Oscillations and irregular persistent firing patterns in a homogeneous network of excitatory stochastic neurons with gap junctions in the mean-field limit (1707.06038v1)

Abstract: We continue the work of a series of previous studies of a mathematical model that describes the mean-field limit behavior of a homogeneous network of excitatory point spiking neurons. Contrary to other models, here noise is intrinsic to the neurons through a membrane-potential dependent spiking rate probability, that we assume to be given by a power law. This allows one to collapse several sources of neural noise into a single function, which aids for a more treatable mathematical and comptuational analysis. In particular, we give pseudo-analytical expressions for the invariant distributions of membrane potentials across the population and study their stability computationally. The neurons are assumed to be connected both by chemical and possibly electrical (gap junction) synapses and can also undergo a leakage of ions with the extracellular medium. The distributions are of compact support whenever leakage or gap junction rates are non-zero and an infinite discontinuity appears at the maximum potential in the population. This happens invariably for the leakage (with no gap junctions) and gap junction (without leakage) cases. However, these discontinuous distributions might only be stable for linear spiking rate functions. It was recently shown how the network can present highly synchronous states of global oscillations in its activity when leakage is not present and gap junctions are strong enough. Here we confirm how oscillations persist also when weak leakage is considered. Thus, these dendritic (rather than axonal) gap junctions could play a role in the high levels of neural synchrony necessary for the development of neural systems at young ages. The model, thus, presents a rich phenomenology and capability to reproduce several biological features of neural networks despite of its mathematical simplicity, hence presenting a powerful tool for high performance computing.