## Xdz  FIGURE 4.4-1 (A) Computed responses of a simulated T-neuron. Traces: 1, Vex (input to RPFM SGL; 2, x2 (input 1 epsp); 3, x4 (input 2 epsp); 4, (RPFM firing threshold); 5, SGL output. Simulation parameters in all scenarios in this figure: a = b = 1; c = 0.5; q>r = 0.355, pulse 2 delay, T = 3 ms; input pulse areas = 1. SGL obviously does not fire. (B) Same conditions except input pulse spacing, T = 2.05 ms. SGL on threshold of firing. (C) Same conditions except input pulse spacing, T = 0.3 ms. SGL fires one pulse at about t = 1.8 ms. Simulation indicates that the SGL will fire for T 3 2.0 ms.

FIGURE 4.4-1 (A) Computed responses of a simulated T-neuron. Traces: 1, Vex (input to RPFM SGL; 2, x2 (input 1 epsp); 3, x4 (input 2 epsp); 4, (RPFM firing threshold); 5, SGL output. Simulation parameters in all scenarios in this figure: a = b = 1; c = 0.5; q>r = 0.355, pulse 2 delay, T = 3 ms; input pulse areas = 1. SGL obviously does not fire. (B) Same conditions except input pulse spacing, T = 2.05 ms. SGL on threshold of firing. (C) Same conditions except input pulse spacing, T = 0.3 ms. SGL fires one pulse at about t = 1.8 ms. Simulation indicates that the SGL will fire for T 3 2.0 ms.

4.4.2 A Theoretical Band-Pass Structure: The Band Detector

Reiss (1964) proposed a theoretical neural structure that displayed selectivity for a range of input frequencies. He called this structure a band-detector. It produced output pulses for a steady-state periodic input pulse train with a frequencies between rmin1 and rmax1 , and also outputs in harmonic bands, 2rmin1 to 2rmax1, 3rmin1 to 3rmax1, etc. Eventually the bands overlap, and there is a continuous output at high input frequencies. Figure 4.4-2 illustrates Reiss's band detector. N1 is the input neuron (source of periodic pulses), N2 is a 1:1 neuron that delays the N1 pulses by D before they synapse on N3, a T-neuron. Both inputs to N3 are excitatory and have equal weight. Each excitatory input has a duration T after the pulse arrives at N3. The input from N1 is not delayed. It is possible to show, assuming strict interpretation of the 1:1 and T-neuron function, that the first pass band has: rmin1 = 1/(D + T), rmax1 = 1/(D - T), and the arithmetic center of the band 1 is rcenter1 = D/(D2 - T2). Thus, the "Q" of the first pass band is FIGURE 4.4-2 Schematic of a band detector neural system.

max1 min1 I

The second passband is rmin 2 = 2/(D + T), rmax2 = 2/(D - T), etc.

One of the problems with the basic band detector is that it passes harmonics of the fundamental passband. To circumvent this problem and to make a pure, single band-pass system, Reiss developed a band-detector with harmonic suppression. The output is active only in the primary passband. Figure 4.4-3 illustrates a harmonic-suppressing band detector. The input pulse train inputs without delay to synapse x on the T-neuron. The output of a 1:1 neuron, driven by the source, is delayed D2 then inputs to the T-neuron through excitatory synapse y. Another 1: 1 neuron driven by the source, is delayed D1, then inputs to the T-neuron through inhibitory synapse w. D2 > D1. The dwell of each excitatory input is T, as in the case of the band detector, and the inhibitory input has dwell I. That is, no output of N3 can occur, FIGURE 4.4-3 Schematic of a band detector with harmonic suppression. An RPFM T-neuron is used.

FIGURE 4.4-3 Schematic of a band detector with harmonic suppression. An RPFM T-neuron is used.

regardless of the enabling condition, if the enabling condition occurs within I seconds following the inhibitory input. Reiss shows that second- and higher-order harmonic bands will not appear in the output if the conditions, D1 > T, (D1 + I) + (n - 1)/r > (D2 + T), n S 2, are met. The primary passband of N3 is given by rmin1 = 1/(D2 + T), and rmax1 = 1/(D2 - T), as in the case of the simple band detector. Determining the conditions on the steady-state input pulse period, 1/r, that will produce an N3 output is best done by trial and error, sketching various cases for the three input waveforms.

To examine the performance of a more realistic neural model band detector with harmonic suppression (BDHS), the author has written a Simnon neural model using an RPFM SGL for N3. The program is given below:

Continuous system BDsuprl " V. 3/04/99

" Use EULER integration with delT = tau. There are 9 states.

" This system is Reiss' BD with harmonic suppression.

STATE noise v1 v3 p1 q1 p2 q2 p3 q3

DER dnoise dv1 dv3 dp1 dq1 dp2 dq2 dp3 dq3

dnoise = -wo*noise + SD*NORM(t) " BW limiting ODE for noise.

Vin = noise + Kr*t + Vo " Noise + ramp + const. drive for N1 " IPFM VFC.

" THE IPFM SPIKE SOURCE:

dvl = Vin - zl " IPFM neuron N1: Integrator Ki = 1. wl = IF v1 > phi1 THEN 1 ELSE 0 " N1 behaves as an ideal VFC. s1 = DELAY(w1, tau) " Pulse generator. x1 = w1 - s1

y1 = IF x1 > 0 THEN x1 ELSE 0 " Pulses are > 0.

z1 = y1*phi1/tau " Pulse resets integrator.

u1 = y1*Do1/tau " Pulse train output, pulse areas = Do1.

y2 = DELAY (y1, D2) " Excit. delay D2 ms. to N3 synapse. u2 = Do2*y2/tau" Thru excit. 1:1 interneuron (xparent).

yi = DELAY(y1, D1)" Inhibitory delay (thru xparent 1:1 neuron), ui = Do3*yi/tau

" SYNAPTIC BALLISIC FILTER ODEs. (There are 3 synapses on N3) dp1 = -a1*p1 + a1*u1 dq1 = -a1*q1 + a1*p1

dp3 = -a3*p3 + a3*ui " BF for ipsp. Determines effective delay, I.

dv3 = -c3*v3 + c3*e3 - z3 z3 = y3*phi3/tau w3 = IF v3 > phi3 THEN 1 ELSE 0

y3 = IF x3 > 0 THEN x3 ELSE 0 e3 = g1*q1 + g2*q2 - g3*q3

" OFFSET SCALED SPIKE OUTPUTS FOR PLOTTING:

 b0 = yi/4 + 