Volume 65, 2019CEMRACS 2017 - Numerical methods for stochastic models: control, uncertainty quantification, mean-field
|Page(s)||445 - 475|
|Published online||02 April 2019|
Network of interacting neurons with random synaptic weights
Mathematics Institute, University of Warwick, Coventry, United Kingdom
2 Department of Mathematics, University of Pisa, Pisa, Italy
3 Université Côte d’Azur, CNRS, I3S, France
4 Laboratoire Jean Kuntzmann, Université Grenoble Alpes (UFR IM2AG), Grenoble, France
5 Université Côte d’Azur, CNRS, LJAD, France
6 Université Côte d’Azur, Inria, France.
Since the pioneering works of Lapicque  and of Hodgkin and Huxley , several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.
© EDP Sciences, SMAI 2019
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