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# The dynamics of resting fluctuations in the brain: metastability and its dynamical cortical core

## Abstract

In the human brain, spontaneous activity during resting state consists of rapid transitions between functional network states over time but the underlying mechanisms are not understood. We use connectome based computational brain network modeling to reveal fundamental principles of how the human brain generates large-scale activity observable by noninvasive neuroimaging. We used structural and functional neuroimaging data to construct whole- brain models. With this novel approach, we reveal that the human brain during resting state operates at maximum metastability, i.e. in a state of maximum network switching. In addition, we investigate cortical heterogeneity across areas. Optimization of the spectral characteristics of each local brain region revealed the dynamical cortical core of the human brain, which is driving the activity of the rest of the whole brain. Brain network modelling goes beyond correlational neuroimaging analysis and reveals non-trivial network mechanisms underlying non-invasive observations. Our novel findings significantly pertain to the important role of computational connectomics in understanding principles of brain function.

## Introduction

“When we take a general view of the wonderful stream of our consciousness, what strikes us first is the different pace of its parts. Like a bird’s life, it seems to be made of an alternation of flights and perchings.” William James 1

Survival remains the perhaps most important problem faced by brains and a key challenge is how to segregate and integrate relevant information over different timescales when faced with hostile, often constantly changing environments2. Reconciling different speeds of information processing, from fast to slow, is especially important, and could be key to the relative evolutionary success of mammals whose sophisticated brains are able to combine prior information from past memories with current stimuli to predict the future and to adapt behaviour accordingly3,4,5.

This was recognized well over a century ago by William James, generally acknowledged as one of the fathers of modern cognitive psychology1. Speaking of this problem using the apt metaphor of the stream of consciousness, James noted that there is a different pace to its parts, comparing it to the life of a bird whose journey consists of an “alternation of flights and perchings”. In the language of today’s dynamical systems, the flights are akin to fast, segregative tendencies and the perchings to slower, integrative tendencies of the dynamic brain in action2, 6, 7. In addition, motivated by recent experimental and modelling work of other labs8, 9, we investigate cortical heterogeneity across areas. By optimizing the spectral characteristics of each local brain node (in the coupled network), this allowed us to discover a dynamical core of the brain, i.e. the set of brain regions, which through their oscillations are driving the rest of the brain. Furthermore, with regards to balancing the different speeds of processing, a large body of psychological research has focused on what is known as dual process theories10, 11, identifying competing fast and slow systems which have to co-exist and function on multiple time-scales in order for the brain to efficiently allocate the resources necessary for survival12, 13.

Yet, the temporal dynamics and underlying neural mechanisms of this temporal processing on multiple timescales are poorly understood. Here we aim to provide a better understanding of the dynamics using computational brain network modelling which has emerged as a powerful tool for investigating the causal dynamics of the human brain, when carefully constrained by functional (FC) and structural connectivity (SC) obtained from empirical neuroimaging data14,15,16,17,18. This theoretical framework has been largely successful in explaining the highly structured dynamics arising from spontaneous brain activity in the so-called resting-state-networks (RSN)19,20,21, even if the resting brain never truly rests20. Efficient task-related brain activity has been shown to rely on metastability of spontaneous brain activity allowing for optimal exploration of the dynamical repertoire22 but it is not known if this metastability is maximally metastable6. In dynamic systems, metastability refers to a state that falls outside the natural equilibrium state of the system but persists for an extended period of time. One example of a metastable dynamical system is a winnerless competition23, however metastability can arise from a number of underlying mechanisms and it is in this broader sense that we use the term metastability. Here, we use metastability to denote the variability of the global synchronization as measured by the Kuramoto order parameter, following the work of Wildie and Shanahan24. This is motivated by previous research which has shown that this variability can be linked to the underlying metastable cluster synchronization25.

We investigated the dynamics of the brain network system through a local node neural mass description based on the most general form of expressing both noisy asynchronous dynamics and oscillations, namely a normal form of a Hopf bifurcation26,27,28. Previous research has shown the usefulness, richness and generality of this type of model for describing EEG dynamics at the local node level27, 28. This normal form allowed us to fit the model to neuroimaging data over time, i.e. not only by fitting the grand average FC but also by fitting the temporal structure of the fluctuations, functional connectivity dynamics29 (FCD, Fig. 1A,B).

We further explored if the optimal working point where FC and FCD are fitted corresponds to a dynamical region where the global metastability of the whole brain is maximized6. In addition, motivated by recent experimental and modelling work of other labs8, 9, we investigate cortical heterogeneity across areas. By optimizing the spectral characteristics of each local brain node (in the coupled network), this allowed us discovering a dynamical core of the brain, i.e. the set of brain regions, which through their oscillations is driving the rest of the brain. As such this investigation was designed to provide an empirical, scientific footing for James’ metaphorical speculations of the flights and perchings of human brain dynamics, and to demonstrate the potential of sophisticated brain network computational modelling to provide new insights into the causal mechanisms of neuroimaging results.

## Results

The results arose from using brain network computational models for the analysis of empirical neuroimaging data characterising the functional and structural connectivity of 24 healthy human participants acquired using standard MRI techniques17 (see Methods). In particular, we were able to gain new insights on the emergence of transiently spatiotemporal structured networks among segregated brain regions by examining a whole-brain network model using a very general neural mass model known as the normal form of a Hopf bifurcation (also known as Landau-Stuart Oscillators), which is the canonical model for studying the transition from noisy to oscillatory dynamics26 (Fig. 2). Here, we extended previous research on local node dynamics27, 28 by studying the whole-brain network dynamics, i.e. by investigating how those local noisy oscillators interact, and how the emerging whole-brain network activity relates to fMRI resting state dynamics. Within this model, each node of the network is modeled by a normal Hopf bifurcation, with an intrinsic frequency ω i in the 0.04–0.07 Hz band (i = 1, …,n). The intrinsic frequencies were estimated directly from the data, as given by the averaged peak frequency of the narrowband BOLD signals of each brain region (see Methods). The state of each node i is determined by its phase, φ i (t), and the interaction between nodes depends both on the structural couplings and the phase difference between the nodes. The model has only two types of control parameters, namely: one single global parameter, G, that represents the global scaling of the anatomical connectivity matrix, and the bifurcation parameters a j for each node (see Fig. 3 and methods for the general structure and strategy of the brain network model).

### Maximal metastability at the optimal working point of model

Using the Hopf model, we were able discern the dynamical properties of the optimal working point of the system that is able to fit the characteristics of the empirical fMRI data. We were able to distinguish the origin of resting activity between the two hypothesized scenarios, namely: 1) noisy excursions at the edge of a critical bifurcation19, 20, 30, 31 or 2) metastable oscillations16. The first scenario refers to the entrainment of noisy dynamics through the underlying anatomical connectivity matrix, i.e. inducing correlations of the local noise because of the underlying SC connections. The second scenario refers to the structuring of metastable cluster synchronizations of the underlying local oscillatory dynamics through the underlying anatomical SC connections. We define metastability as the standard deviation of synchrony at the network level described by order parameter R(t), where R(t) measures the phase uniformity and varies between 0 for a fully desynchronized network and 1 for a fully synchronized network (see methods and Fig. 1C)24. The present model is able to describe both types of dynamics, and the smooth transitions from one to the other, i.e. the transition from noisy to oscillatory dynamics (Fig. 2). In order to distinguish the dynamical scenario, we investigated the capabilities of the model for fitting the grand average FC and also the time dependent characteristics of the RSN as reflected in the FCD in the different dynamical working regions (i.e. as a function of the control parameters). The grand average FC describes the mean spatial structure of the resting activity, whereas the FCD captures the statistical characteristic of the temporal structure of those spatial correlations (see Methods and ref. 29).

Figure 3 shows that the best fit to the empirical data of Hopf model is found at the brink of the Hopf bifurcation. We equalized all local bifurcation parameters to a common value i.e. a j  = a, in order to reduce the investigations to just two parameters, namely global bifurcation parameter (a) and global coupling strength (G). Figure 3 shows how the empirical data are fitted in the Hopf model for different working points. The right column of Fig. 3 shows the level of fitting of the FC, FCD and metastability. As can be seen, the best fitting of the three measures is obtained at the region on the brink of the Hopf bifurcation, i.e. for bifurcation parameter a, at the edge of zero on the negative side, such that the oscillators remain damped still. In this region not only the correlation between the empirical and simulated FC is maximized, but also the statistics of the rapid switching between FC(t) across time (FCD) is minimized in Kolmogorov-Smirnov sense, and the level of metastability of the data is reproduced. The fitting of the FC was measured by the Pearson correlation coefficient between corresponding elements of the upper triangular part of the matrices (see Fig. 1 and Methods). For comparing the FCD statistics, we collected the upper triangular elements of the matrices (over all participants or sessions) and compared the simulated and empirical distribution by means of the Kolmogorov-Smirnov distance between them (see Methods).

Furthermore, the results showed that only in the region at the border between noisy and oscillatory behaviour, is where the signals resembles the data, i.e. like noise with an oscillatory component around 0.05 Hz (Fig. 2). The first three columns of Fig. 3 show the dependence of those measurements as a function of the global scaling parameter G for three specific values of the bifurcation parameter a, namely at the noisy region, at the edge of the bifurcation and at the oscillatory regime. Clearly, the best results are obtained for the second column (at the edge of the bifurcation). The same panel shows that the FCD is the best constraining measure. There is a broad range of G where the FC and the metastability is well fitted, but only a relative narrow range where the FCD statistics is minimal, i.e. maximally fitted. In other words, the spatiotemporal structure of the FC is more informative than the grand average of the FC (i.e. the “classical” RSN). This is important, because until now, brain network models have always been fitted with the grand average FC - but see also the work of Hansen and colleagues29.

We would like to remark that Fig. 3 characterizes some of the bifurcation behaviour of the whole system. Indeed, the metastability for example serves as a network metric and characterizes the variability of this global synchronization as a function of those two control parameters. All three parameter spaces in Fig. 3B, in conjunction, present a full picture of the spatiotemporal organization of the system. The three metrics characterize computationally the bifurcation properties of the full network dynamics.

Perhaps most importantly, as shown in Fig. 3, the brain network model shows maximal metastability at the optimal working point of the model (a = 0 and G = 2.85), where the metastability is reflecting the variability of the synchronization between different nodes, i.e. the fluctuations of the states of phase configurations as a function of time24. Further characterisation of these results is shown in Fig. 4 which shows the optimal working point at the edge of the Hopf bifurcation (i.e. bifurcation parameter a = 0), the FC, FCD and FCD statistics for three levels of global coupling G namely low, optimal and large. For comparison, the same matrices and distributions are plotted on the rightmost column for the empirical data (Fig. 4B). Only the FCD and its statistics (bottom row) are constraining enough for optimizing the working point. Please note that for low G the FCD statistics does not show any switching between states in the RSN and that for very large G there are too much switching between states.

### Dynamical core: contribution of individual brain regions to dynamics

In order to obtain information about the dynamical characteristics of each single brain area and to generate a heterogeneous brain network model (i.e. with different dynamics at each node), we optimized each single bifurcation parameter a j independently by fitting for each value of global coupling G the spectral characteristics of the simulated and empirical BOLD signals at each brain area (see Methods). The main results are plotted in Fig. 5, where Fig. 5A shows the evolution of the fitting of the FC and FCD statistics as a function of G. For large enough value of the global coupling a good fitting of both is obtained, i.e. large correlation between the empirical and simulated grand average FC and low difference in the statistics of the empirical and simulated FCD (Kolmogorov-Smirnov distance). Please note that Fig. 5A is generated in a different way than Fig. 4A (which uses only the optimum fit a = 0 for all regions and G = 2.85). Instead, in Fig. 5A, for each G we optimize the bifurcation value, a, for each region (shown in Fig. 5B). As can be seen at a critical value of G, the bifurcation values remain the same, only scaled. Thus the FCD fit in Fig. 5A will asymptote as G increases.

For optimizing a j values, we use a greedy optimisation strategy, where we iteratively increase or decrease the a j value according to the local power of the signal in a given region j. Greedy algorithms exploit local optima, but often approximate optimal solutions well in reasonable time and produce good results as shown in Fig. 5. The local bifurcation parameters for each region for the uncoupled network (i.e. G = 0) and for the optimal coupling (G = 5.4) can be seen in Fig. 5C. If the network is uncoupled, each single brain area fitted the spectral characteristics of the empirical BOLD signals in a very homogeneous way by local bifurcations parameters at the edge of the local Hopf bifurcation, i.e. at zero. When the brain network is coupled, the “true” intrinsic local dynamics for the profile of optimal local bifurcation parameters a j observed at that point that fit the local empirical BOLD characteristics and the global quantities FC, FCD and metastability (Fig. 5D).

Brain regions, for which best predictions were achieved in an oscillatory mode, i.e. with bifurcation parameters a > 0.1 are visualised in Fig. 6. We found that the dynamical core within this parcellation consisted of eight lateralised brain regions: medial orbitofrontal cortex, posterior cingulate cortex and transverse temporal gyrus in the right hemisphere, and caudal middle frontal gyrus, precentral gyrus, precuneus cortex, rostral anterior cingulate cortex and transverse temporal gyrus in the left hemisphere. Those nodes working at the edge of the bifurcation are highlighted as a “dynamical core” whose perturbations can propagate in an optimal way to the rest of the network.

## Discussion

We provide mechanistic explanations of the complex spatiotemporal dynamics of brain function arising from James’ early speculations1 to much more detailed scientific enquiry2, 32,33,34. This confirms that brain results from complex interactions in a system of non-linearly coupled, non-linear oscillatory processes which display dynamical system phenomena such as multiple stable states, instability, state transitions and metastability, of which the latter has been proposed to form a core dynamical description of coordinated brain and behavioral activity6.

In the 1980s the physicist Hermann Haken suggested to mechanistically interpret brain processes of segregation and integration as a sequence of semistable states, so-called saddle states35. He proposed to view the complex integrative and segregative tendencies as expressions of emergent lower-dimensional behavior of collective variables, which he termed ‘order parameters’. Scott Kelso popularized this concept using the term ‘metastability’ based on his brain-behaviour experiments and drawing inspiration from other researchers including Rodolfo Llinás and Francisco Varela33, 34. He generalized metastability to include the oscillatory states of brain processes found in between complete synchronization and independence32, 36. Later research has formalized these concepts more rigorously, e.g. via the heteroclinic channel37, 38 and Structured Flows on Manifolds (SFM)39, 40.

We shed new causal light on the mechanisms underlying RSNs by extending previous research which has demonstrated the existence of RSNs, i.e. brain networks correlated within the grand average FC during resting state21, 41, 42. FC has become routinely used as a biomarker in various clinical applications, even though its predictive value holds only for group analyses, and not currently for the individual43. This problem arises most likely from the lack of taking time into account, i.e. the non-stationary nature of the resting state dynamics44, 45. Hansen and colleagues demonstrated that the grand average FC is more closely linked to the SC and linear models of FC29. When non-linearities are considered in the network models, the spatiotemporally dynamic repertoire of the network is significantly enhanced and the resting state dynamics shows the non-stationary FCD, which expresses itself as the switching dynamics of the FC. While Hansen and colleagues proposed FCD as a novel biomarker and demonstrated that all known RSNs can be derived from the non-linear network dynamics of FCD, they did not fit the model to the empirical functional time series data. The patterns in the FCD matrix arise from what is essentially a random process and thus different for different measurements. This renders the fitting process for brain network models more complex than fitting with the grand average FC, for which a Pearson correlation across empirical and simulated FC matrices is sufficient.

We have addressed this issue through a systematic fitting approach of the random process in FCD to the empirical data. The conjunction of using sophisticated fitting and systematic parameter analysis allowed us to test the mechanistic hypotheses underlying the resting state, i.e. whether the brain at rest operates close to the edge of a bifurcation and/or occupies a metastable state. Both scenarios can be mechanistically realized by non-linearly coupling Hopf bifurcators26. Hopf oscillators have been used previously in connectome-based modelling of resting state dynamics in EEG/MEG and fMRI14, as well as for the modelling of the detailed temporal dynamics in EEG/MEG27, 28. The usage here though is different from the previous research, since the Hopf oscillators act as the sources of BOLD signal in the connectome based network model. Ghosh and colleagues used the Hopf oscillators as the sources of the electrophysiological signal and employed the Balloon Windkessel to derive the BOLD signal46. Given this interpretation, they needed to include all the signal transmission delays. In our present approach, the oscillation frequencies are significantly slower and thus permit the neglect of the time delays, which simplifies the computational effort of the simulation and thus the computational fitting of the models against empirical data.

Our key finding is the demonstration that the optimal operating regime is at the edge of the local Hopf bifurcation, i.e. a balance of noisy excursions in the oscillatory state. We not only were able to demonstrate that previous findings on the optimal operating point based on grand-average FC hold true if we take into account the temporal dynamics of FC, i.e. FCD. We also demonstrated that a better way of constraining brain network models is by not only fitting the grand average FC, but by also fitting the temporal structure of the fluctuations using the FCD.

Another remarkable and important finding is that high metastability is only present in a narrow range of bifurcation parameter when a is close to the edge of the bifurcation. In other words, the FCD of the spontaneous resting state, in conjunction with brain network modelling provide evidence that the brain at rest is maximally metastable, refining and demonstrating the hypothesis of Tognoli and Kelso6. Note that there is also a region for very small G and positive a (oscillatory regime) where a relatively good fitting is obtained. This dynamic regime was previously observed with a pure oscillatory Kuramoto model of the BOLD signals at the mesoscopic level47. Nevertheless, the level of fitting for the FC, metastability and even FCD is not as good as the one obtained in the region at the edge of the Hopf bifurcation. On the other hand, besides the extreme sensitivity of that working point (ultra-narrow regime of optimality) which means that the result is not so robust, the qualitative description of the BOLD signals is not realistic in the pure oscillatory regime in comparison with the noisy/oscillatory excursions evidenced in the regime of the bifurcation parameter a near zero.

For constructing a heterogeneous brain network model with different local parameter values, we took into account the spectral information of the BOLD data. We addressed the question if the oscillations at the individual nodes play a mechanistic role for the emergence of FC/FCD. In particular, we identified a cortical core of eight brain regions with the optimal fit of bifurcation parameter a close to the edge of bifurcation. We propose to call this the dynamical cortical core of the brain. Interestingly, three of these regions (the medial orbitofrontal cortex, posterior cingulate cortex and precuneus cortex) are part of the default mode network and thus re-experience past events and pre-experience possible future events48, 49. In this vein other regions (parahippocampal and transverse temporal gyrus) have also been implicated in memory processing and may thus perhaps be helping integrate information over different timescales, binding fast and slow processes over time2. This information is always contextual and in the noisy, unpredictable scanner it is perhaps not surprising that the brain is attending to the auditory signals (transverse temporal gyrus). As such this information processing is available for conflict monitoring and selection for action (rostral anterior cingulate cortex and caudal middle frontal gyrus) and motor execution (precentral gyrus)50. Equally, the involvement of the cingulate cortex is interesting given that this region recently has been shown to be part of the common neurobiological substrate for mental illness across across six diverse diagnostic groups (schizophrenia, bipolar disorder, depression, addiction, obsessive-compulsive disorder, and anxiety) based on a meta-analysis of grey matter loss in 193 neuroimaging studies of 15892 individuals51. This reinforces the potential use of brain network computational modelling for understanding the underlying mechanisms of neuropsychiatric disorders52. The right-handed quality of Fig. 6, presumably arises from the specifics of the data used to fit the model and we will be exploring its biological validity in subsequent studies with larger group sizes.

Although the bifurcation parameter does not have a direct biophysical correlate, it seems to be involved in mediating biophysical effects. Another note of caution: we have presented a mesoscopic phenomenological model, i.e. the dynamical equation corresponds directly to the measured BOLD signals and not to the neural signals. It is a phenomenological model since the real coupling between regions does not, of course, occur between the hemodynamic signals, but between the underlying neural activity. Still, by using this simplified approach, we do not need to convolve modelled neural signals with the haemodynamic response function47.

We therefore propose that in future both the global bifurcation parameter as well as the individual parameters could potentially serve as biomarkers for disease. It will be important to explore the changes for different brain diseases, e.g. within a standardized framework for connectome-based modelling such as The Virtual Brain (TVB)53, 54, and applications such as fitting of TVB’s dynamic regime and TVB Processing pipeline17.

Overall, we have shown that neuroimaging data can be causally analysed by constructing a relatively simple brain network computational model using a Hopf bifurcation. This model was shown to be maximally metastable at the optimal fitting with the spatiotemporal dynamics of spontaneous brain activity. This dynamical regime may well allow for the optimal integration and segregation of fast and slow information over different time-scales, the “flights” and “perchings” of the stream of consciousness alluded to by William James over 100 years ago.

## Methods

### Ethics Statement

All participants of this study gave written informed consent before the study, which was performed in compliance with the relevant laws and institutional guidelines and approved by the ethics committee of the Charité University Berlin.

### Empirical MRI Data Collection

Structural data from DTI and resting-state BOLD signal time series were acquired for 24 healthy participants (age between 18 and 33 years old, mean 25.7, 12 females, 12 males). A full description of the generation of SC and FC matrices from those data can be found in ref. 17. Here, we provide a quick overview of the employed methods. Empirical data were acquired at Berlin Center for Advanced Imaging, Charité University Medicine, Berlin, Germany. For simultaneous EEG-fMRI55, 56, participants were asked to stay awake and keep their eyes closed. No other controlled task had to be performed. In addition, a localizer, DTI and T2 sequence were recorded for each participant. MRI was performed using a 3 Tesla Siemens Trim Trio MR scanner and a 12-channel Siemens head coil. Specifications for the employed sequences can be found in ref. 56. For each participant anatomical T1-weighted scans were acquired. DTI and GRE field mapping were measured directly after the anatomical scans. Next, functional MRI (BOLD-sensitive, T2*-weighted, TR 1940 ms, TE 30 ms, FA 78°, 32 transversal slices (3 mm), voxel size 3 × 3 × 3 mm, FoV 192 mm, 64 matrix) was recorded simultaneously to the EEG recording.

### MRI Data Analysis

Processing steps executed by the public Berlin automatized processing pipeline56 comprised 1) preprocessing of T1-weighted scans, cortical reconstruction, tessellation and parcellation, 2) transformation of anatomical masks to diffusion space, 3) processing of diffusion data, 4) transformation of anatomical masks to fMRI space, 5) Processing of fMRI data.

### Anatomical MRI Data Analysis

The highly resolved anatomical images are important to create a precise parcellation of the brain. For each of those parcellated units, empirical functional data time series are spatially aggregated. T1-weighted images are pre-processed using FREESURFER including probabilistic atlas based cortical parcellation, here using Desikan-Killany (DK) atlas57 (Table 1). This generates volumes that contain all cortical and subcortical parcellated regions with corresponding region labels used for fiber-tracking and BOLD time-series extraction.

### Empirical fMRI Data Analysis

In order to generate the FC matrices, FSL’s FEAT pipeline is used to perform the following operations: deleting the first five images of the series to exclude possible saturation effects in the images, high-pass temporal filtering (100 seconds high-pass filter), motion correction, brain extraction and a 6 DOF linear registration to the MNI space. Functional data is registered to the participant’s T1-weighted images and parcellated according to FREESURFER’s cortical segmentation. By inverting the mapping rule found by registration, anatomical segmentations are mapped onto the functional space. Finally, average BOLD signal time series for each region are generated by computing the mean over all voxel time-series for each region. From the region wise aggregated BOLD data, FC matrices are computed within MATLAB using and Pearson’s linear correlation coefficient as FC metrics. We did not perform global signal regression on data.

### Brain Network Model

The brain network model consists of 68 coupled brain areas (nodes) derived from the parcellation explained above. The global dynamics of the brain network model used here results from the mutual interactions of local node dynamics coupled through the underlying empirical anatomical structural connectivity matrix C ij (see Fig. 2). The structural matrix C i denotes the density of fibres between cortical area i and j as extracted from the DTI based tractography (scaled to a maximum value of 0.2). The local dynamics of each individual node is described by the normal form of a supercritical Hopf bifurcation, which is able to describe the transition from asynchronous noisy behavior to full oscillations. Thus, in complex coordinates, each node j is described by following equation:

$$\frac{d{z}_{j}}{dt}=z[{a}_{j}+i{\omega }_{j}-|{z}_{j}^{2}|]+\beta {\eta }_{j}(t)$$
(1)

where

$${z}_{j}={\rho }_{j}{e}^{i{\theta }_{j}}={x}_{j}+i{y}_{j}$$
(2)

and η i (t) is additive Gaussian noise with standard deviation β=0.02. This normal form has a supercritical bifurcation at a j  = 0, so that for a j  < 0 the local dynamics has a stable fixed point at z j  = 0 (which because of the additive noise corresponds to a low activity asynchronous state) and for a j  > 0 there exists a stable limit cycle oscillation with frequency $${f}_{j}={\omega }_{j}/2\pi$$. We insert equation 2 in equation 1 and separate real part in equation 3 and imaginary part in equation 4.

Thus, the whole-brain dynamics is defined by following set of coupled equations:

$$\frac{d{x}_{j}}{dt}=[{a}_{j}-{x}_{j}^{2}-{y}_{j}^{2}]{x}_{j}-{\omega }_{j}{y}_{j}+G\,\sum _{i}{C}_{ij}({x}_{i}-{x}_{j})\beta {\eta }_{j}(t)$$
(3)
$$\frac{d{y}_{j}}{dt}=[{a}_{j}-{x}_{j}^{2}-{y}_{j}^{2}]{y}_{j}+{\omega }_{j}{x}_{j}+G\,\sum _{i}{C}_{ij}({y}_{i}-{y}_{j})\beta {\eta }_{j}(t)$$
(4)

Please note that following the literature from physics, the equations are written in Cartesian, rather than polar coordinates59,60,61,62. We couple the equations using the common difference coupling, which approximates the simplest (linear) part of a general coupling function. These equations are valid in the weakly coupled oscillator limit, in which the coupling preserves the periodic orbit of the uncoupled oscillators. If the linear coupling (following a Taylor expansion of the full coupling) does not exist, the next non-vanishing higher order term should be considered, which is a case we do not address here (please see Kuramoto59 (see Eq 5.3.1) and Pikovsky, Arkady and Kurths60 (see Eq. 8.12) for more detailed analytic treatments of the equations).

In the latter equations, G is a global scaling factor (global conductivity parameter scaling equally all synaptic connections). The global scaling factor G and the bifurcation parameters a j are the control parameters with which we study the optimal dynamical working region where the simulations maximally fit the empirical FC and the FCD. We model with the variables x j the BOLD signal of each node j. The empirical BOLD signals were band-pass filtered within the narrowband 0.04–0.07 Hz. This frequency band has been mapped to the gray matter and it has been shown to be more reliable and functionally relevant than other frequency bands63,64,65,66. Within this model, the intrinsic frequency ω j of each node is in the 0.04–0.07 Hz band (i = 1, …, n). The intrinsic frequencies were estimated from the data, as given by the averaged peak frequency of the narrowband BOLD signals of each brain region.

### Grand average FC and FCD matrices

The grand average FC is defined as the matrix of correlations of the BOLD signals between two brain areas over the whole time window of acquisition. In order to characterize the time dependent structure of the resting fluctuations, we estimate the FCD matrix29 (see Fig. 1). Each full-length BOLD signal of 22 min is split up into M=61 sliding windows of 60 sec, overlapping by 40 sec. For each sliding window, centered at time t, we calculated a separate FC matrix, FC(t). The FCD is a MxM symmetric matrix whose (t1, t2) entry is defined by the Pearson correlation between the upper triangular parts of the two matrices FC(t1) and FC(t2). Epochs of stable FC(t) configurations are reflected around the FCD diagonal in blocks of elevated inter-FC(t) correlations.

The grand average FC and the FCD matrices were estimated for the recordings of each of the 24 participants as well as for 24 simulations of 22 minutes of the computational model. We compared the FC matrices of the model (averaged Fisher’s z-transformed over the 24 sessions) and the empirical data (averaged Fisher’s z-transformed over the 24 participants), adopting as a measure of similarity between the two matrices the Pearson correlation coefficient between corresponding elements of the upper triangular part of the matrices. For comparing the FCD statistics, we collected the upper triangular elements of the matrices (over all participants or sessions) and generated the distribution of them. Then, we compared the simulated and empirical distribution by means of the Kolmogorov-Smirnov distance between them. The Kolmogorov–Smirnov distance quantifies the maximal difference between the cumulative distribution functions of the two samples.

### Metastability

Here, we refer to metastability as a measure of how variable are the states of phase configurations as a function of time, i.e. how the synchronization between the different nodes fluctuates across time24. Thus, we measure the metastability as the standard deviation of the Kuramoto order parameter across time. The Kuramoto order parameter is defined by following equation:

$$R(t)=|\sum _{k=1}^{n}{e}^{i{\phi }_{k}(t)}|/n$$
(5)

where φ k (t) is the instantaneous phase of each narrowband BOLD signal at node k. The Kuramoto order parameter measures the global level of synchronization of the n oscillating signals. Under complete independence, the n phases are uniformly distributed and thus R is nearly zero, whereas R=1 if all phases are equal (full synchronization). Thus, for calculating the metastability of the empirical and simulated BOLD signals, we first band-pass filtered within the narrowband 0.04–0.07 Hz (as previously explained) and computed the instantaneous phase φ k (t) of each narrowband signal k using the Hilbert transform. The Hilbert transform yields the associated analytical signals. The analytic signal represents a narrowband signal, s(t), in the time domain as a rotating vector with an instantaneous phase, φ k (t), and an instantaneous amplitude, A(t), i.e., $$s(t)=A(t)\cos (\phi (t))$$. The phase and the amplitude are given by the argument and the modulus, respectively, of the complex signal z(t), given by $$z(t)=s(t)+i.H[s(t)]$$, where i is the imaginary unit and H[s(t)] is the Hilbert transform of s(t).

### Local Optimization of Brain Nodes

The local optimization of each single bifurcation parameter a j is based on the fitting of the spectral information of the empirical BOLD signals in each node. In particular, we aim to fit the proportion of power in the 0.04–0.07 Hz band with respect to the 0.04–0.25 Hz band (i.e. we remove the smallest frequencies below 0.04 Hz and consider the whole spectra until the Nyquist frequency which is 0.25 Hz)47. For this, we filtered the BOLD signals in the 0.04–0.25 Hz band, and calculated the power spectrum P j (f) for each node j. We define the proportion,

$${p}_{j}=\frac{{\int }_{0.04}^{0.07}{P}_{j}(f){\rm{d}}f}{{\int }_{0.04}^{0.25}{P}_{j}(f){\rm{d}}f}$$
(6)

and update the local bifurcation parameters by a gradient descendent strategy, i.e.:

$${a}_{j}={a}_{j}+\eta ({p}_{j}^{empirical}-{p}_{j}^{simulated})$$
(7)

until convergence. We used here η = 0.1. The updates of the a j values are done in each optimization step in parallel.

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## Acknowledgements

GD is supported by the ERC Advanced Grant DYSTRUCTURE (n. 295129), by the Spanish Research Project PSI2016-75688-P. MLK is supported by the ERC Consolidator Grant: CAREGIVING (n. 615539) and Center for Music in the Brain, funded by the Danish National Research Foundation (DNRF117). VJ and GD are supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement n. 720270 (HBP SGA1). VJ and PR are supported by the James S. McDonnell Foundation (Brain Network Recovery Group JSMF22002082). VJ is supported by FHU EPINEXT [A*MIDEX project (ANR-11-IDEX-0001-02) funded by the ‘Investissements d’Avenir’ French Government]. PR is supported the German Ministry of Education and Research (US-German Collaboration in Computational Neuroscience 100258846 and Bernstein Focus State Dependencies of Learning 01GQ0971-5), the Max-Planck Society and funding from the European Union Horizon 2020 (ERC Consolidator grant BrainModes 683049).

## Author information

G.D., M.K., V.J. & P.R. designed research, contributed data, carried out the analyses, prepared figures, wrote the manuscript.

### Competing Interests

The authors declare that they have no competing interests.

Correspondence to Morten L. Kringelbach.

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