Black Dog Institute and School of Psychiatry, University of New South Wales, Sydney, Australia

Mental Health Research Division, Queensland Institute of Medical Research, Brisbane, Australia

CSIRO Information and Communication Technologies Centre, Sydney, Australia

Department of Psychological and Brain Sciences, Indiana University, Bloomington, USA

Laboratory for Perceptual Dynamics, RIKEN Brain Science Institute, Saitama, Japan

Royal Brisbane and Women's Hospital Mental Health Service, Brisbane, Australia

Abstract

Background

Brain structure and dynamics are interdependent through processes such as activity-dependent neuroplasticity. In this study, we aim to theoretically examine this interdependence in a model of spontaneous cortical activity. To this end, we simulate spontaneous brain dynamics on structural connectivity networks, using coupled nonlinear maps. On slow time scales structural connectivity is gradually adjusted towards the resulting functional patterns via an unsupervised, activity-dependent rewiring rule. The present model has been previously shown to generate cortical-like, modular small-world structural topology from initially random connectivity. We provide further biophysical justification for this model and quantitatively characterize the relationship between structure, function and dynamics that accompanies the ensuing self-organization.

Results

We show that coupled chaotic dynamics generate ordered and modular functional patterns, even on a random underlying structural connectivity. Consequently, structural connectivity becomes more modular as it rewires towards these functional patterns. Functional networks reflect the underlying structural networks on slow time scales, but significantly less so on faster time scales. In spite of ordered functional topology, structural networks remain robustly interconnected – and therefore small-world – due to the presence of central, inter-modular hub nodes. The noisy dynamics of these hubs enable them to persist despite ongoing rewiring and despite their comparative absence in functional networks.

Conclusion

Our results outline a theoretical mechanism by which brain dynamics may facilitate neuroanatomical self-organization. We find time scale dependent differences between structural and functional networks. These differences are likely to arise from the distinct dynamics of central structural nodes.

Background

Modular small-world network topology may represent a basic organizational principle of neuroanatomical connectivity across multiple spatial scales

Cortical structure and dynamics are highly interdependent. On relatively fast time scales, structure enables the emergence of complex dynamics

There hence exists a "symbiotic" relationship between structural brain connectivity and brain activity. Such a relationship is thought to be central to the emergence of complex neuroanatomical connectivity from a relatively unstructured neuropil

The relationship between structural and functional brain connectivity is gaining rapid interest. Recent studies have explored this relationship by simulating neuronal dynamics on large scale neuroanatomical connectivity networks. These studies found that the resulting functional patterns passively reflect the underlying structural connectivity on slow time scales

Several models of complex network growth have been well established in the wider network community. These include the well known preferential attachment model

The nonlinear nature of neuronal dynamics

Functional connectivity of simulated neural mass model dynamics on a random structural network

**Functional connectivity of simulated neural mass model dynamics on a random structural network**. (A). The underlying 256 node random structural network. Here and in the following figures, networks are represented by their square connectivity matrices, where row and column indices correspond to nodes, and matrix entries correspond to connections between individual nodes. (B). The emergent spatiotemporal dynamics: color represents the state of individual dynamical units according to space (horizontal axis) and time (vertical axis). (C). The resulting functional connectivity matrix, derived by linear cross-correlation between the spatiotemporal dynamics from B and reordered to maximize the visual appearance of modules (this reordering was also applied to A, with negligible impact). Each dynamical unit represents the mean state of a local population of densely connected inhibitory and pyramidal neurons, with conductance-based transmembrane ion flows and zero-order synaptic kinetics. Full details of these dynamics are provided in Breakspear et al.

This intuition underlies the activity-dependent model of structural rewiring proposed by Gong and van Leeuwen

Consistent with the approach of Gong and van Leeuwen, the present study approximates neuronal dynamics using an ensemble of coupled chaotic unimodal maps. Such maps are well known to exhibit universal dynamical properties

Dimension reduction of nonlinear neuronal dynamics

**Dimension reduction of nonlinear neuronal dynamics**. (A). Phase space attractor of a three-dimensional neural mass flow. This attractor is an illustration of the dynamics generated by the flow of a neural mass model (see Breakspear et al.

We hence seek a detailed exploration of the nature of this structural self-organization. We observe that, as in Figure

Results

Interdependent evolution of structural and functional networks

Our model consists of an ensemble of chaotic logistic maps, coupled via a directed binary structural connectivity network. The dynamics of these maps generated a series of functional connectivity networks on static structural networks. As the dynamics evolved, structural networks were gradually adjusted towards emergent synchrony patterns: periodically, a node was randomly chosen and its connections were rewired such that it gained a link to a node with which it was most synchronous, and lost a link to a neighbor with which it was least synchronous. We measured synchronization using the absolute difference (Euclidean distance) between individual unit states (see Methods). We began simulations from initially random structural connectivity and proceeded until asymptotic conditions, as characterized by globally invariant structural and functional clustering and closeness.

Figure

Interdependent evolution of structural and functional networks

**Interdependent evolution of structural and functional networks**. Concurrent evolution of clustering (A), closeness (B) and modularity (C) of structural (black) and functional (blue) networks. Metrics derived from surrogate random networks (solid lines) are plotted for comparison. (D) Median, minimum and maximum rewiring rates at each rewiring step. While some nodes cease rewiring at the asymptotic state, others remain highly rewirable – hence rewiring is ongoing despite a stable structural topology. Error bars represent the standard error of the mean, as estimated over 20 simulations.

Figure

Characteristic structural and functional networks at different phases in the evolution

**Characteristic structural and functional networks at different phases in the evolution**. The initial (row 1), evolving (row 2) and asymptotic (row 3) network configurations are illustrated for structural (column 1), fast time scale (column 2) and slow time scale functional (column 3) networks. Fast time scale networks represent the instantaneous patterns of dynamical synchrony, measured as the Euclidean distance between individual unit states. Slow time scale networks are derived by calculating the correlation coefficient of 100 consecutive functional states. Nodes in all networks are reordered to maximize the appearance of modules, via the maximization of modularity (see Methods). Consequently, a network may be reordered differently, at different times in its evolution. However, given the similarity between structural and slow time scale functional networks, pairs (D)-(F) and (G)-(I) have exactly the same ordering in the current figure.

A key difference between structural and functional connectivity is the robust presence of inter-modular links in structural networks, and a relative absence of these links in functional networks. Inter-modular links represent the crucial difference between a structural small-world and a functional lattice

The degree distributions in both structural and functional networks do not evolve toward a scale-free, or broad-scale distribution (Additional file

**Evolution of degree in structural and functional networks**. Minimum and maximum degree, along with the mean and standard deviations (dotted lines) for structural (black) and functional (blue) networks. Error bars represent the standard error of the mean, as estimated over 20 simulations.

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Robustness of structural self-organization

We incorporated spatial constraints into our model by placing nodes randomly on the surface of a sphere, and subsequently restricting potential synapses to the spatially closest 40% of all neighbor pairs. Such an arrangement introduces some local clustering into the initial network topology (Figure

Robustness of structural self-organization

**Robustness of structural self-organization**. Temporal evolution of clustering and closeness of structural (black) and functional (blue) networks. (A) Evolution under spatial constraints (nodes are placed randomly on a three-dimensional sphere). (B) Evolution from an initial lattice structural topology. (C) Evolution under memory guided rewiring. Insets show initial structural connectivity matrices. Compared to Figure 3, the onset of a small-world topology is faster in (B) and (C) (note the difference in time scale). Metrics derived from surrogate random networks (solid lines) are plotted for comparison. Error bars represent the standard error of the mean, as estimated over 20 simulations.

We evaluated the effects of incorporating a memory function into the rewiring rule, therefore effectively rewiring the system towards slow time scale functional networks (Figure

We evaluated the dependence of the model on parameters by systematically varying the coupling parameter

Dependence of structural evolution on parameters

**Dependence of structural evolution on parameters**. Asymptotic values for structural clustering (A) and closeness (B), observed for a range of values of the control parameter

Correlation between structural and functional network metrics

We initially examined correlations between structural networks and averaged fast time scale functional networks. Figure

**Correlation between structural and functional network metrics**. Temporal evolution of the correlation coefficient between structural and functional participation (A), betweenness (B) and degree (C) with illustrative scatter plots (insets) at specified time instants. Functional network metrics are derived by averaging the metrics of fast time scale networks. An alternative approach, emphasizing the instantaneous expression of functional connectivity (see text) results in significantly weaker correlations (solid lines). Error bars represent the standard error of the mean, as estimated over 20 simulations.

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Correlation between structural and functional network metrics

**Correlation between structural and functional network metrics**. Temporal evolution of the correlation coefficient between structural and functional networks (A), as well between node-wise structural and functional clustering (B) and closeness (C), with illustrative scatter plots (insets) at specified time instants. Functional network metrics are derived by analyzing fast time scale functional networks and averaging the resulting metrics (see main text for details). An alternative approach, which averages the correlations of fast time scale networks, results in significantly weaker correlations (solid lines). (C) Correlation between structural and functional clustering at a single structural state, plotted against the number of sampled instantaneous functional networks. A strong correlation emerges as more networks are sampled. Error bars represent the standard error of the mean, as estimated over 20 simulations.

There also exists an alternative approach to extracting correlations from structural and functional networks. This involves exchanging the sequence of our initial analysis by firstly calculating correlations between the structural and fast time scale functional networks, and subsequently temporally averaging these correlations. This second approach emphasizes the instantaneous expression of structure-function correlations. Figure

The dynamics of central and peripheral nodes

Central and peripheral nodes manifest distinctly different dynamics (Figure

**Correlation between degree and the likelihood of link gain or loss**. Temporal evolution of the correlation coefficient between degree and link gain/loss likelihood for all nodes (A), and for central nodes only (B), defined as those nodes with participation of greater than 0.4. Error bars represent the standard error of the mean, as estimated over 20 simulations.

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Correlation between centrality, dynamics and rewiring

**Correlation between centrality, dynamics and rewiring**. Temporal evolution of the correlation between participation and Lyapunov exponent (A), fractal dimension (B), rewiring rate (C), and the likelihood of losing or gaining a link to a rewirable node (D). Scatter plots illustrate typical correlations at the asymptotic state. Participation is a measure of centrality, sensitive for nodes with connections distributed over multiple modules. Note that participation is unreliable at the early stages of evolution, given the weakly modular nature of structural networks. Error bars represent the standard error of the mean, as estimated over 20 simulations.

Continuous network plasticity gradually "mixes" individual structural metrics across the network, even though the network-wide metric averages remain invariant. Figure

Fluctuation of structural centrality metrics at the asymptotic phase

**Fluctuation of structural centrality metrics at the asymptotic phase**. (A) Node-wise autocorrelation of clustering, participation and betweenness at the asymptotic phase as a function of progressive rewiring. Error bars represent the standard error of the mean, as estimated over 20 simulations. (B-D) An illustration of the rapid fluctuations in clustering (B), participation (C) and betweenness (D). Note the shorter time scale, compared to (A). Nodes were rank-ordered by centrality (from lowest to highest) at each rewiring step. Color corresponds to the rank-ordering position at the first sampled rewiring step.

Figures

Maps of four representative nodes

**Maps of four representative nodes**. (A) Low-dimensional chaotic dynamics of a peripheral node, (B) Stochastic high-dimensional cloud of a highly participating hub, (C) An intermediate node, whose dynamics resemble the Poincaré first return map of the neural mass model in Figure 2B, (D) Contracting dynamics of a periodic-like node, perturbed by unsynchronized inputs.

Discussion

The elusive nature and role of structural and functional brain connectivity

Our construction of functional networks is based on the calculation of Euclidean distance between one-dimensional unit states (see Methods), and will necessarily generate ordered fast time scale functional connectivity, no matter how chaotic the dynamics. More importantly, however, functional networks constructed on a slower time scale likewise remain ordered (Figures

On a random structural network, synchrony is likely to be stronger between nodes with chance higher connectivity. It is probable that early in neuroanatomical development, higher connectivity strongly correlates with spatial proximity. We find that such connectivity is subsequently reinforced by activity-dependent rewiring; a process which leads to the emergence of clustered structural modules. Therefore, in our simulations, functional networks emergent on random structural networks, anticipate the asymptotic modular connectivity. Our model illustrates a potential mechanism by which brain-like structural connectivity may emerge in an unsupervised way, without a global search for optimal connectivity. It is known that a global search (testing all possible synapses) is a hard combinatorial optimization problem in a sparse network

We find that slow time scale functional connectivity strongly reflects the underlying structural connectivity, in agreement with recent reports

The present theoretical approach may also be used to interpret functional connectivity findings from empirical studies, by validating structural connectivity patterns against DTI data, and validating functional connectivity patterns against EEG or MEG data (on fast time scales) and fMRI data (on slower timescales). For example, a detailed classification of hubs in mammalian neuroanatomical networks has recently been performed

The role of noise in neural systems is currently a subject of considerable interest

We also explored the influence of slower time scale dynamics on activity-dependent rewiring, by incorporating a memory function into the rewiring rule. Such a function may represent a gradual consolidation of memories in cortical tissue. However, the use of a memory function which linearly decays with time is putatively problematic, given that the resulting slow time scale networks neglect any itinerant dynamics and consequently fail to capture the richness of instantaneous functional states (Additional file

**Relationship between fast time scale and slow time scale functional connectivity**. (A) Five consecutive iterations of spatiotemporal dynamics are shown in the top row, with the corresponding functional networks in the bottom row, ordered by the corresponding structural modular arrangement. Note the complex interplay of intra and inter-modular synchrony, reflecting a mix of segregative and integrative dynamics. (B) Dynamics and functional network obtained by calculating the correlation coefficient for the five iterations in A. The inter-modular synchrony is largely averaged at this slower time scale.

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A clear neurobiological limitation of the present study is the use of the simple unimodal map. We have provided a cursory justification for this by comparing the Poincaré first return map of a detailed neural mass model with the unimodal topology of the logistic map (Figure

Conclusion

We explicitly conceptualized the interdependent relationship between structural and functional brain connectivity, and explored the mechanisms by which this relationship may lead to the emergence of cortical-like structural networks. Our study theoretically reinforces the central role for neuronal dynamics in the emergence of complex brain connectivity. We show that functional connectivity becomes gradually more constrained by the underlying structural connectivity, as functional networks are extracted at increasingly slower time scales. The fluctuations of functional networks at faster time scales may arise from the noisy dynamics of central structural nodes.

Methods

Structural and dynamical components of the model

The model consists of an ensemble of quadratic logistic maps, coupled via a directed binary connectivity matrix. Following established neuroscientific notation, we refer to the coupling matrix as structural connectivity. Correspondingly, we refer to the correlations between dynamical states arising on structural connectivity, as patterns of functional connectivity

Formally, we represent structural connectivity with a directed binary graph _{ij }= 1; the lack of such a connection is denoted by _{ij }= 0 (with self-connections not allowed by definition). Let _{i }represent the set of neighbors (neighborhood) of node _{i }be the number of neighbors (degree) of

The set of nodes **X**; hence each node _{i}. The dynamics of the unit state at discrete time

where the control parameter

where coupling is facilitated through in-connections and _{i }(0) ≤ 1 for all _{i }∈ **X**).

The coupling parameter _{i }effectively rescales the coupling input, and may be thought to represent the mechanisms of homeostatic neuroplasticity

The control parameter

Activity-dependent rewiring rule

We used a rewiring rule which periodically modified the structural connectivity matrix towards emergent patterns of functional connectivity. For each structural network, the dynamics were iterated for 1000 iterations. Following this, a node was randomly chosen and its connections were rewired such that it gained a link to a node with which it was most synchronous, and lost a link to a neighbor with which it was least synchronous. If the most synchronous node was already a neighbor, a different node was chosen until one connection was successfully rewired. This rule exploits the fact that all nodes have identical parameter values so that the Euclidean distance |_{i }- _{j}| accurately captures pair-wise synchronization.

Formally, a node _{i }- _{N}|; that is, when _{i }was chosen, such that _{ik }= 1 and _{ij }= 0. Rewiring alternated between in and out neighbors at consecutive steps.

Extraction of functional networks

For each structural topology, "fast time scale functional networks" were extracted through computing inter-unit synchrony – measured as the Euclidean distance between instantaneous dynamical unit states. A strongest synchrony threshold was applied to convert the resulting synchrony matrices into binary networks, of the same connection density as the structural networks. For a given structural network, we extracted an ensemble of fast time scale functional networks and characterized their properties using network analysis methods. We then averaged the resulting metrics over time to obtain characteristic functional network metrics expressed on a given structural state. Hence we first extracted network metrics from fast time scale networks, and subsequently averaged these metrics. This contrasts with an alternative approach, whereby an ensemble of fast time scale networks is first averaged, and network metrics are subsequently extracted from the resulting "slow time scale networks". The two approaches are not commutative. We focused on the first approach, which emphasizes the average expression of spatiotemporal dynamics in functional connectivity, but also permits incorporating the effects of transient synchrony. Such itinerant effects are averaged out in the slow time scale functional networks.

Formally, fast time scale functional networks were constructed from **X **as _{i }- _{j}|. For a given structural network, one functional network was extracted at every tenth iteration of the dynamics, hence enabling an ensemble of 100 fast time scale functional networks for 1000 iterations. Each network was individually analyzed, and the obtained network metrics were averaged to represent the characteristic functional topology. In the initial random networks, all unit states rapidly synchronized, and the dynamics were hence iterated only while there existed a meaningful difference between states (typically for 400–500 iterations, hence enabling the extraction of 40–50 functional networks). Slow time scale functional networks were extracted by averaging 100 consecutive fast time scale functional networks.

Network analysis methods

We analyzed structural and functional connectivity properties using metrics of local and global network topology, as well as of individual node centrality. All computations were performed in Matlab (The MathWorks, Inc.), using double precision arithmetic. Our network analytic software is available to download from

The clustering coefficient for an individual node, represents the likelihood that any two neighbors of that node will themselves be neighbors

We computed the directed clustering coefficient using the method of Fagiolo

Closeness represents the average distance from one node, to all other nodes in the network

where _{ij }is the shortest path length between nodes

Small-world networks are defined as networks that are significantly more clustered than surrogate random networks (_{random }>> 1), but have approximately the same closeness as random networks (_{random }≈ 1). Surrogate random networks were generated using the degree distribution preserving algorithm of Maslov and Sneppen

Modularity describes the presence of groups of nodes (modules) which have dense intra-group connectivity, but only sparse inter-group connectivity. We subdivided the network into a set of modules

where _{uv }represents the proportion of all links in the network that connect nodes in module

Node centrality was assessed with the participation coefficient _{i }is defined as

**Correlation between participation and other structural network metrics**. Temporal evolution of the correlation coefficient between participation, and betweenness centrality (A), clustering (B), degree (C), and the number of modules interconnected by a node (D). Scatter plots illustrate typical correlations at the asymptotic state. Error bars represent the standard error of the mean, as estimated over 20 simulations.

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where _{iu }is the number of links between node

Nodal rewirability was estimated for each structural network by comparing the network with a corresponding ensemble of functional networks, emergent on that structural connectivity.

Nonlinear dynamical analysis

We characterized the temporal dynamics of individual units by computing their Lyapunov exponents and the fractal dimensions of their corresponding attractors. Taken together, these metrics indicate whether the dynamics are chaotic and low-dimensional (positive Lyapunov exponent and low fractal dimension), or alternatively, due to discordant inputs, are better characterized as high-dimensional and stochastic. Note that, given the deterministic nature of the logistic map, we use the term stochastic heuristically, to invoke the putative impact of multiple uncorrelated chaotic inputs via the coupling term.

Formally, the Lyapunov exponent for an individual unit _{i}, denoted as _{i}, quantitatively determines the average stability of the orbit of the attractor of _{i}. The Lyapunov exponent was approximated as

where

The fractal (correlation) dimension for an individual unit _{i}, denoted as _{i}, estimates the dimension of the attractor of _{i }

where _{i }was approximated by generating 1000 points of the orbit of _{i }and computing

Authors' contributions

MR, OS, CVL, MB designed research. MR, MB performed research. MR, MB analyzed the data. MR, OS, CVL, MB contributed reagents/materials/analysis tools. MR, MB wrote the paper. All authors read and approved the final manuscript.

Acknowledgements

The authors thank S. Knock, M.E.J. Newman and D. van den Berg for helpful comments. MR, OS and MB were supported by Brain NRG JSMF22002082. MR was supported by CSIRO ICT Centre top up scholarship. MB was supported by NHMRC Program Grant 510135 and ARC Thinking Systems Initiative TS 0669860.