Paper Info
Title
Neuronal Networks with Gap Junctions: A Study of Piecewise Linear Planar Neuron Models
Abstract
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 piecewise linear models that can mimic the. ring response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) are 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 states. For a piecewise 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.
Year
DOI
Venue
2008
10.1137/070707579
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
Keywords
Field
DocType
piecewise linear models,gap junctions,Floquet theory,coupled-oscillator theory,phase-density function
Gap junction,Coupling,Mathematical analysis,Control theory,Constant current,Planar,Phase response curve,Periodic graph (geometry),Piecewise linear function,Mathematics,Floquet theory
Journal
Volume
Issue
ISSN
7
3
1536-0040
Citations 
PageRank 
References 
20
1.52
16
Authors
1
Name
Order
Citations
PageRank
Stephen Coombes118418.30