Mathematical Modeling of Neural Dynamics Through Stochastic Fractional FitzHugh–Nagumo Equations: An Inverse Problem Approach

Neural field dynamics in the cerebral cortex exhibit complex spatiotemporal patterns inadequately captured by classical integer-order diffusion models that assume exponentially decaying spatial interactions. This study establishes a stochastic fractional FitzHugh–Nagumo framework incorporating power-law spatial correlations through fractional Laplacian operators, providing explicit parameterization of non-local cortical connectivity characteristics. The inverse problem of estimating fractional orders and model parameters from electroencephalographic data is addressed through multi-objective optimization with rigorous train–test validation. Systematic sensitivity analysis across the parameter space (αu,αv)∈[1.0,2.0]×[1.0,2.0] identifies optimal subdiffusive characteristics at αu=αv=1.5, corresponding to power-law spatial kernels C(x)∼|x|−1.5 consistent with anatomical connectivity measurements. The optimized model achieves out-of-sample performance R2=0.973 on held-out test data, approaching the measurement noise ceiling. While classical FitzHugh–Nagumo models achieve comparable test accuracy, the fractional framework provides enhanced interpretability through explicit spatial interaction parameterization. The fractional orders serve as quantitative biomarkers of cortical network organization, enabling data-driven characterization across brain states and neurological conditions. The methodology establishes computational foundations for clinical applications in epilepsy monitoring, neurodegenerative disease detection, and brain–computer interfaces.