Nonlinear partial differential equations in neuroscience: from modelling to mathematical theory (2501.06015v1)

Abstract: Many systems of partial differential equations have been proposed as simplified representations of complex collective behaviours in large networks of neurons. In this survey, we briefly discuss their derivations and then review the mathematical methods developed to handle the unique features of these models, which are often nonlinear and non-local. The first part focuses on parabolic Fokker-Planck equations: the Nonlinear Noisy Leaky Integrate and Fire neuron model, stochastic neural fields in PDE form with applications to grid cells, and rate-based models for decision-making. The second part concerns hyperbolic transport equations, namely the model of the Time Elapsed since the last discharge and the jump-based Leaky Integrate and Fire model. The last part covers some kinetic mesoscopic models, with particular attention to the kinetic Voltage-Conductance model and FitzHugh-Nagumo kinetic Fokker-Planck systems.

• The paper presents the main contribution of analyzing nonlinear PDE models in neuroscience, detailing stationary states, synchronization phenomena, and oscillatory behavior through models like the NNLIF.
• It employs advanced mathematical techniques such as fixed-point theorems, entropy methods, and spectral analysis to investigate parabolic, hyperbolic, and kinetic equations in neural dynamics.
• The findings offer a rigorous theoretical foundation for computational tools, enhancing our understanding of neural activity patterns and decision-making processes in complex brain networks.

An Overview of the Paper on Nonlinear Partial Differential Equations in Neuroscience

The paper, authored by José A. Carrillo and Pierre Roux, presents a comprehensive review of the mathematical modeling and theoretical analysis of systems of partial differential equations (PDEs) used to represent collective behaviors in large networks of neurons. It primarily focuses on nonlinear and non-local PDEs, which are inherently complex due to their structural challenges. The paper covers various models, including parabolic Fokker-Planck equations, hyperbolic transport equations, and kinetic mesoscopic models, each linked to different aspects of neural dynamics.

Parabolic Fokker-Planck Equations

1. Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) Model: The NNLIF model explains the emergence of self-sustained oscillations in neural networks with strong inhibitory connections. The authors detail the derivation of this model, stemming from simplified neuron dynamics represented by stochastic differential equations. They explore the mathematical framework for analyzing the NNLIF model, discussing the existence of stationary states and their stability. The paper emphasizes the conditions under which blow-up (synchronization) or convergence to a stationary state occurs, with particular attention to the diffusion term's impact.
2. Stochastic Neural Fields and Grid Cells: The authors extend the discussion to neural field models, describing how stochastic influences impact the stability and robustness of hexagonal firing patterns in grid cells, essential for spatial navigation and memory.
3. Rate Models for Decision-Making: The decision-making framework is described using rate models, where neuronal activation is linked to the accumulation of evidence in tasks requiring binary decisions. The models reflect electrophysiological observations in animal studies and relate these to theoretical diffusion models.

Hyperbolic Transport Equations

1. Time Elapsed Model: This section examines transport equations structured by the time since a neuron's last action potential. The model accounts for refractory periods and synaptic delays, seeking to understand oscillatory patterns in neural assemblies. The authors discuss the conditions for the existence and long-term behavior of solutions using characteristics and semigroup theory. The balance between synaptic delay and refractoriness can result in the stability of periodic solutions and attractors.

Kinetic Mesoscopic Models

1. Voltage-Conductance Models: The kinetic framework addresses the interactions between voltage and conductance in neurons, employing McKean-Vlasov equations to model the system's mesoscopic behavior. The complexity arises from the system's variances and the distribution of state variables across the neural assembly.
2. FitzHugh-Nagumo Systems: The kinetic representation of this model incorporates Fokker-Planck systems to capture the dynamics of neuron population behaviors, particularly focusing on the implications of noise and other perturbations.

Mathematical Techniques and Implications

Throughout the paper, various mathematical approaches are employed to analyze the PDE models, including fixed-point theorems, entropy methods, and spectral analysis. The authors highlight the importance of these mathematical tools for understanding key phenomena in neuroscience, such as synchronization, oscillations, and decision-making processes.

Implications and Future Directions: The theoretical insights provided by these models form a foundational understanding for developing computational tools and extending neuroscience research into more complex brain behavior models. Future research might explore more versatile frameworks incorporating additional biological nuances, moving towards integrating more refined descriptions of synaptic plasticity and neuronal adaptation mechanisms.

Overall, the paper serves as a valuable resource for researchers interested in the intersection of mathematical modeling and neurobiology, providing a thorough grounding in the current methodologies and their applications to complex neuronal behaviors.