Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional
This work addresses the challenge of modeling complex networks like brain dynamics with spatially varying connectivity, offering a more accurate continuum approximation for researchers in applied mathematics and neuroscience.
The paper tackles the problem of deriving a continuum limit for ε-graphs with a general connectivity functional, proving that the discrete and continuum energies differ by at most O(ε) with an error bound robust to local fluctuations. As an application, it introduces a neural-network method to reconstruct connectivity density from edge-weight data and embeds this into a brain-dynamics framework, showing that spatially varying diffusion coefficients lead to significantly different dynamics compared to constant-diffusion models.
We derive an energy-based continuum limit for $\varepsilon$-graphs endowed with a general connectivity functional. We prove that the discrete energy and its continuum counterpart differ by at most $O(\varepsilon)$; the prefactor involves only the $W^{1,1}$-norm of the connectivity density as $\varepsilon\to0$, so the error bound remains valid even when that density has strong local fluctuations. As an application, we introduce a neural-network procedure that reconstructs the connectivity density from edge-weight data and then embeds the resulting continuum model into a brain-dynamics framework. In this setting, the usual constant diffusion coefficient is replaced by the spatially varying coefficient produced by the learned density, yielding dynamics that differ significantly from those obtained with conventional constant-diffusion models.