A mean field approach to model levels of consciousness from EEG recordings
Marco Alberto Javarone, Olivia Gosseries, Daniele Marinazzo, Quentin Noirhomme, Vincent Bonhomme, Steven Laureys, Srivas Chennu
AA mean field approach to model levels of consciousness from EEGrecordings
Marco Alberto Javarone ∗ Department of Mathematics, University College London, London, UK
Olivia Gosseries
Coma Science Group, GIGA Consciousness Universityand University Hospital of Liege, Liege, Belgium
Daniele Marinazzo
University of Ghent, Ghent, Belgium
Quentin Noirhomme
Faculty of Psychology and Neuroscience Maastricht University, Maastricht, Netherlands
Vincent Bonhomme
GIGA - Consciousness, Anesthesia and Intensive Care Medicine LaboratoryUniversity and CHU University Hospital of Liege, Liege, BelgiumDepartment of Anesthesia and Intensive Care Medicine CHUUniversity Hospital of Liege and CHR Citadelle, Liege, Belgium
Steven Laureys † Coma Science Group, GIGA Consciousness Universityand University Hospital of Liege, Liege, Belgium
Srivas Chennu † University of Kent, Medway, UKUniversity of Cambridge, Cambridge, UK (Dated: August 5, 2020) a r X i v : . [ q - b i o . N C ] A ug bstract We introduce a mean-field model for analysing the dynamics of human consciousness. In par-ticular, inspired by the Giulio Tononi’s Integrated Information Theory and by the Max Tegmark’srepresentation of consciousness, we study order-disorder phase transitions on Curie-Weiss modelsgenerated by processing EEG signals. The latter have been recorded on healthy individuals un-dergoing deep sedation. Then, we implement a machine learning tool for classifying mental statesusing, as input, the critical temperatures computed in the Curie-Weiss models. Results show that,by the proposed method, it is possible to discriminate between states of awareness and states ofdeep sedation. Besides, we identify a state space for representing the path between mental states,whose dimensions correspond to critical temperatures computed over different frequency bands ofthe EEG signal. Beyond possible theoretical implications in the study of human consciousness,resulting from our model, we deem relevant to emphasise that the proposed method could beexploited for clinical applications. ∗ [email protected] † these two authors contributed equally . INTRODUCTION Consciousness is one of the most complex and fascinating phenomena in the brain, attract-ing the interest of a variety of scholars, spanning from neuroscientists to mathematicians, andfrom physicists to philosophers [1–8]. In addition, consciousness, as well as other complexsystems as those we find in biology, social science, finance and artificial intelligence [9–15],has strongly benefited from the introduction of cross-disciplinary approaches. Despite a hugenumber of investigations, a lot of its aspects and mechanisms still require to be clarified.Given these observations, we focus on the challenge of quantifying consciousness, puttingattention on the transition between mental states. So, we introduce a method for gener-ating Curie-Weiss models [16, 17] from EEG signals, and then we analyse its outcomes bya machine learning classifier. As discussed later, although the spirit of this work is mostlytheoretical, we conceive a framework that could support clinicians in some relevant tasks,e.g. in calibrating the optimal amount of anaesthetic for patients, and in assessing the cogni-tive conditions of unresponsive individuals. Nowadays, a number of devices allow studyingthe brain structure and its dynamics. Usually, the choice of a specific tool reflects bothclinical needs and patient conditions. Here, we use recordings obtained by EEG analyses fortwo main reasons, i.e. its cheaper cost compared to other technologies and its non-invasivenature. At the same time, it is worth to report that the EEG is less advanced than otherdiagnostic tools, as fMRI, that generate images of higher quality (e.g. higher spatial reso-lution). Notwithstanding, stimulated by the above reasons, we aim to improve as much aspossible the value of the information content of EEG signals, concerning the dynamics ofhuman consciousness. Before moving to the proposed method, we very briefly introduce twoseminal works on this topic, the Integrated Information Theory (IIT hereinafter) developedby Giulio Tononi [18] and the Max Tegmark’s manuscript on the physical representationof consciousness as a state of matter [3]. Both works contain ideas an observations thatinspired us during our investigation. The IIT is based on the core concept that humanconsciousness results from integrated information, generated by an ensemble of interactingelements. So, information emerges from collective action, and its content is extremely muchricher than that one can obtain just by a simple summation of the individual contributions,like those provided by the elements belonging to the same ensemble. More in general, theconcept of collective effect pervades the field of complex systems, as effectively explained by3nderson in ’More is different’ [19]. Under that light, Tegmark proposed a computationaldescription of consciousness [3], trying to address the IIT by the language of Physics, anddeveloping both a classical and a quantum representation of this phenomenon. From hiswork [3], we take into account the ’classical’ description of IIT, achieved via the Ising model,where a ’conscious’ regime emerges only within a very restricted range of values. Such re-stricted range refers to the collective phenomena occurring when a spin system gets closeto its critical temperature, and it is in full agreement with the Damasio’s observations [20]on the conditions required for the reaching of homeostasis (i.e. some physical parameters,in the brain, have to be kept within a narrow range of values). Therefore, following ideasand insights of the above-mentioned authors, we propose a method for building a mean-fieldmodel from EEG data. Then, we analyse its behaviour by implementing Monte Carlo sim-ulations, whose outcomes are expected to provide the information we need to quantify thetransitions between mental states of individuals undergoing deep sedation. In particular,we consider the critical temperature computed in the various realisations of the model, e.g.those achieved on varying the frequency of the EEG signal. Details about the proposedmodel and the method for classifying mental states are provided in the next section. Herewe take the opportunity for highlighting that, despite the increasing interest for modellingbrain dynamics by networked approaches (e.g. [21–23, 25–32]), the present investigation isbased on the modelling and the analysis of the distribution of electrical activity recordedin the scalp. Moreover, to each frequency band of the signal, as δ for 1 − θ for4 − α (8 −
12 Hz) band can be useful for quantifying transitions between mental states (e.g. [31]),as well as other signal components. At the same time, most of these works consider networksgenerated by avoiding to include ’weak’ interactions. Notably, to remove weak interactionsa threshold needs to be defined, and such practice has received some fair criticisms [33].Remarkably, to the best of our knowledge, the definition of a suitable threshold is currentlybased only on rules of thumb. So, it is important to remark that the proposed model doesnot require to filter out, or to cut off, weak interactions. Eventually, let us observe thatwhile from a neuroscience perspective the human consciousness might be investigated con-sidering the full set of frequency bands of an EEG signal, those of major interest seem tobe the δ , the α , and the β band. However, due to the influence that has been reported4etween the propofol, i.e. the drug administered in our individuals during the examination,and the behaviour of the β band [34, 35], we decided to take into account only the bands δ and α . A more detailed list of features, for analysing consciousness, can be found in [36].Summarising, our goal is to quantify consciousness, looking also at potential clinical appli-cations. Notably, the EEG signal, as currently processed, provides some information aboutthe state of consciousness of a patient, but it has some limitations. For instance, assessingthe level of unconsciousness, or understanding why some individuals report having beenfully aware (even if, obviously, unable to communicate) during surgery, is currently difficultby inspecting only the EEG signal. Therefore, under the assumption that the latter mightcontain more information than those currently extracted, we propose a method to improveits content (see also [37]). Notably, in mathematical terms, the proposed model can bethought of as a more rich representation of the EEG signal, being mapped to a novel vectorspace that we define state space of mental states. In relation to that, the Curie-Weiss modelrepresents the tool to generate that space of states, by identifying the critical temperaturesof each individual during an examination. Here, while critical temperatures are computedto generate the state space of mental states, they have no meaning with what is occurring inthe brain of individuals. It is also worth to remark that our choice of using a model (i.e. theCurie-Weiss) usually adopted to describe collective phenomena, as phase transitions, aimsto build a direct link with the IIT framework, where the concept of collective behaviouris central. The remainder of the paper is organised as follows: Section II introduces theproposed model. Section III shows results of numerical simulations. Eventually, Section IVends the paper providing an overall discussion on this investigation, from its goal to themain outcomes, and on some possible future developments. II. MODEL
In this section, we describe a framework for classifying mental states, whose variation isrepresented as a phase transition. In statistical mechanics, the most simple representationof phase transitions is achieved by the Ising model, and the latter has been used in [3] forshowing how, in terms of information content, the set of states reachable by that model, atthe critical temperature, contains suitable candidates for representing states of consciousness.Therefore, we aim to evaluate with data (i.e. EEG recordings) whether that theoretical5nsight can be exploited for quantifying consciousness and performing classification tasks. Itis worth to add that the Ising model has been used also by other authors for investigatingdifferent dynamics of the brain (see for instance [21, 23, 24, 38–40]). As above mentioned,the EEG signal can be decomposed into frequency bands and different measures can beadopted for its analysis, usually selected according to specific needs. For our purposes, aparticularly useful parameter is the weighted Phase Lag Index [32] (wPLI hereinafter) thatquantifies the correlation between pairs of sensors in the scalp. In general, considering twotime series a ( t ) and b ( t ), the wPLI is defined as wP LI = | (cid:80) nt = i | im ( P ab,t ) | sgn im ( P ab,t ) (cid:80) nt = i | im ( P ab,t ) | | (1)where sgn indicates the signum function, im indicates the imaginary part, and P ab,t thecomplex cross-spectral density, of a ( t ) and b ( t ), at time t . Notably, given the power spectraldensitities of the two signals P aa and P bb , i.e. the distribution of the power across thefrequency components of a ( t ) and b ( t ), respectively, the cross-spectral density quantifies thecorrelation between them. Then, for each frequency band, the interactions of the resultingCurie-Weiss are computed by scalar multiplication of the wPLI index with the relative powerof the signal. A quick inspection of 1 shows that the wPLI and the power of the signal arestrongly correlated. However, we found beneficial to combine them for realising the mean-field model. In doing so, inspired by the Tegmark’s approach for studying the IIT by a simplephysical system, and remaining in the land of the classical physics, we build a Curie-Weissfrom EEG recordings assigning a spin to each sensor and computing interactions by thecombination of the indexes above mentioned (i.e. wPLI and Power). Thus, starting withrandomly assigned values of spins ( σ ± T αc for the α band. However, since this signal varies over time, node interactions can vary as well. Here,actually, the variation of interactions is expected to be useful for detecting variations ofmental states. For instance, as reported in [31], networks built using the wPLI index showvariations as individuals undergo sedation and then recover to their original conscious state.To tackle this aspect, the EEG signal is sampled into four different points labelled as C , S , DS , R , representing consciousness, sedation, deep sedation, and recovery, respectively.Mental states C and DS are both classified as states of consciousness, in agreement with642], while S and R are labelled as transition states. Also, C and DS can be viewed as twoequilibrium states (although the deep sedation, i.e. DS , in our individuals has been inducedby a drug). Let us now proceed to the formal definition of spin interactions. Recalling thatthe EEG signal is decomposed into 5 main frequency bands and that we extract 4 samples perrecording, for each individual we can generate up to 20 mean-field models. Since the wPLIquantifies the correlation between pairs of nodes, we indicate with wP LI xi,j the correlationbetween sensors i and j in the x th frequency band. Accordingly, the interaction term J reads J xi,j ( s ) = P x ( s ) · wP LI xi,j ( s ) (2)with s sample (or mental state) and P x ( s ) power of the x th band for that specific sample.Thus, the Hamiltonian of the system is H = − N N (cid:88) i,j =1 ,j (cid:54) = i J i,j σ i σ j (3)with N number of sensors and σ spin assigned to them. While the spin is a quantum propertyof particles, it finds large utilisation in non-physical models, typically for representing binaryfeatures. For instance, in social dynamics the spin can represent a binary opinion [43], inevolutionary game theory a strategy [44, 45], and in neural models it can indicate firing (+1)and resting ( −
1) states [46]. The proposed model does not define explicitly a connectionbetween the value of the spin and the underlying neural activity since the spin dynamicsis implemented only to obtain theoretical insights about the activity distribution recordedin the scalp. In particular, the studying of the order-disorder phase transitions, occurringin each model realisation, allows computing the set of critical temperatures T c associatedto every mental state. To this end, for each configuration (i.e. an individual in a givenstate), spin interactions J are considered as quenched , so one can analyse the dynamicsof spins starting from a random distribution. Here, since our efforts are directed towardsquantifying the human consciousness, we put the attention on the α and δ bands that,according to previous clinical studies (e.g. [47, 48]), seem to be quite relevant for investigatingthis complex phenomenon, as well as others as psychotic disorders [49]. The describedmethod allows observing the motion that individuals take in the space of mental states. Suchmotion is defined along a path on a bidimensional plane, whose axes are ( T αc , T δc ). Moreover,since recordings terminate once individuals recover their original cognitive state, the resulting7ath forms a closed cycle. Then, we implement a Machine Learning tool to exploit theresulting paths (or cycles) for classification tasks, as for identifying the correct label of apoint (e.g. DS ) in the mental state space. In particular, a classifier able to assign a label toeach point takes as input vectors with components ( T αc , T δc ). As a huge literature suggests,classifiers can be realised by many algorithms, e.g. neural networks [50]. However, given thesmall size of our dataset, we implemented a Support Vector Machine (SVM hereinafter) [50]—see also A. Summarising, starting from EEG recordings, the proposed model generatesbidimensional vectors, whose entries correspond to the critical temperatures computed inthe α and δ frequency bands, state by state. These vectors constitute the input of an SVMdesigned to discriminate between the two states of consciousness C and DS . III. RESULTS
The proposed model has been tested with a dataset of EEG signals obtained by record-ing 8 healthy volunteers [51] undergoing sedation induced by propofol —see Appendix Bfor details. So, each recording started with individuals in the conscious state and termi-nated after their complete recovery. Four main mental states can be identified: awareness,sedation, deep sedation, and recovery, and for each of them, one sample is extracted fromthe recording. The resulting amount of samples (based on 173 sensors), available for all 5frequency bands, allows generating 20 Curie-Weiss configurations per individual. However,for the reasons reported above, only the α and δ bands are considered, therefore the numberof configurations in this investigation is limited to 8. A model configuration has specificvalues of spin interactions. Let us remind that spins are the mathematical representation ofthe sensors in the scalp, and their value (i.e. ±
1) is randomly assigned as below described.Instead, the interactions are computed by Eq. 2. Before proceeding showing results of thesimulation, we compare the average value of interactions obtained by using the above equa-tion (i.e. Eq. 2) with those (average) values that can be achieved by using the spectral power( P ) and by the wP LI , individually. This comparison is performed over the 4 mental statesand the outcomes have been averaged over all individuals. Results are shown in Figure 1,which reports also the related standard error of the mean, i.e. SEM = σ √ N with σ standarddeviation and N number of individuals. A quick inspection of Figure 1 suggests that ourmethod enhances in good extent the difference between the states C and DS , embedding8he contribution of the spectral power and that of the wPLI index. Let us highlight that our FIG. 1. Average value of the interaction terms, over 4 different mental states, in Curie-Weissbuilt using three different approaches. In particular, the interactions have been defined by using a ) the power spectrum (cid:104) J (cid:105) = (cid:104) P ( s ) (cid:105) , b ) the wPLI index, i.e. the correlation (cid:104) J (cid:105) = (cid:104) wP LI ( s ) (cid:105) between pairs of sensors, and c ) the scalar product of the power with wPLI index (i.e. (cid:104) J (cid:105) = (cid:104) P ( s ) · wP LI ( s ) (cid:105) ). The legend indicates the considered frequency bands: red line for the δ and thegreen line for the α , and the related diamonds and squares represent the average value for eachlabel. The error bars have been calculated using the standard error of the mean (SEM). Then,semi-transparent points (i.e. red stars for δ and green circles for α ) represent the single individuals. goal is now to extract information related to the mental states of individuals by processingthe resulting Curie-Weiss models. So, the next step is focused on the identification of thecritical temperature for the different configurations of the Curie-Weiss model. This processallows us to build a ’state space’ of mental states, where we can quantify, and also visualise,the path followed by individuals across the clinical examination. Then, the paths of mentalstates are used to train a simple machine learning tool whose goal is to classify consciousstates of our individuals. 9 . Mental States I: Building the State Space All numerical simulations have been performed on Curie-Weiss configurations related toevery single individual, i.e. spin interactions have not been averaged as (instead) it hasbeen done for the analysis shown in Figure 1. While spin interactions J are quenched , westudy the dynamics of spins that, at the beginning of each simulation, are randomly setto ±
1. Notably, simulations are implemented for studying order-disorder phase transitions,and more specifically for identifying the critical temperature of each configuration. For thatpurpose, a useful parameter is the absolute value of the average magnetization of the system.The latter strongly depends on the system temperature T , despite its definition does notinclude the temperature explicitly, and it reads (cid:104) M (cid:105) = N (cid:88) i | σ i | N (4)It is worth to observe that beyond identifying the critical temperature T c for each case, it ispossible to assess if T c ≈ (cid:104) J (cid:105) . The latter, as further explained later, can be a relevant featurefor designing clinical applications based on the proposed model. Figure 2 reports the resultsobtained on one individual, randomly chosen among those available. To assess whethera temperature is ’critical’, the variance of M (indicated as σ ) is analysed in function ofthe inverse temperature β since the highest value of σ is reached at T = T c —see insetof Figure 2. Then, once computed the critical temperature for all considered cases, wefocus on the numerical difference between these values and the average interaction term(i.e. (cid:104) J (cid:105) ) of each Curie-Weiss realisation. In doing so, we found that approximating the T c with the (cid:104) J (cid:105) gives small errors, limited to the 12 .
5% of T c (see for instance the linesred and green, indicating the (cid:104) J (cid:105) and the T c , respectively, in figure 2). Finally, resultsof the mean-field model are shown in Figure 3. Notably, plotting the values of the criticaltemperatures computed in the band δ ( T δc ) versus those computed in the band α ( T αc ) allowsus to visualise the path from the initial point C to the point DS , and that of return. So,we have now a state space that contains ’mental paths’, whose evolution (or motion) can befurther analysed. Before proceeding to the classification of mental states, we highlight thatthe order-disorder phase transition studied in the resulting Curie-Weiss model has not abiological meaning in this context. Notably, it is a process simulated to extract informationfrom the resulting model, that we assume can be useful for understanding the dynamics of10 IG. 2. Average magnetization in function of the system temperature. A pictorial of a Curie-Weissmodel is shown close to the main line, while the inset shows the variance of M in function of theinverse temperature β (on a semi-logarithmic scale). The critical temperature is indicated by thegreen line (both in the main picture and in the inset), while the average interaction term for thesystem ( (cid:104) J (cid:105) ) is indicated by a red dotted line (both in the main picture and in the inset). mental states. In addition, the value of critical temperatures has been computed on a finitesize system, while phase transitions occur in the thermodynamic limit. Therefore, furtherinvestigations with bigger systems can be useful also to evaluate if the critical temperatureswe obtained in our analyses are much different from those one would find scaling the size ofthe model. 11 IG. 3. Diagram (cid:104) T αc (cid:105) , (cid:104) T δc (cid:105) , with an inset showing the results obtained on a single individual(randomly chosen). Notably, the points in the diagram (and in the inset) represent the 4 differentmental states during the clinical experiment: C , S , DS , and R , i.e. consciousness, sedation, deepsedation, and recovery, respectively. The black line indicates the path followed by each individual,undergoing sedation and then recovering to the initial state. B. Mental States II: Classification
Each path obtained by the previous method characterises an individual, thus we use thisrepresentation for implementing a Machine Learning tool for classifying mental states. Inour investigation, paths are composed of 4 points in bi-dimensional state space. Therefore,we can identify a vector, whose entries are the critical temperatures computed for the bands α and δ , for each mental state. The goal is to assess whether these vectors are useful forgenerating a confident boundary able to separate different mental states. So, for the sakeof simplicity, we build a classifier for discriminating between the state C and the state DS .Due to the small size of the dataset, we use an SVM implemented by a kernel based radialbasis function (as commonly adopted in classification tasks). The outcomes are shown in12igure 4, where the axes refer to the two critical temperatures of each individual (i.e. oneper frequency band), not to their average values (as in the main plot of Figure 3). Let us FIG. 4. Results of SVM applied to the EEG dataset on the plane T αc vs T δc . Different symbols referto different individuals. The green colour indicates the conscious state ( C ), while the red colourindicates deep sedation ( DS ). The black line identifies the edge between the two classes, i.e. C and DS . briefly describe the procedure for training and testing the SVM. Notably, given a datasetof 8 individuals, the training has been performed on 7 out of them, and the testing on theexcluded one. This process was then repeated excluding each time one different individual.In doing so, we can evaluate 8 different testing phase results. Following the above procedure,now a further analysis compares the results that an SVM achieves when fed with vectorsobtained by three different approaches. Notably, in all cases, vectors resulted from a mean-field model (with the method above described), but its interactions can be defined by Eq. 2,or by the power spectrum and the wPLI, individually. This comparison, shown in Figure 5,is quite relevant because we want to evaluate in which extent the utilisation of Eq. 2 providesan actual benefit for our purposes. Remarkably, we found that our method produced thesmallest error rate in the task of classifying mental states —see Figure 6. In particular, theSVM trained and tested with our method did only one error, over 8 tests (i.e. 12%), while13 IG. 5. Comparison between the results of SVM model built with data coming from three differentmethods: a ) Using spectral power; b ) Using wPLI; c ) Critical temperatures obtained by usingEq. 2. Different symbols refer to different individuals. The green colour indicates the consciousstate ( C ) and red one that of deep sedation ( DS ). The black line identifies the edge between thetwo classes, i.e. C and DS in the mental state space. Then, points with the same symbol, but adifferent colour (e.g. blue and black), indicate an error in the classification process. those trained with data coming from the two other methods did more errors (i.e. 33% and25% by using power spectrum and wPLI, respectively). Before to conclude this section, wedeem important to mention that actual measures might be used to compare outcomes ofdifferent models, as the AUC. At the same time in this investigation, considering the numberof samples, we found beneficial to evaluate the error rate.14 IG. 6. SVM error rate computed by data coming from three different methods: Critical temper-atures ( T c ) obtained by Eq. 2, Spectral power, and wPLI. Note: the lesser the better. IV. DISCUSSION AND CONCLUSION
In this manuscript, we propose a method for quantifying human consciousness and classi-fying mental states using EEG signals (see also [36, 52]). Notably, we introduce a mean-fieldmodel of the distribution of electrical activity in the brain, whose outcomes are used fortraining a Machine Learning classifier. Inspired by the Tegmark’s work [3] about the IIT,we study order-disorder phase transitions occurring in Curie-Weiss models whose interac-tions depend on the phase differences across scalp locations. This analysis allows computingthe critical temperatures achieved for different configurations of the model, i.e. on varyingthe mental state and the considered frequency band of the signal. Here, the interactionsbetween spins are computed by the scalar product of the power spectrum with the wPLIindex, for each specific band. The benefits coming from this choice are reported in Figure 5and Figure 6. Remarkably, the critical temperatures, computed by means of numericalsimulations, allow defining a state space where we can observe the path of mental states.It is worth to mention that, according to Giulio Tononi [42], consciousness emerges alsoduring dreamlike phases of the sleep. Therefore, considering previous investigations statingthat some individuals reported dreamlike activity, during an induced deep sedation [53],15oth awareness and deep sedation in principle should be considered as conscious states (seealso [54]). The path between awareness and deep sedation shows, in the middle, the transi-tion states ( S and R ). Thus, summarising, we have two conscious states and two transitionstates. It is relevant to clarify that we are not using the formal meaning of ’transition state’,as usually adopted in stochastic models (e.g. the voter model [55]), otherwise also the ’deepsedation’ state would be defined a transition state since it lasts only for the duration ofthe drug effect. At the same time, these considerations could be extended further since, forinstance, states of coma could be properly classified as absorbing states, and so on. Hence,our definitions have not that level of formality, despite we find interesting to investigate moreon this. So, once defined a mental state space, we use an SVM for tracing the boundarybetween states C and DS . We remind that the dataset for performing the investigationhas been obtained with 8 individuals wearing EEG sensors. Then, the EEG recordings havebeen used for building the mean-field model (i.e. Curie-Weiss) and generating a trainingdataset. The actual choice of the frequency bands, i.e. α and δ , actually depends also onthe clinical settings of examinations (e.g. the utilisation of propofol for inducing the se-dation). Results suggest that the critical temperatures are useful to perform classificationtasks, discriminating between the two conscious states. Therefore, thinking about the po-tential use of the proposed model, at clinical level, we conceive a framework composed oftwo elements: one devised for computing the average interaction term in different bands,that, as we proved, approximates the critical temperature in the related Curie-Weiss con-figuration, and the other based on an SVM (or on another Machine Learning algorithm).The fact that the average interaction can be approximated by the critical temperature, inprinciple, is not too surprising. Notably, the critical temperature of the 2D Ising model is T c = 1 for J = 1. However, it has been worth, for the reasons above described, to confirmthat hypothesis. Beyond the theoretical analysis, which requires further investigations toconfirm our achievements, the framework that we conceive could support, after appropriatetesting, clinicians in different scenarios. For instance, it could be useful for defining theoptimal amount of drug for sedating a patient, or for classifying the level of unconsciousnessof unresponsive patients. Also, it is interesting to observe that the mean-field model realisedby EEG recordings might be conceptually related to the ’classical’ Tegmark’s description ofconsciousness (i.e. that based on the Ising model). Notably, it would be useful to evaluatehow to obtain only one model, across the different bands, and to study its behaviour at dif-16erent temperatures. Furthermore, we highlight the possibility to extend further this worktrying to improve the connection with the IIT framework, in order to develop mathematicaltools able to analyse the human consciousness from a perspective supported by the Tononi’sinsights. Finally, we deem interesting to provide a comment on the method implementedto represent the path of mental states. Notably, as above reported, each state is describedby a vector of critical temperatures (one per frequency band). So, like for other kinds ofmodels, as those based on network theory, the proposed approach can find application invarious contexts, in particular when the strength of interaction among the elements of asystem is relevant. In our case, the interaction strength clearly shows a time dependency,however that is not a mandatory requirement to implement and analyse a system by meansof a Curie-Weiss model. Moreover, the method might provide interesting insights also whenbuilt considering interactions defined by meaningful semantic relations among elements ofa system, i.e. abstracting from a physical system and therefore increasing its potential ap-plicability to a much wider set of cases. We conclude emphasising that our results indicatethat the EEG signal might be further exploited both for obtaining a deeper understandingof human consciousness, and for implementing novel tools to support clinicians in manycomplex and critical activities. To this aim, there are some important aspects to consider infuture investigations. Firstly, here we focused on two frequency bands ( α and δ ), however itis important to assess if our choice is the optimal one. Notably, although previous literaturesuggests that these two bands are particularly relevant for the human consciousness, findingthe way to exploit also other frequency bands might be useful (as before mentioned, thechoice can depend also on the anesthetic drug). Then, interactions among sensors have beenidentified by means of the wPLI index. However, also other correlation measures could betaken into account. Eventually, we identified transient states, i.e. those states between C and DS . We deem that their role, dynamics and properties need to be further analysed, aswell as a classification algorithm for their detection could be useful. ACKNOWLEDGMENTS
This work was supported by the Belgian National Funds for Scientific Research (FRS-FNRS), the European Unions Horizon 2020 Framework Programme for Research and Inno-vation under the Specific Grant Agreement No. 945539 (Human Brain Project SGA3), the17niversity and University Hospital of Liege, the fund Generet, the King Baudouin Founda-tion, the BIAL Foundation, the AstraZeneca foundation, the Belgian Federal Science PolicyOffice (BELSPO) in the framework of the PRODEX Programme, the Center-TBI project(FP7-HEALTH- 602150), the Public Utility Foundation Universit Europenne du Travail,Fondazione Europea di Ricerca Biomedica, the Mind Science Foundation, and the EuropeanCommission. SL is research director at F.R.S-FNRS. [1] Tononi, G., Edelman, G.M.: Consciousness and Complexity.
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Appendix A: Data Classification
Here we briefly present the algorithm used for classifying mental states, i.e. the SupportVector Machine (SVM). The latter receives as input vectors generated by using the outputof the Curie-Weiss model. In doing so, the input vectors contain the critical temperaturecomputed for each frequency band, across the different mental states. For instance, consid-ering the bands δ and α , a vector representing one individual in the conscious state containsthe two related critical temperatures, i.e. V i ( c ) = [ T c δ , T c α ]. The collection of these vectorsis used for training and testing the SVM. Notably, since an SVM is a supervised learningmodel, it requires a training set for the learning process and, usually, such training set isbased on a fraction (randomly selected) of the dataset under investigation. For instance, ageneric dataset D contains n elements, i.e. D = { ( x , y ) , ( x , y ) , ..., ( x n , y n ) } , so that foreach input vector x i one has the associated output y i , i.e. a label or class of belonging (inour case x i is a temperature vector, while y i is the corresponding mental state). The dimen-sion of the input vector depends on the number of features which describe the system, so inour case corresponds to the number of considered frequency bands. Accordingly, a binaryclassification task can be implemented considering as outputs two possible values, e.g. +1and −
1. Similarly, for a multi-label case, one has to identify proper values for the output.Then, an SVM aims to identify the ’maximum-margin hyperplane’ that separates input vec-tors according to their corresponding output. In the bidimensional case, the hyperplane is asimple line. The SVM can be both linear and non-linear, depending on the function (definedas Kernel) used to separates vectors into the related classes. Also, the closest vectors to thehyperplane are called ’support vectors’. For instance, in the linear case, a bidimensionalhyperplane can be identified by the equation ˆ w ˆ x + b = 0, with ˆ w vector of weights, b (smallconstant) bias, and ˆ x feature space of the input vectors. Therefore, the SVM tries to iden-tify the most suitable ˆ w and b to maximise the margin, i.e. the distance, between vectorsbelonging to the two (or more) different classes. Following the above example, in a binaryclassification, input vectors located above the computed hyperplane belong to one class (e.g.+1), while those located below the hyperplane belong to the other class (e.g. − ppendix B: Clinical Information The full description of the experimental setup implemented to acquire the dataset, usedin this investigation, is reported in [51]. However, for the sake of completeness, here weprovide some relevant details related to the participants and to the experimental protocol.First of all, we emphasise that the investigation was approved by the Ethics Committee ofthe Faculty of Medicine of the University of Liege. Also, the group of participants, withmean age 22 ±±