Emergence of complex dynamics in neuronal and glial cells of the CNS

Neurons and astrocytes show complex behaviors, which are typically associated with specific biological functions. Historically, the study of these phenomena has relied on a combination of mathematical modeling, theoretical analysis, and numerical simulations. This integrated approach has revealed the underlying geometries of such processes, enabling both the understanding and prediction of cellular behaviors. This thesis addresses four neuroscience-related problems through computational and mathematical methodologies, with a focus on modeling single neurons and astrocytes and applying theories of multiple time-scale dynamical systems. Numerical continuation software plays a key role in these applications. Among them, XPPAUT is widely used but presents various limitations, precisely the non-trivial handling of continuation results. To address this issue, the first contribution of this thesis presents the development of XPPLORE, a new tool for parsing continuation data. XPPLORE assists users in tasks of different complexity: from the optimized management of simple bifurcation diagrams and simulations, to advanced tasks such as manifold reconstruction and the computation of averaged nullclines. The remaining chapters focus on the analysis of neural and glial cells of the central nervous system. Layer V cortical neurons can exhibit mixed-mode oscillations (MMOs). Recent experimental findings have suggested the involvement of HCN channels in the generation of these complex oscillations. Through bifurcation theory and geometric analysis, we demonstrate that HCN channels cooperate with M-currents to regulate these dynamics. Furthermore, a reduced version of the new computational model, obtained via time-scale decomposition, considering the two super-slow variables as parameters, highlights the geometric mechanism generating the small oscillations, which corresponds to the presence of a folded-node singularity. The second case study investigates cartwheel interneurons, the main source of inhibition of the dorsal cochlear nucleus, a brainstem region implicated in auditory disorders, e.g. tinnitus and certain forms of spatial hearing loss. Using morphological index maps and averaging theory, we explored the dynamic transitions induced by pharmacological blockade of BK and L-type Ca2+ channels (via iberiotoxin and nifedipine, respectively). Our analyses show that spiking and complex-spiking responses fail to persist following small perturbations in the maximal conductance of these channels. These findings are supported by the stability analysis of the super-slow equilibrium, detected as the intersection between the super-slow averaged nullclines, which highlights that small changes in the maximal conductances can alter the stability of the super-slow equilibrium point, suggesting why channelopathies can affect the electrical behavior of these crucial interneurons. The final project focuses on a novel astrocyte model. Specifically, the analysis of a reduced formulation, neglecting the desensitization subsystem of G-coupled protein receptors, clarifies the geometric mechanisms underlying observed delays: the relative position of a folded-node singularity and the trajectory landing point, modulated by model parameters and initial conditions, finely regulates the delay and the observed small amplitude oscillations during the transient. Among all evidence, the parametric maps reveal that the initial calcium concentration in the endoplasmic reticulum critically determines the transient’s duration. Based on these findings, we design and test in silico an innovative experimental procedure, tailored to exploit the detected folded node, to induce controlled delays in astrocytic calcium time series using TPEN. This study provides new insights to explain the heterogeneity observed in experimentally recorded cytoplasmic calcium responses under glutamate stimulation.