A Novel Mathematical Framework for Multi-Scale Neural Dynamics: Integrating Stochastic Differential Equations with Graph Neural Networks

Abstract

The integration of multi-scale neural dynamics remains a fundamental challenge in computational neuroscience. This paper introduces a novel mathematical framework that integrates stochastic differential equations with graph neural networks to model multi-scale brain dynamics. Our approach treats neural activity, synaptic efficacy, and network topology as co-evolving dynamical variables governed by coupled equations spanning microscopic, mesoscopic, and macroscopic scales. The framework's multi-level architecture combines stochastic differential equations for microscopic neural dynamics with biologically realistic noise; activity-dependent plasticity rules for mesoscopic connectivity evolution; and graph neural networks for macroscopic network property learning. Through rigorous mathematical analysis, we derive stability conditions and identify critical bifurcation points governing cross-scale interactions. We introduce a novel mathematical framework unifying stochastic differential equations with graph neural networks to model hierarchical brain dynamics. Mathematical contributions include: (i) existence and uniqueness theorems for coupled SDE systems with adaptive connectivity; (ii) multi-scale stability analysis revealing emergent criticality; (iii) rigorous numerical analysis of multi-timescale integration; (iv) demonstration of universal scaling laws in neural avalanche statistics. The framework achieves an R² of 0.87 in predicting functional connectivity while achieving a 50× computational speedup via multi-timescale integration. The model exhibits emergent criticality with power-law avalanche distributions (\(\alpha = -1.52 \pm 0.08\)) matching empirical data.Clinical relevance is demonstrated through identified parameter shifts in schizophrenia (reduced inhibition) and depression (impaired plasticity) models. The framework successfully predicts both local neural dynamics and global network reorganization, providing testable hypotheses for experimental validation and offering a comprehensive mathematical foundation for understanding multi-scale brain dynamics in health and disease.