# The smallest value of the axon density for which ‘ignition’ can occur in a random net

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## Abstract

As shown by A. Rapoport (1952), when a very brief stimulation or “instantaneous input” is applied to a random net, the subsequent events are determined by the parameters of the net as follows: If the axon density*a* is sufficiently large and the fraction γ of the neurons initially stimulated exceeds a certain value γ_{1} (the*over-all* threshold of the net for instantaneous stimulation), excitation will spread through the net until a steady state is reached in which a fraction γ_{2} ⩾ γ_{1} of the neurons is firing (“ignition phenomenon”). If γ < γ_{1} the activity in the net dies out. However, if the axon density is too small, the activity will ultimately die out, no matter how large the fraction of initially stimulated neurons. Thus there exists a limiting value*A* of the axon density below which the net cannot “ignite”. This*A* is a function of*h*, the*individual* threshold of the neurons constituting the net (we assume here*h*≥2, since for*h*=1 the situation is essentially different). Geometrically γ_{1} and γ_{2} are determined as the two intersection points of a straight line with a sigmoid curve. When*a<A* the two curves do not intersect and for*a=A* they are tangent.

In this paper the “tangency case” is investigated and the general features of the function*A(h)* are determined. It is shown that*A* increases monotonically with*h* (as one would expect). For all values of*h*>1 we have*A(h)>h*, but the fraction*A(h)/h* and the derivative*dA(h)/dh* approach unity as*h* increases. An analytical expression of the function*A(h)* valid for very large values of*h* is derived.

## Keywords

Neuron Firing Sigmoid Curve Hypergeometric Series Good Approxima Positive Steady State## Preview

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## Literature

- Rapoport, A. 1952. “‘Ignition’ Phenomena in Random Nets.”
*Bull. Math. Biophysics*,**14**, 35–44.MathSciNetGoogle Scholar - Whittaker, E. T. and G. N. Watson. 1945.
*A Course of Modern Analysis*. Amer. Ed. New York: Macmillan Co.Google Scholar