Deterministic, stochastic, and mean-field PDE models in neuroscience.

Large neuronal networks demonstrate complex dynamics across multiple scales, ranging from single-neuron excitability and spike-train variability to mesoscopic rhythms and whole-brain activity. Different types of differential equation models have been developed to comprehend these phenomena, connecting deterministic, stochastic, and mean-field descriptions. At the deterministic level, ordinary differential equation (ODE) models, including conductance-based neuron models, neural-mass systems, and whole-brain networks, summarize neural behavior through a reduced set of macroscopic variables. At the population level, mean-field partial differential equation (PDE) models such as Fokker-Planck, age-structured, kinetic, and neural field equations describe the evolution of probability or population densities over membrane-potentials, synaptic states, and other kinetic variables. These PDEs link single-neuron mechanisms to population-level activity and allow one to analyze bifurcations, oscillations and other collective patterns. Stochastic differential equation (SDE) models and their extensions that include jump-diffusion processes and stochastic PDEs (SPDEs) are widely used to describe random membrane fluctuations, irregular spike trains, synaptic plasticity and large-scale variability in neural activity. These stochastic models are also applied to neural data analysis, for example to quantify noise in electro-physiological recordings and to infer latent neural dynamics. Because variability and noise are central in neural systems, we devote more space to stochastic models but always relate them back to the surrounding ODE and PDE frameworks. This hierarchy of ODE, PDE, and SDE-SPDE models shows that the versatility of differential-equation-based approaches in neuroscience offers unified tools for multiscale modeling, neural signal processing, cognitive modeling, and the analysis of noisy neural systems. We also discuss some known numerical and computational approaches, especially for stochastic models and conclude by outlining open challenges, such as multiscale inference, control-oriented formulations and the integration of differential-equation models with modern machine-learning methods.