A model of neuronal bursting using three coupled first order differential equations

Abstract

A modification to a recent model of the action potential which introduces 2 additional equilibrium points is described. By using stability analysis 1 of these equilibrium points is shown to be a saddle point from which there are 2 separatrices which divide the phase plane into 2 regions. In 1 region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron of the pond snail into one that fires repetitively. Addition of a 3rd equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge.