Phase Control of Coupled Neuron Oscillators

V. Mladenov et al. (Eds.): ICANN 2013, LNCS 8131, pp. 296–303, 2013. © Springer-Verlag Berlin Heidelberg 2013

Phase Control of Coupled Neuron Oscillators

Mayumi Irifune and Robert H. Fujii

Computer Systems Department, University of Aizu, Aizu Wakamatsu City, Fukushima Prefecture, Japan [email protected]

Abstract. The phase response of an Izhikevich neuron integrator/resonator model based oscillator to a weak short-duration external input pulse is used to determine the Izhikevich model dynamic parameter values needed to attain a specified phase difference between coupled neuron oscillators working at the same natural oscillation frequency. The design of a new type of neuron oscillator-chain based artificial central pattern generator for the coordinated four-legged animal walking movement is proposed as an application.

Keywords: Izhikevich neuron model, saddle-node bifurcation, Andronov-Hopf bifurcation, integrator, resonator, stable limit cycle, phase difference, phase response curve, weakly-coupled oscillators.

1 Introduction

Various synchronization phenomena exist in the biological world. For example, the repetitive muscle movements in swimming and cardiac rhythmic movements [2] are believed to result from a system composed of coupled oscillating neurons that act in synchrony while maintaining various phase differences with respect to each other.

In this paper, the Izhikevich neuron model of a cortical neuron [5,6,7,8] is used to model an integrator as well as a resonator type neuron acting with oscillatory behavior. The Izhikevich neuron model has its roots in the Hodgkin-Huxley (HH) neuron model [9] but it is a considerably simpler model; it approximates the action potential function of the HH neuron model and captures the essence of the INa+ IK ionic current dynamics that allow 2D dynamical system phase plane analysis and numerical simulations of the neuron action potential as shown in Figs. 1(a) and (b) respectively. The Izhikevich cortical neuron model [7] can be described by the following equations:

0.04 5 140 (1)

(2)

if v > v peak , then v← c

u← u+ d.

The variable v representspotential recovery variablactivation of K ionic currendescribes a time scale valuesensitivity factor value forreset voltage value for thevalue of the recovery variabportrait (solid curve with arshown in Fig. 1(a). The neu1(b). Since the Izhikevich not the downstroke actionpotential when the membran

Fig. 1. Neuron beh

The resting state is equsystem of the neuron. A pethe neuron limit cycle dyncurrent I is changed, the neneuron parameter value(sbifurcation is said to have othat undergo saddle-node bifurcations (resonator type

Both the integrator and rspecified natural frequency[1,3] represents the amounmagnitude) [8] input pulsoscillation) is applied at thevalue is positive (negative)saddle-node bifurcation neushown in Fig. 2(a) where threpresents the time relative the input stimulus pulse was


s the action potential. The variables u represents the actle that takes into account ionic currents, such as nt and the inactivation Na ionic current. The parametere for the recovery variable u. The parameter 'b' describer the recovery variable u. The parameter 'c' describes e action potential v. The parameter 'd' describes the reble u after an output spike has occurred. The neuron phrrow followed by dark dashed line) in the vu phase planuron action potential as a function of time is shown in Fmodel can generate the upstroke action potential (v2) n potential, the Izhikevich model resets the membrne potential v exceeds vpeak.

avior: (a) vu plane and (b) action potential (v) vs. time

uivalent to the stable equilibrium point in the dynameriodic spiking state (i.e. oscillatory output) correspondnamics. When a neuron model parameter value such

euron state can shift states. When a smooth small change) causes the neuron behavior to suddenly changeoccurred. In this paper, the oscillatory behavior of neur

(integrator type) bifurcations [7] and Andronov-He) [7] are considered. esonator model neurons can be made to oscillate with so

y. A phase response curve or phase resetting curve (PRnt of phase shift that occurs when a weak (i.e. sme stimulus of short duration (relative to the period

e input of the oscillating neuron. If the phase resetting cu, the phase is advanced (delayed). An example PRC fo

uron model with some constant output spiking oscillatiohe y-axis represents the amount of phase shift and the x-ato the oscillator period (period = 40 ms. in the figure) w

s received. A PRC can be defined as follows:

.

297

tion the

r 'a' es a the

eset hase ne is Fig. but

rane

mical s to

h as e in

e, a rons

Hopf

ome RC) mall d of urve or a n is axis

when

(3)

298 M. Irifune and R.H. Fujii

where the phase ⁄ with t denoting the time since the last output spike of the oscillator and T denoting the oscillator neuron period. Tn denotes the time of nth spike when no external stimulus is present (dotted line in Fig. 2(b)), and Tnew denotes the time of the next spike after the external stimulus was received (solid line in Fig. 2 (b)). The overall phase shift over many periods is composed of the current phase of the oscillator (relative to some absolute reference) and the phase shift due to the most recent external stimulus pulse.

(4)

θcurrent describes the phase of the oscillator at the instant when the input stimulus pulse is received. Mod T stands for modulo T. T is the oscillator frequency. The Izhikevich cortical neuron model Equations (1) and (2) were used to obtain the Phase response Curve (PRC) shown in Fig. 2 (a).

Fig. 2. (a) Phase response curve for Izhikevich neuron model with a = 0.01, b = 0, c = - 60, d = 6, I = 35pA, and external input stimulus pulse strength of 0.05pA. (b) Phase shift due to external stimulus. (c) Neuron membrane potential plot showing phase shift due to external stimulus.

The goal of this research was to analyze how neuron oscillators could be made to have absolute/traveling-wave phase differences between them. In order to do so, the behavior of weakly coupled [8] integrator/resonator neuron oscillators operating at identical natural frequencies was analyzed using PRCs under the assumption that each oscillator had a stable limit cycle and that their coupling strengths were small.

2 Phase-Shift Analysis of Single Neuron Oscillators

Izhikevich model parameters a, b, c, d and the current I were selected appropriately so that both the saddle-node and sub-critical Andronov-Hopf bifurcation neurons could be made to oscillate.

θnew= θcurrent+ PRC (θcurrent ) mod T.

Two example PRCs forHopf (AH) bifurcations wpulses are shown in Fig. 3and negative phase shift val

Fig. 3. PRC1 has

Fig. 4. (a) a = 0.01, b= -0.2, from 3.5pA to 37.5pA in incr0.02 in increments of 0.001. (d

PRCs for the Izhikevich Fig. 4; when excitatory (positive (negative) values [


r an Izhikevich neuron undergoing sub-critical Andronwhile receiving excitatory external weak short stimu. It should be noted that the AH PRCs have both positlues. The X-axis range is 0 - 2πradians.

s a = 0.07, b = 0.26 and PRC2 has a = 0.01, b = 0.20

c = -60, d = 6, and I = 17pA. (b) Input stimulus pulse strengrements of 3.5pA applied. (c) Parameter ‘a’ varied from 0.0d) Parameter ‘b’ varied from -0.2 to 0 in increments of 0.05.

neuron undergoing saddle-node bifurcations are shown(inhibitory) stimulus pulses occur, the PRCs have o[3]. As can be seen in Figs. 4 (b), (c), and (d), increas

299

nov-ulus tive

gths 1 to

n in only sing


parameters 'a', 'b', and/or external input stimulus pulse magnitude made the peak of the PRC larger and shifted it to the left. The PRC y-axis represents the phase shift in radians and the PRC x-axis represents the phase of the oscillator at the time the external weak short pulse stimulus is received. The x-axis range is 0 to 2πradians.

3 Phase Analysis of Two-Coupled Oscillators

The behavior of two oscillating neurons whose outputs are mutually coupled to each other can be described using phase shift equations [10,11].

Ω . (5)

Ω . (6)

In the above equations, i = 1 or 2 for the two neuron sytem, Ωi = natural oscillator i frequency, ɛ1= coupling strength of output of oscillator 2 to oscillator 1, ɛ2 = coupling strength of output of oscillator 1 to oscillator 2, Hij = shift in phase of oscillator i due to input from oscillator j, θ1 - θ2 = phase of oscillator 1 when oscillator 2 fires a pulse, θ2 - θ1 = phase of oscillator 2 when oscillator 1 fires a pulse.

The following describes the phase difference between the two oscillators:

. (7)

X = θ1-θ2, w = Ω1-Ω2, G(X) = ɛ1H12 (X) –ɛ2H21 (-X). Thus, G(X) is the graph

obtained as the difference between the PRC1 graph and the mirrored PRC2 graph. Since the two oscillators have the same frequency, w = 0. A constant phase difference between the two oscillators is achieved when dG(X)/dX is negative (stable equilibrium) and dX/dt = G(X) = 0.

Oscillators that have different PRCs can be selected to achieve a desired constant phase difference. Whether the phase difference is a constant absolute phase difference or a constant traveling-wave type phase difference depends on the PRC values where G(X) = 0 occurs. If the PRC values are 0 at G(X) = 0, a constant absolute phase difference can be achieved otherwise, a traveling-wave type phase difference is achieved. For sub-critical Andronov-Hopf (AH) bifurcation oscillators, the G(X) = 0 point usually corresponds to a point where the PRCs are not 0, thus a constant but traveling wave-type phase difference between the two oscillators is achieved. Two different G(X) graphs for sub-critical AH bifurcation oscillators are shown in Fig. 5.

Achievable phase difference ranges for a pair of coupled saddle-node/AH bifurcation oscillators are shown in Tables 1 and 2. Phase difference ranges are larger in Table 2 than in Table 1 because in Table 2 the a, b parameter values and the ɛ values are varied instead of just varying the coupling strength ɛ value.

Fig. 5. Sub-critical AH oscillanegative, the constant phase di

Table 1. Examples of achievabnode and AH bifurcation neuvalues, but same/different coup

Both SN oscillators a = 0.01, b = - 0.2

Both AH oscillators a = 0.1, b= 0.26

Table 2. Achievable phase oscillators. The coupled oscilla

a1=0.01, a2=0.01 to 0.1

b1=0.26, b2= 0.05 to 0.26

b1=0.20, b2= 0.05 to 0.26

b1=0.15, b2= 0.05 to 0.26

b1=0.10, b2= 0.05 to 0.26

b1=0.05, b2= 0.05 to 0.26

4 Chain of Oscilla

With a knowledge of the rcoupled oscillators, it is po


ators G(X) graphs. When PRCs ≠0 at G(X) = 0 and dG(X)/difference is a traveling wave.

ble phase difference ranges for a pair of mutually coupled saduron oscillators. The coupled oscillators have the same a anpling strength ɛ values.

ɛ1 = -1; ɛ2 = -1 to -3 ɛ1 = -1 to -3; ɛ2 = -1

3.14 to 4.21 radians 2.07 to 3.14 radians


difference for a pair of coupled sub-critical AH bifurcaators have various a, b, and ɛ values.

ɛ1=ɛ

2 = -0.1 ɛ

1= -1, ɛ

2=-0.3






ators

range of possible phase differences achievable with twossible develop a chain of two-coupled oscillators that

301

dX=

ddle-nd b

ation

wo-can


be used to design an artificial central pattern generator [10,11] that mimics the synchronized and phase shifted walking leg movements of a four-legged animal. A new type of oscillator chain that can achieve such movements is shown in Fig. 6.

Fig. 6. Chained Oscillator System (COS) comprised of coupled oscillators 1 and 2 chained to coupled oscillators 3 and 4

It is assumed that all oscillator neurons 1 - 4 are AH bifurcation oscillators. A traveling-wave type constant phase shift between coupled oscillators 1 and 2 as well as between coupled oscillators 3 and 4 can each be set to 3.883 rad. by a proper selection of inhibitory (negative) ɛi (i= 1 or 2) values, and ai and bi parameter values (see Table 2). The chaining of the coupled oscillators 1-2 to 3-4 is accomplished with a positive ɛS (i.e. excitatory) coupling strength. The phase shift between oscillators 2 and 3 will be approximately 0 radians because the PRC for oscillator 3 has a negative slope (i.e. a stable point) near 0 and 2π radians. The chaining between oscillators 2 and 3 is needed for synchronization.

Fig. 7. Outputs for chain of oscillators shown in Fig. 6. Top output is for oscillator 1 followed by oscillator 2, oscillator 3, and bottom output for oscillator 4.

Thus, the constant phase difference (either absolute or a traveling-wave-type) between oscillator 1 and oscillator 4 will be 1.5 radians (i.e. (3.883 radians + 3.883

1 2

ε2

ε1

3 4

εS ε2

ε1

Phase Control of Coupled Neuron Oscillators 303

radians) mod. 2π) ≃1.5 radians ≃π/2). If the COS shown in Figure 6 is chained to another COS of identical specification, another phase shift of approximatelyπ/2 radians can be achieved. Hence, to achieve the synchronized phase shifted walking leg movements of a four-legged animal with each leg moving π/2 apart in time, three COSs (12 neurons) will be needed. The phase shifted traveling wave-type of outputs for the COS oscillators 1 (top) through 4 (bottom) in Fig. 6 are shown in Fig. 7. If oscillator 3 was a SN bifurcation oscillator instead of an AH bifurcation oscillator, it would not be possible to have a 0 phase shift difference between oscillators 2 and 3.

5 Conclusions

Phase response curves (PRCs) of Izhikevich neuron model based oscillators were obtained through simulations. Using this database of PRCs, it was possible to determine the range of constant absolute or traveling-wave type phase difference that could be achieved between two mutually coupled oscillators. A manual search of the appropriate neuron parameter values was carried out in order to apply the proposed method for the coordinated walking movement of a four-legged animal. An automated computer search and optimization algorithm that uses simulated PRC data or numerically generated Malkin’s theorem [4,8] based PRC data can be developed.

References

1. Ermentraut, B.: Type I Membrane, Phase Resetting Curves and Synchrony. Neural Computation 8, 979–1001 (1966)

2. Brodfuehrer, P.D., Debski, E.A., O’Gara, B.A., Friesen, W.O.: Neuronal Control of Leech Swimming. J. of Neurobiology 27(3), 403–418 (1995)

3. Gutkin, B.S., Ermentrout, G.B., Reyes, A.D.: Phase-response Curves Give the Responses of Neurons to Transient Input. J. of Neurophysiology 94, 1623–1635 (2005)

4. Malkin, I.J.: Some Problems in the Theory of Nonlinear Oscillations. Moscow (1956); U.S. Atomic Energy Commission, translation AEC-tr-3766, Washington D.C (1959)

5. Izhikevich, E.M.: Simple Model of Spiking Neurons. IEEE Trans. on Neural Networks 14, 1569–1572 (2003)

6. Izhikevich, E.M.: Which Model to Use for Cortical Spiking Neurons? IEEE Trans. on Neural Networks 15, 1063–1070 (2004)

7. Izhikevich, E.M.: Dynamical System in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press, Massachusetts (2007)

8. Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Springer, New York (1997)

9. Hodgkin, A.L., Huxley, A.F.: A Quantitative Description of Membrane Current and Application to Conduction and Excitation, Nerve. J. of Physiology 117, 550–554 (1952)

10. Kopell, N.: Toward a Theory of Modeling Central Pattern Generators. In: Cohen, A., Grillner, S., Rossignol, S. (eds.) Neural Control of Rhythmic Movements in Vertebrates, pp. 369–413. J. Wiley & Sons, New York (1988)

11. Rand, R.H., Cohen, A.H., Holmes, P.J.: System of Coupled Oscillators as Models of Central Pattern Generators. In: Cohen, A., Grillner, S., Rossignol, S. (eds.) Neural Control of Rhythmic Movements in Vertebrates, pp. 333–367. J. Wiley & Sons, New York (1988)

Phase Control of Coupled Neuron Oscillators

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