4.1. The behavior of an IPFM pulse generator can be described by the equations:

where Vg (t) is the generator potential, y is the unit impulse output, and 9 is the firing threshold = 0.025 V. Let Vg (t) = 4e-7t , t S 0.

a. Find a general expression for tk, the time of the kth output pulse.

b. How many output pulses occur?

c. Find the maximum instantaneous pulse frequency output.

d. Find the range of peak input voltage at t = 0 such that only one output pulse occurs.

4.2. An RPFM (leaky integrator) spike generator is described by the equations:

Repeat (a), (b), (c), and (d) of Problem 4.j. A simulation can be used to obtain solutions.

4.3. An RPFM spike generator system is described by the system:

where the SGL input is a train of impulses of area A and occurring at period T. The SGL output is y, and its time constant is t. A = 0.5, 9 = 1.3, t = 0.667 s.

a. Draw a functional block diagram of the system.

b. Find the T value above which the SGL will never fire.

c. Now assume the SGL input is only two impulses: x = A8(t) + A8(t -8). Find the range of instantaneous frequency, r = 8-1, over which a single output pulse is produced. Note: Parts b and c may be solved analytically, or by simulation.

4.4. The analog outputs of two adjacent photoreceptors act on two nonspiking interneurons, (N1 and N2), whose outputs, in turn, control the generator potential of a third, nonspiking interneuron (N3). A spiking interneuron (N4) is modeled by an RPFM system. Receptor A generates a depolarizing (positive) potential, Va, at N3 according to the ODE: Va = -a*Va + K*I*a. Receptor B generates an inhibitory, hyperpolarizing potential, given by: Vb = -b*Vb - K*I*b. The input to N4 is simply Ve = Va - Vb. N4 can generate spikes only if Ve is positive and large enough. In this system, a > b. This system is illustrated in Figure P4.4.

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