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Mathematical Analysis of Neuronal Dynamics – MANDy

Submission summary

Our project aims at developping mathematically rigorous approaches to neuroscience considering single neurons as well as interconnected neuronal populations. Our target is to conduct the mathematical analysis of existing models where there is still much work to be done and to enrich the modelling by proposing new models. The presence of neuroscientists in our project will facilitate the validation of the mathematical models. The members of our project gather internationally renowned competence in mathematics as well as neuroscience. A lot of available studies have been conducted by simulations. Although this approach has certainly been fruitful and must be pursued, we it is limited and new results coming from a profound mathematical analysis are necessary. We center on partial differential equations (pdes) and probability. Even the classical models in neuroscience raise profound mathematical questions at the frontier of present mathematical knowledge ; we plan to conduct their study theoretically as well as numerically. One originality of our project is to integrate the various dynamical levels of the nervous system. For one single neuron the question is to describe large excursions of its electrical membrane potential from its rest value due to ionnic chemical reactions. Two biologists, Hodgkin and Huxley, proposed a four dimensional nonlinear system of reaction-diffusion equations with several time scales. Travelling and standing waves, oscillatory and asymptotic behaviour for this system is still today a challenging problem both theoretically and numerically that we address in this project. It is now clear that stochastic models are necessary to model experimental observations: the intrinsically stochastic ionic channel mechanism, constant synaptic efficacy, synchronization or resonance are observed only in stochastic conditions. We consider multidimensional stochastic models obtained by adding a random perturbation to a deterministic model of Hodgkin-Huxley type disregarding space propagation (or simplified versions in dimension 2 or 4). We search for the law of the first exit time of this process from a domain, for oscillating regimes and resonance induced by noise and investigate long time behaviour. We will have to face highly degenerate non gradient stochastic systems with non globally Lipschitz coefficients. Existing results do not apply. One dimensional Leaky Integrate and Fire (LIF) models will also be considered. Numerical stochastic schemes are necessary for their first passage time above a threshold : no accurate nor fast scheme is presently available. We will address this question. We also aim at developing stochastic models including space propagation. This amounts to introduce stochastic pdes models; it is a challenge both theoretically and numerically. There is no tradition to some reference models for populations of interconnected neurons. We are interested in synchronization, spontaneous activity, decision-making, dynamical probabilistic inference, information processing. We plan to study synchronization through the Kuramoto model in the light of recent results on out-of-equilibrium systems and disordered media. We also aim at undertaking the mathematical study of original models derived experimentally by neuroscientists in our group or foreign collaborators to describe respectively spontaneous activity and decision-making. Specific difficulties in pdes arise: specific non linearities, irrelevance of existing methods, non gradient Fokker-Planck equations. We also plan to develop models for the interaction of cortical micro-colums built on non linear Fourier integral operators. Recent studies lead by one of the neuroscientists in our group have shown that probabilistic models of dynamic inference can accurately account for various aspects of perception. One of the main theoretical difficulties in dynamical probabilistic inference is the exponential growing of necessary memory resources. Our goal is to examine several solutions to simplify the dynamical probabilistic computation on large set of variables. Perception and decision making result from information processing between several groups of neurons. Actually a complete mathematical formalism of information processing by a population of neurons is still missing. We would like to show that brain activities such as perception, memory, action, decision and adaptation can be interpreted in the light of non-equilibrium thermodynamics.

Project coordination

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

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Help of the ANR 0 euros
Beginning and duration of the scientific project: - 0 Months

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