Synchronization dependent on spatial structures of a mesoscopic whole-brain network

Abstract

Complex structural connectivity of the mammalian brain is believed to underlie the versatility of neural computations. Many previous studies have investigated properties of small subsystems or coarse connectivity among large brain regions that are often binarized and lack spatial information. Yet little is known about spatial embedding of the detailed whole-brain connectivity and its functional implications. We focus on closing this gap by analyzing how spatially-constrained neural connectivity shapes synchronization of the brain dynamics based on a system of coupled phase oscillators on a mammalian whole-brain network at the mesoscopic level. This was made possible by the recent development of the Allen Mouse Brain Connectivity Atlas constructed from viral tracing experiments together with a new mapping algorithm. We investigated whether the network can be compactly represented based on the spatial dependence of the network topology. We found that the connectivity has a significant spatial dependence, with spatially close brain regions strongly connected and distal regions weakly connected, following a power law. However, there are a number of residuals above the power-law fit, indicating connections between brain regions that are stronger than predicted by the power-law relationship. By measuring the sensitivity of the network order parameter, we show how these strong connections dispersed across multiple spatial scales of the network promote rapid transitions between partial synchronization and more global synchronization as the global coupling coefficient changes. We further demonstrate the significance of the locations of the residual connections, suggesting a possible link between the network complexity and the brain's exceptional ability to swiftly switch computational states depending on stimulus and behavioral context.

Fig 1. Connectivity matrices constructed from viral injection data and the power-law dependence on distance.

(A) Connectivity matrix from viral tracing data (left); reconstructed connectivity from the power-law dependence on distance between nodes (middle); residual connection strengths of the data-driven network above the power-law distance dependence. We show 244 brain regions divided in to coarser major brain divisions defined in the Allen Mouse Brain Reference Atlas. These divisions are: Isocortex, Olfactory Bulb, Hippocampus, Cortical Subplate, Striatum, Pallidum, Thalamus, Hypothalamus, Midbrain, Pons, Medulla, and Cerebellum. (B) Connection strengths as a function of distance between brain regions (left panels). The connections obtained from experiments (gray) are fit by a power law (red) on the log scale with base 10 (right panels). Inset: Goodness of fit.

Fig 2. Residual connection weights unexplained by the power-law distance dependence.

(A) Distributions of the connection strengths from the data (blue) and the residual connection strengths (red). The x-axes are restricted here to better visualize the positive tails and many of the residuals clustered around zero. (B) Residual connection weights as a function of distance between nodes. (C) Directed pairs of brain regions with large positive residual connections. These represent pairs of regions with connections stronger than predicted by the interregional distance. For reference on the acronyms of the regions, see the Allen Mouse Brain Reference Atlas (https://mouse.brain-map.org/static/atlas).

Fig 3. Local and global synchronization of the data-driven brain network and the power-law network.

(A) Phase differences cos(θ_i − θ_j) of pairs of nodes (i,j) as a function of time (x-axis) and distance between nodes (y-axis) for the data-driven and power-law networks. For both networks, the intrinsic frequencies of the oscillators were at 40 Hz (σ_d = 0) and the standard deviation of the Gaussian white noise was fixed at σ_n = 2. (B) Universal order parameter r for subnetworks at different spatial scales, for the data-driven and the artificial power-law networks with different amounts of global coupling coefficient k. Order parameter r is averaged over 10 simulations. (C) Universal order parameter (solid) and Kuramoto’s original order parameter (dotted) for the whole networks of the data-driven connectivity (red) and the power-law distance-dependent connectivity (blue), as a function of global coupling coefficient k. (D) Sensitivity of the order parameter as a function of the coupling coefficient k, for the data-driven connectivity (red) and the power-law approximated connectivity (blue). Lines and shades correspond to the mean and the standard deviation over multiple simulations.

Fig 4. Order parameter r and its sensitivity for the power-law distance-dependent network with a fraction of the residual connections added.

(A) Whole network order parameter r and (B) the sensitivity of the order parameter Δr/Δk, as a function of global coupling coefficient k for networks constructed by adding different percentiles of positive residual connections to the power-law approximated network.

Fig 5. Synchronization measured when the network structure is altered but the total connection strength remains the same as the data-driven network.

(A) Whole network order parameter r as a function of the global coupling coefficient k, for the networks generated by adding the residual connection weights to random locations (gray), by relocating positive residuals (averaged) to connections between spatially close regions (< 570μm) (black dotted), and by placing positive residuals on connections between distal regions (> 10500μm) (black solid). The order parameter for the data-driven brain network (red) is shown for comparison. (B) Order parameter as a function of the spatial scale of subnetworks, for the networks constructed by shuffling locations of residual connections (left, correspond to gray in panel A), by relocating positive residuals to nearby connections (middle, correspond to dotted black in panel A), and by relocating positive residuals to long-distance connections (right, correspond to black solid in panel A). (C) Order parameter r as a function of coupling coefficient k for networks constructed by adding the positive residual strengths to the longest 1-90% of edges (> 9404, 8830, 8470, 8175, 7946, 7168, 6265, 5626, 5063, 4521, 3992, 3455, 2867, 2147μm).