Recurrent or reciprocal inhibition and feed-forward inhibition are well-documented neural architectures that have been proposed to describe the generation of burst firing in an output neuron, and alternate burst firing in a pair of motor neurons innervating antagonistic muscles (in biomedical engineering speak, this is a two-phase motor output). Recurrent inhibition has been discussed by Eccles (1964) as a possible CNS mechanism to generate phased bursts of neural firing observed in the CNS. Kleinfeld and Sopolinsky (1989) discuss central pattern generators (CPGs) that drive moto-neurons that effect rhythmic activities such as swimming, scratching, chewing, walking, and breathing. These authors state, "The smallest [neural] circuit that can produce a rhythmic output consists of two neurons coupled by reciprocal inhibitory synaptic connections."

Figure 4.5-1 illustrates the basic reciprocally inhibited pair as traditionally configured in the literature. Two output neurons (N1 and N2) each receive some excitatory input. Each output neuron also excites an inhibitory interneuron (N3 and N4), which in turn sends its axon to the opposite output neuron, where inhibitory synaptic contact is made. Conventional wisdom says that this circuit should be a two-phase, burst generator. One output neuron is supposed to fire several pulses, which prevent the other output neuron from firing. Then, the second output neuron fires, inhibiting the first, etc. The problem is that this circuit, as proposed, is not a negative feedback circuit; rather, it has positive feedback. (There are two effective sign inversions around the four-neuron loop. Their product gives an effective positive loop gain.) As discussed below, the basic RI pair is unstable in that once an output neuron fires, its output continually inhibits the other output neuron, and there are no alternating spike bursts generated. If the basic RI pair is run with low loop gain, both output neurons fire continuously at a slower rate, but nearly synchronously; there are no bursts. Wide changes in the synaptic ballistic filter time constants do not change this behavior. (Note that no transport delays were used in this model.)

The following section, examines the behavior of the basic reciprocal inhibitory neural model for its effectiveness in producing two-phase burst outputs, given a common input.

Figure 4.5-2 shows a more-detailed architecture of the basic reciprocal inhibition system discussed in the introduction to this section. A Simnon program, RECIPIN3.t, to simulate this model follows:

Continuous system RECIPIN3" V. 3/16/99 " The Basic 4 Neuron RI model.

STATE v1 v2 v3 v4 p1 p2 p3 p4 p5 p6 va

DER dv1 dv2 dv3 dv4 dp1 dp2 dp3 dp4 dp5 dp6 dva

" IPFM VFC TO GENERATE INPUT SPIKES:

xa = wa - sa ya = IF xa > 0 THEN xa ELSE 0 za = ya*phia/tau ua = ya*Doa/tau

" THE RPFM RI PAIR:

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

FIGURE 4.5-2 A more-detailed model of the RI pair, showing synapses and critical parameters. This model is simulated with the program, RECIPIN3.T. An RPFM model is used for each of the four neurons. A variable-frequency input is generated for the output neurons, Nj and N2. Single-time-constant, low-pass filters are used to simulate epsps and ipsps.

FIGURE 4.5-2 A more-detailed model of the RI pair, showing synapses and critical parameters. This model is simulated with the program, RECIPIN3.T. An RPFM model is used for each of the four neurons. A variable-frequency input is generated for the output neurons, Nj and N2. Single-time-constant, low-pass filters are used to simulate epsps and ipsps.

z1 = y1*phi1/tau u1 = y1*Do1/tau dv2 = -c2 *v2 + c2 *E2 - z2 " Output 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

" 2 RPFM INHIBITORY INTERNEURONS:

dv3 = -c3*v3 + c3*e3 - z3 " Inhibitory interneuron 3. 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 " Inhibitory interneuron 4. 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

" 1 TIME-CONSTANT, SYNAPTIC BFs:

" Synapse 1 ballistic filter (ipsp) " Synapse 2 ballistic filter. q2 is output.

" Inputs to RPFM neurons

E1 = p1 - p6 " epsp - ipsp input to output neuron E2 = p2 - p5 " epsp - ipsp input to output neuron E3 = p3 " epsp input to inhibitory interneuron 3 E4 = p4 " epsp input to inhibitory interneuron 4.

" INPUTS TO IPFM VFCs:

Ea = A" Nonzero input to IPFM VFCa.

tau:0.001" ms a1:1 " All natural frequencies are in r/ms.

a2:1

c2:1

c3:1

c4:1

phia:1

phi1:1

phi2:1

phi3:1

phi4:1

Do1:0.75

Do2:0.75

Do3:1

Do4:1

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