By: Mengsen Zhang, Stanford University

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The brain is a complex, nonlinear system, exhibiting ever-evolving patterns of activities even without external inputs or tasks. Such intrinsic dynamics plays key roles in cognitive functions and psychiatric disorders. A challenge is to link the intrinsic dynamics to the underlying structure, given the nonlinearity. Here we use a biophysically constrained, nonlinear-dynamical model to show how the complexity of intrinsic brain dynamics, manifested as its multistability and temporal diversity, can be sculpted by structural properties across scales. At a local level, multistability and temporal diversity can be induced by sufficient recurrent excitatory connectivity and its heterogeneity. At a global level, such functional complexity can also be created by the synergistic interaction between monostable, locally identical regions. Coordination between model brain regions across attractors in the multistable landscape predicts human functional connectivity. Compared to dynamics near a single attractor, cross-attractor coordination better accounts for functional links uncorrelated with structural connectivity. Energy costs of cross-attractor coordination are modulated by both local and global connectivity, and higher in the Default Mode Network. These findings hint that functional connectivity underscores transitions between alternative patterns of activity in the brain—even more than the patterns themselves. This work provides a systematic framework for characterizing intrinsic brain dynamics as a web of cross-attractor transitions and their energy costs. The framework may be used to predict transitions and energy costs associated with experimental or clinical interventions.


A biophysical network model is used to examine how structural properties across scales constrain brain function (left). Local connections define recurrent connectivity within each brain region (solid lines), long-range connections define anatomical connectivity between brain regions (dashed lines). Structural properties influence function by shaping the dynamic landscape (right), which accommodates multiple stable patterns of activity, or a repertoire of attractors (a1-a4, purple balls), and transitions between attractors (black arrows). Brain regions coordinate with each other during transitions in the landscape, which can be summarized by a cross-attractor coordination matrix. The attractor repertoire and cross-attractor coordination change through bifurcations (a->b, a->c). (Updated preprint will be available at


The change of dynamic landscapes is visualized as a series of bifurcation diagrams. Each colored point in a diagram represents an attractor, which can map to the activity level of a brain region (SE) or average activity level of the whole brain (). As model parameter change (external input IE to a local region, or average long-range connection strength G), an attractor may change smoothly forming a stripe or annihilates with a non-attractor (black). Each stripe corresponds to a discrete pattern of brain activation (circled). When local excitatory connections are weak (lower graphs), there is a very limited number of attractors for a single brain region (left) or the whole brain (middle, right). Though importantly, long-range connections can create multistability (multiple stripes, lower middle, right) synergistically out of monostable brain regions (one stripe, lower left). When local excitatory connects are sufficiently strong (upper graphs), a single brain region acquires multistability (upper left). This would not result in much gain of attractors at the whole-brain level when long-range connectivity is uniform (upper middle). When long-range structural connectivity is human-like, thus spatial heterogeneous, the local complexity is much better amplified (upper right). In short, the combination of strong local excitatory connection and realistic long-range connection gives rise to greater functional complexity. Such complexity is reflected both as the greater size of the attractor repertoire and as the more sophisticated patterns of activation (contrast circled brain patterns in upper middle vs. upper right).


When long-range connections in the model are defined by the human connectome, cross-attractor coordination in the dynamic landscape better captures human functional connectivity patterns observed in fMRI than within-attractor coordination. In particular, cross-attractor coordination captures of the large-scale symmetries of human functional connectivity: the similarity between intra- and inter-hemispheric connectivity. Strong cross-attractor coordination is present where the structural connection is weak, speaking strongly to the nonlinear dependency of function on structure. Within-attractor coordination is more similar to structural connectivity, thus captures more of the linear dependency.


Some of the attractors are oscillatory. Structural connectivity not only affects the overall shape of the dynamic landscape, but also modulates the frequency of oscillations of specific attractors. The cause of frequency diversity can be the heterogeneity in local connection strength, as well as the heterogeneity in long-range connections. This is pronounced in the high-frequency oscillations (a, c). Long-range connections have a stronger effect on low-frequency fluctuations (d) than local connections (b).   

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