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Long time behavior of a mean-field model of interacting neurons
رفتار طولانی مدت از یک مدل میانگین میدانی از سلول های عصبی متقابل-2020 We study the long time behavior of the solution to some McKean–Vlasov stochastic differential
equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic
of the membrane potential of a spiking neuron in a large network. We prove that for a small enough
interaction parameter, any solution converges to the unique (in this case) invariant probability measure.
To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation
satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic
external quantity (we call it the external current). For constant current, we obtain the convergence to
the invariant probability measure. Using a perturbation method, we extend this result to more general
external currents. Finally, we prove the result for the non-linear McKean–Vlasov equation. Keywords: McKean–Vlasov SDE | Long time behavior | Mean-field interaction | Volterra integral equation | Piecewise deterministic Markov process |
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