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4.5.4 A Ring CPG Model with Negative Feedback

Kleinfeld and Sompolinsky also suggested ring oscillators as pattern generators. As in the case of the RI pair, a ring oscillator with positive feedback (all excitatory synapses around the ring) will be unstable. The net frequency around the ring rises to a saturation value set by the refractory periods of the neurons; no stable pattern is seen. If the ring uses negative feedback, as shown in Figure 4.5-10, and the epsp time constants, RPFM time constants, and pulse weights are "tuned," a stable, patterned burst oscillation around the ring will occur.

FIGURE 4.5-10 A ring oscillator model that uses negative feedback. A depolarizing dc drive, Vin, is applied to neuron N1. Sustained circulating bursts of firing are produced.

Four neurons are connected in a ring. N1 excites N2, N2 excites N3, N3 excites N4, and N4 inhibits N1, which is also excited with dc. The ring uses negative feedback and acts as a stable oscillator. The Simnon model for this candidate CPG is given below:

continuous system RINGOSC1 " 3/16/99

" Model for a neural ring of 4 oscillator using NFB.

STATE v1 v2 v3 v4 pi p2 p3 p4 q1 q2 q3 q4

DER dv1 dv2 dv3 dv4 dp1 dp2 dp3 dp4 dq1 dq2 dq3 dq4

TIME t

dv1 = -c1*v1 + c1*E1 - z 1 " Neuron 1. RPFM model. w1 = IF v1 > phi1 THEN 1 ELSE 0 s 1 = DELAY (w1, tau) x1 = w1 - s1

y1 = IF x1 > 0 THEN x1 ELSE 0 z1 = y1*phi1/tau u1 = y1*Do1/tau dv2 = -c2*v2 + c2*E2 - z2 " Neuron 2. RPFM model.

w2 = IF v2 > phi 2 THEN 1 ELSE 0 s2 = DELAY (w2, tau) x2 = w2 - s2

y2 = IF x2 > 0 THEN x2 ELSE 0 z2 = y2*phi2/tau u2 = y2*Do2/tau dv3 = -c3*v3 + c3*E3 - z3 " N3 RPFM model.

w3 = IF v3 > phi 3 THEN 1 ELSE 0 s3 = DELAY (w3, tau) x3 = w3 - s3

y3 = IF x3 > 0 THEN x3 ELSE 0 z3 = y3*phi3/tau u3 = y3*Do3/tau dv4 = -c4*v4 + c4*E4 - z4 " N4 RPFM model inhibits N1. w4 = IF v4 > phi 4 THEN 1 ELSE 0 s4 = DELAY (w4, tau) x4 = w4 - s4

y4 = IF x4 > 0 THEN x4 ELSE 0 z4 = y4*phi4/tau u4 = y4*Do4/tau

" 2 TIME CONSTANT SYNAPSES:

dp4 = -a4*p4 + u4 " Inhibitory synapse dq4 = -b4*q4 + p4*a4*b4

" NEURON INPUTS:

E1 = Vin - q4" N1 has a dc clock drive input, Vin E2 = q1 E3 = q2 E4 = q3

" PLOTTING VARIABLES:

" CONSTANTS:

tau:

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