### Collective phenomena in networks of spiking neurons.

Early Stage Researcher: Federico Devalle

Principle Investigators: Ernest Montbrio (Barcelona, UPF – major institution), Aneta Stefanovska and Peter McClintock (Lancaster, ULANC – partner institution)

When studying the collective dynamics of cortical neurons theoretically, large networks of spiking model neurons have naturally been the benchmark model. Network models incorporate the most fundamental physiological properties of neurons: sub-threshold voltage dynamics, spiking, and discontinuous synaptic interactions. For this reason, networks of spiking neurons are considered to be biologically realistic. However, network models of spiking neurons are typically not amenable to analytical work, and thus constitute above all a computational tool.

Rather, researchers use reduced or simplified models which describe some measure of the mean activity in a population of cells, oftentimes taken as the firing-rate. Firing-rate models are simple, phenomenological models of neuronal activity, generally in the form of continuous, first-order nonlinear differential equations [1]. Such firing-rate models can be exhaustively analyzed using standard techniques of bifurcation theory, and nonlinear dynamics, see e.g. [2]. Nonetheless, firing-rate models do not represent proper mathematical reductions of the original network dynamics but rather are heuristic. As such, there is no theory establishing a link between the microscopic spiking neurons’ parameters with macroscopic quantities such as firing rates.

In this project we will use reduction mathematical techniques to investigate the macroscopic firing rate dynamics of neuronal networks, relating them with the microscopic properties of the neurons. Such techniques have become the subject of intense research in the context of the Kuramoto-like models of coupled oscillators, see e.g [3-9], but their application to investigate neuronal networks is only recently being established [8,9].

The project aims to exhaustively investigate problems of current interest in computational neuroscience using exactly reduced firing-rate models (oscillations and synchronization [2,10,11], neuronal variability and chaos [12-15], balanced states [15]), contributing to further develop a macroscopic theory of neuronal networks.

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