Neurone abstrait : une formalisation de l’intégration dendritique et ses propriétés algébriques

Abstract : Biological neurons communicate by means of electrical impulses, called spikes. Brain functions emerge notably from reception and emission coordination between those spikes. Furthermore, it is widely accepted that the function of each neuron depends on its morphology. In particular, dendrites perform the spatio-temporal integration of received spikes and affect the occurrence of emitted spikes. Dendrites are therefore fundamental for in silico studies of coordination mechanisms, and especially for the study of so-called neuron assemblies. Most of existing neuron models taking into account dendrites are detailed mathematical models, usually based on differential equations, whose simulations require significant computing resources. Moreover, their intrinsic complexity makes difficult the analysis and proofs on such models. In this thesis, we propose an abstract neuron model integrating dendrites. In order to pave the way to formal methods, we establish a rigorous definition of the modeling framework and highlight remarkable algebraic properties of dendritic integration. In particular, we have demonstrated that it is possible to reduce a neuron structure while preserving its input/output function. We have thus revealed equivalence classes with a canonical representative. Based on category theory and thanks to properly defined neuron morphisms, we then analyzed these equivalence classes in more details. A surprising result derives from these properties: simply adding delays in neuron computational models is sufficient to represent an abstract dendritic integration, without explicit tree structure representation of dendrites. At the root of the dendritic tree, soma modeling inevitably contains a differential equation in order to preserve the biological functioning essence. This requires combining an analytical vision with the algebraic vision. Nevertheless, thanks to a preliminary step of temporal discretization, we have also implemented a complete neuron in Lustre which is a formal language allowing proofs by model checking. All in all, we bring in this thesis an encouraging first step towards a complete neuron formalization, with remarkable properties on dendritic integration.
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Ophélie Guinaudeau. Neurone abstrait : une formalisation de l’intégration dendritique et ses propriétés algébriques. Bio-informatique [q-bio.QM]. Université Côte d'Azur, 2019. Français. ⟨NNT : 2019AZUR4001⟩. ⟨tel-02389736⟩

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