Sarah Lemler

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 [1]), 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 [2]). 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 [3]). 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 [4]). 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.

[1] 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.

[2] Dion, C., & Lemler, S. (2019). Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process. Statistical Inference for Stochastic Processes, 1-27

[3] 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

[4] 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.