Abstract In Chapter 6, we studied a simple version of the cable equation, where diffusion term was added to FitzHugh-Nagumo equations. this chapter, will revisit equation and go through derivation model. addition, consider numerical solution for neuronal axon with membrane dynamics modeled by Hodgkin-Huxley

Abstract In this work a new mathematical model for complex neural networks is presented by the partly diffusive FitzHugh-Nagumo equations with ensemble boundary coupling. The dissipative dynamics and coupling of solutions are proved sharp uniform estimates. synchronization kind under condition that synaptic stimulation signal strength exceeds threshold quantitatively expressed.

We provide an asymptotically justified derivation of activity measure evolution equations (AMEE) for a finite size neural network. The approach takes into account the dynamics for each isolated neuron in the network being modeled by a biophysical model, i.e. Hodgkin-Huxley equations or their reductions. By representing the interacting network as self and pairwise interactions, we propose a gene...

This paper gives asymptotic approximations for the probability density functions for delay bifurcations altered by noise. In these processes, very small noise signi cantly reduces the delay until the transition time. The asymptotic approximation describes the dynamics through the transition and is used to calculate mean transition times. The method is applied to a one dimensional canonical mode...