Summed with the excitation is inhibition coupled from the lateral plexus fibers arising from the retinula cells of neighboring ommatidia. The system can be modeled mathematically. For each retinula cell, use the biochemical, kinetic model considered in Problem 2.1. Here, the depolarization of the first retinula cell is given by vm1 = k6 c1, where a1, b1, and c1 are molecular concentrations in the rhabdom of the first retinula cell. Thus,
b1 = a1k1 l°g(1 + I^I0)+ k5c1 - b1 (k2 + k3c1) c1 = b1 (k2 + k3c1)-c1 (k4 + k5 )
The input to the RPFM spike generator of the first eccentric cell is assumed to be given by
The first ommatidium RPFM spike generator is modeled by
Uj = cVg1 - cuj - resetj w1 = IF uj > phi THEN 1 ELSE 0 sj = DELAY (w1, tau) xi = wi - si y1 = IF x1 > 0 THEN x1 ELSE 0 reset1 = y1 phi/tau y1 are the output impulses of the first eccentric cell. (Euler integration must be used with delt = tau if using Simnon to run this simulation.)
Write a program to simulate a three-ommatidium LI system. Plot the output spikes of the three eccentric cells (y1, y2, and y3), the generator potentials Vg1, Vg2, and Vg3, and the inputs, I1(t), I2(t), and I3(t). Arrange the inputs so that they occur singly, overlap in pairs, and occur together. Parameters for the retinula cell depolarization are a(0) = 1, other ICs = 0, k1 = 4, k2 = 0.3, k3 = 40, k4 = 10, k5 = 0.1, k6 = 100, Io = 1. For the RPFM spike generators of the eccentric cells, let phi = 10 mV, c = 1 rad/ms, and all Kjk = 0.3 (assumes the ommatidia are equidistant).
5.10. This problem will consider the properties of lateral inhibition applied between the mechanosensory hair cells that line the interior of an invertebrate statocyst. (See Section 2.3.5 for a description of statocysts.) As in the case of LI applied to visual systems, this problem will examine a mathematical model in which linearity is assumed, and calculations are carried out in continuous form in one dimension. The input variable is a one-dimensional distribution of static, radial force applied to the hair cells; the force is the result of gravity acting on the mass of the statolith (forces from linear and angular acceleration of the statocyst will be neglected). Although the interior of the statocyst is roughly spherical, a major simplification results if the problem is framed in terms of an infinite, linear dimension, x, -x 3 x 3 x, instead of 6, -n 3 6 3 n. In the case of the spherical statcyst, assume that an output neuron (without LI) fires at a frequency, ro, proportional to the radial force exerted on it by the statolith. The same property is assumed for the linear continuous case: e.g., ro(x) = K f(x). Here ro(x) is the frequency of the mechanosensor located at x, f(x) is the radial force at x, and K is a positive constant. Each stimulated receptor is assumed to be inhibited by its neighbors' firings according to the rule:
Where ki(x) is the spatial distribution of inhibition from a sensory cell at x to its neighbors. A schematic of a vertical section through statocyst (not all sensory cells are illustrated) is shown in Figure P5.10A. The one-dimensional, discrete case is illustrated in Figure P5.10B.
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