Bioi. Cybem. 77, 131-140 (1997)

BiologicalCybernetics@ Springer-Verlag 1997

Neural control of interlimb oscillations

I. Human bimanual coordination

Stephen Grossberg*, Christopher Pribe**, Michael A. Cohen***

Center for Adaptive Systems and Department of Cognitive and Neural Systems, Boston University, 677 Beacon Street. Boston. MA 02215. USA

Received: 22 August 1994/ Accepted in revised Conn: 13 May 1997

Abstract. How do humans and other animals accomplishcoordinated movements? How are novel combinations oflimb joints rapidly assembled into new behavioral units thatmove together in in-phase or anti-phase movement patternsduring complex movement tasks? A neural central patterngenerator (CPG) model simulates data from human biman-ual coordination tasks. As in the data, anti-phase oscillationsat low frequencies switch to in-phase oscillations at high fre-quencies, in-phase oscillations occur at both low and highfrequencies, phase fluctuations occur at the anti-phase in-phase transition, a "seagull effect" of larger errors occursat intermediate phases, and oscillations slip toward in-phaseand anti-phase when driven at intermediate phases. Theseoscillations and bifurcations are emergent properties of theCPG model in response to volitional inputs. The CPG modelis a version of the Ellias-Grossberg oscillator. Its neuronsobey Hodgkin-Huxley type equations whose excitatory sig-nals operate on a faster time scale than their inhibitory sig-nals in a recurrent on-center off-surround anatomy. Whenan equal command or GO signal activates both model chan-nels, the model CPG can generate both in-phase and anti-phase oscillations at different GO amplitudes. Phase transi-tions from either in-phase to anti-phase oscillations, or fromanti-phase to in-phase oscillations, can occur in different pa-rameter ranges, as the GO signal increases.

1 In-phase and anti-phase bimanual coordination

Humans and other animals effortlessly control their limbsto accomplish coordinated movements. In particular, novelcombinations of joints can be rapidly assembled into newbehavioral units, or synergies, that are capable of moving to-gether in in-phase or anti-phase movement patterns to carryout complex movement tasks like tool use, dancing, pianoplaying, and the like. In order to study this competence, anexperimental paradigm was previously developed in whichhumans were asked to move fingers from both hands atvariable frequencies and to do so in in-phase or anti-phaserhythms. Data from these experiments exhibit characteristicproperties which provide clues to how new combinations ofjoints can be rapidly bound together to generate coordinatedmovement patterns.

This article describes a neural network model that sug-gests how novel joint combinations can be rapidly boundtogether in rhythmic patterns. These patterns are emergentproperties due to network interactions. They are not explic-itly represented or programmed in the network. The modelsimulates parametric properties of human movement data asemergent, or interactive, properties of nonlinear network in-teractions. This network takes the form of a central patterngenerator (CP) that coordinates the movement across limbjoints when volitional input signals perturb the network.

For example, in a bimanual finger tapping task, Yaman-ishi et al. (1980) required subjects to tap keys in time tovisual cues. The timing of the cues was varied across tenrelative phases: (0.0,0.1,0.2,... 1.0), where 0.0 = 0° and1.0 = 360°. The authors observed two properties in the re-sponses of their subjects. First, the subjects' fingers tendedto slip from intermediate relative phase relationships towardpurely in-phase (0.0 and 1.0) or anti-phase (0.5) relation-ships. Second, the observed in-phase and anti-phase oscil-lations exhibited less variability than intermediate phase re-lationships. That is, when the subjects were asked to syn-chronize to signals whose phase relationships varied from0.0 to 1.0, the standard deviation of the errors was lowestwhen the phase relationship was near in-phase (0.0 and 1.0)or pure anti-phase (0.5). The standard deviation of the errorsincreased as the subjects were required to move away fromthe in-phase or pure anti-phase oscillations. These two prop-~

* Supported in part by the Air Force Office of Scientific Research

(AFOSR F49620-92-J-0499 and AFOSR F49620-92-J-O225), the NationalScience Foundation (NSF IRI-90-24877), and the Office of Naval Research

(ONR NOOOI4-92-J-1309).** Supported in part by the Army Research Office (ARO DAAL03-88-

K-OO88), the Advanced Research Projects Agency (AFOSR 90-0083), theNational Science Foundation (NSF IRI-90-24877), and the Office of NavalResearch (ONR NOOOI4-92-J-1309).*** Supported in part by the Air Force Office of Scientific Research

(AFOSR 90-0128 and AFOSR F49620-92-J-0225).Correspondence to: S. Grossberg, Department of Cognitive and Neural Sys-tems, Boston University, 677 Beacon Street, Room 201, Boston, MA 02215.USATechnical Report CAS/CNS-TR-94-021

132

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Fig. 1. A An example illustrating both the "seagull"effect and the tendency to slip from intermediate phaserelationships toward purely in-phase and anti-phase re-lationships (reprinted with permission from Yamanishiet al. 1980), B The model eJthibits the "seagull" ef-fect: intermediate phase relationships are more variablethan purely in-phase or purely anti-phase relationships,The standard deviation (SD) of the observed relativephases is plotted against the required relative phase.The model exhibits the tendency to slip from interme-diate phase relationships toward purely in-phase andanti-phase relationships. This plot shows the mean ofthe (obser\'ed-required) phase, There are 145 points permean

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(A) (8)

anti-phase to in-phase movements (Kelso 1984; Kelso andScholz 1985).

The CPG model reliably reproduces all four effects inour simulations; see Figs. 2 and 3. In order to simulate thesefour properties, the model was presented with a pulsed waveanti-phase oscillatory input to each channel, as shown inFig. 2A. These pulsed inputs represent the descending voli-tional commands to move the fingers as required. The squarewaves were either equal to a constant input level when on,or set to zero when off. The input level and the duration ofthe "on" portion of the signal were held constant for each ofthe simulations. For each simulation, only the frequency ofthese pulses was varied. The duration of the "on" portion ofthe signals was 2.0 in all simulations. Shorter duration sig-nals did not reliably produce oscillations in both channels.In order to generate .Fig. 2, we computed, for 145 points,the relative phases of the output signals using the times atwhich they exceeded a threshold. As the frequency was var-ied, the model showed a switch from anti-phase (Fig. 2B)to in-phase (Fig. 2D) oscillations. The system also exhibitedfluctuations in which no clear phase relationship dominatesin between these regimes (Fig. 2C). As in the data, the re-verse transition in response to in-phase inputs did not occur

(Fig. 3).

2 The CPG model

The CPG model uses ubiquitously occumng physiologicalmechanisms, notably model nerve cells, or cell populations,that obey membrane equations (Hodgkin 1964), also calledshunting equations (Grossberg 1982). These neurons are con-nected by a recurrent on-center off-surround network, a de-sign that is also ubiquitous in the nervous system (Grossberg1982; Kandel et al. 1991; Kuffler 1953; Ratliff 1965; Yon

erties were also observed by Schoner and Kelso (1988) andby Tuller and Kelso (1989). The appearance of the plot of thestandard deviation of the errors has been called the "seag-ull effect" (Tuller and Kelso 1989); see Fig. lA. The CPGmodel exhibits the seagull effect, as well as the slip towardpure in-phase and pure anti-phase oscillations (Fig. IB).

Kelso (1981) described a related experimental task inbimanual coordination which involved moving fingers orlimbs in in-phase or anti-phase oscillations. For example,adduction of the right index finger simultaneously with ab-duction of the left index finger is an anti-phase movement.Concurrent abduction (or adduction) of both fingers is anin-phase movement. The rate of movement of the fingerswas signaled by a metronome. The following fundamentalqualitative behaviors emerge from the body of the bimanualfinger movement data for normal subjects:

(1) Subjects are capable of producing a variety of rel-ative phases at low frequencies. However, the underlyingoscillation generation mechanism is biased in favor of in-phase and anti-phase relationships (Yamanishi et al. 1980)as shown by the seagull effect described above and by a ten-dency to slip from intermediate phase relationships towardin-phase or anti-phase relationships.

(2) Subjects are capable of performing purely in-phasemovements at both low and high frequencies for biman-ual wrist movements (Kelso 1984) and for bimanual fingermovements (Tuller and Kelso 1989).

(3) Subjects do not have complete conscious control overtheir movements under the conditions of the bimanual co-ordination experiments. In particular, though subjects couldperform anti-phase movements at low frequencies, they ex-hibited a spontaneous switch to an in-phase relationship at

higher frequencies (Kelso 1984).(4) The relative phase of the movement produced by the

subject often fluctuates during a spontaneous switch from

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0.5 ! \, ,"" Fig. 2A-D. Bifurcation from anti-phase to in-phase oscillation in response to anti-phase in-puts of increasing frequency. The anti-phaseinputs Ii in A give rise to the anti-phase os-cillation in B. The input frequency in A islow, 0.1 pulses per unit time (a pulse turnson every 10 time units); C at intermediate in-

put frequencies (0.65), fluctUations occur; Dat high input frequencies (0.85), in-phase os-cillations are obtained. A = 1.0, B = 1.1,C = 2.5, Dii = 0.8, Dij = 0.45, i f j,E = 1.0, F. = 9.0, G. = 3.9, F2 = 0.5,G2 = 0.5. The duration of each pulse was

2.0. The integration step size was 0.00 I. Theinitial conditions were reset to zero beforeeach run. The LSODA numerical integration

package (Petzold and Hindmarsh 1987) pro-vided accurate numerical integration through-

out. LSODA "solves systems * = f withfull or bounded Jacobian when the problemis stiff, but it automatically selects betweennon-stiff (Adams) and stiff (BDF) methods. Ituses the non-stiff method initially, and dy-namically monitors data in order to decidewhich method to use." (Isoda. netlib docu-

mentation)

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ically active nonspecific signal of the same amplitude that isinput equally to all the cells. Such a nonspecific input maybe called an arousal or GO signal. It represents the simplesttype of volitional signal that can activate network oscilla-tions. The model's ability to resolve a temporally changinginput signal is inversely related to its frequency. Supposethat the model exhibits a prescribed phase response to a sus-tained GO signal. Then the output of the system converges tothis response when increasing high-frequency inputs of thesame amplitude are used, irrespective of the phase relation-ships among the inputs. How such a GO signal influencesmodel dynamics is thus studied below.

Before turning to a discussion of GO signal control, aremark about how afferent feedback may alter the presentresults is in order. Including an afferent feedback signal fromthe limbs, say from tactile sensations, proprioception, or jointreceptors, may not necessarily improve the ability of sucha CPG to stay phase-locked to a time-varying input signal.The afferent signal will either overlap in time with the inputsignal or it will not. If it does overlap, suppose to be def-inite that it increases the amplitude of the input. Increasedamplitude has not, in our simulations, improved the abilityof the model to accurately follow the phase of the input.On the other hand, if the efferent signal lags the input, then

Bekesy 1968) and that has been used to explain other typesof motor behavior (Pearson 1993). In particular, the cells ex-cite themselves via fast feedback signals while they inhibitthemselves and other populations via slower feedback sig-nals (Fig. 4). Such slow inhibition is well-known to occur insensory-motor systems; see, for example, Dudel and Kuffler(1961) and Kaczmarek and Levitan (1987). When a subsetof model cells is driven by anti-phase inputs or by in-phaseinputs of increasing frequency, as in Figs. 2 and 3, thenthe network interactions generate the observed properties ofvariable frequency finger movements as emergent proper-ties of the entire network. Our main result is thus to showhow the emergent properties of ubiquitous physiological andanatomical mechanisms give rise to behavioral properties ofmovement. This approach is distinguished from models thatare expressed directly in tenns of operating characteristicsof the data, such as the phase angle of the limbs (Kelso etal. 1988; Schoner et al. 1990; Yamanishi et al. 1980; Yuasaand Ito 1990).

The Kelso data and our simulations suggest the predic-tion that this type of opponent cpa acts as a kind of nonlin-ear low pass filter; that is, at high frequencies of stimulation,the output of the system converges to the response obtainedfrom the network when pulsed inputs are replaced by a ton-

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Fig. 4. The central pattern generator (CPG) is defined by a recurrent on-center off-sun'ound network whose cells obey membrane, or shunting, equa-tions. See text for details

this signal tends to increase the frequency of the total inputto the oscillator, both afferent and efferent, and thus helpsto favor the rhythm that would be generated by a tonicallyactive GO signal.

In the limit of high input frequencies, afferent signalscould alter the dynamics of the system, since the type ofoscillation that is produced by a GO signal does depend uponits amplitude, as will be shown below. This effect, however,does not improve the system's ability to remain phase-lockedto the input, since the amplitude of the GO signal, not thephase of the inputs, would determine the result.

GO signal control has been used in other models of bio-logical motor control, notably ones of how the brain controlsvariable-speed reaching behaviors (Bullock and Grossberg1988, 1991). In these models, the GO signal calibrates thespeed of a phasic reaching movement by a limb such asan ann. In the present example, the GO signal calibratesmovement speed by increasing oscillation frequency. Thesame GO signal can also trigger bifurcations between dif-ferent oscillatory patterns, or gaits. Thus, GO signal con-trol is of interest for understanding both the high-frequencymovements in response to temporally oscillatory inputs aswell as the gaits generated at all frequencies in response totemporally steady inputs. Whenever the volitional signal is

realized by a single GO signal, we call the model a GO gaitgenerator, or G3 model.

3 The ElIias-Grossberg oscillator

The G3 model belongs to a more general class of CPGmodels that is closely related to the model of Ellias andGrossberg (1975). In the Ellias-Grossberg model, the exci-tatory signals but not the inhibitory signals are coupled toa membrane equation, or shunting, interaction. We found it

135

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Fig. S. A plot of the oscillatory regions at different arousal levels for vari-ous choices of inhibitory coefficients. The relative phases were determinedautomatically by an algorithm which compared the relative times when thechannels exceeded an output threshold, set here to 0.35. The initial condi-tions were not reset to 0 as I increased, but only at the beginning of eachrun, when the inhibitory coefficients were changed. The other parameters(A = 1.0, B = 1.1. C = 2.5, E = 1.0, FI = 9.0, GI = 3.9, F1 = 0.5,G2 = 0.5) were chosen as in Fig. 2

term a(xi) and approaches a voltage-dependent asymptotefJ(Xi), both of which increase with voltage Xi.

The notation in (5) and (6) is consistent with the follow-ing biophysical interpretations of (1) and (2). Variable Xicomputes the activity, or potential, of an excitatory neuron,or neuron population, and Yi is the activity, or potential, of aninhibitory interneuron, or interneuron population. Equations(1) and (2) may also be given an intracellular interpreta-tion wherein Yi controls a slow inhibitory intracellular con-ductance, rather than a separate inhibitory interneuron. Asnoted above, the excitatory and inhibitory activities obey amembrane or, shunting, equation (Grossberg 1982; Hodgkin1964). The excitatory and inhibitory feedback signals f(Xi)and g(Xj), respectively, are rectified sigmoids, as in (4). EachXi excites only itself, whereas inhibition may occur via thelateral inhibitory coupling terms Dijg(Yj) in (1). The in-put terms Ii represent volitional input signals. When only ascalar GO signal perturbs the network, all Ii = I.

Oscillations in such a network occur only when the in-hibitory interneuronal rate E in (2) is sufficiently small. In-deed, when E is sufficiently large, Yi tracks Xi in (2). ThenYi may be replaced by [Xi]+ /(1 + [Xi]+) in (I), and the net-work (1) approaches an equilibrium point under very generalconditions on f and 9 if the coefficients Dij are symmet-ric (Cohen and Grossberg 1983; Hirsch 1989). Addition ofthe shunting term -Yi[Xi]+ in (2), that makes a(Xi) voltage-dependent in (6), is needed to generate some gait transitions,such as the transition from the walk to the run in bipeds thatis simulated in Pribe, Grossberg, and Cohen (1997).

4 Simulations of bidirectional phase reversalsas the GO signal increases(1)

and

(2)

where[UJ]+ = max(UJ,O) (3)

and

As noted in Sect. I, coordinated finger movements canswitch from anti-phase to in-phase oscillations as the os-cillation frequency increases. It is known, more generally,that interlimb oscillations can bifurcate between anti-phaseand in-phase oscillations in either direction. As in the case offinger movements, Grillner and Zangger (1979) have shownthat, in the deafferented spinal cat, hind limbs move fromanti-phase to in-phase movement as a function of increasinglevel of stimulation. However, the phase relationship of thetransverse limbs of a free roving quadruped can switch fromin-phase movement to anti-phase movement with increasingspeed, as when a switch from a trot to a pace occurs; seePribe, Grossberg, and Cohen (1997). How can a single CPGgenerate transitions both from in-phase to anti-phase move-ments and from anti-phase to in-phase movements as theoscillation frequency increases with increases in volitionalsignals, particularly a single GO signal?

For this to occur in a quadruped, control of four limbsor movement channels is required. Here, we first show howthis can happen in a simpler two-channel CPG, as in Fig. 4,where I I = h = I = the GO signal. Such a two-channelCPG network can exhibit both in-phase and anti-phase os-cillations such that anti-phase oscillations precede in-phase,or vice versa, in different parameter ranges (Fig. 5) as theoscillation frequency increases. As illustrated in the com-puter simulations shown in Fig. 6, a change in the inhibitorycross-coupling strengths Dij' i # j, coupled with an in-

dd'iYi = a(Xi)[,8(Xi) -Yi] (6)

where a(Xi) = 1 + [Xi]+ and (3(Xi) = [Xi]+ /(1 + [Xi]+). Thus,the slow conductance Yi is gated by a voltage-dependent rate

necessary for both the excitatory and the inhibitory signalsto be coupled to shunting membrane processes to simulateall the data patterns that are presented below and in Pribe,Grossberg, and Cohen (1997). Such a CPG model obeys the

136

Fig. 6. Frequency plots for: A in-phaseto anti-phase oscillations (Dii = 0.8.Dij = 0.45) and B anti-pha.~ to in-phase oscillations (Dii = 1.3. Dij =0.55) as I increases. The initial condi-tions were reset at each I increment.and other parameters are as in Fig. 5.The system approaches an equilibriumpoint between its in-phase and anti-phase regimes in A

(B)(A)

(8)

-(a + x2)[D219(Yt) + D229(Y2)] (9)

and

crease in the self-inhibitory coupling strength Dii tend tomove the system from in-phase-+anti-phase transitions toanti-phase-+in-phase transitions as the GO signal I is para-metrically increased. In addition, there is a tendency in someparameter ranges for anti-phase oscillations to occur at ex-treme values of I which bracket the intermediate I valuesat which in-phase oscillations occur; see Fig. 5.

Figures 7 and 8 illustrate the temporal response of theoscillator to different levels of arousal I. The same values ofI are used in both figures. Each figure illustrates the effectof the inhibitory coefficients, chosen as in Fig. 6A and 6B,respectively, as I is increased. In Fig. 7, in-phase oscillations(Fig. 7 A-C) precede anti-phase oscillations (Fig. 70,E) asarousal frequency increases. In Fig. 8, anti-phase oscillations(Fig. 8A,B) precede in-phase oscillations (Fig. 8C-E). Notethe sharp peaks in the anti-phase waveform in Fig. 8A and Band compare these with the broad plateau waveforms of theanti-phase waveform of Fig. 70 and E. In our simulations,anti-phase oscillations which precede in-phase oscillationsconsistently tend to have sharp peaks and those which oc-cur after in-phase oscillations tend to be plateau-like. Thisproperty illustrates that, in addition to phase and frequency,waveform shape could be used to differentiate and controltransitions between different gaits which have the same rel-ative phase, but different qualitative behavior. This propertyis used in Pribe, Grossberg, and Cohen (1997) to simulatedifferences between a human walk and run, and an elephantamble and walk. This analysis suggests that anti-phase wave-form shape may be a useful observable index for where thesystem lies in parameter space.

5 Oscillations of a two-channel CPGwith asymmetric parameters

d +dtY2 = E[(1 -Y2)[X2] -Y2] (10)

In such a network, each channel excites itself via terms f(Xi)and inhibits the other channel, via terms Dijg(Yj), as wellas itself via term Diig(Yi). A casual inspection of such anopponent organization between channels might have lead tothe erroneous conclusion that it can, at best, generate anti-phase oscillations. As noted in Figs. 6-8, such a G3 modelcan produce both in-phase and anti-phase oscillations as theGO signal I = II = h is increased, and can do so in eitherdirection.

Our analysis of how this can happen was based on themathematical results of Ellias and Grossberg (1975), whostudied a similar system with symmetric inhibitory cou-pling (DII = D22 and DI2 = D21), uniform initial data(Xi(O) = x > 0 and Yi(O) = Y > 0), and uniform inputs(Ii = I). By symmetry, XI = X2 = X and YI = Y2 = Y forall time, so the system behaves like the one-channel networkshown in Fig. 9. Ellias and Grossberg (1975) used the Hopfbifurcation theorem to prove the existence of an oscillatoryregime at intermediate values of I for the one-channel net-work, and thus the existence of in-phase oscillations in thetwo-channel symmetric network. The one-channel networkand two-channel symmetric networks approach equilibriumat smaller and larger I values.

To design a CPG with both in-phase and anti-phase os-cillations, one can use a one-channel oscillator as a buildingblock for constructing a two-channel network that reduces tothe one-channel oscillator when all initial data and parame-ters are symmetric. To accomplish this, choose the inhibitoryweights Dij in (7) and (9) so that }:::j Dij = D, where Dequals the inhibitory coefficient of the one-channel network

The two-channel CPG model in Fig. 4 is defined by the

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Fig. SA-E. In-phase and anti-phase oscillations at different arousal levels with inhibitory coefficients fixed at Dii = 1.3, Dij = 0.55, as in Fig. 6B. Theanti-phase oscillations occur for lower values of [ than do the in-phase oscillations. Note the bimodal anti-phase waveforms in A and B. [ = .1, .25, .5, .95,and 1.15 in A-E, respectively, as in Fig. 6. Other parameters are as in Fig. 5

and rium is re-established. In the symmetric two-channel ver-sion of this network, the variables oscillate in-phase (viz,x = Xl = X2 and Y = Yl = Y2) until an I is reached wherethey converge to a stable equilibrium point. One way to gen-erate a system with both in-phase and anti-phase oscillations

ddtY = E[(l -y)[x]+ -y] (12)

Let I in (II) be increased from the values at which there areone-channel in-phase oscillations to values at which equilib-

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139

out\~ each channel in place of a single arousal or GO signal, asshown in Fig. 2A. These pulsed inputs represent the descend-ing volitional commands to move the fingers as required.The square waves were either equal to a constant input levelwhen on, or set to zero when off. The input level and theduration of the "on" portion of the signal \vere held con-stant for each of the simulations. For each simulation, onlythe frequency of these pulses was varied. The duration ofthe "on" portion of the signals was 2.0 in all simulations.Shorter duration signals did not reliably produce oscillationsin both channels. In order to generate Fig. 1B, we computed,for 145 points, the relative phases of the output signals us-ing the times at which they exceeded a threshold. As thefrequency was varied, the model showed a switch from anti-phase (Fig. 2B) to in-phase (Fig. 2D) oscillations. As in thedata, it did not show the reverse transition in response to in-phase inputs (Fig. 3). The system also exhibited fluctuationsbetween the anti-phase and in-phase regimes (Fig. 2C). Itshould also be noted that parameters can be chosen so thatthe system locks into the anti-phase pattern independent ofthe phase of the pulsed input pattern.

GO

~ ,;!...+\~

is to break the system's symmetry so that it can generateanti-phase oscillations when XI ¥ X2 and YI ¥ Y2 (viz., "offthe diagonal") at I values that are either too small or toolarge to generate symmetric in-phase oscillations.

Several neurophysiologically plausible operations can beused to break symmetry. The first operation makes a slightlyasymmetric choice of inhibitory coefficients Dij, as occursin the bilaterally asymmetric organization of many neu-ral systems (Bradshaw 1989). Such asymmetric coefficientscan bias the system towards generating specific asymmetricgaits. The second operation uses the GO signal, I, to breaksymmetry. This can be done in two ways: (I) Choose oneGO input stronger than the other; that is, let II = I in (7)and h = I + 6 in (9). (2) Choose inputs with equal am-plitudes but slightly asynchronous onset times; that is, letII(t) = I and I2(t) = I(t -6). Mechanism (I) produces aspatial asymmetry in the oscillator, mechanism (2) a tem-poral asymmetry. Both asymmetries are small enough to becaused by random variations in network parameters duringmorphogenesis, if not more pervasive asymmetries in neuralorganization. The temporal asymmetry automatically scaleswith the GO amplitude I. Such a temporal asymmetry can,for example, be robustly designed into the network using anextra interneuron to the cells with delayed signals. We useda temporal asymmetry in the simulations of the two-channeloscillator shown in Figs. 6, 7, and 8 where the lag 6 = 0.00 I.As shown in Fig. 5, this small asynchrony in the GO arrivaltime produces anti-phase oscillations for many values of theparameters. The only parameters that were- varied in thesesimulations were the inhibitory coefficients (Dii and Dij)and the arousal level I. It is shown in Pribe, Grossberg, andCohen (1997) that temporal, but not spatial, asymmetry iscapable of controlling rapid gait transitions in some regimes.Our results thus suggest that measurements which test for thebilateral asymmetry of GO onset times be undertaken in theCPGs that control oscillatory movements.

7 Discussion

The opponent CPG model shows how an ubiquitously occur-ring neural design -a recurrent on-center off-surround net-work whose cells obey membrane equations -can give riseto activation patterns characteristic of coordinated rhythmicmovements. The patterning of inputs organizes the networkto behave as if it possesses special linkages between par-ticular joints, whereas in reality, the inhibitory connectionscan be widespread and nonspecific. The model hereby illus-trates how neural interactions can coordinate novel move-ment combinations that are not specified in the wiring dia-gram of the brain.

The anatomical location of the network that is rate-limiting in transforming the volitional input pulses into os-cillations which exhibit the four properties summarized inSect. I is not yet established. It could, in principle, be lo-cated anywhere on the pathway from the motor cortex tothe spinal cord. In this regard, Jacobs and Donoghue (1991)have reported widespread inhibitory interactions among so-matotopic representations in motor cortex that are consistentwith model properties. If these representations are the gen-erators of the observed pattern, then they would provide anexample of a cortical representation that may be transformedinto a CPG by the patterning of its inputs.

Acknowledgements.. The autho~ wish to thank Carol Jeffe~on and RobinLocke for their valuable assistance in the preparation of the manuscript

References6 Bimanual coordination at variable frequencies

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