Cross-attractor repertoire provides new perspective on structure-function relationship in the brain

Abstract

The brain exhibits complex intrinsic dynamics, i.e., spontaneously arising activity patterns without any external inputs or tasks. Such intrinsic dynamics and their alteration are thought to play crucial roles in typical as well as atypical cognitive functioning. Linking the ever-changing intrinsic dynamics to the rather static anatomy is a challenging endeavor. Dynamical systems models are important tools for understanding how structure and function are linked in the brain. Here, we provide a novel modeling framework to examine how functional connectivity depends on structural connectivity in the brain. Existing modeling frameworks typically focus on noise-driven (or stochastic) dynamics near a single attractor. Complementing existing approaches, we examine deterministic features of the distribution of attractors, in particular, how regional states are correlated across all attractors – cross-attractor coordination. We found that cross-attractor coordination between brain regions better predict human functional connectivity than noise-driven single-attractor dynamics. Importantly, cross-attractor coordination better accounts for the nonlinear dependency of functional connectivity on structural connectivity. Our findings suggest that functional connectivity patterns in the brain may reflect transitions between attractors, which impose an energy cost. The framework may be used to predict transitions and energy costs associated with experimental or clinical interventions.

1. Introduction

A fundamental goal of neuroscience is to understand how the structure of the brain constrains its function [1]. The advent of neuroimaging techniques has enabled detailed, quantitative examination of the structure-function relation, often by comparing the structural and functional connectivity between brain regions [2, 3]. The resting-state functional connectivity is of particular interest, given its relevance to a variety of cognitive functions, neurological diseases, and psychiatric disorders [3, 4]. Though structural connectivity and functional connectivity are correlated, the former does not entirely predict the latter—strong functional coupling exists between regions with only weak or indirect structural connections [2, 5]. Nonlinear dynamical models have been used to provide a mechanistic understanding of the structure-function relation [6–8] and provided many insights (e.g. [9–16]). While successful, previous dynamical system approaches often focus on dynamics near a single stable state, i.e., an attractor. However, biological systems such as the brain are often multistable [17–19], i.e., multiple attractors can coexist in the brain’s dynamical landscape. Such multistability begs the question of whether examining the overall layout of the brain’s attractor states could better inform or complement what we know about the structure-function relation than explorations around a single state.

Intrinsic brain dynamics have long been observed [20, 21] but often treated as a baseline subtracted from task-positive activities. This baseline, however, is more active than meets the eye: it consumes the largest fraction of the brain’s energy resources, while task-related consumption adds little [22]. It constrains task performance and related neural activities across multiple time scales [23–25], and contributes to alteration in neurological and psychiatric disorders [26–30]. In contrast to the restless dynamics is the (relatively) static structure—the anatomical connections between brain regions, which can be estimated non-invasively using large-scale tractography from diffusion-weighted images [31]. How can one compare the ever-changing with the unchanging? From a statistical perspective, one may compute the time-averaged features of the dynamics, such as the correlation between signals generated by two brain regions across time—a common measure of functional connectivity. Such functional connectivity patterns can be directly compared to structural ones through linear correlation [2]. From a dynamics perspective [6, 8, 32], the strength of anatomical connections can be incorporated as constant parameters in a system of differential equations, i.e. a dynamical system. The dynamical system, in turn, describes how the state of a model brain, endowed with realistic anatomy, would evolve over time. The time series generated by the model brain and the derived functional connectivity patterns can then be fitted to that of the real brain. Thus, a dynamical system naturally bridges between the unchanging structure and the ever-changing dynamics.

One popular dynamical systems modeling approach is to simulate the noise-driven dynamics near a chosen attractor, such as the low activity ground state, and compare it to the human resting brain dynamics (see [33] for a summary of different approaches). Such noise-driven exploration of a single attractor has been shown to exhibit key features of human resting brain dynamics, especially near criticality (e.g. [10, 12, 16]). On the other hand, noise-driven exploration beyond a single attractor—across multiple attractors or “ghost” attractors—has been shown to capture non-stationary resting brain dynamics and the switching between different dynamic functional connectivity patterns [11, 14, 15, 34]. The best fit to empirical data is often found near the onset of multistability [14, 34]. These observations suggest that examining the layout of the attractor repertoire over the entire multistable landscape could be crucial for understanding the organization of resting brain dynamics (c.f. [14]).

Complementing existing single-attractor approaches, the present work focuses on the deterministic features of the multistable landscape and examines their empirical relevance. Specifically, we systematically study the organization of the attractor repertoire as a window into the overall shape of the dynamic landscape. We focus on two complementary features of the attractor repertoire: (1) to what extent the states of any two brain regions are correlated across all attractors, which we refer to as cross-attractor coordination, and (2) the distribution of energy gaps between the attractors, which indicates how difficult it is to move across attractors. Mathematically, the former describes in which directions the attractors fall in line with each other in the state space, and the latter describes the spacing between attractors in a predefined direction (such a direction can represent the whole brain or a specific subnetwork). Empirically, we examine how cross-attractor coordination relates to empirically observed functional connectivity between brain regions, and how energy gaps impose a cost on the realization of specific functional connectivity patterns. For constructing the landscape, we use a Wilson-Cowan type biophysical network model that formally combines the reduced Wong-Wang model [12, 13, 35] and the Wilson-Cowan model [36, 37]. We first showed that the model exhibits extensive multistability, i.e. a large repertoire of attractors to serve as landmarks of the landscape. Further, the model also allows us to examine, computationally and analytically, how structural features across scales shape this repertoire. It is important to note that here we used a broader definition of structural features and not only include large-scale structural connectivity between brain regions, but also local recurrent connectivity within regions, and biophysical constraints at the cellular level. Using this modeling framework and a small dataset from the Human Connectome Project (HCP; n=100), we provide evidence with regards to how the cross-attractor relations in the repertoire could better capture key features of human resting functional connectivity and how such features are shaped by structural features across scales. Finally, we provide a novel framework to analyze the energy constraints for such cross-attractor coordination across different local and global structures.

2. Results

2.1. The model

Whole-brain dynamics are modeled as the mean-field activity of neuronal populations in each brain region. We use a Wilson-Cowan type model [36, 37], which is also an adapted version of the Wong-Wang model [13, 35] with a sigmoidal transfer function (equation S11). The adaptation improves the biological plausibility and multistability upon the reduced Wong-Wang model. Here, we briefly introduce the model; an extensive analysis of the numeric and mathematical properties of the model is provided in the Supplementary Materials (Section S4 for numeric results, Section S8–S9 for analytical results). Each model region contains a pair of excitatory (E) and inhibitory (I) populations, whose activity is described by the local model (Figure 1a, left box; equation 1–3) in terms of the state variables S_E and S_I. Physically, S_E and S_I are interpreted as the fraction of open synaptic channels in their respective populations, i.e. the gating variables. Through local connections (w’s), the excitatory population excites itself with strength w_EE and the inhibitory population with strength w_EI, while the inhibitory population inhibits itself with strength w_II and the excitatory population with strength w_IE. Local models further connect to each other through a global network (Figure 1a, dashed lines), giving rise to the global model (right; equation 4–6). For the global model, nodes of the large-scale network correspond to anatomical regions in the human brain based on a 66-region parcellation used in [12, 38] (Figure 1b). Edge weights of the network reflect the strength of long-range structural connectivity between the brain regions (C_ij in equation 6), either estimated using structural data from the Human Connectome Project [39, 40] (Section 4.5.1) or artificially constructed for comparison. The overall strength of long-range connections in the model brain is scaled by a global coupling parameter G (equation 6), which denotes the overall level of inter-regional interaction across the whole-brain. In similar modeling approaches, G is typically treated as a free parameter to be fitted to the empirical fMRI data (e.g., [12, 13, 16, 34]). It was thought to be an indicator of the level of arousal or wakefulness [41]. In the present work, we first examine theoretically how parameter G influences the dynamic landscape of the model (Figure 4) and then fit it to individual subjects’ fMRI data.

Figure 1: A dynamic mean-field model of the human brain.

(a) The model brain (global model) consists of a network of brain regions (local model). The local model (black box) describes the interaction between two local neural populations — one excitatory (E) and one inhibitory (I). The two populations are coupled via two excitatory connections (red; w_EE and w_EI) and two inhibitory connections (blue; w_II and w_IE). The excitatory population of each brain region can further receive input (gray arrow, I_E) from other regions via long-range structural connections (red dashed curves). (b) Nodes in the global model correspond to 66 anatomical regions of the human brain, which can be linked together by the human connectome (see text). Regions are indexed from 1 to 66 (1–33 on the right hemisphere, 34–66 on the left hemisphere in reverse order, following [12]). Specific region names are listed in Table S1.

Figure 4: Local and global structural properties jointly determine the complexity of whole-brain dynamics.

(a-c) show the bifurcation diagrams of the local model for three different types of local excitatory connectivity: (a) w_EE = 0.7 and w_EI = 0.35; (b) w_EE = 2 and w_EI = 1; (c) w_EE = 2.8 and w_EI = 1 (representatives of distinct dynamic regimes of the local model, Figure S1). Overall, local connectivity increases from (a) to (c). The activity of the excitatory population S_E is used as an order parameter, indicating the location of each attractor. The external input I_E is used as a control parameter. Each point in the diagram indicates the location of a particular fixed point. The color denotes the type of each fixed point: non-black points represent attractors, black points unstable fixed points that are not associated with a limit cycle. Horizontal stripes indicate that the attractors are changing continuously with the control parameter for a certain range. All (a)-(c) have an upper stripe and a lower stripe. (b)-(c) have an additional stripe in the middle, where the brain region oscillates. Insets of (b) and (c) show the oscillation frequency of the brain region as a function of the input current. Each stripe corresponds to a discrete level of activation for a single brain region (circled brains in b; color indicates discrete S_E levels, shown in circled legend). (d)-(f) show the corresponding bifurcation diagrams for three uniform global networks, i.e. the large-scale structural connectivity C_ij’s are identical between any two brain regions (equation 6). The average activity of all excitatory populations (${\overline{S}}_E$) is used as an order parameter and the global coupling G (equation 6) as a control parameter. Each attractor stripe corresponds to a pattern of activation over the whole brain (circled brains in (e) show $S_E^{(i)}\text{'}\text{s}$ on the left hemisphere). Similarly, (g)-(i) show the corresponding bifurcation diagrams for three realistic global networks, i.e. C_ij’s reflect the human connectome (see text for details). Here each vertical slice (gray line in h) contains the attractor repertoire of a fixed dynamic landscape shaped by the human connectome. Each attractor repertoire is associated with a matrix describing the coordination between brain regions across attractors (e.g. Figure 5b). See Figure 3 for a cartoon illustration of attractor repertoires and the associated cross-attractor coordination matrices.

In the present work, the local and global models are used in two ways: (1) to compute the repertoire of attractors using zero-finding algorithms (see Section 4.2 for details) and (2) to be numerically integrated to generate simulated brain dynamics. The former is used to characterize the overall organization of the model dynamic landscape. The latter is used to characterized local explorations of the dynamic landscape. Both aspects are compared to the human data to demonstrate the empirically relevant features. Below, we first illustrate the concept of a dynamic landscape and cross-attractor coordination using a toy example, which is followed by more realistic models to examine how structural properties across scales affect the dynamic landscape.

2.2. Multistable landscape of the brain shaped by structural properties across scales

We begin this subsection with a brief introduction to basic dynamical systems concepts necessary for understanding the results. Readers who are highly familiar with dynamical systems concepts such as attractors, phase transitions, and bifurcations should feel free to skip the following paragraph together with Figure 2 and Figure 3.

Figure 2: Conceptual illustration of single-region dynamic landscapes and a bifurcation diagram.

(a-c) Dynamics of a single brain region is here governed by a landscape, where the valleys represent the stable states, or attractors, and the peaks represent unstable states, or repellers. (d) The dynamic landscape of the region can be deformed or controlled by an external parameter (control parameter) such as the current input from other regions. The variation of the dynamic landscape with respect to the control parameter can be studied through the variation of its attractors and repellers, shown as points projected to the bottom plane. This simplified representation, which depicts how the attractors and repellers depend on the continuous changes of the control parameter, is called a bifurcation diagram (e). In the bifurcation diagram, each attractor traces out a stripe (two red lines) under the continuous change of the control parameter. At critical values of the control parameter (±1), one attractor is annihilated by the repeller and disappears. This process is referred to as a bifurcation.

Figure 3: Conceptual illustration of the whole-brain dynamic landscape, bifurcation, and phase transition.

A multistable dynamic landscape (a) contains multiple attractors, shown as troughs occupied by purple balls. Each attractor corresponds to a distinct pattern of activation over the whole brain (i-iv). Influenced by external input or intrinsic noise, the model brain may transition from its current state (attractor i, bright purple ball) to a different one (ii, iii, or iv, dim purple balls), indicated by black arrows. Structural features of a model brain can alter the shape of the landscape, causing some attractors to appear or disappear through a process mathematically named a bifurcation (a→b, a→c, or the reverse). By modifying the repertoire of attractors, bifurcation alters the set of possible transitions and the coordination between regions during transitions. For example, in landscape (a), the left and right hemisphere can be co-activated during a transition (i→iii), or activated separately through other transitions (i→ii, or i→iv). In contrast, in landscape (b), the left and right hemisphere can only be co-activated, and in (c), only activated separately. Numerically, a repertoire of attractors can be represented as a matrix, where each row represents an attractor and each column represents a brain region (repertoire matrix a, b, c, with entries shown as blue/red color blocks). The overall inter-regional coordination across attractors can be estimated by the rank correlation between the columns of the repertoire matrix. The resulted square coordination matrix summarizes how brain regions transition together over the entire landscape, serving as a signature of the landscape (coordination matrix a, b, c, shown to the right of each repertoire). In more complex landscapes (not shown), there are many more attractors, and they correspond to subtler patterns of activation (d,e; see also Figure 4). The coordination between brain regions during a transition is correspondingly more complex (f=e-d), with some regions co-activated (red) while others co-deactivated (blue).

We first introduce basic dynamical systems concepts using a toy model of a single brain region (Figure 2), and then show how inter-regional synchronization arises within this conceptual framework using a toy model of two-region coordination (Figure 3). The dynamics of a single brain region is here governed by a potential landscape (black curves in Figure 2a–c for three different landscapes). When the brain state is in the bottom of a valley (red balls), it is stable and referred to as an attractor. When the brain state is on a peak, it is unstable and referred to as an repeller (black ball). These landscapes can continuously deform into one another under the variation of a control parameter (Figure 2d), for example, the total level of current input from other brain regions. A bifurcation diagram (Figure 2d bottom plane, Figure 2e) keeps track of how the landscape changes with the control parameter via the location of attractors and repellers. At certain points, an attractor disappears by merging into a repeller, which is called a bifurcation. Now, let us consider a slightly augmented toy model that involves two interacting brain regions, i.e., the left and right hemisphere (Figure 3). An attractor in this toy model represents a stable pattern of activation over the whole brain, shown as boxed brains in Figure 3. The dynamic landscape determines a repertoire of attractors, different combinations of regional states, with possible paths of transitions between them. Figure 3a shows such a toy landscape with four attractors (i-iv), each with a different whole-brain activation map. Similar to the single-region landscape, the 3D landscape can be deformed by changes in a control parameter (such as different levels of inter-regional coupling, not shown), which may lead to the creation or destruction of specific attractors through bifurcations (e.g., Figure 3, a to c, a to b). Such creation and destruction of attractors cause a discrete change of the attractor repertoire, and the set of possible transitions. Note that here a transition refers to a change in state, where as a bifurcation refers to a change in the attractor repertoire. A bifurcation can trigger a transition if the system was in a state/attractor that is destabilized during the bifurcation; given sufficient noise, a spontaneous transition can occur without a bifurcation. Further, changes in the attractor repertoire can alter how the brain regions coordinate with each other. Two example bifurcations are shown in Figure 3b–c. In Figure 3b, only two out of four attractors are left, such that the brain can now only transition between attractor (i) and (iii), thereby leading both hemispheres to be in sync (on or off together). This coordination of brain regions (or hemispheres) can be captured by estimating the cross-attractor coordination matrix (shown in Figure 3). Similarly, in Figure 3c, three out of four attractors are left after bifurcation, leading to more complex coordination between brain regions (or hemispheres). Figure 3d–f presents a more complex example, where a finer parcellation is used and each brain region can now take three activation values (instead of just on or off), resulting in more complex spatial activation patterns across the whole brain as well as complex coordination between brain regions. See Section 4.3 for technical details about how the coordination matrix is estimated.

Next, we present a set of more realistic examples to show how local as well as global structural connectivity of the brain can shape the dynamic landscape, its associated repertoire of attractors and their transitions. Here, we depict dynamic landscapes and their changes as bifurcation diagrams (Figure 4; see Section 4.2 for computational details). Figure 4 shows nine different bifurcation diagrams, across its three rows and columns. The rows correspond to bifurcation diagrams from: (first row: a-c) a single brain region (local model); (second row: d-f) the entire brain with uniform connectivity across all brain regions; and (third row: g-i) the entire brain with realistic connectivity across all brain regions. Thus, the second and third rows of Figure 4 aims to depict the effect of changes in global structural connectivity on the dynamical landscape. The columns, on the other hand, in Figure 4 aims to depict the effect of changes in local connectivity, i.e., level of excitation within the individual brain regions, on the dynamic landscape.

In each bifurcation diagram, the y-coordinate of each colored point indicates the position of an attractor: here we use the average activity of all excitatory populations ${\overline{S}}_E$ (Figure 4d–i; black points are repellers). The x-coordinate indicates the value of a control parameter, which modulates the shape of the underlying dynamic landscape: here we use the overall strength of long-range connections—the global coupling G (equation 6). Further, each vertical slice of a bifurcation diagram contains the repertoire of attractors and repellers in a fixed landscape (an example slice is shown in Figure 4h), corresponding to stable and unstable patterns of brain activity respectively. Lastly, an attractor traces out a horizontal “stripe” as it changes continuously with the landscape (examples shown in Figure 4d–i). A colored stripe merges with a black stripe at a bifurcation, where an attractor is annihilated by a repeller. The number of colored stripes indicates the complexity of the landscapes, i.e., more stripes indicate more attractors.

Locally wihtin each model brain region, the dynamics are controlled by local structural connectivity (w’s in Figure 1a; see Section S4 for detailed numeric results and Section S8 for analytical results). In particular, a single model region can switch between a rich set of dynamic regimes by varying the excitatory-to-excitatory connections (w_EE) and the excitatory-to-inhibitory connections (w_EI; Figure S1). Figure 4a–c show the bifurcation diagrams for three local connectivity settings from three distinct dynamic regimes (regime e, d, and a respectively in Figure S1). With an overall increase of local excitatory connectivity from (a) to (c), a single region becomes more complex—more attractors and stronger oscillatory activities.

To understand the effects of global connectivity, we first examined the brain dynamical landscape where all regions are uniformly connected with each other (Figure 4d–f). Here, stronger local excitatory connections (e,f) produce a more complex landscape (3 attractor stripes) than weak ones (d; 2 attractor stripes; see Figure S5 for even weaker local connections). These bifurcation diagrams are very similar to those of a single brain region (Figure 4a–c), in terms of the number of attractors and the presence of oscillation. In fact, the whole brain (Figure 4e) moves up and down together between discrete states of activation, very much like a single region (Figure 4b). Note that, for the global model (equation 4–6) to be multistable, a minimal amount of global coupling is required, i.e., G > 1 (Figure 4d–f). If the brain regions act independently (G = 0), both the individual regions (Figure 4a–c at I_E = 0) and the whole brain (Figure 4d–f at G = 0) are monostable—there is only one stable pattern of activity, where the gating variables are all close to zero. This result indicates that a functionally complex brain can emerge out of the synergistic interaction between simple regions. Additional analytical and numerical results are provided in Section S9 (Multistability) for further validation and generalization.

Next, we show that in addition to the global coupling G, the details of inter-regional connections matter too (C_ij in equation 6). Given a realistic global structural connectivity (human connectome; Figure 4g–i), the complexity of the whole-brain dynamic landscape increases dramatically: 171 attractor stripes in (g), 610 in(h), and 682 in (i) (per single-linkage clustering). Correspondingly, the patterns of activation (Figure 4h) are more complex, with greater differentiation between regions; the coordination between brain regions across attractors is consequently more flexible and subtle (as depicted in Figure 3f). The heterogeneous nature of the human connectome breaks the large-scale spatial symmetry of the model brain, creating more functional differentiation between brain regions and greater functional complexity for the whole brain. In short, the complexity of the global dynamical landscape is a joint product of strong local excitatory connection and complex topology of the large-scale network. See Section S9 for additional analytical supports.

2.3. Cross-attractor coordination reveals large-scale symmetry of human brain functional connectivity

In this section, using data from the Human Connectome Project (HCP) [39] we present both qualitative and quantitative results to show how cross-attractor coordination could better relate with key features of human resting functional connectivity than noise-driven within-attractor coordination. For qualitative results, we used averaged structural and functional connectivity across subjects from a smaller HCP cohort (n=11 individuals), whereas for the quantitative results, we used individual structural and functional connectivity estimates across all 100 unrelated individuals from the HCP data.

Typically, human functional connectivity (FC) is calculated from the fMRI data by estimating co-fluctuations across brain regions. The estimated resting state FC matrix usually reflects large-scale symmetry across the two hemispheres, such that brain regions across the two hemispheres highly co-fluctuate (i.e., high antidiagonal values in Figure 5a). For qualitative results, we used averaged structural and functional connectivity matrices across a small subset of the HCP n=11 unrelated individuals (see Section 4.5 for more details). Using averaged structural connectivity matrix, a dynamical landscape (Figure 4h) was generated and cross-attractor coordination was estimated for a chosen G = 2.2. At the selected G, 97 attractors were found. Mathematically, each attractor is represented by a row vector denoting the activity level of all brain regions (equation 8–9). Thus, for a chosen G, using the (#attractors × #regions) matrix we estimated the cross-attractor coordination between regions (equation 10), such that regions that co-fluctuate across attractors tend to show high cross-attractor coordination (or similarity). See Section 4.2 for mathematical details and Figure 3a–c for an intuition regarding the estimation of cross-attractor coordination matrix. As shown in Figure 5b, the dominant feature of human resting state FC, i.e., large-scale symmetry across brain regions, is well preserved in the attractor cross-coordination matrix. Such inter-hemispheric symmetry is not seen in stochastic within-attractor coordination (Figure 5d). Moreover, the pattern of within-attractor coordination is a closer reflection of the human structural connectivity (Figure 5c) than the human functional connectivity (Figure 5a).

Figure 5: Cross-attractor coordination captures large-scale symmetry of human functional connectivity and its nonlinear dependency on structure better than within-attractor coordination.

An example of human functional connectivity matrix (a) is calculated using the resting fMRI data from the Human Connectome Project [39], averaged over 11 unrelated subjects. The average structural connectivity of the same subjects is shown in (c). Regions (columns and rows) are ordered symmetrically for the left and right hemispheres (see Figure 1b) to reveal the large-scale symmetry of resting brain dynamics (ordering follows [12]; see the complete list of region names in Table S1). White dashed lines delineate the matrix (a) into four blocks, describing the functional connectivity within the right hemisphere (upper left block), within the left hemisphere (lower right), and between two hemispheres (lower left/upper right). Functional connectivity patterns within the hemispheres are similar to each other and similar to inter-hemispheric connectivity patterns. The symmetry between intra- and inter-hemispheric connectivity is well captured by inter-regional coordination in the model brain across attractors (b) (w_EE = 2, w_EI = 1, G = 2.22 for local maximum model-human similarity; c.f. Figure 4h gray slice). Such symmetry is not captured by the coordination within any of the said attractors (d; best fit within-attractor coordination matrix). When examined across individuals (using n=100 unrelated HCP participants), a significantly better model-human similarity was obtained when cross-attractor coordination was used instead of within-attractor coordination (e); the difference between cross-attractor coordination and within-attractor coordination with respect to model-human similarity is even greater when partial correlation is used to control for the contribution of structural connectivity (f). Comparing intra-hemisphere and inter-hemisphere functional connectivity separately (g), we found that cross-attractor coordination captures human intra- and inter-hemisphere functional connectivity equally well (red bars), while within-attractor coordination is better at capturing intra-hemisphere than inter-hemisphere functional connectivity (blue bars). The distributions of correlation coefficients were obtained through a model fitting procedure with the same local parameters (w_EE = 2, w_EI = 1) while allowing the G parameter to vary from 1.7 to 3.0 by steps of 0.1.

Next, to examine the model fit on an individual basis, we ran a quantitative analysis using the n=100 unrelated-individuals cohort of the HCP data [39]. Our results confirm that cross-attractor coordination can better predict human functional connectivity than within-attractor coordination. The maximum model-human correlation for cross-attractor coordination has an average Spearman’s Rho of 0.50 (S.D. 0.09), while the correlation for within-attractor coordination has an average of 0.34 (S.D. 0.06) (Figure 5d). That is, on average, cross-attractor coordination explains 25.0% of the variance in human FC, while within-attractor coordination explains 11.6% of the variance in human FC. Based on within-subject paired t-test, cross-attractor coordination provided significantly better fit of FC (t(99) = 20.2, p < 10⁻³⁶; Figure 5e). To understand how the two types of models relate to the underlying structural connectivity, we calculate the partial correlation between model coordination matrices and human functional connectivity, controlling for the linear contribution of structural connectivity. With the contribution of structural connectivity controlled, the model-human correlation for cross-attractor coordination has an average Spearman’s Rho of 0.39 (S.D. 0.10), which is significantly greater than that of the within-attractor coordination 0.17 (S.D. 0.07) (Figure 5f; t(99) = 21.9, p < 10⁻³⁹). That is, excluding the contribution of structural connectivity, cross-attractor coordination explains 15.2% of the variance in human FC, while within-attractor coordination explains 2.9% of the variance in human FC. 40% of the explanatory power of cross-attractor coordination comes from structural connectivity, while 75% of the explanatory power of within-attractor coordination comes from structural connectivity. Furthermore, the difference between the partial correlation coefficients for the cross-attractor coordination and the within-attractor coordination (0.22 ± 0.10) is significantly greater than that of the regular correlation coefficients (0.16 ± 0.08) with t(99) = 9.5 and p < 10⁻¹⁴ (Figure 5f vs. Figure 5e), which suggests that within-attractor coordination is more linearly dependent on the structural connectivity. Finally, we show that the model-human similarity for inter-hemisphere coordination is significantly lower than that of intra-hemisphere coordination for within-attractor coordination but not for cross-attractor coordination (Figure 5g). In other words, unlike within-attractor coordination, cross-attractor coordination captures human intra- and inter-hemisphere functional connectivity equally well.

To understand how variability in individual model parameters translates into measures of behavior, individual parameters were correlated with a measure of fluid intelligence, an abbreviated version of the Raven’s Progressive Matrix Test (PMAT). Correlating the model parameters with measures of fluid intelligence showed one significant result. Specifically, the strength of the model-human similarity and the number of correct responses on the test was significantly correlated (Spearman’s Rho = 0.21, p = 0.04) after controlling for age and sex. All other correlations with fluid intelligence measures were not significant.

2.4. Structural connectivity defines the energy demands of cross-attractor coordination

In the above section, we computed the cross-attractor coordination matrices (equation 10 in Section 4.3), which only concerns whether two brain regions move up and down together across the dynamic landscape, but not how difficult or metabolically expensive such movements are. In this section, we examine the “energy gaps” between the attractors, and how they are shaped by local and structural properties of the model. Figure 6a gives a conceptual illustration of the relation between attractors and the energy gaps between them. Each attractor is associated with an average level of activity or energy (${\overline{S}}_E$; equation 11) given a fixed parameter G. There is an energy gap between each pair of adjacent attractors (equation 12; see a full technical description in Section 4.3). In the subject-average model (n=11, HCP; Figure 4g–i), the average and maximum energy gap clearly vary with local connectivity (w_EE, w_EI) and global connectivity (G), here summarized in Figure 6b. Quantitatively, stronger local connections reduce the energy gaps (Figure 6c,d), and stronger global connections (G) increase the energy gaps (Figure 6b)—local and global structural connectivity pull the energy cost in different directions.

Figure 6: Local (w_EE, w_EI) and global (G) structural connectivity jointly shape the energy cost of cross-attractor coordination.

(a) gives a conceptual illustration of the relation between attractors and energy gaps with a toy bifurcation diagram. The average and maximum gap sizes are of our interest. (b-d) summarize the size of energy gaps between attractors shown in Figure 4g–i (subject-average model with n=11). (b) Overall, the maximum (solid lines) and average energy gaps (dashed lines) increase with global coupling G, though there is a transient decrease when the maximum energy gap is less than 0.2. Both types of gaps decrease with increasing local connectivity w_EE, w_EI (c,d). (*** p<0.001 with Bonferroni correction. Error bars are standard errors.)

Next, we examine the effect of energy constraints on model fit. Without energy constraints, the computation of the cross-attractor coordination assumes that the brain could traverse arbitrarily large energy gaps, while in reality, crossing very large energy gaps may be unrealistic. To incorporate the effect of energy constraints (see Section 4.3 for detailed methods), we split an ordered attractor repertoire into sub-repertoires between which the energy gap is considered too high. Thus, we obtain a sub-repertoire above the maximum energy gap, equation 13, and a sub-repertoire below the maximum energy gap, equation 14. Cross-attractor coordination matrices computed within the sub-repertoires can be considered as energy-constrained coordination patterns between brain regions. Figure 7a–c shows that such energy-constrained cross-attractor coordination (dashed lines) is more sensitive to different structural features in its ability to capture human functional connectivity. The energy constraint inflicts a greater loss of model-human similarity when the local structural connectivity is weak (area of the shaded region shrinks from Figure 7a to c, and bars in d decreases with strong local connectivity from left to right) and the global structural connectivity is strong (height of shaded regions grows with G in Figure 7a–c). The loss of similarity grows with the maximum gap size (Figure 7d; ρ = 0.96, p<10⁻¹⁰⁰ for w_EE = 0.7 and w_EI = 0.35; ρ = 0.92, p<10⁻¹⁰⁰ for w_EE = 2 and w_EI = 1; ρ = 0.85, p<10⁻¹⁰⁰ for w_EE = 2.8 and w_EI = 1). This indicates that models fitted with a high value of G may implicate an unrealistic level of high-energy-cost transitions. When cross-attractor coordination matrices were fitted to n=100 unrelated HCP subjects (Figure 5e), the optimal G is 2.5 ± 0.28 (Figure 7e). The corresponding maximum energy gaps average to 0.12 ± 0.08 (Figure 7f), and the corresponding mean energy gaps average to 0.009 ± 0.015 (Figure 7f). Thus, the optimal cross-attractor models in the present study are located in a realistic regime, which is least affected by energy constraints (Figure 7d, max energy gap<0.2).

Figure 7: The loss of model-human similarity due to energy constraints depends on local and global structural features.

(a-c) The model-human similarity for cross-attractor coordination (no energy constraint, black solid lines) is stable with respect to varying global coupling G and local excitatory connectivity w_EE and w_EI (a: w_EE = 0.7 and w_EI = 0.35, b: w_EE = 2 and w_EI = 1, c: w_EE = 2.8 and w_EI = 1). Dashed lines indicate the model-human similarity when the model is energy-constrained, i.e., it does not cross the maximum energy gap between attractors. This “energy-constrained” similarity is computed by splitting the attractor repertoire into two subrepertoires, one above the maximum energy gap and one below it (c.f. Figure 6a). The cross-attractor coordination within each subrepertoire is compared to the human functional connectivity through Spearman’s correlation. The dashed lines indicate the greater correlation coefficient (ρ) among the two subrepertoires. The shaded area (Δρ) indicates the loss of model-human similarity due to the energy constraint. (d) Each point in the scatter plot represents the percent loss of model-human similarity for not crossing the maximum energy gap, given a specific combination of global coupling G and local connectivity w_EE and w_EI. Overall, the loss increases with the maximum gap size, which in turn depends on G (Figure 6b). The average loss (bars in d) decreases with increasing local excitatory connectivity (w_EE, w_EI). When the cross-attractor model was fitted to individual subjects (using n=100 unrelated HCP participants; c.f. Figure 5e), the optimal global connectivity G (e) and the corresponding maximum energy gap (f) and mean energy gap (g) are low, where the cross-attractor coordination is least affected by energy constraints. (*** p<0.001 with Bonferroni correction. Error bars are standard errors.)

3. Discussion

The present work examines how the brain’s multistable dynamic landscape can be shaped by structural features across scales and what features of the landscape are relevant to empirical observations. Complementing the previous stochastic noise-driven exploration approach, the present work focuses on the deterministic features of the multistable landscape and examines their empirical relevance. We demonstrate that large-scale symmetries of human functional connectivity patterns and their nonlinear dependency on the structure could be better explained by the relation between attractors in the landscape than the property of any individual attractor. Thus, the present work offers a novel cross-attractor perspective on resting brain dynamics, equipped with a computational framework to produce empirical relevant summaries of the attractor repertoire in full as well as in parts.

The functional complexity of the model brain is controlled by both local and global structural connectivity. At the level of a single isolated brain region, the dynamic repertoire can be effectively controlled by two key local structural properties: local excitatory-to-excitatory connectivity (self-excitation) and local excitatory-to-inhibitory connectivity. In the real brain, local excitatory-to-excitatory connections are particularly abundant [42], and in the model brain, they contribute indispensably to multistability (Section S8). Multistability is a key source of biological complexity from molecular to social levels [18, 19], often tied to self-excitation or positive feedback [43–45]. Manipulating the model’s local excitatory-to-excitatory connections have physical implications. The recurrent excitatory synaptic properties of the present model is determined by the conductance of the N-methyl-D-aspartate (NMDA) receptors [35]. Thus, modulating excitatory-to-excitatory connectivity in the model can be interpreted empirically as modulating the conductance of NMDA receptors in local neuronal populations, using, for example, pharmacological and endogenous antagonists and agonists such as ketamine [46] and dopamine [47]. Such manipulations have been theoretically predicted and shown to affect memory capacity [48–50]. Note that the strength of local excitatory-to-excitatory connections needs to surpass a critical value to induce the transition from a monostable to a multistable regime (equation S30). In the present work, this critical value is a constant (equation S29), which depends on cellular-level properties such as the membrane time constant and the gain of the input-output response. Thus, manipulating such microscopic properties can induce, or remove, multistability from a single brain region.

At the large-scale network level, multistability can be created or amplified by the synergistic interaction between mono- or multi-stable brain regions. Different large-scale network structures have dramatically different capabilities at amplifying local complexity: a realistic global network is much more powerful than a uniform one. The human connectome breaks the spatial symmetry of the global model, whereas symmetry breaking is often a key to complex dynamics [51–56]. On the other hand, the human connectome is endowed with more specific features such as modularity, small-worldness, and multiscale characteristics [1, 57–59]. A systematic study of how these features alter the geometry of the global dynamic landscape is worthy of further theoretical investigation (see Section S9).

Within the multistable landscape sculpted by the human connectome, inter-regional coordination across attractors exhibits key features of human functional connectivity patterns. Such cross-attractor coordination better captures human functional connectivity than within-attractor coordination—synchronization between brain regions within the same basin of attraction. This finding raises the possibility that functional connectivity patterns reflect transitions between stable brain states more than the brain states themselves.

A transition-based, or cross-attractor, view on functional connectivity has several theoretical and empirical implications. First, it provides an explanation for the large-scale symmetry of human FC, i.e., the similarity between intra- and inter-hemispheric connectivity patterns. It has been noted that within-attractor dynamics of similar models lack such symmetry, exhibiting weak inter-hemispheric FC [12, 16]. The weak inter-hemispheric FC has been attributed to an underestimation of structural connections across hemispheres using diffusion-weighted imaging. This explanation is reasonable given that within-attractor dynamics can be approximated by a linear dynamical system, which closely depends on the structural connectivity [12, 13]. Nevertheless, an alternative explanation could be that human FC implicates far-from-equilibrium dynamics where the nonlinearity cannot be ignored [15]. It is a signature of nonlinear systems that a small input does not necessarily produce a small effect. Indeed, strong functional connectivity in humans is known to exist between regions that are not directly connected [5]. As we have shown, cross-attractor coordination takes into account such nonlinear effects. Mathematically, our cross-attractor approach amounts to studying the relation between the zeros of a nonlinear function—a problem that does not admit a linear approximation. The symmetry between intra- and inter-hemispheric connectivity is likely to reflect a symmetry of the set of all zeros, i.e. the (roughly) invariance of the zero set under the exchange of variable indices between the homologous regions of the left and right hemispheres. The invariance of the zero set is, in turn, consequent to the invariance of the differential equations under such a left-right reflection. A full mathematical treatment of the problem is beyond the scope of the present study. Nevertheless, it invites theoretical investigations of the symmetry groups of nonlinear neural dynamical models (c.f. [53]).

The second implication is that the cross-attractor view is compatible with treating functional connectivity as both stationary and dynamical. Cross-attractor coordination, when measured over the entire dynamic landscape, is itself time-invariant. Empirically observed stability and convergence of human functional connectivity [60, 61] may reflect this invariance of the underlying landscape. The static landscape can also support dynamic functional connectivity (dFC) [15, 62]. At any given time, the possible transitions depend on the attractor currently dwelled upon. Thus, in a short time window, cross-attractor coordination is confined to a subset of attractors. In this perspective, dFC reflects the transitions between attractors in a subset of the repertoire. As a consequence, a state-based and a dFC-based representation of neural dynamics may diverge—subsets of attractors that are close in the state space may have distinct patterns of transitions, and subsets of attractors that are far apart in the state space may have similar patterns of transitions. In other words, precaution may be used when treating dFC patterns as brain states.

Finally, the cross-attractor view attaches the concept of energy costs to functional connectivity patterns. In this perspective, the potential for exhibiting, say, normal resting-state functional connectivity patterns may always be there, but the energy costs modulate the difficulty for such potential to be realized. It is especially interesting to consider the potential “costs” of functional connectivity patterns in relation to cognitive functions and psychiatric disorders. The energy gaps in the present framework can be calculated within a subnetwork relevant to a specific cognitive function. An individualized fitting of the model for a subnetwork may help predict how difficult it is for a subject to perform a specific type of tasks. Further, examining the energy gap distributions in people with psychiatric disorders may help elucidate the network mechanisms underlying the pathology. The neural and behavioral correlates of the energy gaps could be further explored. It is reasonable to assume that a greater energy gap between two attractors is associated with a lower probability of transition and thus a lower probability of observing the corresponding cross-attractor coordination. The cross-attractor coordination over pairs of, rather the whole repertoire of, attractors maps may be compared to dynamic functional connectivity in empirical analysis. Extending the present framework to the study of dynamic functional connectivity is a worthy future direction. The use of simultaneous PET-fMRI may provide more direct evidence for metabolic measures. To further examine how the energy gaps may affect cognition, causal manipulations such as non-invasive brain stimulation (e.g., transcranial magnetic stimulation) or pharmacological interventions (e.g., Methylphenidate) are required.

Like all scientific studies, the present work has several limitations. First, although cross-attractor coordination better fits human data than within-attractor coordination, a large portion of the variance in the human FC remains unaccounted for. Incorporating additional individualized model parameters may further improve the model fit: for example, an estimate of the local recurrent connectivity for each brain region and long-range inhibitory connectivity between brain regions. Nevertheless, there may be a fundamental limit to what biologically realistic parameters may be estimated from non-invasive measures of the human brain. Second, the present study has not explored whether increasing the dimensionality of the model will lead to a better model fit. Two obvious ways to increase the dimensionality of the model is to increase the number of local excitatory and inhibitory populations included in each model brain region [14], and/or to use a finer parcellation of the whole brain [16]. Both approaches would likely increase the number of attractors and repellers of the model, providing a more refined estimate of cross-attractor coordination. In the present study, the choice of parcellation and the two-population local model was primarily based on the convention adopted by existing modeling studies using the predecessors of our present model [13, 16, 63]. Identifying a precise scaling law of the number of attractors as a function of the number of parcels and the number of local populations is beyond the scope of the present study but is worthy of further investigation. Future research should also address what minimal complexity/dimensionality of the model is required to achieve a near-optimal fit. Third, the present work only focuses on resting-state fMRI data in a neural typical population. To understand brain functional connectivity in a broader context, the computational framework can be extended to study the changes in functional connectivity driven by external stimulation or due to psychiatric disorders. Future experimental validation of the present framework could utilize fMRI recorded from subjects under non-invasive brain stimulation such as transcranial direct current stimulation (tDCS), transcranial alternating current stimulation (tACS), or transcranial magnetic stimulation (TMS). For a most direct example, tDCS generates a static electric field over the brain, creating different levels of current flow in different brain regions. Simulated regional currents can be incorporated into the model as constants I_E and I_I in equation 1–2. As non-invasive brain stimulation is increasingly used as a treatment for psychiatric disorders such as major depressive disorder [64–66], a modeling framework based on dynamical systems principles may play an important role in the future improvement of treatment designs [67]. Fourth, the present study did not fully explore how features of the model dynamic landscape can help predict behavioral traits or performances of the subjects. One may expect the model to yield better predictions for tasks or behavior that involve interhemispheric coordination. Examples of such tasks include bimanual motor coordination [68, 69] and language processing in involving the non-dominant (right) hemisphere [70]. Future research may explore the correlation between, for example, energy gaps for inter-hemispheric coordination between relevant brain regions with subjects’ motor and language skills.

In summary, the present work examines intrinsic brain dynamics in terms of an underlying landscape and the repertoire of stable activity patterns it affords. Model-based analyses reveal that empirically observed functional connectivity patterns may reflect transitions between activity patterns more than the patterns themselves. The work outlines a modeling framework that emphasizes the relation between stable activity patterns. It is thus suitable for examining systemic changes in the brain that result in interrelated improvement or impairment in multiple cognitive and affective functions, such as in development and psychiatric disorders.

4. Materials and Methods

4.1. The present model

The local model is described by the equations,

$$\frac{\text{d\hspace{0.17em}}{S}_E}{ \text{d\hspace{0.17em}}t}=-\frac{S_E}{{\tau }_E}+\left(1-S_E\right){\gamma }_E{H}_E\left(w_{EE}{S}_E-w_{IE}{S}_I+I_E\right)$$ (1)

$$\frac{\text{d\hspace{0.17em}}{S}_I}{ \text{d\hspace{0.17em}}t}=-\frac{S_I}{{\tau }_I}+\left(1-S_I\right){\gamma }_I{H}_I\left(w_{EI}{S}_E-w_{II}{S}_I+I_I\right).$$ (2)

S_E and S_I are the gating variables of the excitatory and inhibitory population respectively (Figure 1a), which indicate the fraction of open channels in each population. The activity of each population has a natural decay time of τ_E or τ_I, which are the time constants of the NMDA and GABA receptors (Table S2) respectively. The model is a mean-field reduction of a network of leaky-integrate-and-fire neurons, where the slow integration time of the NMDA receptors dominates over the fast dynamics of the AMPA receptors [35] (see also Section S3). Each population’s activity tends to increase with the fraction of closed channels (1 − S_p) and the population firing rate (H_p), scaled by a factor γ_p for p ∈ {E, I}. This is described by the second term on the right-hand-side of equation 1–2. H_E and H_I are transfer functions that map synaptic current input to population firing rate of the excitatory and the inhibitory population respectively. In particular, they are sigmoidal functions of the form

$$H_p(x)=\frac{r_{max}+\frac{a_{p}x-b_p-r_{max}}{1-e^{d_p\left(a_{p}x-b_p-r_{max}\right)}}}{1-e^{-d_p\left(a_{p}x-b_p\right)}},$$ (3)

whose output increases with input monotonically and saturates at r_max—the maximum firing rate limited by the absolute refractory period of neurons (around 2 ms in certain cell types [71, 72]). The specific shape of each transfer function is determined by three additional parameters a_p, b_p and d_p (a_p and b_p determine the location and slope of the near-linear segment in the middle; d_p determines the smoothness of the corners bordering the said near-linear segment). This transfer function is converted from Wong and Wang’s original formulation [35, 73] (a soft rectifier function, equation S6) into a sigmoidal form, while retaining the original value of parameters a_p, b_p, and d_p (shown in Table 1). As a result, the present model is a Wilson-Cowan type model [36, 37]. The parameters were chosen to approximate the average response of a population of spiking pyramidal cells (p = E) and interneurons (p = I) respectively, incorporating physiologically plausible parameters [35, 74]. Interaction between local populations is modulated by four coupling parameters w_pq ⩾ 0 in equation 1–2, indicating the influence from the local population p to q, where p, q ∈ {E, I} (Figure 1 left box). These coupling parameters reflect the local structural connectivity. The local populations are also capable of responding to external current inputs denoted as I_E and I_I in equation 1–2, respectively. Importantly, such input can come from other brain regions in a globally connected network (Figure 1 right panel, dashed lines). This leads us to the global model. Formally, we substitute I_E in the local model (equation 1) with a global input I_G (equation 4),

$$\frac{\text{d\hspace{0.17em}}{S}_E^{(i)}}{\text{d\hspace{0.17em}}t}=-\frac{S_E^{(i)}}{{\tau }_E}+\left(1-S_E^{(i)}\right){\gamma }_E{H}_E\left(w_{EE}^{(i)}{S}_E^{(i)}-w_{IE}^{(i)}{S}_I^{(i)}+I_G^{(i)}\left({\overrightarrow{S}}_E\right)\right)+\sigma {\xi }_E^{(i)}(t)$$ (4)

$$\frac{\text{d\hspace{0.17em}}{S}_I^{(i)}}{\text{d\hspace{0.17em}}t}=-\frac{S_I^{(i)}}{{\tau }_I}+\left(1-S_I^{(i)}\right){\gamma }_I{H}_I\left(w_{EI}^{(i)}{S}_E^{(i)}-w_{II}^{(i)}{S}_I^{(i)}+I_I\right)+\sigma {\xi }_I^{(i)}(t)$$ (5)

where $S_E^{(i)}$ and $S_I^{(i)}$ are the synaptic gating variable of the excitatory and the inhibitory population of the i^th brain region respectively, and ${\xi }_{•}^{(i)}$ is a noise term scaled to an amplitude σ. The state of all excitatory populations is denoted as a vector ${\overrightarrow{S}}_E$, the i^th element of which is $S_E^{(i)}$. The global input to the i^th brain region depends on both its connectivity with, and the ongoing state of, other brain regions,

$$I_G^{(i)}\left({\overrightarrow{S}}_E\right)=G\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^N{C}_{ij}{S}_E^{(j)}$$ (6)

where N denotes the total number of brain areas, C_ij > 0 the long-range structural connectivity from the j^th to the i^th brain region and G is a global coupling parameter that controls the overall level of interaction across brain regions. Since C_ij is only intended to represent long-range connectivity, we let C_ij = 0 for any i = j to preclude recurrent connections. For the effects of G and C_ij to be independently comparable, here we impose a normalization condition on the matrix norm,

$$\Vert \text{C}{\Vert }_{\infty }=\underset{i}{\text{max}}\left(\sum _{j=1}^N\left|C_{ij}\right|\right)\equiv 1.$$ (7)

Table 1: The interpretation and value of model parameters.

Here we summarize the parameters used in equation 1–5. Most parameters assume a fixed value, which was introduced by [35]. A “~” indicates that this parameter is manipulated in the present study to explore the behavior of the model or to fit the model to empirical data. For local connectivity, three pairs of values were used in the present study for w_EE and w_EI, which are (0.7, 0.35), (2, 1), and (2.8, 1). These parameters were chosen from three different regimes of the local model (regime e, d, a in Figure S1–S2), which are dominated by stable fixed points, damped oscillations, and sustained oscillations respectively. w_IE matches w_EE to provide feedback inhibition [13]. These values fall within a range comparable to that of existing studies [13, 16]. The constant excitatory input I_E to the local model ranges between 0 and 1 to demonstrate how the dynamics of a single brain region can be altered by inputs from other regions (Figure 4). When modeling resting brain dynamics using the global model I_E is zero, assuming no external input to the brain. G ranges between 0 and 5, used to explore the effect of global coupling on multistability (Figure 4) and to fit the model to empirical data. The range of G used was consistent with that of existing studies [13, 16]. C_ij is defined by each subject’s structural connectome. The noise amplitude σ is set to zero for verifying the classification of fixed points, is set to 0.01 when calculating within-attractor coordination, following [13].

parameter	interpretation	value
τ E	decay time of NMDA receptor	0.1 (s)
τ I	decay time of GABA receptor	0.01 (s)
γ E	kinetic parameter of excitatory population	0.641
γ I	kinetic parameter of inhibitory population	1
a E	parameter of HE	310 (nC−1)
b E	parameter of HE	125 (Hz)
d E	parameter of HE	0.16 (s)
a I	parameter of HI	615 (nC−1)
b I	parameter of HI	177 (Hz)
d I	parameter of HI	0.087 (s)
r max	maximum firing rate	500 (Hz)
w EE	excitatory-to-excitatory coupling	~ (nA)
w EI	excitatory-to-inhibitory coupling	~ (nA)
w IE	inhibitory-to-excitatory coupling	~ (nA)
w II	inhibitory-to-inhibitory coupling	0.05 (nA)
I E	external input to excitatory population	~ (nA)
I I	external input to inhibitory population	0.1 (nA)
G	global coupling	~ (nA)
C ij	structural connectivity between brain regions	~
σ	noise amplitude	~

Since the global coupling parameter G modulates the level of input to each brain region, one would expect it to have comparable influence on the local dynamics as I_E in the local model (equation 1).

4.2. Computation of attractors and bifurcation diagrams

The repertoire of attractors and bifurcation diagrams (Figure 4) are computed in MATLAB, utilizing the build-in function fsolve. Given a proper initial guess, fsolve finds the coordinates of a nearby fixed point of the dynamical system (e.g., the local model, equation 1–2, or global model, equation 4–5) and calculates the corresponding Jacobian matrix. Given N model brain regions, the spectrum ${\left\{{\lambda }_k\right\}}_{k=1}^{2N}$ of the Jacobian matrix is used to classify the fixed points and identify which ones are attractors. The fixed point is a stable equilibrium if λ_k is real and negative for all k. The fixed point is associated with damped oscillation if Re λ_k < 0 for all k and Im λ_k ≠ 0 for some k. The fixed point is associated with a limit cycle if Re λ_k > 0 and Im λ_k ≠ 0 for some k with the additional criteria that after a small perturbation from the fixed point, the time-average of the solution remains close to the fixed point. The above three types of fixed points—a stable equilibrium, a stable spiral (damped oscillation), a fixed point associated with a limit cycle (sustained oscillation)—represent attractors in the present study. All other types of fixed points are classified as unstable. For damped oscillation and limit cycles in the local model, the frequency of the oscillation (Figure S1) is defined as | Im λ_k |/(2π).

For the local model, a 2D dynamical system, the complete characterization of all fixed points is relatively easy by searching exhaustively through a grid of initial guesses (as for Figure 4a–c). This approach becomes unfeasible when it comes to the global model due to the high dimensionality. Thus, for the global model (Figure 4d–i), we implemented a recursive search: for each value of G, (1) find zeros of equation 4–6 using fsolve given a set of initial guesses that includes, if any, the zeros for G − δG (δG = 0.01 for the present study) in addition to a fixed set of grid points; (2) sort the list of zeros obtained from (1) by the average of $S_E^{(i)}’\text{s}$ denoted as ${\overline{S}}_E$; (3) use the middle points between consecutive zeros in the sorted list as initial guesses; (4) continue to use middle points between past initial guesses as new initial guesses recursively until at least one new zero is found or the recursion has reached a certain depth; (5) append the new zero(s) to the list of zeros and repeat (2)-(5) until the number of identified zeros exceeds a certain value. In the present study, we limit the maximum depth in (4) to 8 and the maximum number of zeros in (5) to 200. The set of zeros so obtained are the fixed points of the dynamical system. Each fixed point is further classified using the respective Jacobian matrix as described above to identify the attractors—a subset of the fixed points forming the attractor repertoire.

For each set of structural parameters (G, C, w_EE, w_EI), we represent the attractor repertoire as a M-by-N matrix,

$$A\left(G,C,w_{EE},w_{EI}\right)=\left[\begin{array}{cccc}{S}_{E,1}^{(1)}& S_{E,1}^{(2)}& \cdots & S_{E,1}^{(N)}\\ S_{E,2}^{(1)}& S_{E,2}^{(2)}& \cdots & S_{E,2}^{(N)}\\ ⋮& ⋮& \ddots & ⋮\\ S_{E,M}^{(1)}& S_{E,M}^{(2)}& \cdots & S_{E,M}^{(N)}\end{array}\right]$$ (8)

where M is the number of attractors, and N is the number of brain regions.

As parameters vary (e.g., G in Figure 4d–i), the attractors form discrete connected components (e.g., stripes in Figure 4) in the product of the state space and the parameter space. Attractors within the same connected component can be considered qualitatively equivalent, as they morph into each other under continuous parameter change. Understanding the relation between these connected components are critical to characterizing phase transitions and bifurcations. It is thus meaningful to define a discretized version of the attractor repertoire,

$$\widehat{A}\left(G,C,w_{EE},w_{EI}\right)=\left[\begin{array}{cccc}{\widehat{S}}_{E,1}^{(1)}& {\widehat{S}}_{E,1}^{(2)}& \cdots & {\widehat{S}}_{E,1}^{(N)}\\ {\widehat{S}}_{E,2}^{(1)}& {\widehat{S}}_{E,2}^{(2)}& \cdots & {\widehat{S}}_{E,2}^{(N)}\\ ⋮& ⋮& \ddots & ⋮\\ {\widehat{S}}_{E,M}^{(1)}& {\widehat{S}}_{E,M}^{(2)}& \cdots & {\widehat{S}}_{E,M}^{(N)}\end{array}\right]$$ (9)

where ${\widehat{S}}_{E,i}^{(j)}$ is positive integer representing a discrete level of activation which $S_{E,i}^{(j)}$ belongs to (see Figure 3 for toy examples). Each row vector in equation 9 gives the multi-index of an attractor connected component. In practice, the mapping between the continuous level $S_{E,i}^{(j)}$ to the discrete levels of ${\widehat{S}}_{E,i}^{(j)}$ can be created by partitioning the continuous interval [0,1] at the minima of the distribution of all $S_{E,i}^{(j)}’\text{s}$ (see Section S7 for examples). Examining the properties of A and  opens the door to systematic characterization of the underlying dynamic landscape.

4.3. Quantifying cross-attractor coordination and energy gaps based on the attractor repertoire

Given a discretized attractor repertoire  (equation 9), the cross-attractor coordination matrix is an N-by-N matrix,

$$P\left(G,C,w_{EE},w_{EI}\right)=\left[\begin{array}{cccc}\rho \left({\widehat{A}}_{\cdot ,1},{\widehat{A}}_{\cdot ,1}\right)& \rho \left({\widehat{A}}_{\cdot ,1},{\widehat{A}}_{\cdot ,2}\right)& \cdots & \rho \left({\widehat{A}}_{\cdot ,1},{\widehat{A}}_{\cdot ,N}\right)\\ \rho \left({\widehat{A}}_{\cdot ,2},{\widehat{A}}_{\cdot ,1}\right)& \rho \left({\widehat{A}}_{\cdot ,2},{\widehat{A}}_{\cdot ,2}\right)& \cdots & \rho \left({\widehat{A}}_{\cdot ,2},{\widehat{A}}_{\cdot ,N}\right)\\ ⋮& ⋮& \ddots & ⋮\\ \rho \left({\widehat{A}}_{\cdot ,N},{\widehat{A}}_{\cdot ,1}\right)& \rho \left({\widehat{A}}_{\cdot ,N},{\widehat{A}}_{\cdot ,2}\right)& \cdots & \rho \left({\widehat{A}}_{\cdot ,N},{\widehat{A}}_{\cdot ,N}\right)\end{array}\right]$$ (10)

where Â._,j denotes the j^th column of  and ρ(x, y) the Spearman’s correlation between variables x and y (see Figure 3 for toy examples, Figure 5b for a more elaborate example). Spearman’s correlation is chosen to reflect the ordinal nature of the variable ${\widehat{S}}_{E,i}^{(j)}$. P_i,j = ρ(Â._,i, Â._,j) gives the level of cross-attractor coordination between model brain region i and j. The use of the discretized attractor repertoire  ensures that coordination matrix P is invariant within the same dynamic regime and only changes during a bifurcation. Thus, matrix P connects the change of brain coordination patterns to dynamical systems concepts such as bifurcation—a qualitative change in the dynamic landscape of the model brain.

The coordination matrix P by itself does not explicitly concern how difficult or energy consuming these cross-attractor movements are. To incorporate energetic properties, we equip each attractor repertoire with a sequence of energy gaps. We first order the rows of the attractor repertoire matrix A so that the row averages descend with the row index. The row averages of the ordered repertoire matrix provide a sequence of energy levels,

$$e\left(G,C,w_{EE},w_{EI}\right)=\left[\begin{array}{c}{\overline{S}}_{E,1}\\ {\overline{S}}_{E,2}\\ ⋮\\ {\overline{S}}_{E,i}\\ ⋮\\ {\overline{S}}_{E,M}\end{array}\right]$$ (11)

where ${\overline{S}}_{E,i}=N^{-1}{\sum }_{j=1}^N{A}_{ij}$, ${\overline{S}}_{E,i}>{\overline{S}}_{E,i+1}$ for any i < M, and M is the number of attractors. The corresponding energy gaps are

$$\Delta e\left(G,C,w_{EE},w_{EI}\right)=\left[\begin{array}{c}{\overline{S}}_{E,1}-{\overline{S}}_{E,2}\\ {\overline{S}}_{E,2}-{\overline{S}}_{E,3}\\ ⋮\\ {\overline{S}}_{E,i-1}-{\overline{S}}_{E,i}\\ ⋮\\ {\overline{S}}_{E,M-1}-{\overline{S}}_{E,M}\end{array}\right],$$ (12)

where $\Delta e_i={\overline{S}}_{E,i-1}-{\overline{S}}_{E,i}$ is the energy gap between the (i − 1)^th and i^th attractor in the repertoire. Physically, each energy gap Δe_i can be interpreted as the energy cost associated with keeping additional x% synaptic channels open. Synaptic transmission is a major energy consumer in the brain [75]. For example, the opening of NMDA receptors are associated with presynaptic, postsynatpic, and astrocytic costs, estimated to be roughtly 70k ATPs per vesicle released and 10⁸ ATPs per action potential generated [76]. The sequence of energy gaps can be used to partition the attractor repertoire into submatrices. For example, if Δe_i is the maximum energy gap, one can split A (and its discretized version Â) into a repertoire above the energy gap

$$A_{+}\left(G,C,w_{EE},w_{EI}\right)=\left[\begin{array}{c}{A}_{1,\cdot }\\ A_{2,\cdot }\\ ⋮\\ A_{i-1,\cdot }\end{array}\right],$$ (13)

and a repertoire below the energy gap

$$A_{-}\left(G,C,w_{EE},w_{EI}\right)=\left[\begin{array}{c}{A}_{i,\cdot }\\ A_{i+1,\cdot }\\ ⋮\\ A_{M,\cdot }\end{array}\right],$$ (14)

where A_i,· is the i^th row of the ordered repertoire matrix A. Each of the sub-repertoires A₊ and A₋ is associated with its own cross-attractor coordination matrix, say, P₊ and P₋ (equation 10). P₊ and P₋ can be considered as the “energy-constrained” coordination patterns where the model brain is not allowed to cross the energy gap Δe_i. The same line of analysis is applicable to a sub-network of the model brain (e.g., the default mode network) by constructing a reduced repertoire matrix that only contains a subset of the columns in A. These selected columns map to the brain regions within the sub-network. This series of analysis applied to attractor repertoire matrices and sub-matrices provides a systematic and simple way to characterize brain dynamic landscape and inter-regional coordination.

4.4. Estimating within-attractor coordination through simulations

Within-attractor coordination is estimated using conventional methods of numeric simulation. In the present work, for each attractor in the repertoire, the dynamical system (equation 4–6) is integrated using stochastic Heun’s method, with a time step of 1 ms, a moderate level of noise σ = 0.01 and a duration T = 864 s (14 min 33 s to match the human data [39]). The exact coordinates of the attractor are used as the initial conditions such that the simulated dynamics reflects the noise-driven exploration near that attractor. Conventional correlation analysis is then applied to the simulated time series of the excitatory populations ($S_E^{(i)}’\text{s}$) to obtain the functional connectivity matrix (e.g., Figure 5c). Spearman’s correlation is used in accordance with the computation of cross-attractor coordination matrix (P, equation 10).

4.5. Data and methods of analysis

4.5.1. Human structural data

The human structural connectome used in the present study is from the S1200 Release from the Human Connectome Project (HCP) [39]. The average connectome of 11 unrelated subjects were used for the first qualitative analysis (it has been shown that averaging over 5 subjects is sufficient [12, 15]), while the individual connectome of 100 subjects obtained from a previous study [40] were used for the second quantitative analysis. For both analyses, the subject-level connectome data are based on the Desikan-Killiany parcellation [77] obtained from [40], retaining the 66 ROIs used in [38] and [12] (Figure 1b). The original diffusion imaging (dMRI) data were obtained using a customized Siemens 3T scanner at Washington University in St. Louis, with a standard 32-channel head coil, with TR = 5520 (ms), TE = 89.5 (ms), 1.25 (mm) isotropic voxels, b=1000, 2000, 3000 (s/mm²). T1 images were obtained using 3D magnetization-prepared rapid gradient echo sequence (MPRAGE) with TR = 2400 (ms), TE = 2.14 (ms), and 0.7 (mm) isotropic voxels. The HCP minimally processed data were further processed using MRtrix3, including bias-field correction, multi-shell multi-tissue constrained spherical deconvolution with a maximum spherical harmonic degree 8. 10 million probabilistic streamlines were generated for each subject using the 2^nd-order Intergration over Fibre Orientation Distributions algorithm (iFOD2) [78] and anatomically-constrained tractography (ACT) [79] (FOD amplitude threshold = 0.06, step size = 0.625 mm). Each streamline was assigned a weight using spherical-deconvolution informed filtering of tractograms (SIFT2) [80]. Connection strengths between ROIs are summed weights of the associated streamlines. Intra-ROI connections are removed. Subjects’ connectivity matrices are normalized according to equation 7 before and after averaging.

4.5.2. Human functional data

Human functional connectivity used in the present study is estimated using the resting-state fMRI (rsfMRI) data of the same subjects from the Human Connectome Project [39] as aforementioned ones from the structural connectivity. rsfMRI scans were acquired using EPI sequences with TR = 720 (ms), TE = 33.1 (ms), flip angle = 52°, voxel size = 2.0 (mm, isotropic), multiband factor = 8. Four runs of rsfMRI scan were obtained from each subject in 2 separate days (2 runs in each day with opposite phase-encoding direction: RL and LR). Each run last 14 min 33 s (1200 TR).

For the first analysis, unprocessed data were downloaded from the Human Connectome Project database (https://db.humanconnectome.org) and preprocessed using fMRIPrep 1.4.0 ([81]; [82]; RRID:SCR_016216), which is based on Nipype 1.2.0 ([83]; [84]; RRID:SCR_002502). First, a reference volume and its skull-stripped version were generated using a custom methodology of fMRIPrep. A deformation field to correct for susceptibility distortions was estimated based on two echo-planar imaging (EPI) references with opposing phase-encoding directions, using 3dQwarp [85] (AFNI 20160207). Based on the estimated susceptibility distortion, an unwarped BOLD reference was calculated for a more accurate co-registration with the anatomical reference. The BOLD reference was then co-registered to the T1w reference using bbregister (FreeSurfer) which implements boundary-based registration [86]. Co-registration was configured with nine degrees of freedom to account for distortions remaining in the BOLD reference. Head-motion parameters with respect to the BOLD reference (transformation matrices, and six corresponding rotation and translation parameters) are estimated before any spatiotemporal filtering using mcflirt [FSL 5.0.9, 87]. The BOLD time-series were resampled to the fsaverage surface space. Several confounding time-series were calculated including framewise displacement (FD), DVARS and three region-wise global signals. FD and DVARS were calculated for each functional run, both using their implementations in Nipype [following the definitions by 88]. The three global signals were extracted within the CSF, the WM, and the whole-brain masks.

Nuance regressions were performed on detrended, preprocessed BOLD time series in the fsaverage space (FreeSurfer), following procedures in [88]. Regressors include 6 motion parameters, CSF signal, WM signal, and their first derivative and second power. Frames with FD>0.2 mm are censored. Spline-interpolated signals are band-pass filtered between 0.009 and 0.08 Hz, and averaged within ROIs based on the Desikan-Killiany parcellation [77]. 66 Regions in [38] are retained and ordered according to [12] (Figure 1b). Functional connectivity between ROIs are estimated using Spearman correlation between z-scored time series for each rsfMRI run of each subject. The connectivity matrices are then averaged across all runs/subjects in Day 1 and in Day 2 separately. The average functional connectivity matrix from Day 1 is used in all comparisons with the model. The average functional connectivity matrix from Day 2 is used to assess the reliability of the estimation.

For the second analysis, an alternative preprocessing pipeline that is standard for individual rsfMRI preprocessing was applied. Specifically, minimally preprocessed rsfMRI volumetric data from the HCP database was downloaded and further denoised through application of a published spatial independent component rejection procedure [89–91]. The resulting rsfMRI timeseries was then converted to ROI space by averaging voxels part of the Desikan-Killiany atlas in volume space [92]. Relative to the list in Table S1, there were six missing ROIs, which are the left and right temporal pole, the left and right frontal pole, and the left and right bank of the superior temporal sulcus. Similar to the case with the 11 subjects, there were 4 runs per subject (2 runs with opposite encoding for 2 separate days). To calculate the functional connectivity, ROI time series were z-scored and then correlated with Spearman correlation between each pair of ROIs. The resulting 4 connectivity matrices were averaged to obtain one final connectivity matrix per subject.

4.5.3. Individual subject simulation and model fitting

To find the optimal human-model fitting on an individual basis, simulations were run with one representative set of local parameters (W_EE = 2, W_EI = 1), while the global coupling parameter was varied from 1.7 to 3.0 with step sizes of 0.1. This set of parameter configurations were chosen with considerations for the run time while capturing the individual variability for the optimal value of G. Additional details of the simulation, including computation of the within and cross attractor coordination, and energy gaps between attractors, were identical to the analysis of the 11 subject averaged data.

Spearman’s correlation was used to quantify the model-human correlation. The optimal G parameter for each individual was determined based on the maximum correlation for the cross-attractor coordination. Within-attractor coordination matrices were determined for individual attractors at that optimal G and used to find the highest model-human correlation for within-attractor coordination. To compare the model fit between within and cross-attractor coordination on an individual level, a paired t-test was applied between the model-human correlation for the two across the 100 individuals.

4.5.4. Correlating with measures of fluid intelligence

An abbreviated 24-items version of the Raven’s Progressive Matrix Test Form A was used for evaluating fluid intelligence Duncan, 2000; Bilker, 2012 and scored with the number of correct responses (PMAT24_A_CR). Spearman’s correlation was calculated between model parameters and PMAT24_A_CR while controlling for age and sex. Model parameters of interest include the maximum correlation value, the associated optimal G parameter, and the maximum energy gap at the optimalG.