Activity Wave in a Neuron Net

We have already noted above that in analyzing excitable systems one has, more often than not, to deal with a parabolic equation with a nonlinear source. In this section we will concern ourselves with an excitable medium of a different type, where the signals are transmitted in the neuron network not by the local currents but by the nervous impulses traveling along the axons. The propagation speed of the activity wave will, if this transmission mode is possible at all, depend not only on the signal transmission speed but also on the other characteristics of nerve cells such as cell body capacitance, conductance, etc. 

To simplify the problem to the maximum possible extent, we will consider a homogeneous isotropic neuron net consisting of cells characterized by the same set of parameters/ The loss in generality will be lavishly compensated for by the considerable simplification of the analysis and the formal clarity in which the results can be obtained. Each neuron will be described as follows. Each unexcited p p ) neuron is a summator of synaptic signals  

C is neuron body capacitance, (p is the potential measured on the neuron body, t is time, ij is total synaptic current of a given neuron, and g is passive conductance of the neuron membrane or leakage conductance (for simplicity the leakage e.m.f. is taken as equal to the resting potential). Assuming the net to be two-dimensional and considering the steady-state excitation mode in it, we have that the variables p and is are functions of the self-similar coordinate  

Let us place the point = 0 on the wave front determined by the condition (p = (psf,- Then the synaptic current to the neuron with the coordinate may be written as  

The function ij(p) has no singular points within the range Re p 0. Therefore, 

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