Role of homeostatic plasticity in critical brain dynamics following focal stroke lesions

Abstract

Can focal brain lesions, such as those caused by stroke, disrupt critical brain dynamics? What biological mechanisms drive its recovery? In a recent study, we showed that focal lesions generate a sub-critical state that recovers over time in parallel with behavior (Rocha et al., Nat. Commun. 13, 2022). The loss of criticality in a cohort of stroke patients was associated with structural brain disconnections, while its recovery was accompanied by the re-modeling of specific white-matter tracts. These results were challenged by Janarek et al. (Sci. Rep. 13, 2023), who proposed an alternative interpretation for the anomalous monotonic decaying of the second cluster size, which is the neural signature originally used to infer loss of criticality. The present study tackles this controversy and provides evidence that the theoretical framework proposed by Janarek et al. cannot explain the anomalous cluster dynamics observed in our patients. Notably, this invalidates the claim that the brain maintains its critical dynamics regardless of the lesion severity. In addition, we explore biological mechanisms beyond white-matter remodeling that may facilitate the recovery of criticality over time. We considered two distinct scenarios: one where we suppress homeostatic plasticity, and another where we increase the excitability of brain regions. We find that suppressing homeostatic plasticity - specifically, the inhibition-excitation balance - disfavors the emergence of criticality. Conversely, increasing brain excitability can help to restore criticality when the latter is disrupted. Our results suggest that normalizing the excitation-inhibition balance is crucial for supporting recovery of critical brain dynamics.

Introduction

In a recent study by our group¹, we tested the hypothesis that focal brain injury can cause a loss of criticality in the intrinsic brain dynamics. Note that spontaneous brain activity (e.g., in resting state) depends on the structural organization of white matter pathways (i.e., the physical connections between brain regions) and its functional significance is firmly established by its relation with both task-related brain activity and behavioral performance (see², for review). We used stroke as the prototypical pathological model of focal brain injury in humans and derived structural connectivity matrices via diffusion-weighted imaging (DWI) and tractography from a cohort of stroke patients and healthy controls. We employed an effective whole-brain model based on stochastic dynamics governed by transition rules that take into account, among other factors, not only the brain topology but also the network excitability, that is the excitation-inhibition balance, as described by the weights of the connectivity matrix. We originally normalized the weights of the connectivity matrix as a way to include homeostatic principles guiding the whole-brain dynamics³.

Through extensive computational simulations, we showed that focal lesions affect neural dynamics criticality through disconnectivity, specifically below a certain level of average degree and connectivity disorder (entropy). Also some patients recover criticality in parallel to behavior, and this improvement is associated with remodeling of specific white matter connections measured directly with diffusion MRI and tractography.

In a recent study Janarek et al⁴ proposed an alternative interpretation for the results described in Rocha et al¹. Based on numerical simulations using synthetic networks as well as ‘lesioned versions’  of (healthy) empirical connectivity matrices, the authors argued that the patients neural dynamics remain critical after stroke, regardless of the severity of brain injury. The authors’ central argument was based on the topology of connectivity matrices. Accordingly, the breakdown of a fully connected structural network into smaller subnetworks would produce false noncritical behavior. Thus, the monotonic decay of the second largest cluster of neural activity ($S_2$) that we previously used as the order parameter to infer lack of criticality would not be a valid signature due to loss of integrity of the brain network. In these situations other order parameters can be used, and the authors suggest, for instance, the standard deviation of the activity ($\sigma _A$) and the first autocorrelation coefficient ($\rho _1$). Clarifying such controversy in the interpretation of the results is important to understand the role of criticality not only in stroke but also for other neurological and psychiatric disorders. The hypothesis of loss of criticality has been described in other clinical contexts, for instance, epilepsy^5,6,7, slow-wave sleep⁸, anesthesia⁹, sustained wakefulness¹⁰, states of (un)consciousness^11,12, Alzheimer’s disease¹³, to name a few (see¹⁴, for review). Despite having distinct etiologies, the aforementioned brain conditions share, to different degrees, changes in structural connectivity and/or abnormalities in the excitation-inhibition balance^{15,16,17,18,19}.

In this study, we revisit the issue of loss of criticality in stroke, providing further evidence that it is indeed found in our patients’ cohort as a consequence of the focal lesions¹. More importantly, we also explore the role of homeostatic plasticity principles in influencing neural dynamics. Indeed, our previous study did not address the specific contribution of homeostatic plasticity to modeling normal and impaired dynamics in stroke. Increasing evidence suggests that neurological disorders, including stroke, affect the excitation-inhibition balance^17,18. Tissue near the lesion or at a distance through short and long-range structural disconnection can show abnormal slow waves that reflect alterations of the excitation-inhibition balance^16,20,21, and that may recover over time¹⁹.

The present results are organized as follows. First, we show that the patients’ connectomes do not conform to the theoretical framework proposed by Janarek et al⁴ (see below), specifically the great majority of individual connectivity matrices (149 out of 159) were fully connected, indicating that they were not partitioned into smaller subnetworks by the lesions, making their conjecture invalid to explain the anomalous behavior of the second cluster size in the present stroke cohort. Therefore, the framework proposed in⁴ is not suitable to claim maintenance of criticality when applied to the real structural connectivity matrices of stroke survivors. Secondly, we show that lack of plasticity mechanisms that balance whole-brain dynamics reduces network heterogeneity (i.e, entropy or connectivity disorder), which in turn hinders the emergence of criticality and the fit to empirical data. Conversely, increasing brain excitability can help restore criticality when it has been disrupted by stroke. We discuss the normalization of the excitation-inhibition balance as a key mechanism to support recovery of critical brain dynamics.

Results

Simulation of large-scale neural dynamics

We modeled the whole-brain activity through a stochastic dynamics based on a discrete cellular automaton as described in previous publications^1,3,22. Individual structural connectivity matrices are the key inputs of the model. Imaging and behavioral data are taken from a large prospective longitudinal stroke study described in previous publications^1,23,24,25. Structural connectivity data was available for 79 patients, acquired three months ($t_1$) and one year ($t_2$) after stroke onset. The same study includes data from 28 healthy controls, acquired twice three months apart (see the original study¹ for details about the stroke dataset, lesion analysis, diffusion weighted imaging (DWI), and resting-state Functional magnetic resonance imaging).

The human connectome is described as a structural network of $N=324$ nodes (i.e., cortical brain regions), linked with symmetric and weighted connections obtained from DWI scans and reconstructed with spherical deconvolution^26,27. The weights of the connectivity matrix, $W_{ij}$, describe the connection density, i.e, the number of white matter fiber tracts connecting a given pair of regions of interest (ROIs) normalized by the product of their average surface and average fiber length²⁸. The ROIs are derived from a functional atlas of the cerebral cortex following Gordon et al parcellation²⁹ (we refer to Table 3 for the community assignments of ROIs within resting state networks (RSNs) and their corresponding MNI coordinates).

Fig. 1

Illustration of the whole-brain dynamics. Neural activity is modeled through a cellular automaton with three states, namely, inactive (I), active (A) and refractory (R). Time evolves in discrete steps from left to right. The temporal evolution of the 7-th inactive node (salmon) is as follows: in $t_1$, it is surrounded by two active nodes (green) and two inactive nodes (salmon); in $t_2$, the incoming excitation from nodes 5 and 6 is propagated (i.e, ${\tilde{W}}_{76}s_6+{\tilde{W}}_{75}s_5>T$); and finally, in $t_3$, it reaches the refractory state (blue). At each time step, one may compute several quantities, in particular, the clusters activity, which are defined as ensembles of nodes that are structurally connected to each other and simultaneously active. For example, in the first time step, the brain state is composed by two clusters; the first cluster ($S_1=2$) formed by nodes 5 and 8 and the second cluster ($S_2=1$) formed by node 6). The brain network represented in this figure is for illustration purposes and does not correspond to the empirical networks used in simulations.

Whole-brain activity is modeled through a discrete cellular automaton with three states, namely, active (A), inactive (I), and refractory (R). The state variable of a given node i, $s_i(t)$, is set to 1 if the node is active and 0 otherwise. The temporal dynamics of the i-th node is governed by the following transition probabilities between pair of states: (i) $I \rightarrow A$ either with a fixed small probability $r_1 \propto N^{-1}$ or with probability 1 if the sum of the connections weights of the active neighbors j, $\sum _j W_{ij}$, is greater than a given threshold T , i.e., $\sum _j W_{ij}s_j >T$, otherwise $I \rightarrow I$, (ii) $A \rightarrow R$ with probability 1, and (iii) $R \rightarrow I$ with a fixed probability $r_2$³. Figure 1 illustrates the node dynamics for an arbitrary network. The state of each node is overwritten only after the whole network is updated. The two parameters $r_1$ and $r_2$ control the time scale of self-activation and recovery to the excited state, while T plays the role of a threshold parameter that regulates the propagation of incoming excitatory activity^1,3.

Next, we normalize the network excitability, so the excitation-inhibition balance, through the following rule,

$$\begin{aligned} {\tilde{W}}_{ij} = W_{ij}/ \sum _j W_{ij}. \end{aligned}$$

(1)

The above normalization fixes the weighted in-degree of all regions to 1, ensuring that each region has at the mesoscopic level a similar contribution on regulating the brain activity, a form of homeostatic plasticity³.

Criticality marks

In Fig. 2 we exemplify the dynamic behavior of the model for fully connected small-world Watts-Strogatz (WS) networks with different topologies by varying the average degree (k) and different activation regimes (by varying the $r_2$ model parameter). Following⁴, we used an exponential random distribution to mimic the weight distribution of the human connectome, $p(w)=\lambda e^{-\lambda w}$, with $\lambda =12.5$. The rewiring probability used to build the synthetic WS networks was $\pi =0.5$. See the methods section for additional information regarding input model parameters. The model experiences distinct dynamic patterns as a function of the activation threshold T. To identify the critical state, we resort to the clusters activity, in particular the average of the first ($\langle S_1\rangle$) and the second ($\langle S_2 \rangle$) largest clusters sizes. Clusters of activity are defined as ensembles of nodes that are structurally connected to each other and simultaneously active^1,3. In this arbitrary example, criticality in the model occurs in the combination of parameters given by $r_2=0.4$ and $k=10$ (Fig. 2, middle). For fully connected brain networks, the critical state is characterized by a sharp peak of $\langle S_2\rangle$ around $T=T_c$ that diverges as a power law when $N \rightarrow \infty$^30,31,32. The first cluster size ($\langle S_1 \rangle$) exhibits a steep change around $T=T_c$ while $\langle S_1\rangle /N$ converges to a finite value when $N\rightarrow \infty$ (see collapse of $\langle S_1\rangle /N$ for varying brain size N for $T<T_c$). In addition, brain activity has the largest variability $\sigma _A$ and largest first autocorrelation coefficient $\rho _1$.

In the other two sets of parameters, the model does not support a critical state for any value of T (Fig. 2, right/left). However, it is interesting to analyse the distinct behavior of dynamic variables. For $k=4$ and $r_2=0.4$ (Fig. 2, left), some neural variables presents monotonic decay, for instance, $\langle S_1\rangle$ (with $\langle S_1\rangle /N \rightarrow 0$ with $N\rightarrow \infty$), $\langle S_2\rangle$ and $\sigma _A$, while others a sharp peak, like $\rho _1$. On the other hand, for $k = 10$ and $r_2=0.1$ (Fig. 2, right), in addition to the peak at $\rho _1$ we also observe the emergence of a sharp peak at $\sigma _A$.

In the study of Janarek et al.⁴, synthetic Watts-Strogatz (WS) networks were used to exemplify the diverse dynamic behaviors (critical vs. noncritical) depending on the topology of the input matrix W in the very same way as we did in Fig. 2. The parameters used in their simulations exemplified clearly discernible noncritical/critical dynamics. Noncritical dynamics was characterized by absence of peaks in $\langle S_2\rangle$, $\sigma _A$ and $\rho _1$, while critical systems have peaks in these same variables (see Fig. 1 in⁴). But caution is required to generalize this behavior. In Fig. 2 (third column) we showed an intermediate case, where a noncritical system displays peaks at $\sigma _A$ and $\rho _1$ but monotonic decaying of $\langle S_2\rangle$ and $\langle S_1\rangle /N \rightarrow 0$ as $N\rightarrow \infty$. Therefore, for fully connected networks, peaks in the standard deviation of activity or in the first autocorrelation coefficient should be interpreted with caution, as they are not sufficient conditions for criticality.

Interestingly, in Fig. 2, we demonstrate that, starting from a fixed Watts-Strogatz topology ($k=10$ and $\pi =0.5$), a decrease (or increase) in $r_2$, which regulates the transition time between refractory and quiescent states, can disrupt (or recover) the critical transition. This simple mechanism may be associated to homeostatic plasticity processes during stroke recovery, enabling the brain to maintain a critical state through increased excitability. We will further explore this concept in the subsequent sections.

Clarifying the controversy

The nature of the divergence in the interpretation of results recently claimed in⁴ concerns the anomalous behavior of the neural variables in stroke, in particular the clusters activity. As we showed in¹, the lesion can affect the normal neural profiles as a function of the threshold parameter T, producing monotonic decay in $\langle S_2\rangle$, flattening/decreasing $\langle S_1\rangle$ as well as functional connectivity (FC) and $\sigma _A$ (these last two showing smoother peaks), forming a scenario qualitatively similar to that illustrated in Fig. 2 (third column). Based on the available theory^30,32, we interpreted this body of evidence as absence of criticality. This interpretation was challenged by Janarek et al⁴ who claim that the critical state is maintained regardless of the monotonic decay of $\langle S_2\rangle$, in other words, the lesion is not capable of disrupting the critical dynamics. Accordingly, the anomalous behavior of $\langle S_2\rangle$ is a consequence of the loss of structural network integrity. Ideally, disruption of the fully connected structural brain network into subnetworks of varying sizes produces false noncritical behavior. In this scenario, neural activity cannot spread throughout the network, becoming trapped in these subnetworks (whose dynamics are independent to each other), generating incorrect cluster size hierarchy (i.e, cluster ordering) so a false anomalous $\langle S_2\rangle$ (noncritical) behavior.

Fig. 2

Behavior of neural variables for fully connected Watts-Strogatz networks for varying topologies and input model parameter supporting critical and noncritical regimes. In this example, the critical transition is mediated by the average degree (k) and the parameter regulating the refractory-to-silent time scale ($r_2$). We fixed the rewiring probability to $\pi =0.5$ and $r_1=7\cdot 10^{-3}$. See legend for details of network size N. The neural variables considered are: the average of the first cluster size normalized by the network size $(\langle S_1\rangle /N)$, the average of the second cluster size $(\langle S_2\rangle )$, the standard deviation of the total activity $(\sigma _A)$ and the first autocorrelation coefficient $(\rho _1)$. First column: example of a noncritical system exhibiting a monotonic decay in $\langle S_2\rangle$ (as well as in $\sigma _A$) but a peak in $\rho _1$. The scaled variable $\langle S_1\rangle /N$ converges to zero with $N\rightarrow \infty$ as expected for noncritical dynamics³². Second column: example of a system supporting criticality; the critical transition is characterized by a peak in $\langle S_2\rangle$ while $\langle S_1\rangle /N$ converges to a finite value for $N\rightarrow \infty$. In the inset of $\langle S_2\rangle$ we simulate the dynamics for disconnected networks with two components of variable size (2 and 298, respectively). The second cluster behaves exactly like the fully connected network, with a peak in the same position ($T=T_c$) of the corresponding fully connected network (dots). Third column: example of noncritical system exhibiting peaks in $\sigma _A$ and $\rho _1$, but monotonic decay of $\langle S_2\rangle$ and $\langle S_1\rangle /N \rightarrow 0$ with $N \rightarrow \infty$. Therefore, for fully connected networks, peaks at $\sigma _A$ and $\rho _1$ are not sufficient conditions of critical phase transitions.

The mechanism for generating the anomalous clusters activity described in⁴ is correct, but its range of validity is very limited and it cannot explain the anomalous $\langle S_2\rangle$ behavior in the present stroke cohort (and most likely in any other cohort). The authors employ a minimal synthetic stroke model, using Hagmann et al.’s average connectome²⁸ as a representative healthy brain (parcellated into $N=998$ regions of interest, which define the brain network’s nodes). The synthetic strokes are built by lesioning inter- resting state connections (inter-RSN), i.e., by gradually removing a fraction of connections from a given RSN (eg. visual network) with other nodes not belonging to the same RSN (eg, auditory, default mode, etc). By varying the fraction of removed nodes, they aimed to parametrize stroke severity in terms of the disconnection of an RSN from the rest of the brain. As we shall see below the assumptions regarding the topology of synthetic stroke networks is too simplistic and it does not accurately describe the empirical patterns of structural disconnection of actual stroke patients^33,34.

In the original study¹, we used some standard measures from graph theory to quantify the topology of the healthy and stroke brain networks, e.g. the average degree, modularity, global efficiency etc. Here, we further characterize their topology by examining the disconnection patterns at the level of resting-state networks (RSN). Accordingly, we compute the total number of inter-RSN connections from the binarized version of the connectivity matrix (W),

$$\begin{aligned} N_\textrm{inter}^k = \sum _{\begin{array}{c} i\in RSN_k \\ j\notin RSN_k \end{array}} W_{i,j}, \end{aligned}$$

(2)

where $k=1,2,\cdots ,13$ is the total number of resting-state networks and $N^k_\textrm{inter}$ counts the number of connections from ROI’s belonging to the kth RSN (eg. visual network) to the remaining ROI’s not belonging to the kth RSN (eg. auditory, cingolo-opercular, default mode, etc). Next we normalize the number of connections by the average across control subjects,

$$\begin{aligned} d_\textrm{inter}^k = \frac{N_\textrm{inter}^k}{\langle N_\textrm{inter}^k \rangle _\mathrm{con.}}, \end{aligned}$$

(3)

where $d_\textrm{inter}<1$ means decreased connectivity compared to healthy controls, $d_\textrm{inter}=1$ is the average across controls subjects and $d_\textrm{inter}>1$ increased connectivity compared to healthy controls.

Fig. 3

Empirical inter-RSN disconnectivity ($d_{inter}$)in stroke and healthy controls. $d_\textrm{inter}$ reflects the fraction of inter-network disconnectivity - thus the loss of connections from a given RSN to the rest of the brain, normalized to healthy controls. Therefore, $d_\textrm{inter}<1$ (decreased connectivity compared to controls average); $d_\textrm{inter}=1$ (controls average); $d_\textrm{inter}>1$ (increased connectivity compared to controls average). In total, 13 resting-state networks were considered. Each individual curve shows the rank of $d_\textrm{inter}$ (from highest disconnectivity to lowest). Panels (a)-(c) show $d_{inter}$ for individual participants in each group (patients at $t_1$/$t_2$ and controls, respectively). Panel (d) shows the mean of the corresponding groups. Panels (e)-(h) show $d_\textrm{inter}$ for artificial stroke newtorks according to⁴. In this particular example, we perform synthetic lesions in the default mode network (composed by 40 brain regions), visual network (38), auditory network (23), and sensory-motor mouth network (8), respectively. The fraction of nodes removed in all RSN is about $80\%$ (see main text). In all panels (a)-(h), the average across control subjects is indicated in blue. Unlike the synthetic networks analyzed in⁴, empirical stroke networks present a more distributed disconnection profile. The individual profiles reveals that $d_\textrm{inter}$ is almost uniformly distributed across RSN (see the smooth slope), i.e., the lesion affects almost uniformly all RSN in contradiction with the artificial strokes used in⁴. In this case, the profiles behave like a step-function, with an accentuated decrease of $d_\textrm{inter}$ in the first RSN. Empirical inter-RSN disconnectivity patterns shows little or no change, as shown in the panel (d), where the lines for $t_1$ and $t_2$ are overlapping for all 13 RSNs (two tailed t-test: $p>0.5$). The asterisks in the plots represent significant two tailed t-tests ($p<0.05$) comparing real connectivity profiles in the control group with real lesions in stroke patients (a-d) and synthetic lesions (e-h).

To have a fair comparison, we construct artificial stroke networks following the same procedure described in⁴. Synthetic lesions were applied to all control subjects by randomly sampling a fixed fraction of nodes (about $80\%$) from a given RSN and then removing their connections to the remaining ROI’s not belonging to the selected RSN. Next, we compute $d_\textrm{inter}$ through (3) for empirical (see Fig. 3, panels a-d) as well as artificial stroke networks (see Fig. 3, panels e-h). In our cohort, stroke lesions are distributed heterogeneously throughout the brain, resulting in nonlinear effects on the structural connectome and affecting different resting state newtorks in varying proportions. Consequently, for each subject, we rank $d^k_\textrm{inter}$ from highest to lowest disconnectivity, so the order of the k-th inter-RSN disconnectivity vary from subject to subject, reflecting an expected inter-individual variability in the empirical connectomes of stroke subjects.

The empirical disconnectivity patterns observed in our study differ significantly from those reported for synthetic lesions in⁴. The empirical disconnection profiles for individual participants reveal that $d_\textrm{inter}$ is almost uniformly distributed across resting state networks (RSNs), as indicated by the smooth slope. In contrast, synthetic lesions primarily impact a single RSN, leading to a step-function behavior where $d_\textrm{inter}$ shows a pronounced decrease in the affected RSN, while other RSNs are minimally impacted and converge toward the average connectivity of controls as the rank decreases. For example, this convergence is evident in networks of varying sizes, such as the default mode network (comprising 40 brain regions), the auditory network (23 regions), and the sensory-motor mouth network (8 regions). However, the visual network (38 regions) demonstrated a more pronounced effect when $80\%$ of its nodes were removed, resembling the empirical data but with reduced separation between averages.

For real lesions, none of the controls exhibited $d_\textrm{inter}<0.25$, indicating that no inter-RSN connections were $75\%$ weaker than the average inter-RSN connectivity. This result was mirrored in patients, with only a few presenting $d_\textrm{inter}<0.25$ (never reaching $d_\textrm{inter}=0$, meaning no complete disconnection of an RSN from the rest of the brain). Specifically, this occurred in just one RSN for three patients at $t_1$ and one patient at $t_2$. Interestingly, Janarek et al. report a monotonic behavior of $\langle S_2\rangle$ for a high fraction of inter-RSN node removals, ranging from $75\%$ to $100\%$ (see Fig.2 in⁴).

We conducted t-tests to examine significant differences between controls and patients, as well as between patients at the two time points. For real lesions, our analysis indicated that all rank comparisons for patients yielded non-significant differences ($p>0.5$), suggesting stable disconnections across time points. In contrast, significant differences were found between controls and patients at both time points ($p<0.05$), for all ranks except the lowest at $t_1$ and for the two lowest ranks at $t_2$. This suggests that resting state networks are weakly impaired at the lower ranks.

When we replicated this analysis for synthetic lesions, we observed a notably different pattern. The RSNs were impaired at higher ranks, with no significant differences in lower ranks compared to the control group. For instance, the default mode network (DMN) showed a significant decrease in $d_\textrm{inter}$ up to the third rank, while the visual network indicated significant differences up to the seventh rank. This variation may be attributed to the connectivity topology of the visual network, which has a greater density of intra-network connections compared to the DMN. These results highlight that the impact of lesions on structural connectivity differs substantially between real and synthetic lesions. Real stroke lesions affect both long-range and short-range connections, resulting in RSN disconnections that are more widespread than those suggested by Janarek et al. Therefore, the minimal stroke model employed in⁴ is not representative of the empirically lesioned networks.

As a final test of Janarek et al.’s conjecture, we can assess whether isolated clusters of activity are actually present within the individual connectomes of stroke patients. Isolated clusters refer to subnetworks created by lesions that cannot interact with ROIs in other subnetworks. Hence we calculated for each participant the number of connected components in the corresponding structural brain network (W). We note in passing that, somewhat surprisingly, this analysis was not performed in⁴. The overwhelming majority of the patients’ networks are fully connected (149 out of 159, see Table 1), i.e. formed by a single giant cluster ($n_c=1$). In particular, for those few networks with more than one cluster ($n_c>1$), the second largest and the third largest subnetworks are of negligible size (subnetwork size $=2$) in all cases. The smallest subnetwork is about $0.5\%$ of the whole system size, one order of magnitude below the scenarios described in⁴ ($\gtrsim 7\%$). These results demonstrate that the theoretical assumptions made in⁴ are not representative of empirical connectivity matrices across the entire stroke cohort, thereby making their conjecture invalid for explaining the anomalous behavior of cluster activity and the consequent claim that criticality is maintained despite brain damage.

Table 1 Number of connected components ($n_c$) in the structural brain networks for all individual participants in Rocha et al¹. The overwhelming majority of the structural connectomes are fully connected (149 out of 159). For those few networks with more than one connected component ($n_c>1$), the second largest and the third largest subnetworks are of negligible size ($=2$) in all cases.

As a control simulation, we tested whether this unbalanced topological configuration could generate monotonic behavior of $\langle S_2\rangle$. We started from a fully connected Watts-Strogatz network with $k=10$, $\pi =0.5$, $r_2=0.4$ and $N=300$. Then, we randomly selected a pair of connected nodes ($i^\star ,j^\star$) and removed their connections with the remaining nodes, thereby forming two subnetworks of variable size (2 and 298, respectively). We simulated the model for several random assignments of ($i^\star ,j^\star$). As expected, the clusters average behaves exactly like the fully connected network (see Fig. 2, middle column, inset of $\langle S_2\rangle$), with a peak in $\langle S_2\rangle$ at $T=T_c$. Therefore, the cluster hierarchy does not play any significant role for such unbalanced distribution of subnetwork sizes. Importantly, this result holds even in the absence of the normalization term (1).

Role of the excitation-inhibition balance and network excitability

The adopted whole-brain model relies on the structural connectivity network (W) to propagate the neural excitability throughout the system. Therefore, brain dynamics are shaped by a trade-off between network topology and connection weight. From a dynamic point of view, we can map the weights of W with the brain regions excitability. Indeed, $\sum _j W_{ij} s_j(t)$ is the total amount of excitation received by region i at a given instant of time t. Therefore, the heterogeneity of the network weights ($W_{ij}$) shapes the spatio-temporal properties of the system, having a crucial impact on the emergence of criticality itself^1,3.

In the original study¹, we normalized the network excitability through Eq. (1). In biological terms, normalization could be viewed as a homeostatic plasticity mechanism aiming to regulate excitation and inhibition through the balancing of weight connections.

Fig. 4

Effect of excitation-inhibition (1) balance in network topology. Panel (a): probability distribution function of the structural connectivity weights in controls and patients before (unnormalized) and after (normalized) homeostatic plasticity (see main text, Eq. (1)). For each group, the non-zero weights of all individual matrices were pooled together and then the histogram was computed leading to a representative PDF for the corresponding group. The black lines are power law fits ($p \sim W_{ij}^{-\alpha }$) in the region of the log-log graph with linear behavior, with exponents obeying $\alpha _{nor} =2,15$ and $\alpha _{un}=4,25$, for the normalized and unnormalized distributions, respectively. Note enhancement of network heterogeneity following homeostatic plasticity. Panel (b): connectivity disorder (i.e., structural entropy, Eq. (12)) for all patients and controls for normalized and unnormalized connectivity matrices. Controls are colored in blue ($n=46$). Brown dots represent patients at $t_1$ (3 months post-stroke, $n=54$), while green dots at $t_2$ (12 months post-stroke, $n=59$). Normalized density plots (histograms) are also shown. Inclusion of excitation-inhibition balance drastically increases both, the connectivity disorder (i.e., entropy) and the separation of the PDFs between controls and stroke patients and between stroke patients at different time-points.

Figure 4-a presents the probability distribution of connection weights for both normalized (1) and unnormalized connectivity matrices. Notably, balancing the excitation input enhances the heterogeneity of network connections, thereby affecting regional excitability. The straight lines represent power law fits ($p \sim W_{ij}^{-\alpha }$) in the linear region of the log-log graph, with exponents of $\alpha _{nor}=2,15$ and $\alpha _{un}=4,25$ for the normalized and unnormalized distributions, respectively. Following the methodology outlined in¹, we quantified connectivity disorder through the entropy of the input matrices (Eq. (12), see Methods). The lack of balance leads to a significant decrease in entropy for all groups compared to their normalized counterparts (Fig. 4-b). The entropy distribution for stroke patients remains relatively stable across both time points, showing no statistically significant difference between the two distributions (unpaired two-tailed t-test: $t=0.18, \,p=0.85$). In contrast, balancing the connections not only increases entropy but also enhances the separation between healthy controls and stroke patients, as well as between stroke patients at both time points (unpaired two-tailed t-test: $t=1.06,\, p=0.28$).

A mixed-design analysis of variance (ANOVA) was performed to assess the differences in entropy between healthy controls and stroke patients across two time points, using time as within-subjects factor. The analysis revealed a significant main effect of group for both normalized and unnormalized distributions ($p=0.002$), indicating that stroke patients exhibited a significantly different level of entropy compared to healthy controls. The main effect of time was not significant for either model, reaffirming that there were no significant differences in entropy across the time points for stroke patients, as previously shown by the t-tests. The interaction between group and time approached significance only for the normalized model ($p=0.019$), suggesting that the change in entropy over time may differ between stroke patients and healthy controls.

Fig. 5

Plausible mechanisms supporting neural dynamics criticality in a representative stroke patient. Analysis of neural activity patterns, $\langle S_1\rangle /N$ and $\langle S_2\rangle$, of a stroke participant. In blue dashed line we show the corresponding control’s group average, while the shaded area corresponds to one standard deviation. For the patient, $t_1$ and $t_2$ correspond to 3 months and 12 months post-stroke. The critical transition appear as a trade-off between balanced excitability across brain regions (i.e., homeostatic plasticity) and overall brain excitation, as regulated by the $r_2$ parameter. In the low $r_2$ regime, we have $r_2=0.36$, while in the high regime, $r_2=0.7$. In low regimes of brain excitability (low $r_2$), unbalanced connectivity weights fail to support a critical state at both time-steps (a-c). Normalizing network excitability brings the dynamics closer to criticality at $t_2$ but not at $t_1$ (b-d). Conversely, increasing brain excitability (high $r_2$) drives the unbalanced network toward criticality, as evidenced by a peak in $\langle S_2\rangle$ at $t_2$ (e-g). The balanced network in the high $r_2$ regime shows a sharper transition at $t_2$ compared to the low $r_2$ regime (f-h).

The network measures are greatly affected by the normalization of the excitation input. However, how do these changes reflect on the dynamic patterns displayed by the model? Figure 5 exemplifies for a representative stroke patient the model’s behavior for different scenarios, where two effects were tested: i) the effect of excitation-inhibition balance (Eq. (1)) and ii) the increase in overall brain excitability (as regulated by the $r_2$ parameter, varying from 0.36 to 0.7). For low regimes of overall brain excitability (low $r_2$), the unnormalized network does not support a critical state for any value of T, as revealed by the monotonic decay of $\langle S_2\rangle$ at both time steps (qualitatively similar to Fig. 2, first/third columns). The patient’s scaled variable $\langle S_1\rangle /N$ is well below the controls average, specially at time $t_1$. However, by imposing the excitation-inhibition balance, the brain network exhibits a smooth peak suggesting a critical state at time step $t_2$ but not at $t_1$. Increasing brain excitability (high $r_2$) renders the unbalanced network critical (i.e, smooth peak at $\langle S_2\rangle$ appears) at $t_2$ but not at $t_1$ (e-g). The balanced network now shows a sharper transition compared to the low $r_2$ regime in $t_2$ (f-h). For this particular stroke participant, the interplay between homeostatic plasticity and increased excitability facilitated the emergence of criticality at $t_2$. However, when only one mechanism – namely, increasing brain excitability through $r_2$ – was applied without homeostatic plasticity, brain criticality was not achieved at $t_1$ due to the network architecture, specifically the decreased connectivity disorder (see Fig. 4). Detailed neural profiles for each stroke patient and healthy control can be found in the Supplementary Information.

Fig. 6

Group-based analysis of neural activity patterns ($S_1/N, S_2, \sigma _A, \rho _1$) as a function of T for all patients and controls in the low $r_2$ scenario described in Fig. 5. The first line presents the unnormalized model, while the second line shows the rescaled model using $T/\langle W\rangle$. The third line displays the results for the normalized model. Brown lines represent patients at $t_1$ (3 months post-stroke, $n=54$), and green lines represent patients at $t_2$ (12 months post-stroke, $n=59$). Thin solid curves indicate individual stroke patients, whereas thick dashed lines represent the group average. The blue dashed line depicts the average across control subjects. Group-level neural patterns exhibit significant differences between the unnormalized and normalized models. In the unnormalized model, the critical point is influenced by individual differences in brain networks³. The second largest cluster shows a peak around $T=0.02$ for healthy controls, but this peak is absent for patients at both time points. When the threshold is rescaled using $T/\langle W\rangle$, a collapse of the neural patterns is observed³. This collapse is particularly noticeable in $\langle S_1\rangle /N$, $\sigma _A$ and $\rho _1$, although it is less pronounced in the second cluster, where patient peaks are absent at both time points. Conversely, a clear trend toward normalization is evident from $t_1$ to $t_2$ in the second largest cluster of the normalized model. Additionally, the variability of activity and the first autocorrelation coefficient are significantly lower compared to the normalized model. The horizontal dashed line in the $\rho _1$ graph illustrates this effect, indicating that the peak of $\rho _1$ in the normalized model exceeds that of the unnormalized model.

Neural patterns at the group level reveal marked differences between the unnormalized and normalized models (see Fig. 6). First, we observe that the critical point is individual dependent (as expected from theory, see³). Surprisingly, the average across individuals of the second largest cluster shows a peak around $T=0.02$ for the controls, while this peak disappears for the patients at both time points. This behavior may be related to the smoothing of the average caused by the individual-dependent peak positions. However, when we rescale the threshold by $T/\langle W \rangle$, a collapse of the neural patterns is anticipated³. This collapse is quite pronounced for $\langle S_1 \rangle /N$, $\sigma _A$, and $\rho _1$, but is less evident in the second cluster. In this cluster, we observe the absence of peaks for the patients at both time points, suggesting that, at the group level, a considerable number of individuals do not exhibit critical behavior. Conversely, there is a noticeable trend toward normalization from $t_1$ to $t_2$ in the second largest cluster of the normalized model. Additionally, we find that the standard deviation of activity and the first autocorrelation coefficient are significantly reduced compared to the normalized model.

Overall these findings show the great importance of weight normalization in increasing entropy and in supporting criticality, as well as in separating stroke from controls and different stages of recovery.

Predicting brain-behavior-functional connectivity relationships

Next, we investigated the role of homeostatic plasticity in predicting the patient’s behavioral and functional connectivity recovery. The patient’s behavioral performance (for more details see¹) was inferred from a neuropsychological battery measuring performance in 8 behavioral domains (motor left / right, language, verbal and spatial memory, attention visual field, attention average performance, attention shifting). We used principal component analysis (PCA), a common data reduction strategy that identifies hidden variables or factors, to capture the possible correlation of behavioral scores across participants. In each domain one component explained the majority of variance across participants (55-77$\%$ depending on the domain). The component scores were normalized with respect to healthy controls (mean = 0 and SD = 1). This normalization allows to express the patients’ scores in units of standard deviations below average – i.e. $B(\textrm{Lang})=-4$ is equivalent to language function 4 SD below the control average. Here, to characterize each patient’s overall performance, we use the average component score obtained from averaging the normalized factor scores across domains, $B=\sum _i B_i/8$.

Following¹, we employ the threshold independent variables, $I_1=\int S_1 dT$ and $I_2=\int S_2 dT$, over the entire range of excitability T, to quantify the relationship between dynamical and behavioral deficits. In the previous study¹ we showed that such variables predicted a large fraction of variability of the behavioral patterns. Here we test if this relationship is influenced by the presence/absence of mechanisms of homeostatic plasticity that mimic excitation-inhibition balance.

Table 2 shows the relevant statistical tests (linear regressions) corrected for multiple comparisons (FDR, $\alpha =0.05$) between $I_1$ and $I_2$ with the composite behavioral score B. The table also shows the respective sample size (n), Pearson correlation ($\rho$) and p-value, corrected for multiple comparisons. To allow a direct comparison, results are shown for the two versions of the model (namely, normalized and unnormalized). The results clearly show significant correlations both at three and twelwe months with dynamical variables $I_1$ and $I_2$, but only when the normalization is applied.

We carried out a similar analysis on the funcional connectivity measures. Previous studies from our group have shown that functional connectivity abnormalities are predictive of behavioral impairment acutely (2 weeks) and recovery of function^24,35,36, specifically, the loss of inter-hemispheric connectivity is strongly predictive of behavior. Hence here we correlated the dynamics variables obtained from normalized vs. unnormalized models with homotopic inter-hemispheric connectivity (homo-FC$_e$) and mean FC (FC$_e$). We confirmed a significant correlation between dynamic variables and homotopic connectivity at both three and twelwe months, but only for normalized models.

Table 2 Role of excitation-inhibition balance in predicting brain-behavior-functional connectivity relationships. Statistical correlates between dynamical, functional and behavioral patterns. In the table we show the variables, the linear correlation ($\rho$), the sample size (n), and the p-value. In bold we show the significant p-values (two-tailed, FDR corrected, $\alpha = 0.05$). Inclusion of homeostatic plasticity greatly improves the prediction of functional and behavioral variability.

Discussion

Several theoretical and experimentally oriented studies have addressed the hypothesis that the brain is a system poised at criticality (see³⁷, for review), a dynamical regimen that provides optimal functional and behavioural advantages. An emerging hypothesis is that critical dynamics would be disrupted in neurological and psychiatric disorders, making this topic of crucial interest in theoretical / computational neuroscience and triggering potential applications at the interface of translational neuroscience¹⁴.

In this context, in a recent study by our group¹, we tested the hypothesis of loss of criticality in stroke and investigated its relationship with the patients functional and behavioural performance. We showed that focal lesions can disrupt the neural dynamics criticality depending on the extent of the brain damage (i.e, the white-matter disconnectivity pattern). We also showed, for some patients, a recovery of criticality over time, accompanied by an improvement in behavior. The latter result highlighted the tight relationship between critical dynamics and behavioral performance. These results were recently challenged by Janarek et al⁴, who proposed an alternative interpretation. Accordingly, the brain maintains its critical dynamics regardless of the lesion severity and the monotonic decaying of $\langle S_2\rangle$. In other words, the reported loss of criticality would be an artifact of loss of network integrity, such that the neural dynamics remain critical, as supported by additional (but questionable) markers of criticality (i.e., the peaks of $\sigma _A$ and $\rho _1$).

The present study addressed the controversy in two different ways. First, we showed that peaks in $\sigma _A$ and $\rho _1$ are not sufficient conditions to critical phase-transitions. Therefore, despite the fact that peaks at $\sigma _A$ and $\rho _1$ are observed in the patient’s dynamics (though with smoother peaks compared to healthy controls), these neural markers do not ensure criticality. As exemplified here and consistent to the available theory³², noncritical dynamics are characterized by monotonic decaying of $\langle S_2\rangle$ and $\langle S_1\rangle /N\rightarrow 0$ with $N\rightarrow \infty$. These last conditions were both met in the original study. We cannot formally access the scaling behavior of $\langle S_1\rangle /N$ as $N\rightarrow \infty$. However, one can easily observe a large decrease of $\langle S_1\rangle /N$ well beyond the inter-individual control variability for those patient’s with disrupted criticality.

Second, we assessed whether the theoretical framework proposed in⁴ is suitable to claim maintenance of criticality in our cohort of stroke patients. The proposed mechanism is correct only in the limit case of disconnected networks (i.e, those composed by $n_c \ge 2$ connected components), where the cluster size hierarchy (i.e, cluster ordering) could play a role. In this regard, we have shown here that the vast majority of empirical structural brain networks are fully connected (149 out of 159). Furthermore, for those few networks with more than one connected component, the smallest subnetworks are of negligible size. The neural activity simulated for systems with such an unbalanced network sizes does not show any departure, in terms of cluster statistics, from a fully connected network. Indeed, $\langle S_2\rangle$ behaved as expected, with a peak at the same position of the corresponding fully connected network.

The monotonic behavior of $\langle S_2\rangle$ (hence, lack of criticality) is not a consequence of the loss of network integrity as proposed in⁴. The emergence or lack of criticality is a consequence of a tight balance between brain topology and brain network excitability. In¹, we established the connection between alterations in network topology (average degree) and associated changes in criticality regime. Loss of criticality occurred for patients with a decrease below a certain critical value of the network average degree and connectivity disorder.

Beyond the correctness of the theoretical framework of⁴, its biological plausibility remains questionable even in the limit condition of two (sufficiently large) disconnected networks. Besides its misapplication to the case of stroke, we cannot identify any clinical condition in which their conjecture might be valid. As noted before, the latter requires a very specific topological configuration of the structural brain network, that is, one in which a sizable subnetwork is entirely disconnected from the rest of the brain. Disconnection of one of the resting-state networks (RSNs) seems an ideal candidate (also see Janarek et al⁴), simply because RSNs are characterized by dense intra-network connectivity and sparse inter-network connectivity mediated by few hubs^38,39. However, it is extremely unlikely that a focal lesion can disconnect a RSN because most of the RSNs (and certainly those that are most important for higher-order behavior and cognition) are spatially distributed across the brain^33,34,39,40.

Finally, we note a further fallacy in Janarek et al.’s hypothesis: if the anomalous behavior of $S_2$ is simply the byproduct of the loss of network integrity (rather than loss of criticality), how can we explain the recovery (normalization) of the typical $S_2$ behavior (criticality) at $t_2$? The only solution that is consistent with their conjecture is that there is massive recovery of structural network integrity at the level of inter-RSN connectivity, something not supported by the literature⁴¹ nor by our results. Indeed, inter-RSN disconnectivity patterns shows little or no change, as shown in the last panel of Fig. 3, where the lines for $t_1$ and $t_2$ are overlapping for all 13 RSNs.

In this study, we also tackled the role of homeostatic plasticity in stroke recovery and criticality. Our findings indicate that the absence of normalization negatively impacts the emergence of criticality in certain patients. Stroke lesions disrupt both short- and long-range connections²⁰, ultimately decreasing the probability of excitation (from I to A) as indicated by $\sum W_{ij}s_j$. The normalization term plays a crucial role in balancing network excitability, thereby facilitating activity propagation and the emergence of criticality. Moreover, lesions increase network modularity¹. Although the network remains fully connected, the altered distribution of weights and connections due to lesion can hinder the spread of activity, making it more challenging to achieve criticality. The heterogenization of network weights induced by Equation 1 mitigates this side effect, positively influencing the emergence of criticality. Regarding the effect of $r_2$, our findings align with the theoretical predictions illustrated in Fig. 2. Specifically, increasing $r_2$ from 0.36 to 0.7 enhances network excitability and facilitates the recovery of critical transitions in some patients; however, this recovery is not universal. We hypothesize that this variability is attributable to the underlying network architecture, which may not adequately support critical dynamics even with elevated parameter values. Factors such as increased modularity, decreased entropy, and reduced average degree could limit the network’s ability to achieve criticality³.

Our results also demonstrate the relevance of homeostatic plasticity mechanisms in supporting the functional and behavioral recovery of stroke patients, as effectively captured by the normalized model across the two time-points (note the increase in $\rho$ from $t_1$ to $t_2$). Importantly, by suppressing the balancing mechanism, the initially large correlation coefficients are significantly reduced, failing to reach significance in most of the regressions tested. However, some exceptions do exist. The index $I_1$ evaluated at $t_2$ shows a significant correlation with both B and homo-FC at $t_2$. In contrast, $I_2$ does not exhibit a significant correlation with any of the empirical variables tested. In summary, the inclusion of homeostatic plasticity normalizes the excitation-inhibition balance, which has crucial implications for predicting behavioral performance and functional connectivity patterns.

We interpret our results through the lens of homeostatic plasticity. Several studies have indicated that post-stroke recovery is significantly influenced by the excitation-inhibition (E-I) balance within cortical networks^16,19,20,21. Following a stroke, there are persistent long-term increases in excitability⁴². Homeostatic mechanisms are believed to play a critical role in maintaining E-I balance⁴³, facilitating adaptations in cortical networks that can counteract initial decreases in excitatory input following a lesion. These mechanisms, which include synaptic scaling of excitatory and inhibitory inputs⁴⁴, operate over extended timescales and may help explain the observed long-term changes in excitability and functional connectivity (FC). While existing studies emphasize the role of E-I homeostasis in cortical dynamics⁴³ and functional recovery⁴⁵, further exploration is necessary to understand its influence on excitability changes across the brain and its potential links to criticality⁴⁶.

Our approach is based on effective modeling of large-scale brain dynamics. Among the class of large-scale models featuring critical dynamics, our model strikes a good balance between simplicity (with only three parameters, all set a priori) and its capacity to capture a variety of empirically observed brain patterns. While neural mass models⁴⁷ are valuable for studying phenomena such as synchronization, oscillations, and functional connectivity, there is no straightforward correlation between model complexity and predictive power. Resting-state whole-brain models vary in terms of biological realism and testing methodologies. Contrary to the expectation that more complex models would better replicate real functional data, simpler linear models often outperform their more sophisticated nonlinear counterparts in assessing functional connectivity (FC) from BOLD fMRI time series^48,49,50. A recent study by Nozari et al.⁴⁹ found that linear autoregressive models provide a superior fit for whole-brain activity, evaluated through three performance metrics: predictive power, computational complexity, and the extent of unexplained residual dynamics. The authors suggest that microscopic nonlinear dynamics may be masked by factors related to macroscopic dynamics, including spatial and temporal averaging, observation noise, and limited data samples. In summary, simpler linear models can effectively and accurately capture macroscopic brain dynamics during the resting state, highlighting that the choice of model is intrinsically linked to the hypotheses being tested, rather than being solely a matter of performance.

Our modeling framework has potential for refinement. In future research, we want to investigate how lesion effects scale with network size and their influence on neural dynamics and criticality. Additionally, we intend to explore the effects of additional parameters, such as a continuous transfer gain function and local dynamics that influence the node’s threshold $T_i(t)$, which represents a form of time-dependent plasticity. Currently, our phenomenological model does not account for biologically relevant factors like time delays in signal propagation^47,51, which are crucial for communication between brain regions.

In conclusion, our findings suggest that criticality could serve as a unifying principle for understanding brain function, offering insights into behavior, functional connectivity, and neurological disorders.

Methods

We have characterized the simulated brain activity through the following state variables:

• the mean network activity,

$$\begin{aligned} \langle A \rangle = \frac{1}{t_s}\sum _{t=1}^{t_s} A(t), \end{aligned}$$

(4)

• the standard deviation of A(t),

$$\begin{aligned} \sigma _A= \sqrt{\frac{1}{t_s}\sum _{t=1}^{t_s}\big ( A(t)-\langle A \rangle \big )^2}, \end{aligned}$$

(5)

where $A(t)=\sum _{i=1}^{N}s_i(t)$ is the instantaneous network activity and $t_s$ is the simulated total time, that we fixed to $t_s=5000$ time-steps;

• Clusters were defined as ensembles of nodes that are structurally connected to each other and simultaneously active. A rigourous definition can be formulated as follows. Consider the instantaneous state vector $v(t)=(s_1(t),s_2(t),\cdots ,s_N(t))^T$, where T is the transpose operation, $s_i(t)=1$ if the i-th node is active at time-step t and $s_i(t)=0$ otherwise. Next, we build the symmetric square matrix P(t),

  $$\begin{aligned} P(t)=v(t)v(t)^T= \begin{bmatrix} s_1s_1 & s_1s_2 & \cdots & s_1s_N \\ s_2s_1 & s_2s_2 & \cdots & s_2s_N \\ \vdots & \vdots & \ddots & \vdots \\ s_Ns_1 & s_Ns_2 & \cdots & s_Ns_N \end{bmatrix}. \end{aligned}$$

  (6)

  The above matrix contains the product $s_is_j$ of all possible combinations of pairs of brain regions, where $P_{ij}=1$ if the pair (i, j) is simultaneously active and zero otherwise. Finally we perform the element-wise product of P with the binarized version of the connectivity matrix W:

  $$\begin{aligned} C(t)= \begin{bmatrix} s_1s_1 W_{11} & s_1s_2 W_{12} & \cdots & s_1s_N W_{1N} \\ s_2s_1 W_{21} & s_2s_2 W_{22} & \cdots & s_2s_N W_{2N}\\ \vdots & \vdots & \ddots & \vdots \\ s_Ns_1 W_{N1} & s_Ns_2 W_{N2} & \cdots & s_Ns_N W_{NN}, \end{bmatrix}, \end{aligned}$$

  (7)

  where it is implicit we are using the binarized version of W in the above expression. By definition, C(t) is a binary (sparse) matrix with diagonal elements igual to zero (because $W_{ii}=0 , \forall i$). For $C_{ij}=1$ the pair of brain regions is simultaneously active and structurally connected to each other and $C_{ij}=0$ otherwise. The instantaneous clusters sizes $S_i(t)$ (where $i=1$ stands for the largest cluster size and $i=2$ for the second largest) are obtained by computing the connected components of the graph formed by C(t). We used arpack and lapack libraries in fortran to perform the numerical computation of the network (graph) components. We computed the average clusters sizes, the largest $S_1$ and the second largest $S_2$,

  $$\begin{aligned} \langle S_i\rangle =\frac{1}{t_s}\sum _{t=1}^{t_s} S_i(t), \hspace{3cm} i=1,2, \end{aligned}$$

  (8)

  where $t_s$ is the total simulation time and $S_i(t)$ is the size of the momentary i-th largest cluster found at time t.

• the first (lag 1) autocorrelation coefficient:

$$\begin{aligned} \rho _1 = \frac{\sum _{t=1}^{t_s-1} (A(t)-\langle A \rangle )( A(t+1)-\langle A \rangle ) }{\sum _{t=1}^{ts} (A(t)-\langle A \rangle )^2}. \end{aligned}$$

(9)

• the functional connectivity matrix ($\text {FC}_{ij}$) is defined through Pearson correlation:

  $$\begin{aligned} \text {FC}_{ij}=\frac{\langle s_i s_j \rangle - \langle s_i \rangle \langle s_j \rangle }{\sigma _i \sigma _j}, \end{aligned}$$

  (10)

  where $\sigma _i=\sqrt{\langle s_i^2 \rangle - \langle s_i \rangle ^2}$ is the standard deviation and $\langle \cdot \rangle$ is the temporal average of the nodes time series $s_i(t)$.

• the average of the functional connectivity (only the upper triangular elements are considered in the average):

$$\begin{aligned} \langle \text {FC}\rangle = \frac{2}{N(N-1)} \sum _i^N \sum _{j>i}^N |\text {FC}_{ij}|. \end{aligned}$$

(11)

• the connectivity disorder (i.e., structural entropy):

  $$\begin{aligned} H_{SC} = - \sum _{i=1}^m p_i \log p_i / \log m, \end{aligned}$$

  (12)

  where m is the number of bins used to construct the probability distribution function of all the elements of the connectivity matrices (W or ${\tilde{W}}$). The normalization factor in the denominator ensures that H is normalized between 0 and 1. The distributions were partitioned with $m=100$ bins.

The model parameters were set to $r_1=7\cdot 10^{-3}$, $r_2=\{0.1,0.36,0.7\}$ (for details, see figure captions), and we vary the activation threshold $T \in [0,0.3]$. The numerical results presented were averages over 30 initial random configurations of A, I and R states. We discarded the initial transient dynamics (first 500 time-steps) to calculate the state variables ($S_1$, $S_2$, $\sigma _A$ and $\rho _1$). We used the function rand in fortran as the pseudo-random number generator for the uniform distribution between 0 and 1.

Table 3 Community assignments of ROIs within resting state networks (RSNs) and their corresponding MNI coordinates.