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FIGURE 4.3-2 Block diagram of a system producing RPFM. It is the same as the IPFM system except that the ideal integrator is replaced by a single real-pole, low-pass filter. When an output pulse occurs, it is fed back to reset the LPF output voltage to zero. The RPFM pulse generator is also called a "leaky integrator" pulse generator by some computational neurobiologists.

FIGURE 4.3-2 Block diagram of a system producing RPFM. It is the same as the IPFM system except that the ideal integrator is replaced by a single real-pole, low-pass filter. When an output pulse occurs, it is fed back to reset the LPF output voltage to zero. The RPFM pulse generator is also called a "leaky integrator" pulse generator by some computational neurobiologists.

FIGURE 4.3-3 A Simnon simulation of an RPFM pulse generator given a step input of Eo = 3.5 V dc. The rising exponential trace is the LPF output when the firing threshold, 9 > Eo = 3.5 V. When the threshold is set to 9 = 2 V, output spikes occur, and the LPF output is reset to zero at each spike. Euler integration was used with dt = 0.001, c = 0.5 r/ms.

FIGURE 4.3-3 A Simnon simulation of an RPFM pulse generator given a step input of Eo = 3.5 V dc. The rising exponential trace is the LPF output when the firing threshold, 9 > Eo = 3.5 V. When the threshold is set to 9 = 2 V, output spikes occur, and the LPF output is reset to zero at each spike. Euler integration was used with dt = 0.001, c = 0.5 r/ms.

Example 4.3-1

As a first example, examine the steady-state firing rate of an RPFM SGL model for a constant e = Eo U(t), where Eo > 9. The RPFM low-pass filter output (if not reset) can be shown to be v(t) = Eo (1 - e-ct ), 0 3 t 3 x.

The first output pulse occurs at t1 when v(t1) = <r. v(t) is reset to 0 by the feedback pulse, and again begins to charge exponentially toward Eo. The second pulse occurs at t2 such that (t2 - t1) = t1 = t, the third pulse will be at t3 so that (t3 - t2) = t, etc. This behavior is illustrated in Figure 4.3-3. It is easy to see that the RPFM output pulse period will be constant and equal to t = t1 when the dc input is Eo > <r. One can write an algebraic expression for the RPFM SGL steady-state output frequency as r(Eo) = 1/t1. First, v(t1) = < = E0 (1 - e-ct1) 4.3-8

By algebraic manipulation, the steady-state period can be written as:

For example, inspection of Equation 4.3-10 reveals that for E = 2<r, the firing rate of the RPFM SGL equals c/ln(2) = 1.443c pps; for E = 10<, r = c/ln(10/9) = 9.49c pps. Of course, for Eo < <r, the system does not fire (here c = 1, <r = 2).

A Simnon model for an RPFM SGL is given below. Note that only the first line of code differs from the IPFM model.

dv = -c*v + c*e - z " Analog LPF with inputs e and z. w = IF v > phir THEN 1 ELSE 0 s = DELAY (w, tau) x = w - s y = IF x > 0 THEN x ELSE 0 " y(t) is triangular: height =1, base = 2tau. z = y*phir/tau " The pulse z resets v to 0. u = y*Do/tau " RPFM SGL output pulses

In summary, the RPFM model for a SGL is "more biological." It possesses a dead zone, mimicking subthreshold stimulation, and also exhibits lossy memory, where the effect of a singular epsp input decays toward zero with time constant, 1/c s. As noted in the preceding section, it can easily be given a relative refractory period. Figure 4.3-4 illustrates the RPFM response of the neural model to impulse inputs to its LPF. It fires on the ninth input pulse, which causes vr to S <r, then resets its LPF to zero.

As an example of subthreshold behavior of an RPFM neuron, calculate the minimum frequency of a periodic input pulse train that will not cause the RPFM neuron to fire. Let each periodic input pulse have unit area; set the firing threshold to < = 15, and let c = 10 r/s. The LPF output v(t) will have a sawtooth appearance as shown in Figure 4.3-5. It is easy to see that the first peak is at v1pk = Do c V. By using superposition, the second peak can be shown to be at v2pk = Do c (1 + e-cT); the third peak will be at v3pk = Do c (1 + e-cT + e-2cT). In general the nth peak can be written as v(n)pk = Docly^ 4.3-11

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