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

1984/03/22 by J. L. Hindmarsh, R. M. Rose · 1,951 citations
Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #Bursting #Computer science #Depolarization #Differential equation #Equilibrium point #Geometry #Limit (mathematics) #Limit cycle #Mathematical analysis #Mathematical optimization #Mathematics #Neural dynamics and brain function #Neuroscience #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Phase (matter) #Phase plane #Physics #Plane (geometry) #Point (geometry) #Saddle #Saddle point #Stability (learning theory) #Statistical physics #stochastic dynamics and bifurcation

paper · doi:10.1098/rspb.1984.0024

published in Proceedings of the Royal Society B Biological Sciences 221(1222), 87-102 (Royal Society)

openalex publication_date 1984/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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.

Citations

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