A One World dynamics seminar
Speaker: Olivier Faugeras
Title: Two examples of thermodynamic limits in neuroscience
Abstract: The human brain contains billions of neurones and glial cells that are tightly interconnected. Describing their electrical and chemical activity is mind-boggling hence the idea of studying the thermodynamic limit of the equations that describe these activities, i.e. to look at what happens when the number of cells grows arbitrarily large. It turns out that under reasonable hypotheses the number of equations to deal with drops down sharply from millions to a handful, albeit more complex. There are many different approaches to this which are usually called mean-field analyses. I present two mathematical methods to illustrate these approaches. They both enjoy the feature that they propagate chaos, a notion I connect to physiological measurements of the correlations between neuronal activities. In the first method, the limit equations can be read off the network equations and methods "à la Sznitman" can be used to prove convergence and propagation of chaos as in the case of a network of biologically plausible neurone models. The second method requires more sophisticated tools such as large deviations to identify the limit and do the rest of the job, as in the case of networks of Hopfield neurones such as those present in the trendy deep neural networks
Speaker: Romain Veltz
Title: Study of a mean field of a network of 2d spiking neurons
Abstract: In this talk, I will present some results regarding the dynamics of a network of stochastic spiking neurons akin to the "generalised linear model". This network is an elaboration of the one introduced in [De Masi et al. 2014] by generalising the dynamics of the individual neurons. This allows to capture most of the known intrinsic neuronal spiking, like bursting for example, and thus to study the effect of the neuron dynamics on the macroscopic one.
The model presents some challenges. It is a nonlinear Piecewise Deterministic Markov process with explosive flow and unbounded total rate function. I will present some theoretical results regarding the linearised equation (well posedness, ergodicity) and will highlight the difficulties associated to the nonlinear one. In particular, I will prove the existence of invariant distributions for the nonlinear process. Some numerical applications will be provided.
Join Zoom Meeting: https://u-paris.zoom.us/j/81207145368?pwd=WmRoWDh5VmtNdXIrcmR5QjBiZGJpQT09, Meeting ID: 812 0714 5368, Passcode: 845075