x − X| + pi X, t − v

(3.4)

(3.5)

where in the latter Kirchhoff’s law was used. External input pj (x, t) is realized such that afferent fibers make synaptic connections and thus pj (x, t) appears only in equations 3.4 and 3.5. Here ae , ai are constant parameters denoting synaptic weights, v the propagation velocity, and S, Sj the sigmoid functions of a class j ensemble. Let us now make the following considerations. We are interested in a spatial scale of several cm and temporal scale of 100 msec, which is relevant in EEG and MEG. Intracortical fibers (excitatory and inhibitory) typically have a length of 0.1 cm; corticocortical (only excitatory) fiber lengths range from about 1 cm to 20 cm (Nunez, 1995). Cortical propagation velocities have a wide range from 0.2 m/sec (Miller, 1987) up to 6–9 m/sec (Nunez, 1995). With an average velocity of 1 m/sec, this yields propagation delays of 1 msec for the intracortical fibers and 10 msec to 200 msec for the corticocortical fibers. Synaptic delays and refractory times are of the order 1 msec; the neuronal membrane constant is in the range of several msec (Braitenberg & Schuz, ¨ 1991). From this brief summary, we see that the spatial and temporal scales vary considerably. The distribution of the intracortical fibers is very homogeneous (Braitenberg & Schuz, ¨ 1991), but the distribution of the corticocortical fibers is not (estimates are that 40% of all possible corticortical connections are realized for the visual areas in the primate cerebral cortex (Felleman & Van Essen, 1991). We assume the corticocortical fiber distributions to be homogeneous as a first approximation. Using the discussed temporal and spatial hierarchies, the dynamics of the system (see equations 3.1–3.5) can be systematically reduced (see Jirsa & Haken, 1996a, 1996b, 1997 for details): the fast dynamics (¿ 100msec) becomes either instantaneous or can be eliminated, and the spatial scales smaller than 1 cm become pointlike. Then the entire dynamics of the system can be described in terms of the slowest variable ψe (x, t) and a modified external input now denoted by

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p(x, t). The dynamics of ψe (x, t) is given by Z ψe (x, t) = ae

dX fe (x − X) ¶ µ ¶¸ · µ |x − X| |x − X| + p X, t − , · Se ρ ψe X, t − v v 0

(3.6)

where ρ is a density of excitatory fibers, modified due to the elimination of the other variables. Note that from the equations 3.1 through 3.5, the models by Wilson and Cowan (1972, 1973) in terms of pulse activities and by Nunez (1974, 1995) in terms of wave activities can be derived and are connected by our approach. Until now, the dimension of the cortical surface has been kept general. Here we want to specify n = 2, meaning that 0 represents a closed onedimensional loop. Such a geometry has been reported by Nunez (1995) to be a good approximation when macroscopic EEG dynamics is considered under more qualitative aspects of dynamics, like changes of dispersion relations. In the following sections, we will perform a low-dimensional mode decomposition, in which case the chosen geometry suffices for a discussion of the temporal mode dynamics. Using the method of Green’s functions (Jirsa & Haken, 1997) the above integral equation (3.6) can be rewritten as a nonlinear partial differential equation, ¶ µ ∂ 2 2 2 ˙ ¨ · S[ρ ψ(x, t)+p(x, t)], ψ +(ω0 −v 4) ψ +2ω0 ψ = ae ω0 +ω0 ∂t

(3.7)

where ω0 = v/σ and we dropped the indices e. Here we call ψ(x, t) the neural field. The interaction of functional units with the cortical sheet P 0 is represented by the external input signals pj (x, t), where p(x, t) = j pj (x, t) and the output signals ψ¯ j (t). A functional unit can include subcortical structures such as the projections of the cerebellum on the cortex or specific functional areas like the motor cortex. Anatomically these areas are obviously defined by their afferent and efferent fibers connecting to the cortical sheet. In the context of the present theory dealing with dynamics on a larger spatiotemporal scale, that is, wavelengths in the regime of several centimeters, it is more appropriate to identify the spatial localizations of the functional input units with the spatial structures that are generated by the time-dependent input signals zj (t), open to observation in the EEG/MEG. In the case of a finger movement, this spatial structure βj (x) corresponds to the well-known dipolar mode in the EEG/MEG located over the contralateral motor areas. Thus, such a functional input unit is defined as pj (t) = βj (x) zj (t) .

(3.8)


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Similarly, an output signal ψ¯ j (t) sent to noncortical areas is picked up from the cortical sheet according to ψ¯ j (t) =

Z 0

dx βj (x)ψ(x, t) ,

(3.9)

where βj (x) defines the spatial localization of the jth functional output unit. In summary, the field-theoretical approach presented here aims at a description of the spatiotemporal brain dynamics on the scale of several centimeters and 100 msec. These scales emphasize the corticocortical connections and allow the derivation of equation 3.6 in one field variable ψ(x, t) governing the spatiotemporal dynamics. Focussing on the dynamical aspects of the interaction of only a few modes in the following sections, a cortical representation of a closed strip is chosen. 3.2 Neural Field Theory of Bimanual Coordination. The neural areas subserving bimanual coordination are numerous and diverse. The cortex, through intracortical connections and long loop, reciprocal pathways to the basal ganglia and cerebellum, obviously plays a crucial role. Propriospinal and brainstem networks are also involved. Wiesendanger, Wicki, and Rouiller (1994), in a review of lesion studies in humans and nonhuman primates, implicate lateral premotor cortex, supplementary motor area, parietal association cortex, and the anterior corpus callosum (among others) in goal-directed bimanual coordination. Although many kinds of cortical lesions can affect bimanual movements, objective measures of spatiotemporal organization are rare in studies of patient populations. In the context of the work presented in this article, Tuller and Kelso (1989) showed that inphase and antiphase movements of the fingers were preserved in split-brain patients. Other phase relations were much more difficult for split brains to produce compared to normal subjects. Anatomical and physiological evidence for bilateral control of each cortical area may explain why callosal damage and unilateral cortical lesions tend to produce only transient disturbances of bimanual coordination (Wiesendanger et al., 1994). We consider a simplified scheme in which cortical areas interact in a cooperative fashion to produce goal-directed bimanual coordination (see Figure 2). Evidence for bilateral activation of primary motor cortices during a bimanual task in which both index fingers are simultaneously moved (see, e.g., Kristeva, Cheyne, & Deecke, 1991), is consistent with our double representation of “motor areas” in Figure 2. Similarly, the presence of movement-evoked fields in both postcentral cortices corresponding to reafferent activity from the periphery during bimanual movements (Kristeva et al., 1991) justifies the two “sensorimotor area” in our model. Thus, motor signals are conveyed from the motor areas in the cortical sheet to the individual fingers; sensorimotor signals carrying information about the finger movements are conveyed to the sensorimotor areas of the brain. Note that

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Figure 2: Two input units localized at βls (x) (sensorimotor, left hemisphere), βrs (x) (sensorimotor, right hemisphere) and two output units localized at βlm (x) (motor, left hemisphere), βrm (x) (motor, right hemisphere) are embedded into a one-dimensional closed neural strip whose activity is described by the field ψ(x, t).

we assign the same index l (r) to the left (right) finger and its contralateral hemisphere in order to keep the notation in the mathematical treatment as simple as possible. Here we deal with the following situation as shown in Figure 2: two input units localized at βls (x) (left hemisphere, sensorimotor), βrs (x) (right hemisphere, sensorimotor), and two output units localized at βlm (x) (left hemisphere, motor), βrm (x) (right hemisphere, motor) are embedded into a one-dimensional closed neural strip. The origin of the underlying coordinate system is located between the two hemispheres, where L is the length of the neural strip. Anatomical considerations imply as a first approximation the following symmetries: βls (x) βlm (x)

= =

βrs (−x) βrm (−x).

(3.10)

The output signal, here the motor field, is defined according to equation 3.9 and is conveyed to the corresponding finger where zl (t) = x1 (t), zr (t) = x2 (t) denote the extensions of the left and right finger movements, respectively.


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The motor movement zj (t) with j = l, r shall be described phenomenologically as a function f of the motor field ψ¯ j (t) and, in order to take phase shifts into account, its derivative ψ˙¯ j (t): zj (t) = f (ψ¯ j (t), ψ˙¯ j (t)) ≈ f0 + f1 · ψ¯ j (t) + f2 · ψ˙¯ j (t) + · · · ,

(3.11)

where f (ψ¯ j (t), ψ˙¯ j (t)) denotes a nonlinear function, which we expanded into a Taylor function in terms of ψ¯ j (t) and ψ˙¯ j (t) and truncated after the linear terms as an approximation. The constant f0 describes a constant amplitude shift and can be set to zero for a rhythmic movement. (See also Jirsa & Haken, 1997, for a treatment of the sensorimotor feedback loop in terms of oscillators.) In the following discussion we will consider two limiting cases: first f1 6= 0, f2 = 0, and later return to the case f1 = 0, f2 6= 0. With equations 3.9 and 3.11, and f1 6= 0, f2 = 0, the sensorimotor feedback is now given by, Z pj (x, t) = βjs (x)zj (t) = c0 βjs (x)

L/2

−L/2

βjm (x)ψ(x, t) dx,

(3.12)

and the feedback loops of the motor and sensorimotor units are closed. We are now in a position to specify the field equation (3.7) as follows: ¶ µ ∂ ψ¨ + (ω02 − v2 4)ψ + 2ω0 ψ˙ = ae ω02 + ω0 ∂t (3.13) · S[ρψ + pl (x, t) + pr (x, t)] . The field ψ(x, t) can also always be written in terms of symmetric and antisymmetric contributions, ψ(x, t) =

1 1 (ψ(x, t) + ψ(−x, t)) + (ψ(x, t) − ψ(−x, t)) . 2 | {z } |2 {z } ψ + (x,t)

(3.14)

ψ − (x,t)

Next, we make an assumption about the spatial pattern underlying the temporal dynamics of ψ + (x, t) and ψ − (x, t). In the bimanual coordination experiment, a transition from one pattern to another is observed on the behavioral level. In this case the theory of dynamical systems, in particular synergetics (Haken, 1983, 1987), predicts low-dimensional behavior of the system under consideration, and we expect to observe low-dimensional transition phenomena on the brain level too. Further, in previous analyses of brain-behavior experiments (Fuchs et al., 1992; Jirsa et al., 1995) involving behavioral transitions during coordination with external stimuli, low-dimensional spatiotemporal brain dynamics was found and could be described in terms of two spatial modes. Here we deal with the different situation of the coordination of two limbs. But since a similar phase transition

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in behavior has also been observed, we assume to a first approximation that each contribution ψ + (x, t) and ψ − (x, t) is dominated by one spatial pattern and factorizes ψ + (x, t) ≈ g+ (x) · ψ+ (t) ψ − (x, t) ≈ g− (x) · ψ− (t)

(3.15)

if only standing waves are present. The assumption of two dominating spatial modes is crucial for the following mathematical analysis and experimentally easy to test. If this assumption is confirmed, higher-order structures of the dynamics can be included following the lines of Jirsa et al. (1995) in which the reproduction of the experimental spatiotemporal signal could be improved by adding more spatial modes whose temporal dynamics depends only on the dynamics of the prior modes. The ansatz (see equation 3.15) can also be expanded for higher dimensions or traveling waves, but, this will increase the complexity of the analytical treatment considerably and in most cases will not lead to the same results as in the case of two dominating spatial modes. For these reasons the hypothesis, equation 3.15, is the first to be tested experimentally. A complex system whose dynamics is governed by a nonlinear evolution law may perform phase transitions from one stationary state to another when a control parameter is varied. Close to the transition point, the dynamics of this system is governed by the first leading orders of the nonlinearities (Haken, 1983, 1987). Here we express the sigmoid function S[n] by the logistic function S[n] =

1 1 − , 1 + exp (−4an) 1 + exp (4a)

(3.16)

where a denotes the sensitivity coefficient of response of the corresponding neural population. We expand equation 3.16 into a Taylor series up to third order in n and obtain 4 S[n] ≈ an − a3 n3 , 3

(3.17)

which is a good approximation for values an smaller than 1. Projecting the neural field equation (3.13) onto g+ (x) and g− (x) following the lines in Jirsa and Haken (1996a, 1997), we obtain ¶ µ ∂ ψ¨ + + 2ω0 ψ˙ + + Ä2+ ψ+ = ae ω02 + ω0 ∂t ¸ · 4 3 3 2 · af10 ψ+ − a ( f30 ψ+ + 3 f12 ψ− ψ+ ) (3.18) 3


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¶ µ ∂ ψ¨ − + 2ω0 ψ˙ − + Ä2− ψ− = ae ω02 + ω0 ∂t ¸ · 4 3 3 2 · af01 ψ− − a ( f03 ψ− + 3 f21 ψ+ ψ− ) (3.19) 3 where Z Ä2i = ω02 − v2 ·

L/2 −L/2

4gi (x) dx

with

i = +, −

(3.20)

and the terms fij with i, j = 0, . . . , 3 are constant parameters, which are given in the appendix. So far we have tackled the level of the brain. Let us now traverse the scales of organization by referring back to the behavioral level where the equations governing the behavioral dynamics are known from equation 2.10. Here we take these equations as a guide to obtain conditions that restrict the solution space of equations 3.18 and 3.19. In equation 2.8 we expressed the behavioral modes in terms of finger displacements. With equation 3.11, we can express the behavioral modes in terms of the neural field: ψ˜ + (t) = zl (t) + zr (t) = c0

ψ˜ − (t) = zl (t) − zr (t) = c0

Z

L/2

L/2

Z

L/2

L/2

(βlm (x) + βrm (x)) · ψ(x, t) dx

(3.21)

(βlm (x) − βrm (x)) · ψ(x, t) dx.

(3.22)

If we take our hypothesis, equation 3.15, of two dominating spatial modes into account, the behavioral modes can be expressed as: ψ˜ + (t) = c0 ψ+ (t)

ψ˜ − (t) = c0 ψ− (t)

Z

L/2

L/2

Z

L/2

L/2

(βlm (x) + βrm (x)) · g+ (x) dx = c+ ψ+ (t)

(3.23)

(βlm (x) − βrm (x)) · g− (x) dx = c− ψ− (t),

(3.24)

where c+ , c− are constant. In the present framework, it turns out that in the bimanual coordination case, the symmetric (antisymmetric) behavioral mode is proportional to the symmetric (antisymmetric) brain mode. As a consequence the dynamical system (see equation 2.10) of the behavioral modes and the system (see equations 3.18 and 3.19) of the brain modes should be equivalent. This requires ω02 ¿ ω0 ∂/∂t, which implies on the considered time scales (see also Jirsa and Haken, 1997, where this limit was

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used) that the mean corticocortical fiber length is large. We rewrite equations 3.18 and 3.19 as follows: ∂ ( f30 ψ+3 + 3 f12 ψ−2 ψ+ ) = 0 ∂t ∂ ψ¨ − − γ ψ˙ − + Ä2− ψ− + b ( f30 ψ−3 + 3 f21 ψ+2 ψ− ) ∂t = 2ψ˙ − · (α + βψ−2 ) ψ¨ + − γ ψ˙ + + Ä2+ ψ+ + b

(3.25)

where b = 4/3ae a3 ω0

γ = ae aω0 f10 − 2ω0

α = 1/2ae aω0 ( f01 − f10 )

β = −2ae a3 ω0 ( f03 − f30 ).

(3.26)

The system (see equation 3.25) is structurally equivalent to the dynamical system (see equation 3.10) for the behavioral modes. Note that in the latter system the Rayleigh terms with the parameter B were introduced in order to obtain a frequency dependence of the oscillator amplitude (see equation 2.6). An alternative way to achieve this dependence is to introduce frequencydependent parameters in equation 2.10, for example, A(Ä) = A + 3BÄ2 or γ = γ (Ä), which yields the same results as a Rayleigh term. If f12 = f21 and Ä+ = Ä− , then the left-hand side of equation 3.25 represents a fully symmetric system with respect to the coupling terms. The right-hand side of the second equation in equation 3.25 is a consequence of the difference between the spatial overlap of functional units–symmetric brain mode and functional units–antisymmetric brain mode. From the behavioral mode system in equation 2.10, we know the necessary condition α < 0, β > 0, which leads to the nontrivial restriction f01 < f10 , f03 < f30 and implies a greater spatial overlap of the considered functional units with the symmetric brain mode. We examine this possibility in experiment shortly. Let us turn to the second limit case f1 = 0, f2 6= 0 in equation 3.11. The subsequent treatment of the field equation, 3.13, is in full analogy to the prior limit case, f1 6= 0, f2 = 0. The main difference is that the sensorimotor ˙ feedback in equation 3.12 now follows the first derivative ψ(x, t) of the neural field. Performing the same calculations as before we obtain, ψ¨ + − γ ψ˙ + + Ä2+ ψ+ + b · ( f30 ψ˙ +3 + 3 f12 ψ˙ −2 ψ˙ + ) = 0 ψ¨ − − γ ψ˙ − + Ä2− ψ− + b · ( f30 ψ˙ −3 + 3 f21 ψ˙ +2 ψ˙ − ) = 2ψ˙ − · (α + β ψ˙ −2 )

(3.27)

with b = 4/3ae a3 ω02

γ = ae aω02 f10 − 2ω0

α = 1/2ae aω02 ( f01 − f10 )

β = −2/3ae a3 ω02 ( f03 − f30 )

(3.28)


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and the two conditions ρ ¿ c0 and Ä∂/∂t ¿ ω02 implying a shorter mean corticocortical fiber length than in the prior case. Under these conditions we have only Rayleigh terms present in the nonlinearities of the left-hand side in equation 3.27. The right-hand side is a modified HKB coupling where the nonlinear coupling term is given by the first derivative ψ˙ − (t). This modification preserves the bistability of the system (see equation 3.27) but leads to an effective parameter β dependent on the frequency Ä such that the critical transition frequency cannot be reached for constant parameters in equation 3.28. Summarizing, both limiting cases in equation 3.10 lead to slightly modified mode equations, 3.25 and 3.27, in comparison to equation 3.10. The first equations provide the correct coupling terms, but the Rayleigh terms are absent. The second provide the Rayleigh terms, but not the correct HKB coupling. Both problems can be overcome by the introduction of frequencydependent parameters. Further, a combination of the two limiting cases in equation 3.11— f1 6= 0, f2 6= 0—introduces Rayleigh terms and the correct HKB coupling, but also further terms. Here, as always in nonlinear complex systems, the choice of the parameters determines what kind of behavior will be observed. 3.3 Numerical Treatment. In order to illustrate our more general results of the previous section, we choose a simple example for a specific set of brain modes and localized functional units. The motor and sensorimotor areas on the right hemisphere are localized as follows: ½ βrm (x) = βrs (x) =

2π L

0

−² ≤ x ≤ L4 , otherwise

(3.29)

− L4 ≤ x ≤ ² , otherwise

(3.30)

and on the left hemisphere, ½ βlm (x) = βls (x) =

2π L

0

and satisfy the required anatomical symmetry of equation 3.10. For reasons of simplicity, we choose the motor and sensorimotor units on the same hemisphere to be identical in the numerical treatment. The sensorimotor feedback is specified for the first limiting case, f1 6= 0, f2 = 0, according to equation 3.11. We introduce a frequency-dependent function γ0 (Ä) into the linear damping G0 (Ä) = 2ω0 + γ0 (Ä) on the left-hand side of equation 3.13, which causes a frequency dependence of the wave amplitude and thus a frequency dependence of the finger movements as experimentally observed. For γ0 (Ä) = 0 the original field equation is present. In our specific ¯ example, equations 3.29 and 3.30, the mean field ψ(x, t), equivalent to the homogeneous mode, has to remain constant. To ensure this, we introduce a ˙¯ t) into equation 3.13. Using a semi-implicit linear mean-field damping ψ(x,

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forward-time-central-space procedure, we integrate the field equation (3.13) numerically. The functional units are localized according to equations 3.29 and 3.30, and the edges of the localization functions were smoothed for reasons of numerical stability. Periodic boundaries were used. The parameters used in the numerical simulations are: ω0 = 2π · 0.1, v = 0.152, ae = 1, a = 0.4, ρ = 0.5, c0 = 2, extension of the neural strip L = 1, spatial overlap of the localization functions ² = 0.025. Here the space unit corresponds to 1 m and the time unit to 100 msec. This parameter range is realistic: the corticocortical propagation velocity v is in the 1 m/sec range, the extension of the neural sheet L is in the 1 m range, and the long-range connectivity σ = v/ω0 is in the 10 cm range. In the bottom left corner of Figure 3 the localization of the functions βli (x), βri (x) with i = m, s is shown within the neural strip; above that, the symmetric mode g+ (x) and the antisymmetric mode g− (x) are shown. Here the space unit used is m. On the right-hand side of Figure 3, four rows each consisting of a space-time plot and time series are given. In each space-time plot, the field ψ(x, t) is plotted where the spatial domain x is vertical and the temporal domain t horizontal, as indicated by the arrows in the top left corner. In the graphs under the space-time plots, the field ψ(x, t) is projected onto the symmetric mode g+ (x) (dashed line) and the antisymmetric mode g− (x) (solid line) plotted over time t (in sec). The first two rows describe the situation before the pretransition, with γ0 (Ä) = 0. Two possible states are shown. In the top row, the antisymmetric mode g− (x) dominates; in the second row, the symmetric mode g+ (x) dominates. Increasing the damping to γ0 (Ä) = 0.3, the antisymmetric mode becomes unstable and performs a transition to the symmetric state. This transition is shown in the third row. The symmetric mode remains stable in the posttransition regime, as can be seen in the bottom row. Hence in the case for ² > 0, bistability is present in the pretransition regime and monostability in the posttransition regime. For ² = 0 no transition is observed, and bistability is preserved for the entire control parameter regime. 3.4 Experimental Predictions. From our results in the previous sections, we can predict some of the gross features to be observed in the macroscopic spatiotemporal brain dynamics measured by EEG or MEG in a bimanual coordination experiment: • The main brain dynamics should be dominated by one spatial pattern in the in-phase situation, as well as in the antiphase situation (see equation 3.15). • In the in-phase situation, the spatial pattern will be symmetric with respect to reflection at the plane between the two hemispheres; in the antiphase situation, it will be antisymmetric. Note that here the terms symmetric and antisymmetric apply to current distributions and hence to patterns to be observed in the EEG. Current flowing in apical dendrites


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Figure 3: The overlap of the localization functions βij (x) is shown in the bottom left corner. The activity of the field ψ(x, t) is plotted over the time t (horizontal) and the space x (vertical), together with the time series of the symmetric mode g+ (x) (dashed line) and the antisymmetric mode g− (x) (solid line). The top two rows correspond to the pretransition region γ0 (Ä) = 0, the bottom two rows to the posttransition region γ0 (Ä) = 0.3.

of pyramidal cells generates a magnetic field in a plane orthogonal to the currents. Thus, spatial antisymmetric (symmetric) patterns in the MEG correspond to symmetric (antisymmetric) current distributions. • By increasing the cycling frequency of the finger movements, the antiphase pattern should remain stable up to the critical frequency, where the transition in the finger movements is observed. Here a transition from the antisymmetric to the symmetric pattern should be observed. Beyond the critical frequency, the symmetric spatial pattern is stable. • Below and beyond the critical frequency, the temporal dynamics of the symmetric and the antisymmetric pattern should be dominated by an oscillation with the cycling frequency of the finger movements. Higher-odd harmonics of the cycling frequency should be visible in

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the frequency spectra of the brain signals. Even harmonics might also be observed if the Taylor expansion of the sigmoid function in equation 3.17 yields even order terms. • The parameters f01 , f10 , f30 , and f03 defined in the appendix have to satisfy the conditions f01 < f10 and f03 < f30 . Note that these conditions are inverted for the MEG. 3.5 Preliminary Experimental Test of Theoretical Predictions. A brainbehavior-experiment has been performed by Kelso, Fuchs, Holroyd, Cheyne, and Weinberg (1994) in which subjects moved their left and right index fingers in time with an auditory metronome presented to both ears in ascending frequency plateaus of 10 cycles each. The initial metronome frequency was 2.0 Hz and increased by 0.2 Hz for each plateau, with a total number of eight plateaus. The subject was instructed to move the right finger antiphase and the left finger in-phase with the metronome, and switched spontaneously to a movement pattern with both fingers in phase as the metronome frequency increased to the fifth plateau (2.8 Hz). During the experiment, magnetic field measurements of brain activity were obtained with a whole-head, 64-channel SQuID magnetometer at a sampling frequency of 250 Hz with a 40 Hz low-pass filter. Each subject performed 5 blocks of 10 runs each. In the following, we check these experimental data against the theoretical predictions. Note that the experimental set-up (auditory metronome) and the subject’s instructions differ somewhat from the experimental conditions of this article. Thus, the following experimental results have to be considered a preliminary test of the presented theory. In order to test the predictions in section 3.4, we perform a KarhunenLo`eve decomposition (KL) (see, e.g., Fuchs et al., 1992) of the MEG data on each frequency plateau separately. A KL decomposition decomposes a spatiotemporal signal ψ(x, t) into orthogonal spatial modes and corresponding time-dependent amplitudes such that a least-square error E is minimized and the KL modes have maximum variance (see Makeig, Bell, Jung, & Sejnowski, 1996) for higher-order approaches—independent component analysis—using mutual information). The normalized KL eigenvalue λ = 1 − E is a measure for the contribution of a KL mode to the entire spatiotemporal signal. Figure 4a shows the first KL mode (top row) and the second KL mode (bottom row) together with the frequency spectra (below the modes) of their corresponding time series for each plateau. The plateau number n increases from left to right. The orientation of the modes is such that the nose is on the top, and their color coding (after normalization) is given on the right of Figure 4. The vertical dotted lines in the frequency spectra denote the movement frequency and its higher harmonics. It turns out that in almost all these spectra (except plateau 1), the temporal dynamics is strongly dominated by the movement frequency. Further, except for the first plateau, we observe a spatial structure in the two KL modes, which


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Figure 4: (a) The first two KL modes of the entire spatiotemporal brain signal over the eight plateaus are shown together with their frequency spectra below in arbitrary units. (b) The first KL modes of the symmetric and antisymmetric expansion of the brain signals are shown over all plateaus, together with their frequency spectra. (c) Top: First KL modes from Figure 4b on plateau 8 and 2. Bottom: Sensorimotor-motor modes for self-paced left and right finger movement.

is constant before the transition (plateau 5). After the behavioral transition (plateaus 6 to 8), a different structure (increased activity on the right-hand side of the mode contributing to a more antisymmetric shape) is observed in the first KL mode, which is similar over the posttransition plateaus. In the pretransition region, the first KL mode is about 70% antisymmetric and 30% symmetric, whereas after the transition, the antisymmetric contribution increases to 90% and the symmetric contribution decreases to 10%. The contributions λ1 of the two KL modes to the entire spatiotemporal signal are plotted over the eight plateaus in Figure 5a (top left). Here the eigenvalue of the first KL mode is represented by a dotted line, the one of the

2040


second KL mode by a solid line and the contribution of the sum of the two KL modes by a bold line. Approximately 70% of the entire spatiotemporal signal can be represented by the sum of the first two KL modes, both before and after the transition. Before the transition, this representation is given as a superposition of the two modes; after the transition, the first KL mode carries most of the variance (about 60%) and dominates the dynamics. The spatial structure in the posttransition regime seems to show a more symmetric current distribution than in the pretransition regime. In order to test this in more detail, we decompose the entire spatiotemporal signal into symmetric contributions ψ + (x, t) and antisymmetric contributions ψ − (x, t) according to equation 3.14. We decompose these two signals into their KL modes and plot the first KL mode of each signal over the eight plateaus (see Figure 4b), where the top row shows the first KL mode of ψ − (x, t) and the bottom row the one of ψ + (x, t) together with their frequency spectra. These KL modes of ψ − (x, t), ψ + (x, t) are entirely dominated by the movement frequency. Note that the temporal activity of the symmetric contributions ψ + (x, t) decreases to almost zero, whereas the temporal activity of ψ − (x, t) shows an increase after the transition. In Figure 5a (bottom left) their KL eigenvalues λ3 are plotted over the plateaus (dotted: antisymmetric, solid: symmetric). Here we see that both KL eigenvalues are around 50% before the transition. After the transition, the KL eigenvalue of the first symmetric KL mode drops to 30%, and the antisymmetric KL mode increases to 75%. A similar behavior can be seen in the graph next to it, where λ4 describes the contribution of these KL modes to the entire spatiotemporal signal ψ(x, t) (dotted: antisymmetric, solid: symmetric, bold: antisymmetric + symmetric). In order to test how well ψ(x, t) can be reconstructed by two spatial modes that are fixed over all plateaus (see equation 3.15), we choose two representatives: the first KL mode g− (x) of ψ − (x, t) on plateau 8 and the first mode g+ (x) of ψ + (x, t) on plateau 2. These two modes are plotted again in Figure 4c (top). The contribution λ2 of these two fixed modes to the entire spatiotemporal signal ψ(x, t) is plotted in Figure 5a (top right, dotted: antisymmetric, solid: symmetric, bold: both). The representation of ψ(x, t) by these two fixed modes is about 30% in the pretransition region, where both modes have similar contributions. In the posttransition region, the mode g− (x) dominates and contributes about 50% to ψ(x, t). The degree of symmetry s of the first KL mode (in Figure 4a) is quantified and plotted over the plateau number n (bold: antisymmetric, solid: symmetric) in Figure 5b. In Figure 5c the time series of the first KL mode (bold line) of ψ(x, t) are plotted for plateau 2 (top row) and plateau 8 (bottom row). The auditory metronome (dotted line) is also plotted over time t. Both time series oscillate in phase with the movement frequency before and after the transition, which is consistent with our theoretical predictions, since the transition phenomena are captured by the transition of the spatial modes. Additionally, the finger movements (solid line) are plotted in the same graph where the left finger movement is always in phase with the metronome according to the


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Figure 5: (a) The contributions λ of the single modes to the brain signals are shown in dependence of the plateau number n. λ1 : KL eigenvalues from Figure 4a (dotted: first KL mode, solid: second KL mode, bold: both). λ2 : contributions from the two fixed spatial modes in Figure 4c (dotted: antisymmetric, solid: symmetric, bold: both). λ3 : KL eigenvalues from Figure 4b (dotted: antisymmetric, solid: symmetric). λ4 : contribution of the KL modes from Figure 4b to the entire brain signals (dotted: antisymmetric, solid: symmetric, bold: both). (b) The degree of symmetry s of the first KL mode is quantified and plotted over the plateau number n (bold: antisymmetric, solid: symmetric). (c) Time series on plateau 2 (top) and plateau 8 (bottom) (bold: first KL mode from Figure 4a, dotted: auditory stimulus, solid: finger movements). The amplitudes are scaled in arbitrary units; the time units are seconds.

subject’s instructions. The right finger movement is antiphase before and in-phase after the transition. From these results, we conclude that our first prediction is partially confirmed: the spatiotemporal dynamics of the brain signals in the pretransition region is equally dominated by the two first KL modes and not by one spatial mode only; after the transition, the first KL mode is dominant as ex-

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pected. The second, third, and fourth predictions are confirmed. We observe a transition-like behavior in the brain signals where the contribution of a symmetric current distribution in the posttransition regime is significantly larger. Further, the brain signals are strongly dominated by the movement frequency of the fingers. In a control condition, the subject performed self-paced rhythmic finger movements with each finger separately at a frequency of 1.5 Hz. During these movements, the magnetic brain activity was also recorded. The spatiotemporal brain dynamics is such that with movement onset, a spatial dipolar structure arises contralaterally and decays after an oscillation with a time period of about 200 msec. Here we do not have the possibility of distinguishing between sensorimotor and motor units, since both are involved in this task, and further controls were not available. Due to the proximity of the sensorimotor and motor units and the large spatial scale (of centimeters) under consideration, we approximate both units by the same spatial structure for this first preliminary analysis. We identify this spatial structure with the dipolar mode apparent during the self-paced movement. Figure 4c (bottom) shows these modes represented by their first KL mode, which contributes 70% to the entire brain signal. Here the left mode shows a strong dipolar structure on the right-hand side corresponding to a left finger movement, and, by analogy, the right mode on the left-hand side corresponding to a right finger movement. We identify these modes with the functional units involved (left mode: βrs (x), βrm (x), right mode: βls (x), βlm (x)). With the above identifications of g+ (x), g− (x) we can determine the parameters defined in the appendix: f01 = 0.57, f10 = 0.20, f30 = 3.5 · 10−4 , and f03 = 0.02 for c0 = 1, ρ = 0.2. These parameters satisfy the required condition f01 > f10 and f03 > f30 for the MEG. This condition appears to be robust against small changes of the spatial modes and functional units; however, it is violated by larger changes, such as rotation of g− (x) by π/2. 4 Summary The main point of this article is to show how it is possible to derive the phenomenological nonlinear laws at the behavioral level from models describing brain activity. For the paradigmatic case of bimanual coordination, we briefly reviewed the collective and component level of description of the behavioral dynamics. We proceeded by transforming the behavioral model on the component level onto a model describing the dynamics of the behavioral modes. The behavioral level was connected to the brain level by deriving the behavioral model on the mode level from a recently developed field-theoretical description of brain activity. For the derivation, the crucial points were the assumption of bimodal brain dynamics and the interplay between functional input and output units in the neural sheet. Here the comparison of behavioral and brain level serves as a guide to consistency of both descriptions. We made theoretical predictions about the global brain


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dynamics and presented a preliminary experimental test that gives strong indications that the predicted spatiotemporal dynamics is present during bimanual coordination tasks. Appendix: Explicit Forms of Coupling Integrals fij We define Z f+ (x) = ρg+ (x) + c0 (βls (x) + βrs (x)) ·

−L/2

Z f− (x) = ρg− (x) + c0 (βls (x) − βrs (x)) ·

L/2

L/2

−L/2

βlm (x)g+ (x) dx

(A.1)

βlm (x)g− (x) dx

(A.2)

and give the explicit forms of the coupling integrals fij in equations 3.18 and 3.19 as Z L/2 Z L/2 g+ (x) · f+ (x) dx f01 = g− (x) · f− (x) dx (A.3) f10 = −L/2

Z f30 =

−L/2

Z f12 = f21 =

L/2

L/2

−L/2 Z L/2 −L/2

−L/2

Z g+ (x) · f+ (x)3 dx

f03 =

L/2

−L/2

g− (x) · f− (x)3 dx (A.4)

g+ (x) · f+ (x) f− (x)2 dx g− (x) · f+ (x)2 f− (x) dx.

(A.5)

Acknowledgments This research was supported by NIMH (Neurosciences Research Branch) grant MH42900, KO5 MH01386, and the Human Frontiers Science Program. Further, we wish to thank Hermann Haken for many interesting discussions. V. K. J. gratefully acknowledges a fellowship from the Deutsche Forschungsgemeinschaft. References Abeles, M. (1991). Corticonics. New York: Cambridge University Press. Braitenberg, V., & Schuz, ¨ A. (1991). Anatomy of the cortex: Statistics and geometry. Berlin: Springer-Verlag. Buchanan, J. J., Kelso, J. A. S., & de Guzman, G. C. (1997). The self-organization of trajectory formation: I. Experimental evidence. Biol. Cybern., 76, 257–273.

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