02:09:09
publiée le 21 mai 2025
Dimension Dependence of Critical Phenomena in Percolation
De Thomas Hutchcroft
Spontaneous oscillations in a pure excitatory mean field networks of neurons
Apparaît dans la collection : 2022 - T1 - WS2 - Mathematical modeling and statistical analysis in neuroscience
We consider a model of network of interacting neurons based on jump processes. Briefly, the membrane potential $V^i_t$ of each individual neuron evolves according to a one-dimensional ODE. Neuron i spikes at rate which only depends on its membrane potential, $f(V^i_t)$. After a spike, $V^i_t$ is reset to a fixed value $V^{rest}$. Simultaneously, the membrane potentials of any (post-synaptic) neuron $j$ connected to the neuron $i$ receives a $kick$ of value $J^{i,j}$.
We study the limit (mean-field) equation obtained where the number of neurons goes to infinity. In this talk, we describe the long time behaviour of the solution. Depending on the intensity of the interactions, we observe convergence of the distribution to a unique invariant measure (small interactions) or we characterize the occurrence of spontaneous oscillations for interactions in the neighbourhood of critical values.
The talk is based on joint works with Quentin Cormier (Princeton) and Romain Veltz (Inria)
