A Differential Equation Based Framework for Modeling Dynamic Functional Connectivity in Brain Networks

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DOI: https://doi.org/10.28924/2291-8639-24-2026-182
Published: Jun 15, 2026
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Md Taufiq Nasseef, A Differential Equation Based Framework for Modeling Dynamic Functional Connectivity in Brain Networks, Int. J. Anal. Appl., 24 (2026), 182.

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Md Taufiq Nasseef

Abstract

The dynamic nature of functional connectivity in brain networks presents a fundamental challenge for neuroscience, as traditional correlation-based methods assume temporal stationarity and fail to capture the intrinsic time-varying properties of neural interactions. To address this, we introduce a novel mathematical framework that models the co-evolution of neural activity and functional connectivity using a system of differential equations. The model’s key innovation is treating connectivity not as a static measure but as a dynamical variable governed by activity-dependent plasticity rules, including Hebbian strengthening, saturation effects, and spatial smoothing. We formulate a system of coupled partial differential equations describing the spatiotemporal evolution of neural activity and connectivity on a cortical surface. Through analytical investigation using bifurcation theory and geometric singular perturbation analysis, we identify critical stability conditions and parameter regimes corresponding to different dynamical states monostable, bistable, and oscillatory observed in empirical studies. The model successfully reproduces essential features of time-varying functional connectivity, including metastability with characteristic dwell times (20–50 seconds), spontaneous state transitions driven by stochastic fluctuations, and emergent oscillations at frequencies (0.01–0.1 Hz) that match empirical BOLD signals. Numerical implementation demonstrates the framework’s computational feasibility, while integration with the Balloon-Windkessel hemodynamic model enables direct comparison with fMRI data. This approach provides a theoretical foundation for understanding dynamic brain organization, bridges multiple scales from local neural dynamics to large-scale network interactions, and offers testable predictions for experimental validation in both healthy and diseased states.

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