Ipfm

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

FIGURE 4.3-1 Block diagram of a system producing IPFM. When an output pulse occurs, it is fed back to reset the integrator output voltage to zero.

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.