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

TLDR

The equations, originally developed to explain isolated triggered bursts, provide one of the simplest models of oscillatory burst discharge. We describe a modification to our recent action‑potential model that introduces two additional equilibrium points. Stability analysis shows one equilibrium is a saddle point with two separatrices dividing the phase plane, and adding a third equation restricts firing to either an isolated burst or a depolarizing afterpotential. In one region phase paths approach a limit cycle while in the other they converge to a stable equilibrium, a brief depolarizing pulse converts a silent neuron into a repetitively firing one, and steady depolarizing current produces periodic bursting.

Abstract

We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one 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 into one that fires repetitively. Addition of a third 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.