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

Abstract

Tools from the field of graph signal processing, in particular the graph Laplacian operator, have recently been successfully applied to the investigation of structure-function relationships in the human brain. The eigenvectors of the human connectome graph Laplacian, dubbed "connectome harmonics", have been shown to relate to the functionally relevant resting-state networks. Whole-brain modelling of brain activity combines structural connectivity with local dynamical models to provide insight into the large-scale functional organization of the human brain. In this study, we employ the graph Laplacian and its properties to define and implement a large class of neural activity models directly on the human connectome. These models, consisting of systems of stochastic integrodifferential equations on graphs, are dubbed graph neural fields, in analogy with the well-established continuous neural fields. We obtain analytic predictions for harmonic and temporal power spectra, as well as functional connectivity and coherence matrices, of graph neural fields, with a technique dubbed CHAOSS (shorthand for Connectome-Harmonic Analysis Of Spatiotemporal Spectra). Combining graph neural fields with appropriate observation models allows for estimating model parameters from experimental data as obtained from electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI). As an example application, we study a stochastic Wilson-Cowan graph neural field model on a high-resolution connectome graph constructed from diffusion tensor imaging (DTI) and structural MRI data. We show that the model equilibrium fluctuations can reproduce the empirically observed harmonic power spectrum of resting-state fMRI data, and predict its functional connectivity, with a high level of detail. Graph neural fields natively allow the inclusion of important features of cortical anatomy and fast computations of observable quantities for comparison with multimodal empirical data. They thus appear particularly suitable for modelling whole-brain activity at mesoscopic scales, and opening new potential avenues for connectome-graph-based investigations of structure-function relationships.

Fig 1. Effects of distance-weighting on graph neural field dynamics.

(A) The harmonic and (B) temporal power spectra of Excitatory activity equilibrium fluctuations in the one-dimensional graph for vertex spacing h = 10⁻⁴ m. A larger vertex spacing, for example h = 2 ⋅ 10⁻⁴ m, renders the steady state unstable. The dashed black lines correspond to the theoretical prediction and the red lines are obtained through numerical simulations. (C) The Excitatory activity functional connectivity obtained by analytic predictions and numerical simulations.

Fig 2. Suppression of oscillatory resonance by non-local connectivity.

(A) The harmonic and (B) temporal power spectra of Excitatory activity equilibrium fluctuations in the one-dimensional graph for vertex spacing h = 10⁻⁴ m, after the addition of a non-local edge between vertices 250 and 750. The dashed black lines correspond to the theoretical prediction and the red lines are obtained through numerical simulations. (C) The Excitatory activity functional connectivity obtained by analytic predictions and numerical simulations. Compare with Fig 1 to note the visible suppression of oscillatory resonance in the temporal power spectrum, and the change in functional connectivity engendered by a single non-local edge.

Fig 3. Abortion of pathological oscillations by non-local connectivity.

(A) The harmonic and (B) temporal power spectra of Excitatory activity equilibrium fluctuations in the one-dimensional graph for vertex spacing h = 2 ⋅ 10⁻⁴ m, after the addition of a non-local edge between vertices 250 and 750. Without the addition, the model dynamics is placed in an unstable limit-cycle regime. The dashed black lines correspond to the theoretical prediction and the red lines are obtained through numerical simulations. (C) The Excitatory activity functional connectivity obtained by analytic predictions and numerical simulations.

Fig 4. Emergence of multiple temporal power peaks by long-range inhibition.

(A) The harmonic and (B) temporal power spectra of Excitatory activity equilibrium fluctuations in the one-dimensional graph, with the size of the Gaussian kernel controlling Inhibitory to Excitatory interactions σ_IE increased by a factory of 20, and everything else unchanged with respect to Fig 3. Allowing Inhibitory activity to exert its influence over larger distances here leads to the emergence of multiple peaks in the temporal power spectrum of Excitatory activity. The dashed black lines correspond to the theoretical prediction and the red lines are obtained through numerical simulations. (C) The Excitatory activity functional connectivity obtained by analytic predictions and numerical simulations.

Fig 5. Stochastic Wilson-Cowan graph neural field model captures the resting-state fMRI harmonic power spectrum.

The theoretical (dashed black line) and numerical (red line) predictions from the stochastic Wilson-Cowan graph neural field model, with the parameters of S2 Table, are in excellent agreement with the empirically observed fMRI harmonic spectrum (cyan line). The numerical spectrum was obtained by taking the median of three independent simulations.

Fig 6. Resting-state fMRI functional connectivity matrix.

Connectome-wide, vertex-wise, single-subject, resting-state fMRI functional connectivity matrix. Zoom in to appreciate the patterns present in the data, in particular the two large blocks (top-left and bottom-right) corresponding to the two hemispheres, and the many intra-hemispheric patterns. Compare with the functional connectivity predicted by the stochastic Wilson-Cowan graph neural field (Fig 7). The light-blue and light-green rectangles indicate the insets visualized in Figs 8 and 9.

Fig 7. Stochastic Wilson-Cowan graph neural field model predicts the experimental functional connectivity matrix.

The CHAOSS prediction for the connectome-wide, vertex-wise, single-subject functional connectivity matrix of the stochastic Wilson-Cowan graph neural field model with the parameters of S2 Table. Compare with Fig 6 to appreciate how the model predicts the patterns of functional connectivity observed in the fMRI data. The light-blue and light-green rectangles indicate the insets visualized in Figs 8 and 9. Note that we did not fit the fMRI functional connectivity of the model to the data, but only the harmonic power spectrum.

Fig 8. Stochastic Wilson-Cowan graph neural field model predicts the experimental functional connectivity matrix (inset 1).

(A) An inset of the vertex-wise, resting-state fMRI functional connectivity matrix for a single subject. (B) The same inset for the Wilson-Cowan graph neural field model with the parameters of S2 Table.

Fig 9. Stochastic Wilson-Cowan graph neural field model predicts the experimental functional connectivity matrix (inset 2).

(A) An inset of the vertex-wise, resting-state fMRI functional connectivity matrix for a single subject. (B) The same inset for the Wilson-Cowan graph neural field model with the parameters of S2 Table.