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| In a first step towards the comprehension of neural activity, one should focus on the stability of
the various dynamical states. Even the characterization of idealized regimes, such as a perfectly
periodic spiking activity, reveals unexpected difficulties. In this talk we discuss a general approach
to linear stability of pulse-coupled neural networks for generic phase-response curves and post-
synaptic response functions. In particular, we present: (i) a mean-field approach developed under
the hypothesis of an infinite network and small synaptic conductances; (ii) a microscopic approach
which applies to finite but large networks. As a result, we find that no matter how large is a
neural network, its response to most of the perturbations depends on the system size. There exists,
however, also a second class of perturbations, whose evolution typically covers an increasingly wide
range of time scales. The analysis of perfectly regular, asynchronous, states reveals that their
stability depends crucially on the smoothness of both the phase-response curve and the transmitted
post-synaptic pulse. The general validity of this scenarion is confirmed by numerical simulations of
systems that are not amenable to a perturbative approach.
[1] S. Olmi, A.Politi, and A. Torcini, Stability of the splay state in networks of pulse-coupled neurons, J. Mathematical Neuroscience 2:12 (2012)
[2] S. Olmi, A.Politi, and A. Torcini, Linear Stability in networks of pulse-coupled neurons, Frontiers in Computational neuroscience 8:8 (2014) |
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