Synchronisation phenomena in complex neuronal networks

The phenomenon of neural synchronisation, a simultaneous and repeated firing of clusters of neurons, underlies many physiological functions and pathological manifestations in the brain of humans and animals, ranging from information encoding to epileptic seizures. Neural synchronisation, as a general phenomenon, can be approached theoretically in the framework of Dynamical Systems on Networks. In the present work, we do so by considering complex networks of FitzHugh-Nagumo model neurons. In the first part we consider the most understood models where each neuron treats its presynaptic neurons all on an equal footing, normalising signals with its in-degree. We study the stability of the synchronous state by devising an algorithm that destabilises it by selecting and removing links from the network, so to obtain a bipartite network. The selection is performed using a perturbative expression, which can be regarded as a specialisation of a previously introduced Spectral Centrality measure. The algorithm is tested on Erdős-Renyi, Watts-Strogatz and Barabási-Albert networks, and its behaviour is assessed from a dynamical and from a structural point of view. In the second part we consider the less studied case in which each neuron divides equally its output among the postsynaptic neurons, so to reproduce schematically the situation where a fixed quantity of neurotransmitter is subdivided between several efferent neurons. In this context a self-consistent approach is formulated and its limitations are explored. In order to extend its application to larger networks, a Mean Field Approximation is presented. The predictivity of the Mean Field Approach is then tested on the different random network models, and the results are discussed in terms of the original network properties