Riaz ud din shah, Sayed (2025) Multiscale Kinetic Modeling of Heterogeneous Neuronal Networks. [Tesi di dottorato]

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Abstract

This PhD thesis presents the development of a multiscale kinetic modeling framework aimed at analyzing the impact of functional and structural heterogeneity on the collective activity within excitatory-inhibitory neuronal networks organized in slice-like architectures. The primary modeling approach employs a microscopic binary-state description, i.e. active/inactive, wherein activity transitions are governed by stochastic rules and constrained by heterogeneous connectivity represented through weighted adjacency matrices. From this microscopic interaction framework, mesoscopic evolution equations are derived for slice- and type-specifc active-neuron counts (or fractions), facilitating a systematic examination of excitation-inhibition interactions while maintaining computational feasibility for multi-slice scenarios. To aid in analysis and parameter exploration, a reduced, slice-aggregated kinetic model is introduced, condensing connectivity and transition information into efective slice-level parameters. This results in a lower-dimensional nonlinear system of ordinary diferential equations, directly expressed in terms of the active counts of inhibitory and excitatory neurons. Analytical results, including equilibrium analysis and classifcation of regimes into excitation-dominated, inhibition-dominated, and balanced dynamics, are obtained for low-dimensional cases, particularly the single-slice model. Numerical simulations encompassing one-, two-, and four-slice confgurations, including driven scenarios with external inputs, demonstrate how functional and structural heterogeneity infuence equilibria, transient dynamics, and activity patterns depending on external stimuli. For the numerical simualtions a specifc Python code is developed. The thesis concludes by outlining future research directions. While we have so far analyzed several properties and behaviors of this new multiscale kinetic modeling framework for functionally and structurally heterogeneous neural networks—sometimes only at a numerical level—many perspectives remain open. Among them, a key direction will be the calibration of the proposed models against population-level data, as well as the establishment of principled links with biophysically grounded point-neuron frameworks, such as A-GLIF, in order to interpret and estimate efective kinetic parameters, thereby bridging the gap between mesoscopic kinetic descriptions and biophysically detailed neuronal models. The results presented in this thesis are part of the published article [1].

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