Probabilistic models for systems of interacting spiking neurons and some words about their mean field limits
E. Löcherbach
I will give an overview of recent results about mean field limits for systems of interacting point processes modeling spiking (biological) neurons. I will start with a short introduction to the functioning of neurons and the modeling of their spiking activity by stochastic integrate and fire models and then focus on a particular class of models which are stochastic intensity based processes. Then we will discuss mean field limits and propagation of chaos results for large homogeneous systems of neurons and see how the limit is described by a McKean-Vlasov type equation driven by Poisson random measure. A second part of the talk is devoted to the study of systems with random synaptic weights in a diffusive scaling. We will see how this setting leads to conditional propagation of chaos and how the convergence can be obtained by means of a new martingale problem. Finally, if time permits, I will also discuss the longtime behavior both of the finite and the limit system of neurons.
The second part of the talk is based on joint work with Xavier Erny and Dasha Loukianova.
Strong conditional propagation of chaos for systems of interacting particles with nearly stable jumps
E. Marini
Scaling limits of a tagged soliton in the randomized box-ball system
Hayate Suda (Tokyo Institute of Technology)
The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior. In recent years, with the rapid progress in the study of the hydrodynamics of integrable systems, there has been a growing interest in BBS with random initial distribution. In this talk, we consider the scaling limits for a tagged soliton in the BBS starting from certain stationary distribution.
This talk is based on a joint work with Stefano Olla and Makiko Sasada.
Probabilistic aspects of integrable systems
Makiko Sasada (University of Tokyo)
The KdV equation and the Toda lattice are classical integrable systems, with the box-ball system (BBS) as their ultra-discrete analogue. The BBS has been studied from various viewpoints such as tropical geometry, combinatorics, and cellular-automaton. Recently, probabilistic approaches to the BBS—such as the Pitman transform and analysis of i.i.d.-type invariant measures—have advanced rapidly and are also applicable to discrete KdV and Toda systems. Connections to Yang-Baxter maps have further deepened this research topic. In this talk, I will provide an overview of these developments.
This talk is based on joint works with David Croydon, Tsuyoshi Kato, Satoshi Tsujimoto, Ryosuke Uozumi, Hiroki Kondo and Sachiko Nakajima.
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