Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional

TL;DR

This paper establishes a rigorous continuum limit for energy-based models on epsilon-graphs with general connectivity, enabling more accurate neural and brain-dynamics modeling with spatially varying diffusion coefficients.

Contribution

It derives an energy-based continuum limit for epsilon-graphs with a general connectivity functional, including error bounds and a neural-network method to reconstruct connectivity density.

Findings

Discrete and continuum energies differ by at most O(ε)

The reconstructed density allows for spatially varying diffusion in brain models

The approach remains valid with strong local fluctuations in connectivity density

Abstract

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.