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Modelling large neural networks via Hawkes processes

Abstract : How does the brain compute complex tasks? Is it possible to create en artificial brain? In order to answer these questions, a key step is to build mathematical models for information processing in the brain. Hence this manuscript focuses on biological neural networks and their modelling. This thesis lies in between three domains of mathematics - the study of partial differential equations (PDE), probabilities and statistics - and deals with their application to neuroscience. On the one hand, the bridges between two neural network models, involving two different scales, are highlighted. At a microscopic scale, the electrical activity of each neuron is described by a temporal point process. At a larger scale, an age structured system of PDE gives the global activity. There are two ways to derive the macroscopic model (PDE system) starting from the microscopic one: by studying the mean dynamics of one typical neuron or by investigating the dynamics of a mean-field network of n neurons when n goes to infinity. In the second case, we furthermore prove the convergence towards an explicit limit dynamics and inspect the fluctuations of the microscopic dynamics around its limit. On the other hand, a method to detect synchronisations between two or more neurons is proposed. To do so, tests of independence between temporal point processes are constructed. The level of the tests are theoretically controlled and the practical validity of the method is illustrated by a simulation study. Finally, the method is applied on real data
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Julien Chevallier. Modelling large neural networks via Hawkes processes. General Mathematics [math.GM]. COMUE Université Côte d'Azur (2015 - 2019), 2016. English. ⟨NNT : 2016AZUR4051⟩. ⟨tel-01415622v2⟩

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