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

Area of Science:

• Neuroscience
• Computational Neuroscience
• Mathematical Biology

Background:

• Neuronal networks exhibit complex dynamics across single-neuron, mesoscopic, and whole-brain scales.
• Various differential equation models exist, including deterministic (ODE), mean-field (PDE), and stochastic (SDE) approaches.

Purpose of the Study:

• To review and connect different differential equation models (ODE, PDE, SDE/SPDE) used in neuroscience.
• To highlight their application in multiscale modeling, neural signal processing, and cognitive modeling.
• To discuss numerical methods and future challenges in integrating these models.

Main Methods:

• Review of ordinary differential equation (ODE) models for deterministic neural behavior.
• Analysis of mean-field partial differential equation (PDE) models for population-level dynamics.
• Exploration of stochastic differential equation (SDE) and stochastic PDE (SPDE) models for neural variability and noise.

Main Results:

• A hierarchy of ODE, PDE, and SDE-SPDE models provides unified tools for neuroscience.
• These models link single-neuron mechanisms to population activity and analyze collective patterns.
• Stochastic models are crucial for understanding neural variability and are applied in neural data analysis.

Conclusions:

• Differential equation-based approaches offer versatile tools for multiscale modeling in neuroscience.
• Further research is needed in multiscale inference, control-oriented formulations, and machine learning integration.
• These models are essential for neural signal processing, cognitive modeling, and analyzing noisy neural systems.