Neural Networks

Abstract

Consider a neuron with its membrane potential near a threshold value. When an external input drives the potential to the threshold, the neuron’s activity experiences a bifurcation: The equilibrium corresponding to the rest potential loses stability or disappears, and the neuron fires. This bifurcation is local, but it results in a nonlocal event — the generation of an action potential, or spike. These are observable global phenomena. To model them requires knowledge about global features of neuron dynamics, which usually is not available. However, to predict the onset of such phenomena, one needs only local information about the behavior of the neuron near the rest potential. Thus, one can obtain some global information about behavior of a system by performing local analysis. Our nonhyperbolic neural network approach uses this observation.