Macroscopic quantities of collective brain activity during wakefulness and anesthesia

The study of states of arousal is key to understand the principles of consciousness. Yet, how different brain states emerge from the collective activity of brain regions remains unknown. Here, we studied the fMRI brain activity of monkeys during wakefulness and anesthesia-induced loss of consciousness. Using maximum entropy models, we derived collective, macroscopic properties that quantify the system’s capabilities to produce work, to contain information and to transmit it, and that indicate a phase transition from critical awake dynamics to supercritical anesthetized states. Moreover, information-theoretic measures identified those parameters that impacted the most the network dynamics. We found that changes in brain state and in state of consciousness primarily depended on changes in network couplings of insular, cingulate, and parietal cortices. Our findings suggest that the brain state transition underlying the loss of consciousness is predominantly driven by the uncoupling of specific brain regions from the rest of the network.

108 where ≠ ( ) is the sum activity of all but ROI : ≠ ( ) = ∑ ( ) ≠ . Recent findings showed that 109 propofol anesthesia affects the coupling to global signal in human and rats (28). In the following we 110 showed that the statistics = [ 1 , … , ] provides, with only parameters, a compact description of 111 the binary collective activity and can be used to classify the brain states.

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The couplings to the population were highly predictive of the functional correlations ( Fig. 2A-C). 120 Indeed, the product = × highly correlated with the functional correlation (FC) between the

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To examine which ROIs contributed the most to distinguish between the awake state and anesthesia 169 based on , we performed PCA on the collection of z-scored vectors . The first principal component 170 was sufficient to separate the awake and anesthesia conditions (Fig. 3D). This component had strong 171 coefficients for brain regions located in the cingulate, parietal, intraparietal, insular cortices, and the 172 hippocampus (Fig. 3E). Overall, changes in average couplings to the population with respect to 173 awake values were similar for all anesthetics (Fig. 3F). We next asked how these changes affect the 174 collective properties of brain dynamics.

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Modelling collective activity using maximum entropy models 177 Collective activity is ultimately described by the probability of each of the binary patterns = MEMs the model parameters were inferred from the data using maximum likelihood (29). Notably, 199 for the coupling-MEMs the partition function can be calculated directlysomething that is generally 200 not the case for most MEMs, since its calculation involves summing over all possible states. 201 We used these coupling-MEMs to fit the binary single-scan fMRI data for the different experimental 202 conditions. The models accurately estimated the distribution of population activity ( ) (average 203 Jensen-Shannon divergence between the model and data distributions: < 10 -6 for both the 204 non-linear and linear coupling-MEM; Fig. 4A and Fig. S2). Moreover, the models were able to 205 moderately predict the covariances of the data (Fig. 4B,C), which were not used to constrain the 206 models. Across the different datasets, the average correlation between the data and predicted 207 covariances was = 0.28 ± 0.03 for the linear coupling model and reached 0.40 ± 0.02 for the non-208 linear coupling model (see also Fig. S2). Furthermore, scan-classification based on parameters 209 yielded 86% and 45% correct classifications between awake and anesthetized conditions and among 210 the six experimental conditions, respectively (Fig. S3A,B). Using parameters the classifier 211 performance decreased to 75% and 28%, respectively (Fig. S3B). Thus, the learned linear coupling-

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Collective activity indicated reduced free energy, susceptibility, and heat capacity under 229 anesthesia 230 We can learn interesting features of collective activity using the estimated models. One important 231 quantity is the system's Helmholtz free energy, which is given by the difference between the average 232 energy (〈 〉) and the entropy ( ), i.e., = 〈 〉 − . The free energy quantifies the useful energy that 9 is obtainable from the system. Using the Boltzmann distribution, the free energy can be directly 234 obtained from the partition function as = −ln(Z). Thus, since is tractable for the coupling-MEMs, 235 we can directly estimate . We found that the free energy was significantly higher for the awake state 236 compared to all anesthetized conditions for both the non-linear (Fig. 4D) and the linear (Fig S4A) 237 coupling-MEM (p < 0.001, one-way ANOVA followed by Tukey's post hoc analysis). This result is 238 both interesting and reasonable because it indicates that more useful energy can be extracted from the 239 awake state than from the anesthetized state. and a larger repertoire of energy states than the system under anesthesia. 249 We next tested whether the same differences in these statistical quantities were found using MEM (4) 255 In this pairwise-MEM, the parameter ℎ , called intrinsic bias, represents the intrinsic tendency of ROI was not used to constrain the model. We found that biases and couplings parameters were changed 264 for different states, with some parameters increasing or decreasing, and with a reduction of the 265 variance of couplings in the anesthetized states ( Fig. S5A-D). Moreover, coupling parameters showed 266 a higher correlation with the anatomical connectivity (or brain connectome) in the anesthetized states 267 than in the awake state (Fig. S5E).
Using this model, we calculated the collective statistical quantities for the different experimental temperatures lead to a low ℎ . However, for a specific temperature max , order and disorder coexist 291 in the system and ℎ is maximal as expected for critical dynamics (33, 34). Thus, a maximal heat 292 capacity at max = 1 (corresponding to the model learned from the data) suggests that the system 293 operated close to a critical state (whereas max < 1 and max > 1 indicates super-critical and sub-294 critical dynamics, respectively). 295 We found that the heat capacity curve was maximal for a temperature equal to 1 for the awake state, 296 while it peaked at max < 1 for the anesthetized conditions (Fig. 4M). These results suggest that the 297 awake state displayed critical dynamics, while dynamics under anesthesia were super-critical, which 298 indicates that the anesthetics had a disconnection effect.

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Couplings to population relate to the sensitive parameters of the system 301 We next evaluated how the different parameters affected the model's collective behavior. In general, 302 changes in parameters can differently affect the system's behavior, with some parameters (called that contributed the most to the FIM were the parameters (Fig 5A). This explains how changes in 312 couplings to population, as observed between awake and anesthetized states, effectively change the 313 collective state of the system, leading to the observed shift from critical to supercritical dynamics. To evaluate the importance of each of the parameters, we defined the parameter's sensitivity as its 331 absolute contribution to the first eigenvector of the FIM (see Methods). The regions with the largest 332 associated sensitivity for parameter were located in the cingulate, parietal, and insular cortices (Fig.   333 5B). Those that contributed the least were visual and prefrontal cortices. Interestingly, the regions 334 presenting larger reductions of between awake and anesthesia tended to be those with higher 335 associated sensitivity (corr: 0.74, p < 0.001; Fig. 5C). 336 Finally, we further examined how changes in pairwise correlations between awake and anesthesia 337 related to changes in parameters of different sensitivity. We analyzed the average difference of 338 correlation (∆ ) between awake and anesthesia. Two groups of ROIs were clearly separated 339 according to their positive or negative contribution to the first eigenvector of the matrix ∆ , 340 respectively (Fig. 5D). Those that contributed positively were prefrontal and visual cortices, and those 341 that contributed negatively were the cingulate, parietal, and insular cortices (Fig. 5E). Both groups 342 presented a reduction of correlations under anesthesia, but prefrontal and visual cortices were related 343 to parameters of low sensitivity (Fig. 5F-G). Hence, although prefrontal and visual areas changed 344 their correlations, these changes were related to parameters that had a low impact on collective 345 dynamics.

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In this study, we analyzed the fMRI binary collective activity of monkeys during wakefulness and 349 under anesthesia. We showed that the coupling between each brain region and the rest of the 350 population provides an efficient statistic that discriminates between awake and anesthetized states.

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We built MEMs based on this and other statistics to derive macroscopic properties that described the 352 different brain states, such as the free energy , the susceptibility , and the heat capacity ℎ . All 353 these quantities were maximized in the awake state. By studying the heat capacity curve ℎ ( ) as a 354 function of scaling parameter, controlling the disorder of the system, we showed that awake critical We used k-means clustering to classify the scans based on different statistics. Let ( ) be a vector 484 calculated from scan , e.g., the vector containing all pairwise correlations among the ROIs. We used

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Fisher information matrix 549 We were interested in detecting which parameters have the strongest effect on the collective activity.  i.e., = 〈 〉⁄ (see Fig S7). Here, we were interested in the so-called zero-field susceptibility

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This parameter acts as a scaling factor for all model parameters as → / . The "temperature" 806 controls the level of disorder and its effects can be understood by examining the system's energy 807 levels (Fig. S8). This creates a family of scaled models in which = 1 corresponds to the MEM 808 that was fitted to the data. The heat capacity as a function of is given by ℎ ( ) = var( )/ 2 and 809 provides useful features of the system. Indeed, it is known that a maximum of the heat capacity close 810 to = 1 suggests that the observed system is likely to be close to a critical state, whereas max < 1 811 and max > 1 indicate super-critical and sub-critical dynamics, respectively (31, 33, 34). Hence, the 812 curve ℎ ( ) can be used as a tool to assess criticality. The heat capacity measures the size of the 813 dynamic repertoire of the system. It not only provides a measure of the system's complexity, but also 814 assess whether the complexity is maximized and whether any reduction of complexity is due to a Fisher information matrix and free energy 833 We were interested in detecting which parameters have the strongest effect on the collective activity.  This was done by updating the biases and couplings as: Where denotes the updating iteration (up to 5.10 3 ) and 〈. 〉 mf are the expected values using the 876 mean-field approximation: where ( ) is the activity of ROI (taking values 1 or −1).