Kernel Reconstruction for Delayed Neural Field Equations

Abstract

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for the inverse problem. We employ spectral regularization techniques for its stable solution. A sensitivity analysis of the regularized kernel reconstruction with respect to the input signal u is carried out, investigating the Fréchet differentiability of the kernel with respect to the signal. Finally, we use numerical examples to show the feasibility of the approach for kernel reconstruction, including numerical sensitivity tests, which show that the integral equation approach is a very stable and promising approach for practical computational neuroscience.

Graph neural fields: A framework for spatiotemporal dynamical models on the human connectome

“…For ease of exposition, here we have focused mainly on neural field models with purely spatial kernels. Although this might be sufficient for modelling wide-band activity such as BOLD fMRI, the large-scale organization of oscillatory activity as recorded with EEG and MEG sensitively depends on the propagation delays of action potentials through white-matter fiber tracts [48][49][50]. To model such delays, spatiotemporal kernels have been used in continues neural field models [32,36,51,52].…”

Dynamics of neural fields with exponential temporal kernel

“…Neural fields have seen significant progress in both theoretical and numerical studies over the recent years (Alswaihli et al. 2018 ; Abbassian et al. 2012 ; Bressloff 2011 ; Haken 2007 ; Karbowski and Kopell 2000 ; Morelli et al.…”

Online estimation of Hilbert-Schmidt operators and application to kernel reconstruction of neural fields

“…The specific class of systems we consider is motivated by an application to kernel reconstruction in neural fields. The offline estimation of these kernels is now a classical issue in inverse problems for neuroscience (see [1,17] and references therein), that can be addressed for instance using a Tikhonov regularization. We instead rely on adaptive observer strategies to address the online estimation problem.…”