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The authors have declared that no competing interests exist.

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.

In a previous study, a data-driven large-scale model of mouse brain connectivity was constructed. This mouse brain connectivity model is estimated by a simplified model which only takes in account anatomy and distance dependence of connection strength which is best fit by a power law. The distance dependence model captures the connection strengths of the mouse whole-brain network well. But can it capture the dynamics? In this study, we show that a small number of connections which are missed by the simple spatial model lead to significant differences in dynamics. The presence of a small number of strong connections over longer distances increases sensitivity of synchronization to perturbations. Unlike the data-driven network, the network without these long-range connections, as well as the network in which these long range connections are shuffled, lose global synchronization while maintaining localized synchrony, underlining the significance of the exact topology of these distal connections in the data-driven brain network.

Structural neural connectivity and its implications for brain function have been a long-sought subject in neuroscience. Many previous studies have been limited either to small networks of few cells or coarser connectivity among larger brain regions [

Biological networks are inherently spatially constrained. Recent studies have shown that geographic constraints play a critical role in generating graph properties of real-world neuronal networks [

By analyzing the latest connectivity data from a new mapping algorithm, we find that the network connectivity strongly depends on its spatial embedding, with spatially close brain regions strongly connected and distal regions weakly connected. We study the precise relationship between connectivity and distance, and investigate possible computational roles of positive residual connection strengths that are not captured by the spatial dependence. To probe the possible implications of the residual connections on the network dynamics, we construct a network of phase oscillators with the data-driven adjacency matrix and compare its dynamics to those of the oscillator network with the connections strictly dependent on distance. We analyze spatial structures of synchronization by measuring the order parameter for varied amounts of global coupling coefficient. We further examine the strong connections between distal brain regions by studying network dynamics when fractions of the strong residual connections are added to the spatially constrained network. Finally, we relocate the positive residuals either to connections between nearby brain regions or to different fractions of longest-range connections, thus increasing the connection strengths for the spatially close or distal brain regions while eliminating sparse, strong connections spread across different edge lengths. The networks restructured this way maintain overall connection strength of the brain network but have a connectivity topology different from that of the brain network. By comparing dynamics of such restructured networks and the data-driven whole brain network, we show that the spatial locations of the strong positive residuals are important. Specifically, our study reveals that strong connections distributed over the brain network across many length-scales enhance the capability of the system to switch between asynchronous and synchronous states, underlining the significance of the existence of these connections. The network without these long-range connections, as well as the network in which these long-range connections are shuffled, when pushed by perturbations or low coupling coefficient, lose global synchronization but maintain local synchronization over small spatial scales. In the same conditions, the data-driven network loses synchronization over all spatial scales. It is interesting to speculate that this phenomenon is necessary for the integrative processes necessary for global cognitive functions.

The mesoscopic mouse whole-brain connectivity was constructed based on viral tracing experiments available on the Allen Mouse Brain Connectivity Atlas [

(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.

We analyzed the relationship between connection strength and spatial distance between brain regions in the data set. In accordance with previous studies on brain networks [

We constructed adjacency matrices for the ipsilateral and the contralateral networks based on the power-law relationship, as shown in

To understand the structure and effects of the residual connection weights that are not captured by the power-law dependence on distance, we had a closer look at these residuals. For both ipsilateral and contralateral connections, a long, positive tail is observed in the distribution of residual connection weights, suggesting strong distal connections above the power-law dependence on distance (

(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 (

Do these positive residual connections between distal regions have any computational significance? In other words, can we capture the full computational capacity of the mesoscopic brain network with connectivity governed by strictly distance-dependent rules, with the residuals removed? To test this, we compare dynamics of the data-driven brain network to those of an artificial, strictly distance-dependent network generated by the power-law relationship. Specifically, we built a network of coupled phase oscillators whose coupling strengths are described by the weighted adjacency matrix of the data-driven brain network or the power-law distance-dependent connectivity. Each of these Kuramoto-type phase oscillators corresponds to a brain area. Kuramoto-type coupled phase oscillators have been widely used to model oscillatory brain dynamics [_{i}, is described by:
_{i} denotes the natural frequency, and _{ij} is the adjacency matrix of the network. For the case of the data-driven brain network, _{ij} = _{ij} where _{ij} indicates the adjacency matrix obtained from viral tracing data, for both ipsilateral and contralateral connections. For simulations of the artificial, distance-dependent network, _{ij} = _{ij} indicates the adjacency matrix constructed by making the connection weights strictly follow the power-law dependence on distance. The last term _{i}(_{i}(_{t} = 0) and variance _{ij} is the Kronecker delta and _{n} is in radians and _{i} are randomly chosen from a symmetric, unimodal distribution _{d} for

Numerous previous studies have shown the importance of distance-dependent delays in networks of oscillators [_{ij}, which is computed by dividing the Euclidean distance _{ij} between nodes

We investigated the dynamics of the data-driven network and the power-law generated network using

Obtaining an explicit, analytical relationship between the order parameter and generalized network structures has been a challenging problem in studies of phase oscillators on complex networks [

Phases were initialized randomly, and ^{−4} (s) for 4 seconds (_{t} = 40000 steps), until a stationary state is reached. In our simulations, the time step size Δ^{−4} (s) satisfies the condition _{t}/2 steps are discarded in measuring the order parameter. The order parameter, representing network coherence, can be modulated by the global coupling coefficient _{d} of the intrinsic frequency distribution, and the standard deviation _{n} in the additive Gaussian white noise. In this paper, we computed the order parameter using _{d} = 0 (Hz) and the standard deviation of the Gaussian noise fixed at _{n} = 2 (rad). For each value of coupling coefficient _{d} = 0.2 (Hz) and _{n} = 2 (rad), to offset different effects of each configuataion of the intrinsic frequencies due to the nonzero _{d}.

(A) Phase differences cos(_{i} − _{j}) of pairs of nodes (_{d} = 0) and the standard deviation of the Gaussian white noise was fixed at _{n} = 2. (B) Universal order parameter

When the standard deviation of the white noise is held constant, increasing the coupling coefficient _{d} has qualitatively the same effect as decreasing _{d} with _{d} determines the network coherence. The same is true for decreasing the amount of _{n}. We show that varying _{n} and _{d} produces the same qualitative results as with varying _{n} is varied, the intrinsic frequency distribution and the coupling coefficient are held constant, at _{d} = 0 and _{d} is varied, the other two parameters are fixed at _{n} = 0 and _{n}, or _{d}, we show that the observed trend in the data-driven brain network and the power-law approximated network is robust.

In _{i} − _{j}) for pairs of nodes (

At a finer scale, we also observe a small amount of initial increase in the order parameter for the shortest-range connections (110-346

In the data-driven brain network, increasing the coupling coefficient

Such trends can be also visualized in the order parameter for the whole network. The overall universal order parameter increases with global coupling coefficient in both the data-driven and the power-law networks (

These trends are more clearly portrayed by plotting the sensitivity of the order parameter (Δ

This result on order parameter can be manifested by a couple of simple measures we use here. To compare the maximum sensitivity of the order parameter to changes in the global coupling coefficient _{n} and _{d}), we introduce a measure of the maximum sensitivity of synchronizability:

For the power-law network, the averaged maximum sensitivity of the order parameter is Γ_{k} = 0.1144 ± 0.0214. The maximum sensitivity of the order parameter is higher in the data-driven mouse brain network, at Γ_{k} = 0.3172 ± 0.0829. The higher value of the sensitivity measure Γ_{k} for the data-driven brain network indicates that a small amount of change in the coupling coefficient can induce a significant change in the network’s coherence state, in particular, within the range of

To evaluate spatial dependence of the order parameter, we use another measure that quantifies the difference between the order parameter for short-range subnetworks and the order parameter for the whole network. This measure is defined as:
_{shortest} is the distance less than 570 (_{longest} is 11955 (_{k} denotes averaging across varied coupling coefficient _{d} = 0.1851 ± 0.0706 and Γ_{d} = 0.5383 ± 0.0234, respectively. The larger Γ_{d} of the power-law network depicts a larger drop in coherence as the region of interest expands from the spatial vicinity to the whole network in the power-law network. In other words, the power-law network exhibits more localized coherence throughout a range of varied coupling coefficients.

We also confirmed that such difference between the data-driven brain network and the strictly distance-dependent, power-law network remains unchanged when the natural frequencies of the nodes are moved to 8 Hz and 20 Hz, which are in the ranges of theta (6-12 Hz) and beta (10-30 Hz) oscillations, respectively. Like gamma oscillations, theta and beta oscillations are frequently observed in the large-scale brain dynamics. While gamma oscillations are thought to be linked to cognitive processing and sensing, theta rhythms are observed in hippocampal LFP and thus believed to underlie memory formation. On the other hand, beta rhythms have been associated with movement preparation and motor coordination [

Our results indicate that in the real brain network, a small change in the global coupling coefficient induces a rapid transition between partial network synchrony and a more globally synchronized state, while in the network with connections strictly following a power-law dependence on distance, such a rapid transition to synchronization is not observed. We get qualitatively the same results when we vary parameters other than the coupling coefficient _{d} and _{n}, to modulate the network synchronizability. The order parameter is more sensitive to changes in the dispersion of intrinsic frequencies (_{d}) and the standard deviation in the additive white noise (_{n}) in the data-driven brain network than in the power-law governed network as well (Supporting Information

We next examined what aspects of the residual connection strengths confer the network’s ability to span a wide range of coherence states. In previous studies on coupled oscillators, it has been found that even a small fraction of shortcuts in a small-world network significantly improves synchronization of the network [_{k} = 0.1423 ± 0.0036, Γ_{k} = 0.1617 ± 0.1266, and Γ_{k} = 0.2785 ± 0.0506, respectively for top 5, 20, 40% of the positive residuals added to the power-law network, on the same edges as in the original data-driven whole-brain network.

(A) Whole network order parameter

Does the location of these strong connections have any significance in emergence of the rapid phase transition? To test whether the sensitivity of the network coherence to coupling coefficient can be recovered by simply adding the positive residuals anywhere to increase the overall connection strength of the power-law network, we studied the dynamics of the network constructed by relocating the positive residuals. We generated three networks with positive residuals relocated. In one of them, the positive residuals above the power-law relationship were positioned at random locations on the network (shuffled). In the other two, the positive residuals were preferentially relocated to the shortest 0.2% or to the longest 0.2% connections of the total edges. For the proximal-relocated network, the positive residual connections were added to connections between spatially close regions, by distributing the total positive residual connection strength among the connections between nodes within 570

When the locations of the positive residuals are randomized and thus there are strong connection weights across multiple spatial scales, the dependence of network synchronization on _{d} = 0.4091 ± 0.00017 for the network with randomized positive residuals, compared to the data-driven brain network (Γ_{d} = 0.1851 ± 0.0706).

(A) Whole network order parameter

When the positive residuals are relocated to proximal connections, the network coherence is no longer as sensitive to small changes in the global coupling coefficient as in the whole-brain network (

In addition, we observe that when positive residuals are relocated to proximal connections, the overall order parameters across the spatial scales are higher (_{i} − _{j})〉_{t} is weighted by the connection strength between the pair of oscillators _{ij}. When positive residuals are placed on proximal connections, the influence of the phase differences between nearby nodes, which increases the overall network order parameter, is emphasized more by larger connection strengths _{ij}. On the other hand, when the positive residuals are relocated to distal connections, although distal nodes are now more strongly coupled than before, the phase differences between distal nodes are still quite large. Therefore, in this case, the large phase differences between distal nodes which lower the overall order parameter, are strongly weighted by _{ij}, and thus, the overall network order parameters are maintained at low values. We also note that the order parameter rapidly increases at large distances in the power-law network with the residuals preferentially added to the longest edges (_{ij}) which induce large values of 〈cos(_{i} − _{j})〉_{t} between distal regions _{ij}〈cos(_{i} − _{j})〉_{t} terms.

To further examine the relationship between the spatial spread of the strong connections and the sensitivity of synchronizability, we measured the order parameter in networks generated from the power-law approximation by placing the positive residual strengths to different fractions of the longest edges. In

Our results show that the location of strong connections above the power-law dependence on distance is critical for generating a steep change in the order parameter. While the precise positions of the strong connections do not have to match those of the data-driven network to produce highly sensitive order parameter to the coupling coefficient, there should be a sufficient amount of strong connections across a range of spatial scales. Precise locations of the strong residuals, however, determine the order parameter’s dependence on the spatial scale, modulating spatial coherence patterns. In sum, the spatial structure of the network connectivity plays a key role in maintaining the brain’s ability to change its computational states with small perturbations, and such sensitivity cannot be achieved by simply matching the total network connection strengths. The structure does not have to precisely match that of the real brain network to maintain the high sensitivity. What is critical to maintain, rather, is some connections stronger than the simple distance-dependence distributed over the network. However, the precise connectivity structure is important for generating specific spatial coherence patterns in the network dynamics.

In this paper, we studied synchronization of a spatially constrained model of a weighted whole-brain network at the mesoscale, constructed from viral tracing experiments. The importance of linking connectivity structure and large-scale brain dynamics have been noted in previous studies [

We hypothesize that the sharp transition in synchronization in the data-driven network, which is absent in the spatially-constrained power-law model, may underlie the brain’s ability to rapidly switch computational states [

The increased sensitivity of the network synchronizability induced by strong long-range connections further implicates a tradeoff between cost-efficiency and high functional capacity in the brain network. Such tradeoff between wiring cost and computational capacity has been suggested as a network-generating principle in a number of previous studies [

In this paper, we infer the dynamics of the mesoscopic brain network by constructing a network of phase oscillators with the coupling strengths determined by the structural connectivity obtained by viral tracing experiments. Thus, while the structural connectivity is based on actual data, the dynamics we conferred on the network are arbitrary. Building a more realistic, data-driven dynamic network based on imaging experiments such as calcium-imaging, ECoG, LFP, and MEG will be a crucial future extension of our study of connecting the network structures to the network dynamics. Furthermore, for future studies, more biophysically-motivated neural mass models [

The mesoscopic mouse whole-brain connectivity was obtained from the Allen Mouse Brain Connectivity Atlas (

We fitted connection strengths as a function of interregional distance, where the distance between each pair of nodes was determined by computing the Euclidean distance in 3-dimensional coordinates between the centroids of the brain regions. Specifically, power-law functions for relationships between connection strength and interregional distance were fitted by using least squares on the log scale. For each of the ipsilateral and contralateral connectivity matrices, we found _{ij} indicates the distance between nodes _{ij} is the residual error. We obtained ^{6} and ^{5} and

We also investigated the power-law constrained network where the relationship between connection strength and interregional distance was found on the real scale, using nonlinear least squares (Levenberg-Marquardt algorithm), which has a poorer explanatory power than linear least squares on the log-scale (r-square: 0.264 vs 0.157 (ipsilateral) / 0.167 vs 0.131 (contralateral)). While this method generated a different power-law function from the one found by least squares on the log-log scale, the dynamics on the power-law network obtained by using nonlinear least squares maintained the same core characteristics, distinct from the data-driven brain network– the order parameter is less sensitive to changes in the global coupling coefficient.

In this section, we describe order parameters that were proposed previously [

In order to quantify network coherence in the original model of phase oscillators with all-to-all connectivity, Kuramoto introduced the complex order parameter [_{Kuramoto}, as the measure of the averaged phase differences of all pairs of oscillators:
_{t} denotes the average over time. However, this unweighted order parameter is not a good measure when comparing collective synchronizations in two networks described by different connectivity matrices, as it does not capture the topology of the networks.

To extend the use of order parameter to more general, weighted networks of oscillators, Restrepo et al [

Furthermore, degree of coherence as a function of spatial extent can be obtained by computing the order parameter for subnetworks of different spatial scales. The order parameter _{i} = ∑_{j∈γ(i,d)} _{ij} is the total connection strength of node

All of the MATLAB code used to numerically compute time-series data of coupled oscillators and the order parameters on the mouse whole-brain network from [

Universal order parameter over a range of global coupling coefficient

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The order parameters for the data-driven mouse brain network (red) and the power-law constrained network (blue) are plotted as a function of either (A) the dispersion in the intrinsic frequency distribution (_{d}) or (B) the standard deviation of the additive white noise (_{n}). The white noise is fixed at 0 (_{n} = 0) while the frequency dispersion (_{d}) is varied. Homogeneous frequencies across the network are assumed (_{d} = 0) while the amount of the white noise (_{n}) is varied. (C) The order parameters for the data-driven network (red) and the power-law constrained network (blue) as a function of the coupling coefficient _{d} = 0.2) and an additive white noise (_{n} = 2). The order parameters are averaged over 100 repeats of simulations.

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The order parameters for the data-driven mouse brain network (red) and the power-law constrained network (blue) are plotted for varied coupling coefficient _{i} = 8(_{i} = 20(

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The same simulations as with _{d} is varied while _{n} = 0 and

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We thank Joseph Knox for providing the whole-brain connectivity matrix used in this analysis. We also thank Kameron Harris for many helpful comments and suggestions. We wish to thank the Allen Institute for Brain Science founder, Paul G. Allen, for his vision, encouragement and support.