Table 1.
Table of variables used in Methods and Results.
Fig 1.
Network of interacting neural populations.
A Schematic diagram of the ND model: the neural populations (purple circles) receive noisy inputs as well as recurrent feedback (gray and red connections, respectively). The topology is non-random: 2 groups of interconnected populations (green dashed boxes) are linked by hubs (cyan dashed box), but the hubs are not connected to each other. B Connectivity matrix C for a network of 50 populations, corresponding to the strengths of the red connections in A. The indices for the two groups are indicated by the green bars, and hubs by the cyan bar. C Example time series between two populations in the network, one from the large group in A and a hub. In the right panel, the darker curve is shifted by τ = 1 s. D Empirical covariance matrices between all neural populations, evaluated from the time series in C with the two time shifts: τ = 0 for and τ = 1 s for
; see expression for Qτ in Eq (2). The color scale is adjusted to focus on off-diagonal elements. E Logarithm of autocovariance
for each node i as a function of τ. The mean over all nodes (blue curve) has a slope close to 1/τx (black line). F Asymmetry of theoretical Qτ as a function of the asymmetry of C. The asymmetry index in Eq (24) scales from 0 for symmetric to 1 for antisymmetric matrices. Three time shifts τ are displayed (lighter to darker blue), each cross represents a cluster-hub network similar to A and the corresponding line indicates the linear fit. G Network effect indicated by the weak match between C weights and Q0 values over all connections. H Schematic representation of a step in the Lyapunov optimization (LO) procedure. The model covariances Q0,τ are evaluated for the current network parameters C and Σ. From the comparison between the model Qτ/0 and their objective counterparts
, the desired updates ΔC and ΔΣ for the model parameters are calculated.
Table 2.
Table of simulation, network and optimization parameters for the ND artificial network model.
Table 3.
Parameters for the dynamic mean-field (DMF) model and hemodynamic response function (HRF).
Fig 2.
Iterative optimization to recover C from empirical Q.
A Estimation based on theoretical covariances. Evolution of Q0 and Qτ errors between the model and theoretical objectives (blue curves), as well as C error between the original and model connectivity (red). The theoretical objectives are calculated using Eqs (6) and (7). Errors are calculated using the normalized matrix distance in Eq (25). The optimization relies on Eq (15) with time shift τest = 1 s. B Details of the match of the model and objective matrices for Qτ (blue dots) and C (red crosses) at three stages of the optimization in A. The black line indicates a perfect match. C Similar plot to A with empirical Q as objectives, evaluated using Eq (2) for 50 simulations of the same network as in A. D Similar plot to B for the optimization in C.
Fig 3.
Influence of choice for τest on accuracy of recovered C.
A Best estimated matrices C for τest = 0, 0.1, 1 and 5 s. For τest = 1, the C matrix at the end of the optimization (step 10000) is also shown. B Evolution of Q errors for the optimizations with various τest in A. bf C C errors corresponding to the best matrices in A (red solid curve), as well as last matrices of the optimization (red dashed curve). The green curve indicates the error of the best C with the symmetric part of the original C. The blue curve indicates the error between the empirical and the corresponding theoretical matrix. D Pearson correlation coefficient between the best and original C (red) as well as reconstructed and objective Q0/τ. between
and each corresponding theoretical Qτ matrix for the same τ = τest in C. E Comparison between LO with empirical
calculated with Eq (2) (red solid curve) and LO with theoretical
obtained from Eq (7) (red dashed curve). The blue curve indicates the inaccuracy between the empirical and theoretical Qτ matrices. F Same as B for the Q error with theoretical objectives
.
Fig 4.
Need for tuning intrinsic noise Σ for each node in addition to C.
A Match between the recovered and the original C (red crosses) and Σ (purple triangles) for LO based on Eqs (15) and (20) with theoretical objectives and τest = 1 s. The original network has inhomogeneous noise Σii. B Comparison of Q error, C error and Pearson correlation coefficient for non-zero weights of the original C for four optimizations: ‘opt’ where both C and Σ are tuned as in A; ‘low’ for the tuning of C with a fixed Σ with the minimum value of the original Σ; likewise, ‘med’ and ‘high’ with a fixed Σ with the mean and maximum value of the original Σ. C Match between the original and recovered C for the optimization with fixed homogeneous Σ for three distinct values: the minimum, mean and maximum of the values in the original Σ. They are to be compared with the left panel in A.
Fig 5.
Robustness of the estimation procedure of C and Σ on artificial networks.
A Left: Comparison of the C error (red curves) and Σ error (purple curve) for optimization based on several τest. For the C error, the cluster-hub topology in Fig 1A corresponds to lighter red and the random connectivity to darker red. The plot is similar to Fig 3C; the error bars correspond to the standard deviation over 40 network configurations. Right: Similar plot for the error of asymC for the two network topologies. B Pearson correlation coefficient between the matrix elements of the whole recovered and original C matrices in red; idem but limited to non-zero elements of the original C in yellow; and between the recovered C and the symmetric part of the original C in green. The plot concerns 50 networks of either topology for which the asymmetry of original C is larger than 0.5; LO is performed using τest = 1 s. C C and Q errors (in red and blue, respectively) plotted against several parameters for the network and LO: the number of simulations used to calculate the empirical , the network size N and the asymmetry of the original C. Each cross/dot corresponds to one of the 80 simulations of either topology in A for τest = 1 s. D Comparison of the estimation performance of LO wit the direct and heuristic methods for the C error (left), the Σ error (middle) and asymmetry error (right). For each case, the left boxplot correspond to 40 cluster-hub networks and the right one to 40 randomly connected networks; the optimizations are performed with τest = 1 s. Contrary to others, the asymmetry error is not normalized.
Fig 6.
Information conveyed by spatio-temporal FC and hemodynamic response function (HRF).
A Schematic representation of the properties of the two dynamic cortical model: ND network whose activity directly models the BOLD signals; dynamic mean-field (DMF) model whose synaptic activity is processed by the HRF to obtain the BOLD signal. B Two types of neural connectivity used with the DMF: random and SC matrix obtained from dwMRI. In the matrix, darker pixels indicate a higher probability of existing fibers between cortical areas. C Autocovariances of the two models for 50 different simulations. The y-axis has a log-scale. For each simulation, the curves are centered vertically with respect of the mean over all nodes to focus on the slope. The black line represents the exponential decay fitted on the mean experimental BOLD with time constant 5.3 s. D Left: Similarity between neural and BOLD covariances (excluding variances) for the two considered DMF models, as measured by the Pearson correlation coefficient. Right: Performance of EC estimation from BOLD FC for the DMF models. The performance is measured by the Pearson correlation coefficient. For each simulation, the objective is the average BOLD FC taken from 50 simulations of the same network. E Example of a typical mapping between neural and BOLD covariances for a DMF/SC network. The covariances have been rescaled. F Variability of the EC estimated from individual FCs. The grey dots represent the match of four EC each estimated for a single simulation of 300 s. The red dots correspond to the average over 50 estimated EC for 50 simulations of the same DMF/SC network. G Uncertainty of the estimated EC as a function of the estimated weight for each neural connection. The y-axis is the standard deviation over the 50 optimizations in F divided by the mean.
Fig 7.
Fitting fMRI data to infer cortico-cortical connectivity.
A Empirical FC, namely and
, averaged over 25 subjects for BOLD signal recorded during rest. B Autocovariances (grey curves, with a log y-axis) as a function of the time shift τ. From the mean curve (red) the time constant for the ND model is estimated: τx = 5.3 s. The inset represents the autocovariances with a standard y-axis. C Variability of similar decay time constants to τx in B for individual cortical area. D SC mask of weights to tune (black pixels), corresponding to the strongest connections of the dwMRI matrix in Fig 6B thresholded to obtain 32% density. E Evolution of the Q error during the optimization. The black line indicates the rate of change of C. F EC corresponding to the estimated C matrix. G Match between empirical and reconstructed Q matrices for the optimization with τest = 4 s while the connections for the mask in D. For both Q0 and Qτ, the blue dots correspond to tuned connections, the black crosses to non-diagonal untuned connections and the cyan dots to diagonal connections. The Pearson correlation coefficient between matrix elements are given above each plot. H Weak correlation between the symmetric part of the estimated C and
, indicating a strong network effect.
Fig 8.
Robustness of estimated C and Σ from FC with different time shifts τest.
A Match between C for τest = 4 s (x-axis) and C for τest = 2, 6 and 8 s (y-axis). Each plotted point corresponds to an EC connection and the black line indicates a perfect match. B Same as B for the EC estimated for τest = 0 on the y-axis. C Similar plot to A for the diagonal elements of the estimated noise matrix Σ. D Similar plot to B for the EC estimated using the direct method. E Agreement as measured by the Pearson correlation coefficient between the pairs of EC and Σ plotted in B and D for the LO (red), heuristic (purple) and direct (cyan) methods. The green error bar on the left represents the same between EC with τest = 0 and the four τest > 0. F Comparison of the models tuned with LO using different time shifts. For each model C obtained with τest from 0 to 6 s, the curve represents the Qτ error for all τmod on the x-axis.
Fig 9.
Robustness of estimated EC with respect to the EC density and individual FC.
A Fit performance as a function of the density of EC. The blue curve indicates the Q error for optimizations with several masks for C (x-axis); the error bar correspond to the 4 τest > 0. The red curve indicates the similarity between the estimated C and the empirical . B Similarity of estimated C measured by the Pearson correlation for the different masks in A and τest = 4 s. C Input-output asymmetry of the estimated C obtained for various optimization masks (error bars) and τest (x-axis). The red curve corresponds to LO and the purple curve to the heuristic method. The blue curve indicates the asymmetry of the empirical FCτ. D Q Error for LO based on individual FCs with τest = 2 and 4 s with 32% density. The horizontal dashed line corresponds to the Q error in Fig 7. E Similarity between the individual ECs and the EC obtained from the mean FC in Fig 7F. Similarity is measured by the Pearson correlation between the non-zero EC weights for the 32% density mask. F EC asymmetry of the individual ECs. The horizontal dashed line indicated the asymmetry of the EC in Fig 7F. G Agreement between the mean EC over individual FCs and the EC estimated from the mean FC.
Fig 10.
Interpretation of estimated cortical interactions and intrinsic noise.
A Weak match between EC and SC values. Each red cross corresponds to a connection. B Plot of the sum of incoming C weights versus the sum of outgoing C weights (x-axis and y-axis, resp.) for each cortical area (red cross). C 3D mapped representations on the cortex of the variances of the empirical BOLD signal, sums of incoming weights in EC, and intrinsic noise (related to spontaneous noisy activity). Red corresponds to larger values and blue to small values.