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| We will investigate the role of connectivity disorder in the emergence of synchronized activity in large neuronal networks, possibly subject to noise. We will start by presenting the limiting behavior of these system as their size tend to infinity. With random coupling, the resulting equation is a non-Markovian stochastic equation, an implicit equation in the space of stochastic processes. Thorough analysis of the solutions, and a fruitful analogy with deterministically coupled networks, allow to infer the qualitative dynamics of such systems. We will focus on a surprising transition from stationary states to synchronized activity as disorder is increased. This theory breaks down for balanced networks (in which the net input received by a neuron vanishes). We will discuss how these networks with such balanced connectivity may be analyzed, and will show that random transitions occur in these systems, in relationship with the type of eigenvalue with largest real part of the random coupling matrix. |
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