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The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states. (English) Zbl 1455.35277

A mean-field approach is taken to describe a neural network dynamics that is emerging from the interaction of spiking cells. The authors prove for a moderate coupling regime that their model is globally well-posed in the space of measures, and also that there exist stationary solutions. In the case of a weak network connectivity (so away from blowup, degeneneracies) they show the uniqueness of the steady state and its global exponential stability.

MSC:

35R06 PDEs with measure
92B20 Neural networks for/in biological studies, artificial life and related topics
35B40 Asymptotic behavior of solutions to PDEs
60J76 Jump processes on general state spaces
35B35 Stability in context of PDEs
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