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Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

Coombes, Stephen



The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.


Coombes, S. (2008). Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models. SIAM Journal on Applied Dynamical Systems, 7(3), 1101–1129.

Journal Article Type Article
Acceptance Date Jun 3, 2008
Online Publication Date Sep 25, 2008
Publication Date Oct 1, 2008
Deposit Date Jun 5, 2008
Publicly Available Date Sep 25, 2008
Journal SIAM Journal on Applied Dynamical Systems
Print ISSN 1536-0040
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 7
Issue 3
Pages 1101–1129
Public URL
Publisher URL


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