Table 1.
Meaning of symbols in the Wilson-Cowan equations.
Table 2.
Spatial convolution kernels in Euclidean, Fourier, and graph domains.
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 σIE 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.