Towards a neural field theory for spiking neuron networks with electrical synapses
A major challenge in statistical physics, nonlinear dynamics and theoretical neuroscience over the last half century has been to understand the self-organizing principles governing the dynamics of large networks of neurons. Physicists and applied mathematicians have proposed simple mean-field descriptions of spatially-extended neural networks in terms of a relevant macroscopic observable, the firing rate. This approach has been particularly successful and so-called Neural Field Models (NFM) have become an extremely popular mathematical tool in neuroscience, physics and applied mathematics. Yet, to date, mean-field theories describe networks with chemical synapses, but it remains a major theoretical challenge to incorporate electrical synaptic interactions in such mathematical descriptions. Recently, a mean-field theory for large networks of spiking neurons has been proposed, which exactly links the dynamics of single neurons with that of two mean-field variables: The firing rate and the mean membrane potential. Remarkably, this theory permits to incorporate electrical interactions, but the mathematical derivation and the analysis of the dynamics of the first NFM is lagging. This project proposes the formal mathematical derivation of such NFM, as well as the thorough analysis of its dynamics and bifurcations. Towards this goal, at the host institution UPF in Barcelona, the ER will apply mean-field methods and nonlinear dynamical systems theory to derive the novel NFM (which we conjecture is of reaction-diffusion type). During a secondment at VU Amsterdam, the ER will be trained to become an expert in numerical analysis of partial differential equations, which will further allow him to perform extensive state-of-the-art computer simulations. The expected results will provide completely novel mechanistic insights on the emergence of complex spatio-temporal patterns of neuronal activity due to the intricate interplay between chemical and electrical synapses.