Hierarchical Heterogeneity across Human Cortex Shapes Large-Scale Neural Dynamics

Abstract

The large-scale organization of dynamical neural activity across cortex emerges through long-range interactions among local circuits. We hypothesized that large-scale dynamics are also shaped by heterogeneity of intrinsic local properties across cortical areas. One key axis along which microcircuit properties are specialized relates to hierarchical levels of cortical organization. We developed a large-scale dynamical circuit model of human cortex that incorporates heterogeneity of local synaptic strengths, following a hierarchical axis inferred from magnetic resonance imaging (MRI)-derived T1- to T2-weighted (T1w/T2w) mapping and fit the model using multimodal neuroimaging data. We found that incorporating hierarchical heterogeneity substantially improves the model fit to functional MRI (fMRI)-measured resting-state functional connectivity and captures sensory-association organization of multiple fMRI features. The model predicts hierarchically organized higher-frequency spectral power, which we tested with resting-state magnetoencephalography. These findings suggest circuit-level mechanisms linking spatiotemporal levels of analysis and highlight the importance of local properties and their hierarchical specialization on the large-scale organization of human cortical dynamics.

Keywords: brain networks; computational model; cortical gradients; cortical hierarchy; functional connectivity; large-scale modeling; magnetoencephalography; resting-state fMRI; structural connectivity.

Figure 1.. Large-Scale Model of Human Cortex with Heterogeneous Local Circuit Properties

(A) Model framework. Each parcellated cortical area is modeled as coupled excitatory (E) and inhibitory (I) populations. Areas interact through long-range projections following dMRI-derived intra-hemispheric structural connectivity (SC). Fit model parameters comprised recurrent excitatory strength (W^EE), excitatory-to-inhibitory strength (W^EI), and a global coupling parameter scaling the strength of long-range connections (g). Inhibitory-to-excitatory strengths (W^IE) were adjusted to maintain a uniform baseline excitatory firing rate across areas. Dynamics of synaptic gating variables (S^E) are transformed into a simulated BOLD signal via the Balloon-Windkessel hemodynamic model. For computational tractability of model fitting, model BOLD FC matrices were calculated via linearization of the extended dynamical equations around the fixed point of the system. Model parameters were fit to maximize the similarity between model and empirical FC matrices. (B) Parametrizing local properties via a heterogeneity map. In the homogeneous model, the parameters (W^EI and W^EE) were identical across cortical regions. In the heterogeneous model, the parameters (W^EI and W^EE) varied across cortical areas based on a heterogeneity map h, whose minimum and maximum value is 0 and 1, respectively. For each region (i), the parameter values were set by an affine function of the heterogeneity map values {h_i}, characterized by an intercept W_min and scale factor W_scale: W_i = W_min + W_scaleh_i. (C) Cortical T1w/T2w map. The median (n = 334 subjects) cortical T1w/T2w map values of each parcellated cortical area (180 per hemisphere). (D) Network assignments. Cortical areas were assigned to eight functional resting-state networks (RSNs) comprising three sensory (AUD, auditory; VIS, visual; and SOM, somatomotor) and five association (DAN, dorsal attention; FPN, frontoparietal; VAN, ventral attention; DMN, default mode; and CON, cinguloopercular) networks. (E) T1w/T2w map values per RSN, averaged across areas. T1w/T2w values are significantly lower in association RSNs than in sensory RSNs (p < 0.003, Wilcoxon signed-rank test, difference between sensory and association T1w/T2w across subjects). Error bars indicate the SD across areas within an RSN.

Figure 2.. Hierarchical Heterogeneity Improves the Model Fit to rs-FC

(A and B) Structural connectivity (SC) (A) and empirical FC (B) matrices (left hemisphere only), averaged across subjects. Colored bars (top and left of matrices) denote resting-state network assignments (colored as in Figure 1). (C and D) Model FC of the homogeneous (C) and heterogeneous (D) models (left hemisphere only), averaged across particles (***,p < 10⁻³). (E and F) Correlation between average empirical FC and average model FC for the homogeneous (E) and heterogeneous (F) models. (G) Goodness of fit (i.e., fraction of explained variance r²) between the average empirical FC and the SC (gray), homogeneous model FC (blue), and heterogeneous model FC (red). The fit for the heterogenous model is greater than that of the homogeneous model, which is greater than that of the SC (p < 10⁻⁵ for each, dependent correlation test). (H) The best-fit values for recurrent excitatory parameters for the models, with regions ordered by increasing values of the T1w/T2w-derived hierarchical heterogeneity map. Shaded regions show SD across particles.

Figure 3.. Surrogate Heterogeneity Maps Show that the T1w/T2w Map Provides a Preferential Axis of Specialization

(A) The T1w/T2w-based hierarchical heterogeneity map, for the left hemisphere, and example surrogate heterogeneity maps with matched spatial autocorrelations. (B) Spatial autocorrelations of the Box-Cox-transformed T1w/T2w map (black) and surrogate heterogeneity maps (gray) as a function of geodesic distance. (C) Histogram of spatial correlations (Spearman rank) between all pairs of random surrogate maps. (D) Histogram of the best fit (correlation between empirical and model FC) of random surrogates. The T1w/T2w map gradient fit is significantly higher than random surrogates (p = 0.008). (E) The correlation between hierarchical heterogeneity-surrogate map similarity (i.e., absolute values of correlation) and model performance (i.e., model-empirical FC similarity). The model-empirical FC similarities for surrogate maps increase with the absolute value of the correlation with the hierarchical heterogeneity map (r = 0.73, p < 10⁻³).

Figure 4.. Model Fits across Resting-State Networks Are Network Specific

(A and D) Schematic of within- and across-network fits. Correlations between empirical and model FC within or between RSNs were calculated for homogeneous and heterogeneous models. (B and E) Within-network (B) and across-network (E) fits of the models. The heterogeneous model showed substantial improvements compared to the homogeneous model, for within- and across-networks fit in all networks. Across-network fit improvements were distributed across sensory and association networks. Within-network fit improvements were preferentially in association networks. (C and F) Topography of the improvement in fit for within-network (C) and across-network (F). Values are shown for each RSN.

Figure 5.. Hierarchical Topography of Cortical GBC

(A) GBC of each region is calculated as the average FC of that region with all other cortical regions. (B) The areal topography of empirical GBC. (C) GBC of sensory areas is significantly larger than that of association areas (p < 0.001, Wilcoxon signed-rank test). (D–F) The correlation between empirical and model GBC is significantly larger in the heterogeneous(D) model than for the homogeneous (E) model (p < 10⁻⁴, dependent correlation test; ***, p < 10⁻³).

Figure 6.. Hierarchical Topography of Inter-individual Dissimilarity of FC

(A) Dissimilarity calculated for FC patterns across subjects (n = 334) in the empirical dataset and across particles (n = 1, 000) in the model fitting framework. The dissimilarity for area i, V_i, is given by V_i = E(1 - Corr(F_i(s_p),F_i(s_q)), where E(…). is the mean across subject pairs and F_i(s_p) is the FC of area i for subject s_p. To compare the areal topographies of empirical and model dissimilarity maps, we standardize the values through Z score. (B and C) Topography of empirical inter-individual dissimilarity (B) and heterogeneous model inter-particle dissimilarity (C). The inter-particle dissimilarity for the homogeneous model was not depicted due to lack of spatial patterns. (D) Inter-individual dissimilarity is higher for association areas than for sensory areas (p < 0.003, Wilcoxon signed-rank test). The heterogeneous model exhibits a similar hierarchical differentiation in inter-particle dissimilarity. The association-sensory difference is larger in the heterogeneous model than homogeneous (p < 10⁻⁴, 2×2 ANOVA, z = 5.3). (E) Similarity between empirical and model SD of FC across subjects (particles). (F and H) Correlations between parameters, across particles in the approximate posterior distributions, for the homogeneous (F) and heterogeneous (H) models. Homogeneous model parameters are very strongly correlated with each other. (G and I) PCA applied to the distribution of particles drawn from the posterior, with their parameter values normalized by the population mean for each parameter. Plotted is the fraction of explained variance by the top PCs. 100% of the variation in homogeneous model parameters is explained by a single dimension (G). The variation in the heterogeneous model is explained by four components (I).

Figure 7.. Intrinsic Dynamics of a Local Microcircuit Model Vary with Recurrent Strengths

(A) The phase diagram of a local microcircuit model (i.e., one node in the large-scale network) without external input. The black line indicates the critical points beyond which the baseline state is unstable, and the green lines indicate the boundaries at which the system exhibits a transition to oscillatory dynamics. For excitatory-to-inhibitory synaptic strengths (W^EI) smaller than 1.35, the system exhibits a pitchfork bifurcation (in asynchronous dynamics), and for larger W^EI values, the system exhibits a transition to damped oscillatory dynamics. (B and C) A representative example of optimal homogeneous (B) and heterogeneous (C) model parameters projected onto the phase diagram. The external input is adjusted to provide the same mean long-range input as in the fit large-scale model from other areas. (D and E) The power spectral densities (PSDs) of the synaptic gating variables (S^E) of the homogeneous (D) and heterogeneous (E) models. The colors indicate the hierarchical level of each area based on its T1w/T2w map value (light, sensory; dark, association).

Figure 8.. Hierarchical Topography of Spectral Power in MEG

(A and B) The empirical PSD derived from raw MEG (A) and with removal of alpha-band (α) and beta-band (β) Lorentzians (B). The shading of lines indicates values of the T1w/T2w-based hierarchical heterogeneity map. (C and D) PCA applied to empirical MEG PSDs, with removal of α and β Lorentzians. The first principal axis is the spectral pattern that captures the most spectral variation across areas (C) inset. The first principal component (PC-1) is the areal map whose values are the loading of the spectral variation for each area onto the first principal axis (D). PC-1 captures 67.5% of the total spectral variance (C). (E and F) PCA results for the homogeneous (E) and heterogeneous (F) models and comparison to the empirical PC-1 map. Left: spatial topography of the model MEG PSD PC-1. Right, top: correlation between model and empirical PC-1 map topographies. Right, bottom: principal axis 1 (**,p < 10⁻²; ***, p < 10⁻³). (G) Spearman correlation of the empirical MEG PSD PC-1 map with T1w/T2w, and model PSD PC-1 maps. The empirical MEG PSD PC-1 map is significantly correlated with T1w/T2w (r_s = 0.631,p = 0.002, against 500 randomized surrogate maps). The correlation of the model PSD PC-1 map with the T1w/T2w map is higher for the heterogeneous model than the homogeneous model.