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Whole brain functional connectivity: Insights from next generation neural mass modelling incorporating electrical synapses [1]

['Michael Forrester', 'Centre For Mathematical Medicine', 'Biology', 'School Of Mathematical Sciences', 'University Of Nottingham', 'Nottingham', 'United Kingdom', 'Sammy Petros', 'Oliver Cattell', 'Yi Ming Lai']

Date: 2024-12

Structural information alone is limited in its ability to explain observed FC data, with significant disparity between proxy and empirical FC ( Fig 4(a) , orange line). In contrast, employing eigenmodes obtained from the network Jacobian, obtained via a linear analysis as described above and hence embedding both structural and dynamical features, provides significant additional improvement. These results highlight how a linear analysis of the dynamical model provides additional explanatory power in understanding empirical FC patterns, in comparison to employing connectomic information alone, but also indicates that the prediction obtained is nonetheless limited in its fidelity.

(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.

Specifically, proxy FC is constructed by linear combination (optimising on the coefficients via MATLAB’s nonlinfit function) of a subset of eigenmodes of either the SC or the Network Jacobian. The order of their addition to the composite FC proxy is as follows: (i) structural eigenmodes, added uniformly at random; (ii) structural eigenmodes, added according to the decreasing size of their corresponding eigenvalue; (iii) structural eigenmodes, in an order chosen for which the step-wise decrease in error between proxy FC and MEG FC is maximised; and (iv) eigenmodes of the network Jacobian (using only the component corresponding to R E ), in order of decreasing size of the corresponding eigenvalue. Fig 4(a) shows the incremental improvement in accuracy in comparison to empirical FC, while panels (b–d) compare the example FC proxy matrices resulting from structural or Jacobian eigenmodes (process (ii) and (iv) described previously, in the case of all eigenmodes employed) with empirical data.

Predictions based only on structural information would only be expected to have limited explanatory power, neglecting as they do the influence of local neural dynamical states on emergent FC pattern [ 35 ]. To illustrate this, Fig 4 compares the accuracy as measured by Pearson distance from MEG FC obtained from α-band activity (see section Functional connectivity—MEG ) of FC proxies constructed from outer products of eigenvectors of the structural connectivity matrix, or of a reduced network Jacobian, in a similar manner to those presented in Fig 3 . Regarding the latter, we compute the eigenvectors of the NM × NM network Jacobian defined in section Linear stability analysis and then construct a projection to by considering only the elements corresponding to R E .

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.

Following the linear arguments above, in section Linear stability analysis , a starting point to investigate the FC patterns naturally supported by the underlying structure is to compute eigenvectors of the structural connectivity matrix w ij , and construct an FC proxy by the outer product of an eigenvector with itself. Emergent FC can potentially be expected to be formed from a combination of such underlying structural eigenmodes [ 41 ]. Fig 3 shows representations of networks constructed in this way, employing each of the first 6 leading eigenvectors (ordered by decreasing size of the corresponding eigenvalue).

Numerical exploration of nonlinear network model

The preceding results highlight the limitations of employing structural data and/or linear analysis in predicting empirical FC. Here we consider the rich detail in neural activity and associated FC patterns supported by the network model (7)–(10) (together with connectomic and delay data described in Section Structural connectivity and path-length data) that are not accessible in the linear regime. These are highly likely to be important in supporting the wide variety of functional states that underpin higher brain function.

Fig 5 provides exemplar time series of activity (R(t) and V(t)) and underlying spiking coherence of the population (|Z(t)|) from selected nodes in the network, obtained from direct simulations. These highlight both the complex oscillatory patterns generated over short timescales (b, d, f), and the longer timescale variation in waveform amplitude (a, c, e) that the network supports. This activity stands in contrast to simpler neural mass models, such as the popular Wilson–Cowan model in which behaviour is largely limited to sinusoidal-type oscillations, or other more complex examples such as Jansen–Rit for which notions of within-population coherence are not available, and for each of which features of key biological and neurophysiological relevance such as gap junction coupling are not accommodated. From the perspective of “intrinsic coupling modes”, MEG is used in this manuscript to assess FC by maximization of imaginary coherency on fast time scales, whereas correlations between BOLD responses attributed to relatively slower time scales should be understood as envelope correlations, see, e.g., [72]. Given this, there ought to be, and typically there is, some relation between these two modalities, e.g., when considering the envelope of MEG time series as shown in Fig 5(g), which shows a synthetic BOLD signal computed via (17) from the R E (t) timeseries.

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TIFF original image Download: Fig 5. Example timeseries for local excitatory population variables, V E , R E and synchrony |Z E | 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 R E (t) timeseries. https://doi.org/10.1371/journal.pcbi.1012647.g005

In Fig 6 we show that by using the PLV measure the phase coherence between the mean phase of each population given by arg(Z) is closely matched by the PLV derived from the corresponding simulated MEG signal. These show qualitative differences to the network derived using the MIM measure. However, the use of MIM might be considered a more appropriate measure of FC as it has been developed as a real-world signal processing tool for MEG data. Thus despite the computational simplicity and ease of constructing static PLV measures of FC within the modelling framework we would advocate for MIM instead.

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TIFF original image Download: 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(Z E ) and panel (c) shows the FC from the simulated MEG signal using the MIM method described in section Linear stability analysis. https://doi.org/10.1371/journal.pcbi.1012647.g006

The key influence of gap junction coupling on emergent network behaviour across the various frequency bands of importance in the analysis of neuroimaging data is made evident in Fig 7 in which we employ a selection of metrics to analyse the dynamical features supported by the model in the presence and absence of such coupling. These aspects are of especial importance in the context of recapitulating the dynamical repertoire observed in functional patterns both in the context of task-switching [73] and fluctuations in resting state [74] and are not captured by the averaged (static) FC patterns considered above. Specifically, we compute dynamic FC (dFC) matrices from the R E component of simulated data in two different ways. First, simulated MEG dFC (obtained as described in Section Functional connectivity) is computed for various frequency bands over sliding windows of width 10 seconds with a 9 second overlap. The resulting FC patterns are interrogated via the network-averaged structure-function clustering coefficient (19), providing a convenient metric to visualise the influence of anatomical structure on evoked activity patterns and how congruence between structure and function (or its absence) evolves over time. This is presented in Fig 7A. Secondly, we follow [75] and employ synthetic BOLD signals via (17). The instantaneous phase of these signals is computed via Hilbert transform, and the cosine of pairwise phase differences between cortical areas provides a dynamic FC matrix for each time point. To analyse these patterns, the leading eigenvector and the vector of upper triangular values are extracted from these matrices, and their autocorrelation computed via the Pearson metric, as presented in Fig 7B.

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TIFF original image Download: 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 R E 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. https://doi.org/10.1371/journal.pcbi.1012647.g007

We see that the divergence between connectivity structure and evoked function, as measured by the network averaged structure-function clustering coefficient differs significantly between frequency bands. Moreover, while small fluctuations are evident in each frequency band in panel A(a), in the presence of gap junctions, intermittent periods of strong SC-FC disparity are induced in all frequency bands (panel A(b)). Similarly, Fig 7Ba and 7Bb highlight that a rather richer correlation structure in the time-evolution of dFC is supported by gap junction coupling, compared to that obtained in its absence.

S1 and S2 Videos provide a further exposition of the influence of gap junctions on network behaviour. Here, we project the local structure-function clustering coefficient C wsf (i) that arises in the network in the presence and absence of gap junction coupling (computed via simulated MEG dFC and filtering in the α band; see Section Functional Connectivity) onto a reference cortical surface. Cortical surface visualisations in the videos were made using BrainNet Viewer [76].

Fig 8 considers frequency band-filtered FC patterns obtained from the model in more detail. Here, simulated MEG FC is computed for the entire timeseries (as described in Section Functional connectivity) highlighting that a rich diversity of FC patterns across bands, as observed empirically, is not available in the linear regime (namely the regime close to a bifurcation, where one typically expects a linearisation to be good approximation of the full dynamics, at least for a supercritical bifurcation). In this regime, the linear analysis presented earlier applies, and a reasonable prediction of FC is available. However, the FC patterns presented in panel (a), for which the model is poised in a neighbourhood of a Hopf bifurcation which gives rise to network oscillations, show relatively little variation across frequency bands as would be expected empirically and so this predictive power is arguably of limited utility. In contrast, when nonlinearities play a dominant role (panel (b)), in which variation increased of global coupling strength places the model in a regime of larger oscillations and more complex network dynamics, stronger variation in FC is observed across frequency bands.

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TIFF original image Download: 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 (k ext = 0.2); (b) in the nonlinear regime (k ext = 0.5) in which larger oscillations and more complex dynamics are obtained, supporting a range FC patterns across frequency bands. https://doi.org/10.1371/journal.pcbi.1012647.g008

Correspondence between simulated and empirical data is further considered in Fig 9. Here, we return to the importance of gap junction coupling on the model dynamics and present exemplar results highlighting that inclusion of such coupling facilitates improved fits to resting-state MEG FC (see Section Functional connectivity) in the α band. Distinct differences in FC are observed in simulated FC as gap junction strength is increased between panels (a–c) further underscoring its importance in mediating FC patterns. Moreover, the observed goodness-of-fit to empirical data first improves and then worsens as coupling strength is increased. This local minimum is identified with only small manual increase in and , suggesting that these parameters provide a natural choice for more comprehensive future fitting studies leveraging recent advances in parameter optimisation for whole brain models based on covariance matrix adaptation evolution strategies and Bayesian optimization [77].

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TIFF original image Download: 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. https://doi.org/10.1371/journal.pcbi.1012647.g009

Lastly, Fig 10 brings together some of the ideas discussed above to compare the performance of our next generation neural mass model in fitting to empirical MEG FC (as shown in Fig 9(b)) and that of the phenomenological model of Jansen and Rit [16]. For the latter, the parameter values are chosen to be consistent with [35]. This simple example serves to illustrate how the more complex dynamics generated by this new modelling framework supports the generation of patterns of brain network activity more reminiscent of MEG data in the sense that a lower value of the Pearson distance between empirical and simulated FC is readily obtained.

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1012647

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