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Mutually exciting Hawkes process for neurons
Résumé : This talk will present a unified framework to analyze neuronal activity from data extraction to statistical inference. Our approach relies on two types of data : extracellular recordings of multiple spikes trains and intracellular recordings of the membrane potential of a central neuron. We propose a new jump-diffusion model with jumps driven by a multi-dimensional Hawkes process to model such complex data. We have established ergodicity results (in collaboration with Charlotte Dion and Eva Löcherbach, see ), allowing us to make statistical inferences on the model parameters. We proposed a drift estimation procedure and established oracle inequalities to guarantee the theoretical performances of our estimator (in collaboration with Charlotte Dion, see ). In a second work, we were interested in estimating the model’s volatility term and jump function (in collaboration with Chiara Amorino, Charlotte Dion, and Arnaud Gloter ). Finally, we have studied the real data obtained by measuring the membrane potential of a fixed neuron of a turtle as well as the spike trains from a large number of neurons around the fixed neuron and looked at how to apply our jump-diffusion model to these data (in collaboration with Charlotte Dion and Anna Bonnet ). This modeling makes it possible to use all data available to us and represent the link between both signals (intra- and extra-cellular) received by a neuron and its electrical potential.
 Dion, C., Lemler, S., & Löcherbach, E. (2019). Exponential ergodicity for diffusions with jumps driven by a Hawkes process. arXiv preprint arXiv:1904.06051, to appear in Theory of Probability and Mathematical Statistics.
 Dion, C., & Lemler, S. (2019). Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process. Statistical Inference for Stochastic Processes, 1-27
 Amorino, C., Dion C., Gloter A., Lemler, S. (2020). On the nonparametric inference of coefficients of self-exciting jump-diffusion, arXiv preprint arXiv:2011.12387
 Bonnet, A., Dion, C., Gindraud, F., & Lemler, S. (2021). Neuronal Network Inference and Membrane Potential Model using Multivariate Hawkes Processes. arXiv preprint arXiv:2108.00758.