The IPFM model for SGL action is an ideal linear VFC that generates a pulse train output whose average frequency is equal to the average input voltage. Figure 4.3-1 illustrates a block diagram of an IPFM system. The input to an ideal integrator is a continuous positive analog voltage, e. e is integrated until the integrator output v exceeds the pulse generation threshold, 9, at t = tk. As 9 is exceeded, an impulse (delta function) of area Do is produced at tk at the IPFM generator output, and, simultaneously, an impulse of area -9 is added to e at the integrator input. This feedback pulse resets the integrator output to v = 0 at tk+, and the integration of e

continues. This process may be described mathematically by the set of integral equations:

Jtk-i where Ki is the integrator gain. e(t) is the input; e(t) = 0 for t < 0. tk is the time the kth pulse is emitted. Equation 4.3-1 can be rewritten as

Here rk is the kth element of instantaneous frequency, defined as the reciprocal of the interval between the kth and (k - 1)th output pulses, xk = (tk - tk-1). So the kth element of instantaneous frequency is given by rk - VTk = (Ki/9){eKt 4.3-3

Simulating an IPFM SGL using Simnon can use the following subroutine:

dv = e - z " Integrator with K± = 1. v is a state, dv is its derivative, w = IF v > phi THEN 1 ELSE 0 "}

s = DELAY(w, tau) "} ^ Pulse generator, x = w - s "}

IF x > 0 THEN x ELSE 0 " Pulses are > 0. y*phi/tau " Pulse resets integrator from phi to 0. y*Do/tau " Pulse train output, pulse areas = Do.

Note that Euler (or rectangular) integration must be used with AT = t. By using Simnon, the actual, simulated "unit" pulses, b, are triangular with height 1 and base width 2t. Thus pulses y(t) have areas A = 1(2t)/2 = t. To reset the integrator from v = 9 to v = 0, the unit impulse y is given area 9 by multiplying it by (9/T), and then subtracting this pulse from the input, e.

The IPFM model is a simple linear VFC, responding only to v > 0. It provides an "a-emulation" of a neural SGL. The IPFM SGL has infinite memory for past subthreshold inputs, which is decidedly unbiological. A relative refractory period can be easily added to the Simnon program for IPFM voltage-to-frequency conversion by manipulating the firing threshold, 9. Define:

phi is the instantaneous firing threshold. phi0 is the steady-state value of phi when the IPFM system has not produced a spike in a long time, and the relative refractory period is made by inputting the output pulse, z, occurring at t = tk, into a low-pass filter. Thus, the first-order ODE makes a low-pass filter with time constant 1/a:

(y/tau) is a unit impulse, so the output of the low-pass filter is the simple exponential decay, which is added to 9(t) at t = tk:

Note that the RPFM system below can use the same scheme to give its SGL a refractory period.

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