Gap Junctions and Emergent Rhythms

Abstract

Gap junction coupling is ubiquitous in the brain, particularly between the dendritic trees of inhibitory interneurons. Such direct non-synaptic interaction allows for direct electrical communication between cells. Unlike spike-time driven synaptic neural network models, which are event based, any model with gap junctions must necessarily involve a single neuron model that can represent the shape of an action potential. Indeed, not only do neurons communicating via gaps feel super-threshold spikes, but they also experience, and respond to, sub-threshold voltage signals. In this chapter we show that the so-called absolute integrate-and-fire model is ideally suited to such studies. At the single neuron level voltage traces for the model may be obtained in closed form, and are shown to mimic those of fastspiking inhibitory neurons. Interestingly in the presence of a slow spike adaptation current the model is shown to support periodic bursting oscillations. For both tonic and bursting modes the phase response curve can be calculated in closed form. At the network level we focus on global gap junction coupling and show how to analyze the asynchronous firing state in large networks. Importantly, we are able to determine the emergence of non-trivial network rhythms due to strong coupling instabilities. To illustrate the use of our theoretical techniques (particularly the phase-density formalism used to determine stability) we focus on a spike adaptation induced transition from asynchronous tonic activity to synchronous bursting in a gap-junction coupled network.

Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities

“…While a large number of theoretical investigations have been devoted to the effects of chemical synapses on synchrony, electrical synapses have received less attention (Sherman and Rinzel 1992;Chow and Kopell 2000;Lewis and Rinzel 2003;Pfeuty et al 2003Pfeuty et al , 2005Kopell and Ermentrout 2004;Bem et al 2005;Schneider et al 2006;Coombes and Zachariou 2008;Coombes 2008). In particular, these studies mostly considered homogeneous networks in the low-noise regime, where all neurons fire in a regular fashion at an identical firing rate.…”

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

“…Indeed the techniques for constructing the relevant saltation matrices for handling synaptic dynamics with pulsatile forcing as well as discontinuous reset in non-linear integrate-and-fire systems have recently been described by Ladenbauer et al [33]. The combination of this with pwl caricatures of Izhikevich style integrate-and-fire models [46], such as described in [47], opens the door for a very thorough examination of network states in biologically realistic spiking networks.…”

Neural Activity Measures and Their Dynamics

“…Existence of a synchronous attractor was shown to exist in general network architectures with sufficiently strong connections. Examples include a network of oscillators with linear connections [41], neural piecewise linear models connected by gap junctions [42,43] and in general models with gap junctions interactions that obey the semi-diffusivity property [44]. Here, Fig.…”