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as the local positive voltage makes successive cell-membrane locations more permeable to sodium ions from the outside The propagated pulse eventually produces chemical output by releasing transmitter substance at an output synapse The membrane voltage subsides as positive potassium ions leave the cell, and the cell relaxes to restore equilibrium ion densities The model in Figure 6-14a has an integrate-and-fire block producing pulses y(t) shaped like those in Figures 6-14b and c, and one or more delay sections modeling pulse propagation toward the output synapse The pulse generator integrates a positive input x until the output y reaches a threshold value, and then produces a finite-duration pulse y(t) returns to its initial value in the course of a refractory period determined by ion-motion delays If the positive input persists, the process repeats and generates a pulse train The pulse frequency increases with the input amplitude but is limited to about 1 kHz by the refractory periods The simplest delay sections are represented by differential equations
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(d/dt) qout = qin qout/Tau
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whose time constants Tau depend on ion exchanges and myelin-sheath insulation in successive membrane sections More complicated differentialequation systems can be substituted Each pulsed neuron, then, is represented by a few differential equations that involve switching functions Once one decides on such a neuron model, it is easy to simulate layers and groups of neurons by replacing differential equations and defined-variable assignments with DESIRE vector differential equations and vector assignments (Section 6-25) Multiple neuron inputs can be readily modeled by replacing the input x with a matrix-vector product A * x or with sigmoid(A * x) The resulting pulsed-neuron models can be used in networks such as those in the preceding sections, or in entirely new combinations, especially for biological modeling 6-25 A Simple Integrate and Fire Model In the DYNAMIC-segment program of Figure 6-15, the neuron input x can be a weighted sum of excitatory positive inputs and inhibitory negative inputs; only positive values of x can fire the neuron A neuron pulse y(t) with positive rest value y0 and peak value peak > y0 is produced by an integrator modeled with
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d/dt y = ydot
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The effective integrator input ydot is produced by a combination of switching functions preceded by a step operator that ensures proper numerical integration (Sections 2-9 and 2-11) The crucial point is the use of a difference-equation state variable (Section 2-16) z defined by the recursive assignment (difference equation)
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Pulsed-neuron Replication
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inputs INTERGRATE-AND-FIRE PULSE GENERATOR
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FIGURE 6-14 (a) Pulsed-neuron model, and (b,c) integrate-and-fire time histories
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z = swtch(y peak + bb * z)
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[bb = (1 y0) * peak]
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We start with the integrator output y at its rest value y = y0 (Figs 6-14a and 6-15) y0 is small, so that z is negative and the relay-comparator function produces ydot = c * ydot1 Then for a positive neuron input x: 1 The neuron integrates its input x with ydot = c * ydot1 = c * x until y reaches the firing threshold fire1 2 y then rises rapidly with ydot = c * ydot1 = c * b1, until y = fire2 > fire1 3 y next rises less rapidly with ydot = c * ydot1 = c * b2, until y reaches the peak value y = peak > fire2 At this point, z becomes positive 4 Now, the relay comparator produces ydot = c * ydot2, and y decays with ydot = c * ydot2 = r * c * y, until y returns to its rest value y0
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z then becomes negative again, and the process repeats if x > 0
Step 3 sets the pulse width and can be omitted for about 10% extra speed if thin, pointed pulses (Fig 6-14c) are satisfactory As noted in Section 6-24, the integrate-and-fire model will be followed by linear or nonlinear delay sections representing the pulse propagation along the neuron membrane, for example,