Analytic results on the Hodgkin-Huxley neural network: Spikes annihilation

Abstract

Various families of Neural Networks (NN) have been used in the study and development of the field of Artificial Intelligence (AI). More recently, the Hodgkin-Huxley (HH) has attracted a fair bit of attention. Apart from the HH neuron possessing desirable "computing" properties, it also can be used to reasonably model biological phenomena, and in particular, in modeling neurons which axe "synchronized/desynchronized1". This paper, which we believe is a of pioneering sort, derives some of the analytic/learning properties of the HH neuron, and neural network. The HH Neuron is a nonlinear system with two equilibrium states: A fixed point and a limit cycle. Both of them co-exist and are stable. The behavior of this neuron can be switched between these two equilibria, namely spiking and resting respectively, by using a perturbation method. The change from spiking to resting is referred to as Spike Annihilation. In this paper, we analytically prove the existence of a brief excitation (input) which, when delivered to the HH neuron during its repetitively firing state, annihilates its spikes. We also formally derive the characteristics of this brief excitation. We thus believe that our analysis of the HH neuron has practical implications in clinical applications, especially in the case of the desynchronization of neuronal populations. © Springer-Verlag Berlin Heidelberg 2007.