Fig 1.
A schematic overview of the model used in this work.
From left to right, the underlying components of the model are presented at increasing spatial scales. On the far left is a network of spiking neurons, with interactions between pre- (vpre) and post-synaptic (vpost) cells. There are two forms of neural interaction: gap-junctions (modelled with a fixed resistance κv, which we refer to as gap-junction coupling strength) and synapses (modelled as a variable resistor with conductivity g with a synaptic reversal potential vsyn). Via a reduction methodology that invokes the Ott-Antonsen ansatz [39], a low-dimensional description of the spiking neurons’ activity can be formulated, in the infinite-neuron limit, allowing for a computationally tractable model of large-scale neural populations. For a macroscopic region of the cortex, we consider two such populations, one excitatory and one inhibitory, that have both reciprocal and self-coupled synapses and gap-junctions. Using the reduced model, we run forward simulations of average, or mean-field, population statistics: membrane potential, firing rate and Kuramoto order parameter of synchrony (see Eq (6)). Each E–I pair is embedded in a whole-brain network of 68 cortical regions and we allow long-range synaptic connections between excitatory populations, with connectivity strengths and conduction delays derived from white-matter diffusion MRI data. Network behaviour is quantified using pairwise-correlation analogous to methods used to compute functional connectivity from empirical MEG/EEG time series.
Fig 2.
An overview of data types discussed in this paper.
In all cases, the data presented is an average over 10 subjects’ datasets from the Human Connectome Project database, and is downsampled onto a 68 node network using the Desikan–Killiany atlas. A MEG FC matrices computed via the ‘Multivariate Interaction Measure’ (12) within 8 different bands, which fall into the classical frequency bands α to γ, where β and γ are further divided into 2 and 3 sub-bands respectively. B BOLD FC is computed by z-scoring parcellated BOLD time series data before computing the pairwise Pearson correlation for all node pairs. C Structural data is constructed by applying a probabilistic tractography process to diffusion MRI data. The data is then normalised by row-sum. Visualisations of the structural network are provided for the two hemispheres.
Fig 3.
Exemplar proxy FC computed from eigenvectors of the structural connectivity matrix wij, computed by taking the outer product of an eigenvector with itself.
Panels (A-F) show FC matrices constructed in this way from structural eigenmodes that correspond to the largest six eigenvalues, ordered by decreasing size. Visualisations on a cortical surface are coloured according to the value of normalised eigenvector components, with warmer colours indicating higher values.
Fig 4.
Proxy FC provides limited explanatory power in understanding empirical FC patterns.
(a) Accuracy of proxy FC constructed from outer products of eigenvectors of the structural connectivity matrix, or of the network Jacobian, in a similar manner to those presented in Fig 3. Accuracy is measured by Pearson distance from MEG FC obtained from α-band activity. FC proxies are computed via iterative linear combination of increasingly many eigenmode FC patterns, with accuracy measured after each subsequent addition; specifically: structural eigenvectors added at random (blue), according to the decreasing size of the corresponding eigenvalue (red), in an order chosen for which the step-wise decrease in error is maximised (orange); and eigenmodes of the network Jacobian, in order of decreasing size of the corresponding eigenvalue (purple). Panels (b–d) provide visual comparison of the most accurate FC proxies obtained from structural eigenmodes and network Jacobian eigenmodes with empirical FC.
Fig 5.
Example timeseries for local excitatory population variables, VE, RE and synchrony |ZE| obtained via direct simulation of the network model (7)–(10), employing connectomic and delay data described in Section Structural connectivity and path-length data. Note only results from 3 selected nodes are shown, for clarity. The left pericalcarine cortex [node 20] (blue), left supramarginal gyrus [node 30] (red) and right fusiform gyrus [node 40] (yellow). (a), (c) and (e) show the amplitude envelope of the whole timeseries for each variable, given by the absolute value of the Hilbert-transformed signal. Within the time intervals indicated by the inset purple boxes, (b), (d) and (f) show a sample of the raw timeseries. Panel (g) shows a synthetic BOLD signal, computed via (17) from the RE(t) timeseries.
Fig 6.
Comparsion of methods to compute functional connectivity panel (a) shows the PLV matrix computed from the simulated MEG signal using the Hilbert transform to the phase 16, panel (b) shows the corresponding PLV using the mean phase of each population given by arg(ZE) and panel (c) shows the FC from the simulated MEG signal using the MIM method described in section Linear stability analysis.
Fig 7.
Gap junction coupling supports rich and dynamic neural activity.
Direct simulations of the network model (7)–(10) (together with connectomic and delay data described in Section Structural connectivity and path-length data) provide simulated data in the absence () and presence (
and
) of gap junction coupling; dynamic FC (dFC) matrices are obtained by employing the RE component of node activity. A Network averaged structure-function clustering coefficient
(19) computed via simulated MEG dFC (see Section Functional Connectivity) for each of the listed frequency bands using a sliding time window of width 10 s and 90% overlap. B Following [75], the instantaneous phase of synthetic BOLD signals (17) is computed with the Hilbert transform and used to compute dFC matrices whose entries comprise the cosine of the pairwise phase differences. To interrogate their time-variation, the leading eigenvector (that corresponding to the largest eigenvalue) and the vector of upper triangular values is extracted and time-correlation assessed via Pearson correlation.
Fig 8.
The importance of non-linearities in the system in generating simulated frequency-band filtered MEG FC more reminiscent of empirical data.
MEG FC obtained as described in Section Functional connectivity for parameter values in which the system is (a) poised in the neighbourhood of a Hopf bifurcation (kext = 0.2); (b) in the nonlinear regime (kext = 0.5) in which larger oscillations and more complex dynamics are obtained, supporting a range FC patterns across frequency bands.
Fig 9.
Gap junction coupling facilitates improved fits to empirical data.
Panels (a–c) present simulated α band MEG FC, and its similarity to empirical resting-state data (see Section Functional connectivity) for three different values of gap junction coupling strength. Similarity to empirical FC is measured by the Pearson distance d.
Fig 10.
Comparison of empirical MEG FC (a) and simulated FC obtained from (b) the Jansen-Rit model [16], (c) the next generation neural mass model. Panel (c) is reproduced from Fig 9(b); parameter values in the Jansen–Rit model are taken from [35]. Similarity to empirical FC is measured by the Pearson distance d.