The initial parcellation does not influence the dimension of the latent space.

To determine the optimal dimension of the latent space, we trained Auto-Encoders (AE, Fig 1A, purple) with different bottleneck dimensions ranging from 2 to 25. Briefly, an AE is an artificial neural network composed of successive layers of reducing dimension (the encoder) until reaching the bottleneck, or latent dimension, and a symmetric group of layers of increasing dimension (the decoder) that brings the data back to its original dimension. The AE is then trained to minimize the difference between input and output. The optimal latent space dimension is considered to be where the curve of the mean squared reconstruction error (MSE) vs latent dimension presents an elbow: indeed, when the latent dimension is too small, there is a loss of information and thus the reconstruction is bad, but then, increasing the latent dimension reduces the error until reaching the actual dimension needed to carry all the information contained in the data and, from this point forward, adding more dimensions does not improve the reconstruction (see methods Sect Auto-encoders for more details).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Latent space exploration.

A: Example subjects’ initial rs-BOLD time-series (left) color-coded per condition (blue: CNT, magenta: MCS, cyan: UWS) were projected into a reduced latent space (center) either using an AE (center left, purple) or PCA (center right, red) and then decoded (for the AE) trying to minimize the reconstruction error (right). B: MSE Reconstruction error for the AE with respect to the latent space dimension for different initial parcellations with and without sub-cortical regions (shades of purple). C: MSE vs latent dimension for initial parcellation Schaefer 100, divided per subjects condition. D: Comparison of the classification accuracy between conditions for AE (purple) vs PCA (red) for all latent dimensions, compared to accuracy in the natural space (black solid line). Grey dotted line indicates chance level.

https://doi.org/10.1371/journal.pone.0328219.g001

Here, to determine whether the original parcellation influenced the results, we performed this analysis on the same data (see methods Sect Participants for description of the dataset) that we parcellated using the Schaefer parcellation [55] with 100 regions (Fig 1B light purple), then, we added the subcortical regions using the Tian parcellation [56] with 16 regions (pink), and finally we tried the Schaefer parcellation with 1000 regions (deep purple). We found no difference in the MSE between the different parcellations. Furthermore, the elbow was reached in all cases around dimension 15. Therefore, for the rest of the results of this study, we used the Schaefer 100 parcellation. This was done using different datasets, from two different centers (in Paris and Liege) of fMRI data from DoC patients with UWS and MCS diagnosis and healthy controls (see Methods Sects Participants and Datasets combination for details).

We also checked whether the patient’s diagnosis would influence the elbow’s position by using the previously trained AEs (trained on a balanced mix of subjects from all diagnosis and healthy controls (CNT)). We then split the subjects of the test set by condition (i.e. patients’ diagnosis: UWS or MCS, or healthy controls) and looked at the MSE by group vs the latent-dimension (Fig 1C control (CNT), blue, MCS, magenta, UWS, cyan), and found that there was no difference in the optimal latent-dimension based on the subjects’ condition.

The latent dimension minimizes the MSE and allows the classification between conditions better than in the natural space.

To make sure that the dimensionality reduction did not lead to a loss of information, we decided to use a second criterion: given that we already knew that, in the natural space we can distinguish between clinical conditions based on the FC matrices of the subjects [13], we wanted to make sure that, in the latent space we were still able to do it, at least as well. To do so, we trained a support vector machine (SVM) classifier in the source space (of dimension 100) to distinguish between the empirical FC of all our subjects that are grouped into three categories based on the clinical exams: UWS, MCS, and CNT. We could obtain a very good accuracy of over 82%. We then used a similar training procedure on the reduced data for all latent dimensions between 2 and 25 and for the two dimensionality-reduction methods (PCA and AE). The training of the SVMs in the reduced dimensions was done on the latent empirical FCs, i.e. the Pearson correlation matrices of the reduced signals, i.e. for each reduced dimension we obtained latent FC matrices of dimension on which we trained the SVM classifier. The rationale was that the latent dimension should be the smallest one that allows for similar, or even better, classification accuracy than in the source space. We found that dimension d_opt = 15 was the optimum for both methods. The fact that both methods gave the same results is a good sign of robustness, showing that the intrinsic level of information contained in the data is of this optimal dimension, regardless of the dimensionality reduction method used.

To conclude, we found that both AE and PCA give similar results regarding the classification of the FC matrices between clinical conditions. However, PCA is a linear process while we know that the processes underlying the generation of the fMRI signals that we observe are intrinsically non-linear. For this reason we decided to pursue our study using the AE. Finally, we projected all the available fMRI data into this latent space of optimal dimension 15, to start working on the modeling step that we describe in the next section.

Individual patient fitting procedure.

For both models, there are two types of parameters: (i) the local, or node parameters, e.g., the bifurcation parameter (often denoted a) for the Hopf model, which determines the type of activity (random fluctuations or synchronized oscillations); or the parameters relative to AHP in the case of the second model. (ii) The connectivity matrix between the nodes, also known as the Effective Connectivity (EC) matrix, which gives the actual level of influence each node of the model has on the remaining ones.

We implemented the model fitting procedure (Fig 2A) as follows: (1) We first fitted the node parameters by minimizing the distance between the empirical and simulated FC, i.e., the Pearson correlation matrices between the time series of the 15 nodes. We then computed the structural similarity index (SSIM) [61] between the empirical and simulated FC, which we maximized as a fitting criterion (a SSIM of 1 means perfect similarity and 0 none). Indeed, previous studies have shown that the SSIM offers a good trade-off between absolute (e.g MSE) and relative (e.g. correlation) assessment of the differences between two matrices [62]. The parameter space was then sampled randomly until a threshold was reached, or until the maximum number of simulations N_max was reached. We kept the parameter set that maximized the SSIM. (2) The second step of the procedure was to iteratively fit the generative effective connectivity (GEC) for each patient by comparing the empirical FC to the simulated one in a similar manner as described above, using the previously fitted node parameters (see Methods, Sect Fitting algorithm, for more details).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Models fitting.

A: Iterative fitting procedure: the node parameters were randomly initialized and the connectivity was initialized by the empirical FC (step 1, left). Then, the subject’s latent time series were simulated and the empirical and simulated FC were compared to maximize the SSIM (step 2, center) over the fitting iterations with random exploration of the parameter space (step 3). Then, with the best node parameters fixed, the GEC was fitted iteratively to maximize the SSIM between simulated and empirical FC and shifted FC (steps 4 and 5, right). The procedure was iterated at step 2, to fit the node parameters using the fitted GEC. B: Fitting results: SSIM between the simulated and empirical FC after various fitting iterations per condition (blue: CNT, magenta: MCS, cyan: UWS) for the Hopt (left) and AHP (right) models. C: 1st level MBBs per condition, obtained from the fitted node parameters (each dot is one subject, top left to bottom right: noise intensity σ, facilitation resting-state value X_eq, depression resting-state value Y_eq, hyperpolarization depth T_h, all dimensionless, medium and slow AHP time rates (in Hz), each dot is one patient.

https://doi.org/10.1371/journal.pone.0328219.g002

We found that the fit quality plateaued around 0.6 for the Hopf model (Fig 2B, left) and was constant over iterations. Note that in [62], a SSIM of 0.4 is already considered as a good fit, here, we considerably increased it. In contrast, the SSIM of the AHP model (Fig 2B, right) increased over the first iterations before stabilizing around 0.80-0.86 (depending on the group, the UWS had the best-fit quality, although the difference was small).

Since we have more node parameters (6 vs 2) in the case of the AHP model compared to the Hopf model, it is of no surprise that the fit would be of better quality. Therefore, in the following sections, we present the results obtained from this model. However, all analyses were performed on both models, with the results for the Hopf model being available in the supplementary information (SI). We used this double analysis as a control to see whether we get coherent results with both models to make sure that we did not see model artifacts but robust results. In brief, the AHP model provided us with a more precise fit and interpretation than the Hopf model, but both models point to similar results, as we shall see in the next sections.

Node fitting results: Model-based biomarkers.

Each patient’s set of fitted node parameters constitutes model-based biomarkers (MBB, Fig 2C). For the AHP model, we obtained a vector of dimension 6 for each patient (the fitted node parameters are, from top left to bottom right: the noise intensity σ, the facilitation resting state X_eq, the depression resting state Y_eq, and the three AHP parameters: hyperpolarization depth T_h, and medium and slow timerates and ). We first checked whether these parameters revealed differences between the three groups (CNT in blue, MCS in magenta, and UWS in cyan) but, apart from the slow AHP time rate that was significantly increased (i.e. indicating a shorter AHP duration) in the MCS group compared to the controls, as well as a close to significant difference (p=0.07) between MCS and UWS, the other parameters didn’t show any clear difference based on the clinical state. We found the same results for the Hopf model MBBs, which are only 2 (noise intensity σ and the bifurcation parameter a, see S1 Fig). However, each distribution was quite spread out thus revealing a wide within-group variability. At this point, we concluded that the node MBBs do not carry much information about the clinical state, and therefore do not allow to distinguish between conditions but that they may hold other type of information about the patients’ conditions such as their etiology, age, underlying affected mechanisms, and other clinical variables, as we explore in the last section of this study. But first, we still needed to recover and explain the difference between the conditions we knew we could distinguish from (Fig 1D).

A two-clique structure for the GEC.

To analyze the difference between the groups we thus turned to the GEC matrices. Fig 3A shows the averaged fitted GEC matrix per condition. However, this visualization makes it hard to see the differences between them. Thus, to further explore the structure of these GEC matrices, we converted each GEC matrix into a weighted graph, where the GEC is the connectivity matrix of the graph. Then, for a clearer visualization, we split these graphs into two subgraphs: one with positive edges (Fig 3B, upper line) and another one with negative edges (lower line). The first striking observation was that, for all groups, these graphs are constituted of two groups of nodes (or cliques), strongly positively connected within each group and negatively projecting to the other clique. At first glance, we seemed to have a stronger within-clique positive density for the CNT (0.66), than for the MCS (0.65) and lowest for the UWS (0.60), while we had the opposite trend for the negative inter-clique projections (CNT: 0.49, MCS: 0.50 and UWS: 0.54).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. GEC-based biomarkers for the AHP model.

A: Average over all subjects of the fitted GEC matrix per condition (left to right, CNT, MCS, and UWS). B: Extracted graphs from the average GEC matrices (top: positive connections, bottom: negative connections) showing two subgroups (or cliques) of nodes positively connected between them and projecting negatively on the other subgroup, for each condition. C: Inter-clique metrics: mean and standard deviation for positive (left) and negative (right) edges. D: Graph metrics: node centrality and Small-world clustering parameter. E. GEC extracted measures: left to right: sparsity, asymmetry, mean and standard deviation values. Statistical tests: pairwise Kolmogorov-Smirnov tests, ****: , ***:, **:, *:.

https://doi.org/10.1371/journal.pone.0328219.g003

We thus decided to further quantify the clique interactions (Fig 3C). To do so, we used the networkx package to automatically compute the two major cliques based on the selection of each node, and computed the mean and standard deviation of the inter-clique connectivity for the positive (Fig 3C, left), and negative subgraphs (right). We found that these measures, not only allowed us to differentiate the controls from the patients, but also between the two patients groups with high significance.

We also looked at other graph measurements (Fig 3D), namely the mean node centrality for the positive subgraph (first left), which displayed a small difference (p=0.07) between CNT and MCS but not between the others; and the mean node centrality of the negative subgraph (second left), which allowed to distinguish between CNT and MCS but not between the others. The small-world clustering (Fig 3D, left) for the positive edges showed a significant difference between CNT and MCS and a fairly significant difference (p=0.08) between CNT and UWS, but failed to distinguish between MCS and UWS, and the same for the negative edges, where it could only differentiate between CNT and UWS.

Finally we also directly quantified values from the GEC matrices (Fig 3E): from left to right: the sparsity (quantified as the proportion of connections <0.015), the asymmetry (difference between the GEC matrix and its transpose), and the mean connectivity and its standard deviation (i.e., the mean and std values of the matrix components). Similarly, we found no clear difference between the two DoC groups. However, the mean connectivity seemed higher in both DoC groups than for the CNT, indicating a more synchronized activity, as expected. We saw the same trends using the Hopf model (see Fig S2).

In conclusion, it seems that the difference between conditions is mostly carried by this two-clique structure we observe in the GEC graphs, with stronger negative inter-clique and weaker positive inter-clique projections in the least conscious patients than in the more conscious ones and the controls.

Decoding the latent dimensions correlates to the resting state networks.

To further understand the meaning of the two-clique structure observed in the GEC matrices (Fig 3B), we decoded the signals from the latent GEC cliques and each latent dimension to see how they correlate to the natural structures in the brain. In particular, we investigated the link between GEC cliques (resp. latent dimensions) and the 7 Yeo Resting State Networks (RSN) [54], with a similar approach to the one used in [51] (see Methods, Sect Decoding of the GEC cliques and latent dimensions).

Briefly, we decoded signals with activity (from the control subjects) on only one of the two GEC cliques. We computed the resulting FC matrix (Fig 4A, left, and 4B) for the average over all control subjects. We then compared these two FC matrices with those corresponding to activity on each of the 7 RSNs (Fig 4A, right). For both cliques, we computed the SSIM between the clique FC and each RSN FC (Fig 4C). We found that the visual network (light gray) projected most strongly on Clique 0, followed by the Dorsal Attention network (blue). In contrast the default mode network (DMN, beige) and the Ventral Attention network (purple) were more strongly associated with Clique 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. GEC cliques and latent dimensions connection to the Resting State Networks.

A: FC matrices from the decoded signals for both GEC cliques (left), and FC matrices corresponding to the 7 RSNs in the natural space. B: GEC cliques from the average CNT GEC graphs. C: SSIM between the decoded FC matrices from each GEC clique compared to each RSN. D: Same as C but for the decoded signals from each latent dimension.

https://doi.org/10.1371/journal.pone.0328219.g004

We then decided to investigate the connection between each latent dimension and the 7 RSNs. Using a similar approach, we compared the FC matrices from decoded signals with activity on only one of the LD at a time to the RSN FC matrices (Fig 4D). We found that the Somatomotor network (light blue), was most strongly associated with latent dimensions LD13 and 2, while the DMN strongly projected onto LD1 and 9. We also noted that LD11 was mainly related to the Ventral attention network. Generally, most networks were associated with a combination of latent dimensions rather than one in particular.

This could mean that the lower within-clique density observed in UWS patients is related to lower or less consistent activity on specific RSNs. However, a more detailed analysis would be required to confirm such a hypothesis.

Latent space dynamics and structural interpretation.

A growing body of literature shows evidence of a low-dimensional manifold that encompasses the brain’s resting-state activity. The dimension of this latent space is always found to be of the same order of magnitude: around 10 to 15, across various studies, both from our group and from others working on different datasets and conditions [48–53]. Here, using rs-fMRI recording from DoC patients, we confirmed that finding and showed that we could work in a latent space of dimension 15 without losing the information contained in our data, or even our ability to explain it. This latent space description is not only convenient regarding the actual working dimension for explaining the data, but it also allows us to avoid inherent difficulties linked to fMRI preprocessing, parcellation choices, and denoising, as we showed in Fig 1. This reveals that the intrinsic amount of information contained in rs-fMRI data, from two different datasets, can be explained in such tractable dimensions, regardless of the condition studied.

However, the question of the biological interpretation of the latent space remains open. We believe that it should be interpreted in terms of interacting networks. In a recent study, we showed how the dimensions of the latent space (also called latent modes) project onto the resting state networks (RSNs) [51]. Here, we deepened this analysis of the connection between the latent dimensions and the RSNs (Fig 4). We found that each RSN seemed to be encoded by a combination of latent-dimensions, confirming that the results observed in the latent space have a biological interpretation. However, given the complexity and the amount of possible combinations, further research is still needed to decipher fully the mapping between the latent and natural spaces. Meanwhile, other works are unveiling the tight relation between spatial and temporal activity in the brain, or in other terms, how structure and function are intrinsically linked. For example, using harmonic decomposition, Atasoy and colleagues showed in several studies how the RSNs overlapped with functional task-based networks [65], thus revealing harmonic brain modes that could be compared to the latent modes. This framework recently allowed us to reveal and quantify the effect of hallucinogenic drugs such as DMT [66] and ketamine [67]. In a similar attempt to connect the latent functional modes to the brain’s structure, we recently showed that it was possible to extract the low-dimensional manifold information corresponding to the structural networks using Schroedinger’s equation [68]. Similarly, Dan and colleagues developed deep-learning methods to link structural and functional brain dynamics [69,70]. Altogether, these results show that latent space dimensions are directly connected to structural brain networks, and thus ensure the biological interpretability of our latent whole-brain models, although more work is needed to get a more direct mapping between the two. Further developments in this direction shall shed even more light on our model interpretation shortly.

Latent, biologically relevant, whole-brain models.

We implemented two different models with different levels of readability: the Hopf and the AHP models. This choice was motivated by the fact that we wanted to make sure that our results would not be model artifacts. This was confirmed given that they are consistent regardless of the model chosen at the node level. However, we also showed that using the AHP model leads to a better-fit quality (Fig 2B) while offering a straightforward biological interpretation for each of the fitted parameters (see Figs 2C and 5B, and the last paragraph of Sect The node MBBs reveal transverse clusters that correlate to etiology and outcome). In particular, this new model is paving the way for the development of neuron-glia whole-brain models. Indeed, in the past decades, it has become evident that neuron-only approaches were not sufficient to grasp and explain the complexities of the brain as well as to investigate its disorders. For example, recent modeling efforts have shown that astrocyte calcium activity could explain the neurovascular coupling leading to the BOLD response [33]. Initiating a direction of biologically motivated neuron-glia models at the whole-brain level is, to the best of our knowledge, still a niche approach. However, regarding the importance of glial cell regulation of brain activity, it seems fundamental. In this work, we showed that adding basic astrocyte regulatory mechanisms into our model already could increase our model quality and the data’s explainability. In the future, it would be interesting to push this approach forward and work on implementing models that account more explicitly for astrocytes, and other types of glial cells, such as oligodendrocytes and microglia [71,72].

Finally, we point out that the fMRI processing pipeline provided here could also be used to study different datasets or conditions. Moreover, the modular structure of the provided code and latent whole brain models could even allow one to plug in any other model at the node level that would account for the mechanism one is interested in studying, without having to change the rest of the pipeline. We thus propose here a versatile analysis toolkit that could have multiple applications.

The GEC carries the information on the condition.

Functional connectivity matrices were already known to be able to distinguish between UWS and MCS [13], and later works on effective connectivity confirmed these findings and showed that even better distinction could be made, in particular in the latent space [51]. The effective connectivity framework also allows us to distinguish between tasks such as rest vs movie watching [73]. In this study, we fitted Generative Effective Connectivity matrices, in a latent space, to analyze how they carried information about the severity of the DoC condition (Fig 3). In particular, using graph theory, we showed that our subjects’ GEC presented two cliques of nodes, strongly connected positively but negatively projecting on the other subgroup (Fig 3B). Although this structure was similar in different conditions, we found significant differences between the three groups of subjects that we had. This analysis is thus a new step towards understanding in more detail the meso-scale dynamics that are underlying consciousness, and how the GEC matrices carry them. This approach can be related to the circuit models of consciousness [10–12,74].

Local node information correlates to prognosis and patient’s etiology.

In the last part of this paper, we showed that looking into the local node parameters (node MBBs) revealed novel information about our patients’ conditions (Fig 5). In particular, we found that these node MBBs, allowed us to group the patients into transverse clusters that are agnostic about the severity of the condition but are correlated to the etiology and evolution of the patients’ condition, thus paving the way for novel prognosis tools or biomarkers. Moreover, the cluster separability analysis revealed which parameters were leading to belong to each cluster. Given the direct biological interpretation of the AHP model’s parameters, this provides a new hypothesis regarding which mechanism is affected in which group of patients.

Finally, this work opens many possibilities for the investigation of potential treatments. Indeed, it would be very interesting to use the fitted models to see how modifying one specific node parameter affects the cluster to which a patient belongs. For example, if, by affecting one specific parameter (i.e., mechanisms or pharmacological pathway), we can move one patient from a least favorable outcome cluster to a more favorable one, this could be very informative about the treatments to explore for this specific patient. This type of analysis could thus lead to a more precision medicine approach to DoC treatments. Indeed, although many studies exist regarding treatment options both through external neuronal stimulations [19] and pharmacological approaches [11], the wide variability of lesions, and underlying causes for the DoC patients remains a barrier to generalization of such treatments.

Paris.

The dataset included 77 brain injury patients hospitalized at Paris Pitié-Salpêtrière. The patients’ states of consciousness were evaluated using the Coma Recovery Scale-Revised (CRS-R) by trained clinicians. Patients were classified as being in an unresponsive wakefulness state (UWS) if they demonstrated arousal (such as eye-opening) but lacked any signs of awareness, such as voluntary movement. In contrast, those diagnosed with a minimally conscious state (MCS) exhibited behaviours suggestive of awareness, including visual tracking, response to pain, or consistent command-following. Exclusions from the study were made for individuals with errors in T1 image acquisition (n = 5), significant motion artefacts (n = 7), registration errors (n = 4), or large focal brain lesions (n = 4). As a result, the final cohort comprised 33 patients with MCS (11 females, mean age ± SD: 47.25±20.76 years), 24 patients with UWS (10 females, mean age ± SD: 39.25±16.30 years), and 13 healthy controls (7 females, mean age ± SD: 42.54±13.64 years). This research was approved by the local ethics committee, Comité de Protection des Personnes Ile de France 1 (Paris, France), under the protocol code ’Recherche en soins courants’ (NEURODOC protocol, n 2013-A01385-40). Informed consent was obtained from the patient’s families, and all procedures were conducted under the Declaration of Helsinki and French regulations.

Liege.

The dataset involved 35 healthy controls (14 females, mean age±SD: 40±14 years) and 48 patients with DoC. Diagnoses were established following at least five Coma CRS-R assessments conducted by trained clinicians. The final diagnosis, confirmed using Positron Emission Tomography (PET), was determined by the highest level of consciousness recorded. Patients diagnosed with a minimally conscious state (MCS) showed relatively preserved metabolism in the fronto-parietal network, whereas those in an unresponsive wakefulness state (UWS) exhibited bilateral hypometabolism in the same network. The cohort ultimately included 33 patients in MCS (9 females, mean age±SD: 45±16 years) and 15 patients in UWS (6 females, mean age±SD: 47±16 years). The Ethics Committee of the Faculty of Medicine at the University of Liege approved the study protocol, and the research was conducted in compliance with the Helsinki Declaration. Written informed consent was obtained from controls and the legal representatives of the patients.

Paris.

MRI images were obtained using two distinct acquisition protocols. The first protocol involved 26 patients and 13 healthy controls, with data collected on a 3T General Electric Signa System. Resting-state T2*-weighted images of the entire brain were acquired using a gradient-echo EPI sequence in an axial orientation (200 volumes, 48 slices, slice thickness: 3 mm, TR/TE: 2400 ms/30 ms, voxel size: 3.4375 3.4375 3.4375 mm, flip angle: 90^°, FOV: 220 mm²). Additionally, an anatomical T1-weighted MPRAGE sequence was acquired during the same session (154 slices, slice thickness: 1.2 mm, TR/TE: 7.112 ms/3.084 ms, voxel size: 1 1 1 mm, flip angle: 15^°). The second protocol involved MRI data acquisition from 51 patients on a 3T Siemens Skyra System. In this case, T2*-weighted whole-brain resting-state images were recorded with a gradient-echo EPI sequence in an axial orientation (180 volumes, 62 slices, slice thickness: 2.5 mm, TR/TE: 2000 ms/30 ms, voxel size: 2 2 2 mm, flip angle: 90^°, FOV: 240 mm², multiband factor: 2). An anatomical volume was also acquired during the same session using a T1-weighted MPRAGE sequence (208 slices, slice thickness: 1.2 mm, TR/TE: 1800 ms/2.35 ms, voxel size: 0.85 0.85 0.85 mm, flip angle: 8^°).

Liege.

MRI images were obtained using a Siemens 3T Trio scanner (Siemens Inc, Munich, Germany). The acquisition protocol included a gradient echo-planar imaging (EPI) sequence with 32 transverse slices and 300 volumes (TR/TE: 2000 ms/30 ms, flip angle: 78^°, voxel size: 3 3 3 mm, FOV: 192 mm). A structural T1-weighted scan was also acquired with 120 transverse slices (TR: 2300 ms, voxel size: 1.0 1.0 1.2 mm, flip angle: 9^°, FOV: 256 mm).

Datasets combination.

In all that follows we combined the datasets from Paris and Liege described above in Sects Participants to Data pre-processing.

To overcome the problem of different time resolutions between the different datasets, we performed a simple time-interpolation step on the subsets with larger TR to obtain time-series that all have the same TR = 2s. This was done using the interp1d function from the scipy.interpolate package.

The decision of combining the different datasets was taken for the sake of robustness. Indeed, any classification algorithm trained on only one dataset would likely pick up some artifacts or features specific to the scanner used. This could result in overfitting to this specific dataset. Mixing different datasets allows us to avoid this pitfall and ensure the generalizability of our methods, regardless of the center and technical setup where the data was recorded. Furthermore, the preprocessing was strictly the same for all recordings from both centers, thus ensuring no differentiation between the datasets at this stage. Moreover, combining the datasets greatly increases the number of patients in each group and thus the statistical power and significance of any group-related analysis. We recall that the group classification (between MCS and UWS) used in the manuscript comes from the clinical diagnosis performed by the expert clinicians in Paris and Liege based on the CRS-R method which is a universal framework for clinical diagnosis developed for that purpose to ensure that diagnosis of DoC patients is consistent between different centers. Finally, the dimensionality reduction also reduces possible differences due to the differences in data recording.

Auto-encoders.

We used auto-encoders with three dense (i.e., fully connected) layers of dimension , and before the bottleneck layer, where is the dimension of the natural space data depending on the initial parcellation (Schaeffer 100 and 1000 [55] and Schaeffer 100 with Tian 16 for the subcortical regions [56]). For each dimension of the bottleneck layer , we split the subjects into a 90-10% training-test split, then each data point given as input to the AE is a vector of dimension d_init which corresponds to one time-point in the data time series of one patient. The mean squared reconstruction error (MSE) is then calculated as the MSE between the input and output vectors on the test set. For each d_latent we performed a k-fold cross-validation using the Kfold function of the Python library Sckit-learn (sklearn v1.2.2) with k = 10.

The auto-encoders were implemented using the Keras library of Tensorflow v2.11 and trained using cuda 11.8 for GPU processing. Training hyperparameters are , , EPOCHS = 15.

Principal component analysis.

The PCA was implemented in Python using native Python functions and Numpy.

SVM classifier.

We used the SVC method from the sklearn Python library v1.2.2 and leave-one-out cross-validation using the KFold method from sklearn. We used a degree 3 polynomial kernel with hyperparameters C = 0.1 and . The algorithm was trained to distinguish the empirical Functional Connectivity matrices (FC) of the subjects. To enhance the classification accuracy, due to the limited number of subjects available, we previously split the data time-series into n_splits = 4 and then computed the FC of the split time-series which led us to N = n_splits N_sub = 556 FC matrices to train the SVM classifier, and 4 for the test (we do the Leave-one-out split on the subjects to avoid overfitting by training on some FC matrices of a subject and testing on the same subject).

Hopf model.

The classical Hopf model is composed, in each node of the model, of the normal form of a supercritical Hopf bifurcation (Stuart-Landau oscillator), close to its bifurcation point. Here we have nodes in the latent space. The nodes are then connected via a connectivity matrix C, which we describe in Sect Fitting algorithm. The model equations at node i are given by:

(1)

where x is the real part of the system and y its imaginary part, G denotes the global coupling of the system, that we keep set to G = 1, C is the connectivity matrix, is the intrinsic frequency at node i (here we take the Spectral Edge Frequency at 50% (SEF50) value of the spectrogram of the signal) and is an additive Gaussian noise term at each node. We recall that d_opt = 15 is the dimension of the latent space and thus corresponds to the number of nodes in our model. Finally, a, the bifurcation parameter, and σ the noise intensity are the parameters that we will fit for each patient (see Sect Fitting algorithm).

We chose to keep the global coupling parameter set to G = 1 because we fitted the entire connectivity matrix C (Sect Fitting algorithm); thus, any changes in the global coupling intensity are integrated in the fitting of the connectivity matrix. From a mathematical point of view, G being a constant it can be included in the sum and this is equivalent to fitting a connectivity matrix .

Local AHP model.

For the AHP model we implemented a whole-brain version of the model introduced and fully described in [37,58] which will describe the activity of each node of the model. Here, we recall the model’s equations for one node. The model has three variables, the mean population voltage h, the short-term synaptic facilitation x, and the short-term synaptic depression y:

(2)

where h⁺ = max(h,0) is the population mean firing rate [83,84], J is the average local connectivity parameter [85,86], i.e., within one local population (or node), the influence of the other nodes will be added below in a similar way (see Eq. 3). The parameters K and L describe how the firing rate influences the duration and probability of vesicular release respectively. The time scales (for facilitation) and (for depression) define the recovery rates of a synapse from the local network activity and X_eq is the resting-state value for the facilitation, which can be associated to the presynaptic calcium concentration. is an additive Gaussian noise and σ its amplitude. Finally, we accounted for AHP with two features: 1) a varying equilibrium state T₀ representing hyperpolarization at the end of a burst of neuronal activity 2) two timescales for the medium and slow AHP recovery rates.

The resting membrane potential T₀ and the recovery time constant of the voltage h are defined piece-wise as follows:

1. – and T₀ = 0 when {y>Y_AHP and , represents the fluctuations around the resting membrane potential and the bursting dynamics.
2. – and for and y<Y_h} when the hyperpolarizing currents at the end of the burst become dominant and force the voltage to hyperpolarize (medium AHP period).
3. – and T₀ = 0 for and (Y_AHP<y or h<H_AHP)}, which represents the slow recovery to resting membrane potential (slow AHP period).

The threshold parameters defining the three phases are Y_h = 0.5, Y_AHP = 0.85 and H_AHP = −7.5.

Generalization to a whole-brain model.

To generalize this model to a whole-brain version, we just need to add the long-range connections, i.e., the influence of the other nodes of the model, as follows. For node i we now have:

(3)

Similarly as for the Hopf model, the intrinsic timescale τ is extracted directly from the data using the Spectral Edge Frequency at 50% (SEF50) from the spectrogram of the signal. Similarly as for the Hopf model, we directly fit a connectivity matrix , integrating the global coupling into the fitted connectivity matrix. Note that we have replaced the resting-state value of the depression that was left fixed at 1 in [37,58,84,87] by a parameter Y_eq that we will fit to the data to be able to grasp potentially depressed synapses in some patients. The fixed model parameters are taken from [58] and are given in Table 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Fixed AHP model parameters.

https://doi.org/10.1371/journal.pone.0328219.t002

Balloon-Windkessel model.

To simulate the BOLD signal, we add a Balloon-Windkessel transformation as described in [59,60]. We used the implementation from github.com/dagush/WholeBrain/

Models implementation and numerical integration.

All models are implemented in Python, with Numba (v0.56.4), a just-in-time compiler, to speed up the computations. The numerical integration for the Hopf and AHP model is done with the Runge-Kutta 4 scheme with a time step of dt_sim = 0.05s.

SSIM.

The SSIM index is a measure of similarity between two images (or matrices) of the same size [61]. It is computed by comparing common-size windows of corresponding zones of both matrices and then aggregating the results. The formula used for the comparison of two matrices A and B is:

(4)

where:

is the pixel empirical mean of matrix and its empirical variance;

the covariance of the two matrices;

c₁ and c₂ are two variables introduced to stabilize the division with a weak denominator, we used the classical default values:

and where L is the dynamic range of pixel values in A and B, which in our case is L = 1.0.

Node parameters fitting.

First we fit the node parameters, that is for the Hopf model and for the AHP model. The goal of this step is to find the choice of node parameters that will maximize the SSIM between empirical and simulated functional connectivity matrices for each patient. To do so, we keep the connectivity matrix fixed and vary the node parameters exploring the parameter space.

We explore randomly the parameter space in the manner of a multi-scale grid search, i.e., we first explore a larger range of parameters and at each iteration, we refine the search around the best-found value. In practice, at the first iteration n_iter = 0, the connectivity matrix C is defined as the empirical functional connectivity matrix, FC_emp, taken from the data, and for the following iterations, we take , where iter is the iteration step of the fitting procedure. Here, we stopped at iter_max = 4 for the Hopf model and iter_max = 6 for the AHP model, when it was clear that the SSIM stopped increasing with fit iterations.

At each iteration iter, we explored N_max = (4 − 1000 (for and then N_max = 1000 for the following iterations) values for the node parameters . Since we have a stochastic model we need to run many simulations of the model at each iteration. Thus, for each value of we ran  +  15 (for and then N_sim = 50 for the following iterations) simulations of duration dur = 300s (same as our recordings) and then averaged them before taking the correlation matrix to obtain the simulated FC FC_sim. The reason why the number of simulations ran depends on the iteration step iter is to mimic the approach of multi-level grid searches where we first explore a wider range of parameters (i.e., a bigger N_max) with less precision (i.e., a smaller N_sim) and then progressively explore fewer parameters (decreasing N_max) with more precision (increasing N_sim) while keeping reasonable computation times.

The explored parameter ranges are summarized in Table 3.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Authorized parameter ranges.

https://doi.org/10.1371/journal.pone.0328219.t003

Generative Effective Connectivity fitting.

Briefly, the generative effective connectivity matrix (GEC) is a matrix that allows for the reproduction of both the empirical correlation matrix of the data and a forward time-shifted correlation matrix. This is done to grasp the asymmetries in the effective connectivity between the nodes and leads to an asymmetric matrix which can reproduce non-equilibrium dynamics (see, for example, [51,88] for more details about GEC matrices). In practice, we fitted the GEC by keeping the node parameters from the previous step fixed and adjusting the connectivity matrix to maximize the SSIM between the empirical vs simulated matrices, as well as between the shifted matrices –tau and –tau. The shifted matrices were obtained by shifting in time each of the time series of a small time step τ and taking its correlation with the unshifted other dimensions. This process is done to grasp the asymmetries in the effective connectivity between the nodes. The GEC was fitted iteratively using a pseudo gradient descent algorithm according to Eq. 5 below, which weighs both the fitting of the direct and shifted FC matrices, as also explained in [51]:

(5)

The learning rate is and the shift in time TR = 6s [51]. Similarly, as in the node fitting step, we need to run several simulations at each step to account for the stochasticity, here we have N_sim = 5. For each iteration, this is repeated until the SSIM is larger than 0.95 or for N_max = 1000 iterations.

Outcome categorization.

To analyze the link between the MBB clusters and the patients metadata, we defined five outcome categories for the patients based on the successive CRS-R diagnosis available:

1. Dead;
2. Decreasing: when the evolution of the CRS-R diagnosis went towards less conscious over successive checks but did not lead to death;
3. Stable: when the CRS-R diagnosis did not change over time;
4. Increasing: when the diagnosis ameliorated without leading to consciousness;
5. Conscious or Emerged.

The final outcome available in our dataset is the CRS-R diagnosis after two years unless the patient died or regained consciousness before that.

We refer to outcomes 1 to 3 as negative outcomes, and outcomes 4 and 5 as positive or favorable outcomes.

UMAPS and HDBSCAN clustering.

To obtain the clusters we used Uniform Manifold Approximation and Projection for Dimension Reduction (UMAP) [63] on the 6-dimensional (resp. 2-d) vectors of fitted node parameters (node MBBs) from the AHP (resp. Hopf) model, using the UMAP Python package (umap-learn v0.5.5), we used the parameters and , to project in 2D, the rest are the default parameters.

The clustering was then done on the UMAP projections using the HDBScan clustering algorithm [64] implemented for Python in the HDBSCAN package (hdbscan v0.8.33). We used parameters and , and the rest take default values.