Structural and functional mean connectivities.

<p>Empirical mean empFC was obtained from resting-state fMRI BOLD time-series (top, middle) and used to generate the linear analytically inferred aSC (bottom, left). Structural data from DTI (bottom, middle) were the base to generate both the linear analytically predicted aFC (top, left) and to simulate time series through a model exhibiting non-linear dynamics to produce a simulated sFC (top, right). The latter has then been used to analytically infer again SC (bottom, right) to show how the non-linearities present in the model used to generate sFC do not prevent the linear retrieving of SC. The range of values in the FCs goes from zero to one. We used the same scale for the SCs (correlations between connectivities are independent from the scaling factor).</p

Decomposition of resting-state fluctuations.

<p>We performed single subjects sliding-window analysis (window size 60s, step 2s) of the empirical data and in this set of figures we show the results for the two subjects A and B. The total empCov can be written as a linear combination of its 66 projectors each strictly linked to one of the eigenvectors. Each window empCov_w can then be decomposed in a linear combination of projectors of empCov and a residual part accounting for mixed eigenvectors terms. (a,d) Time-evolution of the coefficients used in the linear combination of projectors and mixed terms appears to strongly fluctuate in time. A small number of terms (the first three projectors and three mixed terms) have a higher weight in the linear combination than the others, and also seem to exhibit a higher variability in time. (b,e) The correlation between each window’s covariance and the linear combination of the first three projectors and three mixed terms is plotted, together with the correlation of the window covariance with the linear combination of the other projectors and mixed terms. We also show the correlation between the window covariance and the total covariance (c,f) For each window we computed the window’s Cov and its eigenvectors. We ranked the eigenvectors according to the value of their eigenvalues and selected up to five of them as a basis for a subspace of the measure’s space. We built subspaces of dimension one through five (matrices from left to right). For each dimension, we show a matrix of which the entries represent a measure of similarity between the subspaces of different windows. The canonical correlatioin is used as a measure of similarity, where 1 means that the subspaces are identical and 0 that there is no similarity.</p

Role for global coupling parameter <i>c</i>.

<p>(a) Correlation between the empirical FC (mean across 14 subjects) and the linear analytical one (again, mean across the 14 subjects), for various values of the global coupling parameter c. The measure used for FC is Pearson Correlation Coefficients. (b) For each of the 14 subjects is shown the difference between the critical value of the parameter and the best fitting one versus the gain in correlation (the difference between the correlation obtained for the best fitting parameter and that for <i>c</i> ∼ 0). The closer the best value of the parameter to the critical value the higher the gain in correlation. (c) Results for the sliding-windows analysis for two subjects, labeled A and B: the analytical operator to obtain aFC has been applied to each window. Solid line shows the behavior of the best fitting parameter and the dashed one indicates the correlation reached between empirical and analytical FCs. Dotted lines indicate the value of <i>c</i>_{<i>critic</i>}.</p

Across subjects variability.

<p>In the three matrices we show correlation between connectivities of the 14 subjects: (a) correlation between empSC and empFC; (b) correlation between empFC and aFC (derived from empSC); (c) correlation between empSC and aSC (derived from empFC) in the right panel. It is not possible to link subject specific empirical connectivities and variability remains between the individual analytic and empirical connectivities.</p

Eigenvectors, projectors and dimension reduction.

<p>Two visualization tools are given in this figure. (a) The eigenvectors of the empCov (which, in our model, are the same as aSC) are plotted on the brain surface in arbitrary units. (b) The portion of empCov (or aSC), which is strictly dependent on each eigenvector is pictured at its side through the use of projectors. (c) The average empCov. (d) The mixed terms relative to the first three eigenvectors. (e) For each single subject, we computed the goodness of fit between the complete time-series (spanning a 66 dimensional space) and a version of them projected on a smaller subspace composed of only a limited amount of the empCov eigenvectors. For each number of eigenvectors considered, we plotted the mean goodness of fit and the standard deviation across subjects. It appears that a three dimensional subspace is enough to account for almost 94% of the data.</p

Alpha-phase dependency of the firing rate.

<p>Top panel shows alpha phase vs. MUA behaviour within the cortical node. Bottom panel shows a control condition: alpha phase in the cortical node vs. MUA behaviour from a disconnected node (CSF = 0.6).</p

Correlation coefficients for the relationship between cortical alpha power and hemodynamic response of the individual nodes as well as the correlation of cortical alpha power and MUA across all nodes (CSF = 0.6).

<p>Correlation coefficients for the relationship between cortical alpha power and hemodynamic response of the individual nodes as well as the correlation of cortical alpha power and MUA across all nodes (CSF = 0.6).</p

<i>A</i>. The effect of global connectivity strength (CSF) on the correlation between alpha-band power and firing rate.

<p>Effect in cortical node. <b><i>B</i>.</b> Effect in thalamus. <b><i>C</i>.</b> Effect in reticular nucleus. The inverse relationship remains stable across a broad range of connectivity levels and is not highly sensitive to the general power within the alpha band (which decreases with increasing CSF, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004352#pcbi.1004352.g004" target="_blank">Fig 4</a>). Error bars indicate standard deviation across repetitions. For most scenarios, apart from the hyper-excitation scenarios, we see a negative correlation between alpha-band power and firing rate as estimated by MUA.</p

<i>A</i>. The general large-scale connectivity of the thalamo-cortical model is shown across the nodes.

<p>The model comprises the three network nodes (cortex, reticular nucleus and thalamus) known to be major processing units of the visual system. The model accounts for inhibitory and excitatory connections (here, red is inhibitory, green is excitatory) and their directionality as well as anatomical distances. <b><i>B</i>.</b> The intrinsic connectivity across the 3 modes of the SJ3D within a single node is shown (with <i>ξ</i> being the main state variable for excitatory activity and <i>α</i> for inhibitory activity).</p

Data_Sheet_1_A deep learning approach to estimating initial conditions of Brain Network Models in reference to measured fMRI data.pdf

IntroductionBrain Network Models (BNMs) are mathematical models that simulate the activity of the entire brain. These models use neural mass models to represent local activity in different brain regions that interact with each other via a global structural network. Researchers have been interested in using these models to explain measured brain activity, particularly resting state functional magnetic resonance imaging (rs-fMRI). BNMs have shown to produce similar properties as measured data computed over longer periods of time such as average functional connectivity (FC), but it is unclear how well simulated trajectories compare to empirical trajectories on a timepoint-by-timepoint basis. During task fMRI, the relevant processes pertaining to task occur over the time frame of the hemodynamic response function, and thus it is important to understand how BNMs capture these dynamics over these short periods.MethodsTo test the nature of BNMs’ short-term trajectories, we used a deep learning technique called Neural ODE to simulate short trajectories from estimated initial conditions based on observed fMRI measurements. To compare to previous methods, we solved for the parameterization of a specific BNM, the Firing Rate Model, using these short-term trajectories as a metric.ResultsOur results show an agreement between parameterization of using previous long-term metrics with the novel short term metrics exists if also considering other factors such as the sensitivity in accuracy with relative to changes in structural connectivity, and the presence of noise.DiscussionTherefore, we conclude that there is evidence that by using Neural ODE, BNMs can be simulated in a meaningful way when comparing against measured data trajectories, although future studies are necessary to establish how BNM activity relate to behavioral variables or to faster neural processes during this time period.</p