Whole-brain models to explore altered states of consciousness from the bottom up
Rodrigo Cofré, Rubén Herzog, Pedro A.M. Mediano, Juan Piccinini, Fernando E. Rosas, Yonatan Sanz Perl, Enzo Tagliazucchi
AArticle
Whole-brain models to explore altered states ofconsciousness from the bottom up
Rodrigo Cofré ∗ , Rubén Herzog , Pedro A.M. Mediano , Juan Piccinini ,Fernando E. Rosas , Yonatan Sanz Perl , Enzo Tagliazucchi CIMFAV-Ingemat, Facultad de Ingeniería 2340000, Universidad de Valparaíso, Valparaíso, Chile Centro Interdisciplinario de Neurociencia de Valparaíso 2360103, Universidad de Valparaíso, Valparaíso, Chile Department of Psychology, University of Cambridge, Cambridge CB2 3EB, UK National Scientific and Technical Research Council, Buenos Aires, Argentina Buenos Aires Physics Institute and Physics Department, University of Buenos Aires, Buenos Aires, Argentina Centre for Psychedelic Research, Department of Brain Science, Imperial College London, London SW7 2DD, UK Data Science Institute, Imperial College London, London SW7 2AZ, UK Centre for Complexity Science, Imperial College London, London SW7 2AZ, UK Universidad de San Andrés, Buenos Aires, Argentina * Correspondence: [email protected]
Abstract:
The scope of human consciousness includes states departing from what most of us experienceas ordinary wakefulness. These altered states of consciousness constitute a prime opportunity to studyhow global changes in brain activity relate to different varieties of subjective experience. We considerthe problem of explaining how global signatures of altered consciousness arise from the interplaybetween large-scale connectivity and local dynamical rules that can be traced to known properties ofneural tissue. For this purpose, we advocate a research program aimed at bridging the gap betweenbottom-up generative models of whole-brain activity and the top-down signatures proposed by theoriesof consciousness. Throughout this paper, we define altered states of consciousness, discuss relevantsignatures of consciousness observed in brain activity, and introduce whole-brain models to explorethe mechanisms of altered consciousness from the bottom-up. We discuss the potential of our proposalin view of the current state of the art, give specific examples of how this research agenda might playout, and emphasise how a systematic investigation of altered states of consciousness via bottom-upmodelling may help us better understand the biophysical, informational, and dynamical underpinningsof consciousness.
Keywords: whole-brain models; altered states of consciousness; signatures of consciousness; integratedinformation theory; psychedelics
1. Introduction
Consciousness has been for centuries a puzzle beyond the scope of natural science; however,the significant progress seen during the last 30 years of research suggests that a rigorous scientificunderstanding of consciousness is possible [1–3]. The dawn of the modern scientific approach toconsciousness can be traced back to Crick and Koch’s proposal for identifying the neural correlates ofconsciousness (NCC) [4,5], understood as the minimal set of neural events associated with certain subjectiveexperience. The key intuition that fuels this proposal is that careful experimentation should suffice to revealbrain events that are systematically associated with conscious (as opposed to unconscious or subliminal)perception. Needless to say, the methodological challenges associated with this idea are vast – particularlyconcerning the determination of what constitutes conscious content (e.g. must content be explicitly a r X i v : . [ q - b i o . N C ] A ug of 28 reported, or are other less direct forms of inference equally valid? [6,7]). Despite these problems, theprogram put forward by Crick and Koch succeeded to jump-start contemporary consciousness research. While the quest for the NCC aims to provide answers to where and when consciousness occurs inthe brain, subsequent theoretical efforts have attempted to discover systematic signatures within thoseNCC that could reflect key mechanisms underlying the emergence of consciousness. In other words,these efforts try to answer how consciousness emerges from the processes that give rise to the NCC[12,13]. Hence, theoretical models of consciousness strive to "compress" our empirical knowledge of theNCC, i.e. to provide rules that can predict when and where from how . The nature of those rules, in turn,determines the kind of explanation offered by a theoretical model of consciousness. Here we considertwo possible approaches: top-down and bottom-up [14]. On the one hand, top-down approaches start byidentifying high-level signatures of consciousness, and then try to narrow down low-level biophysicalmechanisms compatible with those signatures. On the other hand, bottom-up approaches build fromdynamical rules of elementary units (such as neurons or groups of neurons [15]), and attempt to providequantitative predictions by exploring the aggregated consequences of these rules across various temporaland spatial scales. We further subdivide explanations into those addressing conscious information access(e.g. perception in different sensory modalities) and those concerning consciousness as a temporallyextended state, such as wakefulness, sleep, anaesthesia, and the altered states that can be elicited bypharmacological manipulation [16–22].Our objective is to put forward a research program for the development of bottom-up explanations forthe relationship between brain activity and states of consciousness, which we claim is underrepresentedboth in past and current research. Theories that rely heavily on a top-down perspective risk beingunder-determined in the reductive sense; i.e. they could be compatible with multiple and potentiallydivergent lower-level biological and physical mechanisms [23]. While we do not know whetherconsciousness may be instantiated in other physical systems, we certainly do know that it is instantiatedin the human brain, and therefore all theoretical models of consciousness should be consistent withthe low-level biophysical details of the brain to be considered acceptable. In light of this potentialunder-determination, it is difficult to decide whether the different theories currently dominating the fieldare competing (in the sense of predicting mutually contradictory empirical findings) or convergent (inspite of being formulated from disparate perspectives). Without investigating theories of consciousnessfrom the bottom-up, it could be simply too early for proposals of an experimentum crucis to decide betweencandidates [24].In this paper we posit that computational models can play a crucial role in determining the low-levelphysical and biological mechanisms fulfilling the high-level phenomenological and computationalconstraints of theoretical models of consciousness. The idea that consciousness is intrinsically dependenton the dynamics of neural activity is not new, and in this sense we follow the trail of pioneers such as WalterJ. Freeman [25], Francisco Varela [26], and Gerald Edelman [27], among others. However, our proposalreaches further than these previous attempts by building upon the technological and conceptual advancesaccumulated over the last decades. In particular, the widespread availability of non-invasive neuroimagingmethods (fMRI, DTI, MEG) has expanded our knowledge of the functional and structural aspects ofthe brain, while the development of connectomics has revealed the intricate meso- and macroscopicconnectivity patterns that wire cortical and subcortical structures together [28]. Moreover, for the first timethere is sufficient empirical data and computational power available to construct whole-brain models withreal predictive power [15,29,30], which represents a radical improvement over past research efforts. We For recent reviews on the empirical search for NCC see Ref. [8], for a theoretical examination of the concept of NCC see Ref. [9],and for criticism to the concept of NCC see Refs. [10,11]. of 28 expect that these advances will enable us to compare the predictions of theories of consciousness by meansof whole-brain computational models that can be directly contrasted with empirical results.In the following, we adopt and explore the consequences of this perspective. Our proposal andits justification are structured as follows. First, Section 2 describes several examples of altered statesof consciousness and briefly discusses some proposed general definitions. Next, Section 3 introducestop-down approaches for quantifying and classifying states of consciousness solely from functionaldata. Then, Section 4 introduces the main technical ideas underlying the development of whole-braincomputational models, highlighting novel results with special emphasis on those informing research onaltered states of consciousness. Section 5 discusses how computational models can contribute to overcomeopen challenges and conceptual difficulties, thus providing new insights into the study of altered states ofconsciousness. Finally, Section 6.1 elaborates on possible future directions of research stemming from ourproposal.
2. What is an altered state of consciousness? Examples and defining features
A basic distinction is commonly drawn between phenomenal and access consciousness [31]. Thefirst represents the subjective experience of sensory perception, emotion, thoughts, etc.; in other words,what it feels like to perceive something, undergo a certain emotion, or engage in a certain thought process.The second represents the global availability of conscious content for cognitive functions such as speech,reasoning, and decision-making, enabling the capacity to issue first-person reports.The term "consciousness" is also used in reference to a third concept whose definition is comparativelymore elusive: that of temporally extended and qualitatively distinct modes or states of consciousness[16–22]. This concept is perhaps best introduced by listing examples, such as our ordinary state of consciouswakefulness, the different phases of the wake-sleep cycle, dreaming during rapid eye movement (REM)sleep, sedation and general anaesthesia, post-comatose disorders such as the unresponsive wakefulnesssyndrome, the acute effects of certain drugs (mainly serotonergic psychedelics and glutamatergicdissociatives), the state achieved in some contemplative traditions by means of meditation, hypnosis, andshamanic trance, among others. Following Ludwig [20] and Tart [32], we refer to these as "altered states ofconsciousness", adopting this term to emphasise their dissimilarity to ordinary conscious wakefulness.Let us describe commonalities shared by altered states of consciousness, which point towards apotential general definition. First, altered states of consciousness are temporally extended and typically(but not always) reversible. Second, they are not defined by the presence of specific subjective experiences,but instead by general and qualitative modifications to the contents of consciousness, including theirexperienced intensity [17]. Third, at least some states can be ordered along a hierarchy of levels, fromstates of "reduced" consciousness (e.g. general anaesthesia, sleep) to others considered "richer" (e.g. certainstates achieved during meditation or induced by pharmacological means) [33].A proper definition of what constitutes an altered state of consciousness is, unfortunately, moreelusive than suggested by the examination of these examples. If states of consciousness are transient, thenwhat is their minimum accepted length? Do qualitative modifications of conscious content apply only tothe sensory domain, or encompass other forms of subjective experience as well? Does a déjà-vu (a briefepisode of eerie familiarity with an unknown past event) qualify as an altered state of consciousness?What about an orgasm, or the state of pain caused by hitting one’s finger with a hammer? Without doubt,these examples modify in one way or another the general contents of consciousness, but they are notcommonly considered as altered states of consciousness.The intuitive notion of "levels" of consciousness is also problematic [34]. We are familiar with the factthat some states appear to be "more conscious" than others; for instance, ordinary wakefulness wouldhave a higher conscious level than deep sleep or an absence seizure. But in what sense is deep sleep of 28
Table 1.
Categories of altered states of consciousness
Category Examples Reversibility
Natural or endogenous deep sleepdreaming transitoryPharmacological general anaesthesiapsychedelic state transitoryInduced by other means meditationhypnosis transitoryPathological epilepsypsychotic episodes transitory or permanent more or less conscious than an absence seizure? Following this logic, how should dreaming, the acuteeffects of psychedelic drugs, and the state achieved by expert meditators be ordered along a hypotheticaluni-dimensional hierarchy of levels of consciousness? It seems that altered states of consciousness canonly be subject to partial ordering, with comparisons between certain pairs of states being questionable oroutright meaningless.These difficulties relate to two main problems. The first problem is granularity: how long is longenough to qualify as an altered state of consciousness? The second is compositeness: instead of a singlelevel of intensity, multiple dimensions are likely required for an unambiguous characterisation; however,it is unclear how many dimensions are needed and how they should be determined [34,35]. A subsidiaryissue related to the granularity problem is whether altered states of consciousness represent discrete stateswith sharply defined boundaries, or are more adequately understood as continuous transitions.Several proposals have been put forward to circumvent these issues and define altered states ofconsciousness [16–22]. Here, we adopt perforce a more pragmatic stance: we are interested in alteredstates of consciousness lasting enough to be investigated by modern neuroimaging techniques (>10 min).At the same time, we strive to show that whole-brain models can be sufficiently rich to transcend theunidimensional characterisation of consciousness in terms of "levels".For the purposes of this article, we divide altered states of consciousness into the following (neitherexhaustive nor mutually exclusive) categories: natural or endogenous (e.g. the states within the sleepcycle), induced by pharmacological means (e.g. general anaesthesia, the psychedelic state), inducedby other means (e.g. meditation, hypnosis), caused by pathological processes, either neurological orpsychiatric (e.g. disorders of consciousness, epilepsy, psychotic episodes), and transitory vs. permanent.
3. Top-down signatures of consciousness from brain signals
A major challenge in the study of altered states of consciousness has been to establish empiricalsignatures in brain signals that are characteristic of different states, thus allowing us to identify them "fromthe outside" – i.e. not depending on self-report or behavioural tasks [13]. Establishing and validatingthese signatures also carries significance from a clinical perspective, since they could lead to efficientand specific biomarkers for certain neuropsychiatric conditions [36,37]. Furthermore, when interpretedwithin a broader theory, some of these signatures may also provide new insights about the nature of thecorresponding conscious states, advancing our fundamental understanding of consciousness itself.In the following, we first provide a broad overview of general aspects of theories of consciousness,and then illustrate what a signature of consciousness is by reviewing two well-known examples. of 28
When we consider the most prominent contemporary theories of consciousness, we find that theymainly differ in what they take as valid empirical data to be explained by the theory. There are essentiallytwo positions on this matter, which can be related to the influential division between functionalistand non-functionalist positions on the mind-brain problem. For a functionalist, the subjective qualityof conscious experience is rejected as a valid target of scientific explanation. According to this view,most famously articulated by Daniel Dennett in
Consciousness Explained [38], only third-person objectivemeasurements fall into the scope of a science of consciousness. This data is limited to observable behaviourand neural activity recordings; for instance, whenever an experimental subject claims to be experiencing acertain shade of blue, the neuroscientist is not tasked with finding how a physical process in the braincan cause a subjective feeling of blue, but with determining the mechanisms leading the subject to declaresuch experience [39]. Non-functionalists, on the other hand, reject this position as a sophisticated form ofbehaviourism [40]. According to this view, introspection plays a crucial role in the scientific explanationof consciousness, because it reveals the very nature of the explanandum itself; any other kind of datarepresents, at best, an indirect approximation [41–43]. It is one of the defining features of consciousness,argue the defenders of this position, that it cannot be illusory [44] since being conscious of something isprecisely what bears that conscious experience into existence [45,46].When translated into the domain of neuroscience, these positions inform the two most influentialcontemporary models of consciousness. The global neuronal workspace theory (GNW) [47,48] linksconsciousness with the widespread and sustained propagation of activity in the cortex, serving thecomputational function of broadcasting information to be processed by specialised modules [49]. Thistheory was developed to explain the neural signatures of consciousness seen in cognitive neuroscienceexperiments – in other words, to explain third-person objective data. On the contrary, integratedinformation theory (IIT) [50–52] is based on certain first-person qualities of subjective experience, whichare accessed by introspection and can be taken as "postulates" or "axioms" for the theory [52]. This theorystrives to provide a quantitative characterisation of consciousness, as well as to determine the neuralcorrelates of conscious contents from first principles only (even though concrete predictions may becomputationally intractable [53]). Both theories have been the target of intense criticism [6,54–58], whichcan be taken as a sign that the scientific problem of consciousness remains unsolved.While GWT and IIT are frequently pitted against each other, their predictions for human brainsmay still be mutually compatible [59,60]. For our purpose, what these two theories have in commonis that they follow a top-down approach, in the sense that they both focus on abstract computationalor information-theoretical principles, without necessarily specifying how these principles arise as aconsequence of local dynamics within the underlying neural substrate. We argue that it is via detailedwhole-brain modelling that the points of agreement and divergence between theories, and how they relateto the neurophysiology of the human brain, can and should be studied ahead of possible experiments.
Since the conception of NCC, neuroscientists have turned to every available neuroimaging technologyin the search for signatures of consciousness [4,5]. Although many kinds of signatures have been explored(including some related to metabolic consumption [61] or cortical connectivity [62]), for the purposes ofthis article we will focus on signatures measurable with functional neuroimaging tools like MEG, EEGand fMRI (which can be simulated with the models described in Section 4). In the sequel, we illustrate thenature and application of signatures of consciousness by elaborating on two well-known examples. of 28 of the correspondingbrain signals. Following this rationale, the level of consciousness should be proportional (at least withinreasonable range) to the entropy of brain signals.An effective tool to estimate the entropy rate of a signal is the Lempel-Ziv complexity (LZc) [63–65],originally conceived as a lossless compression algorithm. The LZc of brain signals has proven to be anextremely robust signature of consciousness, and has been tested in a breadth of scenarios includinganaesthesia [66], coma [67], sleep [68], epilepsy [69], meditation [70] and the psychedelic state [71,72].More recently, it has also been used to assess fluctuations of consciousness during normal wakefulnessdue to cognitive tasks [73], stress [74], fatigue [75], and music performance or listening [76].With its impressive track record and wide applicability, LZc stands as a prominent signature ofconsciousness to compare across biological and simulated brains. Furthermore, LZc can be used intandem with transcranial magnetic stimulation to compute the perturbational complexity index [77], aclinically-tested marker of consciousness, which can also be used as a test measure for whole-brain models.3.2.2. Integrated information theoryA strong limitation of standard brain entropy analyses is that they consider only the entropy ofindividual signals, without acknowledging the multivariate structure of brain dynamics. An attractiveway of studying interdependencies between brain signals is with tools drawn from integrated informationtheory (IIT) [78]. IIT proposes an intimate relationship between consciousness and the ability of a physicalsystem to be integrated in such a way that is "more that the sum of its parts" – i.e. to display dynamicalproperties in the whole that are not observed in any of its parts.IIT builds on key information-theoretic ideas first presented in the seminal early work of Tononi,Sporns, and Edelman [79], and has been subject of continuous development since [50–52,80]. FollowingMediano et al. [81], we distinguish between empirical IIT and fundamentalist IIT as two separate branches ofthe theory. While fundamentalist IIT has been highly controversial and subject of extensive criticism [53,82–84], multiple efforts in empirical IIT have been made to overcome the computational challenges of thetheory [85–87].At the core of empirical applications of IIT is a quantitative measure of integrated information,typically denoted by Φ . There is currently no agreed-upon Φ measure, although multiple proposalshave been put forward [81] and can be used to understand and compare the dynamical structure ofsystems of interest. Detailed procedures describing how to compute different versions of Φ can be foundin Ref. [81]. Although the evidence supporting IIT as a fundamental theory of consciousness has beencontested [88], measures inspired by empirical IIT have proven useful in analysing both empirical [89,90]as well as simulated [87,91] neural data. Altogether, the family of information-theoretic measures inspiredby empirical IIT provides a valuable toolkit to study the multivariate dynamics of whole-brain models. While the entropy estimates the average uncertainty in a signal, the entropy rate estimates how hard is to predict the nexttime-point given its history. of 28
4. Bottom-up whole-brain models
While human neuroscience research has been increasingly dominated by imaging experiments,an important complement to this research is provided by computational neuroscience [92]. In effect,neuroimaging data is usually insufficient to inform the underlying mechanisms at play behind neuralphenomena unfolding at different spatial and temporal scales [93]. Also, since ethical considerationsseverely limit direct causal manipulation of human brain activity, most of the neuroimaging literature islimited to correlational studies.The application of computational models to neuroimaging data with the purpose of making causaland mechanistic assertions has been proposed and developed in parallel with different objectives. Forinstance, deep neural networks can be used to model information-processing in the brain [94] by comparingtheir representational content via second-order isomorphisms (e.g. representational similarity analysis)[95]. These models can be used to investigate the plausibility of different computational architectureswithin cognitive neuroscience [96]. Another example is dynamic causal modelling (DCM), which wasdeveloped to make model-based causal inferences from neuroimaging experiments [97]. DCM is basedon simulating brain signals under the assumption of different causal interactions and then performingmodel comparison and selection. Finally, whole-brain models are based on dynamical systems coupled bylarge-scale anatomical connectivity networks, and are developed to reproduce the statistics of empiricalbrain signals at multiple scales [98]. We also distinguish whole-brain models from attempts to produceextremely detailed reproductions of large neural circuits (e.g. cortical columns) [99], mainly due todifferences in model complexity.Whole-brain models provide a practical, ethical, and inexpensive "digital scalpel", which allowsresearchers to explore the counterfactual consequences of modifying structural or dynamical aspects of thebrain. More generally, whole-brain models build a bridge from local networked dynamics to the large-scalepatterns of activity that are addressed by theoretical signatures of consciousness. As such, they represent avaluable tool to narrow the space of mechanistic explanations compatible with the observed neuroimagingdata, including data acquired from subjects undergoing different altered states of consciousness.In this section, we provide a brief introduction to whole-brain models to the unfamiliar reader,discussing their various types and the principles behind their tuning to empirical data. Additionally, wereview recent articles where these models have been used to shed light on the neurobiological mechanismsunderlying different altered states of consciousness.
Whole-brain models are sets of equations that describe the dynamics and interactions between neuralpopulations in different brain regions. These models typically focus on the joint evolution of a set of keybiophysical variables using systems of coupled differential equations (although discrete time step modelscan also be used, as will be discussed below). These equations can be built from knowledge concerningthe biophysical mechanisms underlying different forms of brain activity, or as phenomenological modelschosen by the kind of dynamics they produce. Then, local dynamics are combined by in vivo estimatesof anatomical connectivity networks. In particular, fMRI, EEG and MEG signals can be used to definethe statistical observables, diffusion tensor imaging (DTI) can provide information about the structuralconnectivity between brain regions by means of whole-brain tractography, and PET imaging can informon metabolism and produce receptor density maps for a given neuromodulator.Most whole-brain models are structured around three basic elements:
A. Brain parcellation:
A brain parcellation determines the number of regions and the spatial resolutionat which the brain dynamics take place. The parcellation may include cortical, sub-cortical, and of 28
Bottom-upWhole-Brain Model forConscious WakefulnessInputs Model Optimization to Fit Empirical Features Whole-Brain Models for Altered States + TractographyParcellationLocal Dynamics
PharmacologicalAltered StateEndogenousAltered StatePathologicalAltered State
Altered StatesBrain Recordings
Figure 1.
Workflow describing the construction of whole-brain models. First, model inputs are determinedbased on anatomical connectivity, a brain parcellation (representing a certain coarse graining), and thelocal dynamics (left). Each region defined by the parcellation is endowed with a specific connectivityprofile and local dynamics. Then, the model can be optimised to generate data as similar as possible to thebrain activity observed during conscious wakefulness. Generally, this similarity is determined by certainstatistical properties of the empirical brain signals, which constitute the target observable. The same oranother observable is obtained from subjects during altered states of consciousness and used again as thetarget of an optimisation algorithm to infer model parameters. Following a given working hypothesis, themodel for wakeful consciousness can be perturbed in such a way that optimises the similarity between thetarget observable for the altered state of consciousness and the data generated by the model. In this way, awhole-brain model for an altered state of consciousness can be used to test working hypotheses about itsmechanistic underpinnings. cerebellar regions. Examples of well-known parcellations are the Hagmann parcellation [100], andthe automated anatomical labeling (AAL) atlas [101].
B. Anatomical connectivity matrix:
This matrix defines the network of connections between brainregions. Most studies are based on the human connectome, obtained by estimating the number ofwhite-matter fibers connecting brain areas from DTI data combined with probabilistic tractography[28]. For control purposes, randomized versions of the connectome (null hypothesis networks) mayalso be employed.
C. Local dynamics:
The activity of each brain region is typically determined by the chosen localdynamics plus interaction terms with other regions. A variety of approaches have been proposed tomodel whole-brain dynamics, including cellular automata [102,103], the Ising spin model [104–106],autoregressive models [107], stochastic linear models [108], non-linear oscillators [109,110], neuralfield theory [111,112], neural mass models [113,114], and dynamic mean field models [115–117]. Adetailed review of the different models that can be explored within this context can be found in[15,29]. of 28
The first two items are guided by available experimental data. In contrast, the choice of local dynamicsis usually driven by the phenomena under study and the epistemological context at which the modellingeffort takes place. Because of this hybrid nature, whole-brain models constructed following this processare sometimes called semi-empirical
We showcase two models that have been frequently used to assess mechanistic hypotheses behindboth pharmacologically and physiologically-induced altered states of consciousness: the dynamic meanfield model [115,116,118], and the model comprised by Stuart-Landau non-linear coupled oscillators[109,110,119]. These examples are chosen to represent a biologically realistic model (dynamic mean field)and a phenomenological model (Stuart-Landau oscillators); moreover, these models have been applied todifferent states of consciousness, making them pertinent in the context of the present discussion.4.2.1. Dynamic mean field (DMF) modelIn this approach, the neuronal activity in a given brain region is represented by a set of differentialequations describing the interaction between inhibitory and excitatory pools of neurons [120]. The DMFpresents three variables for each population: the synaptic current, the firing rate, and the synaptic gating,where the excitatory coupling is mediated by NMDA receptors and the inhibitory by GABA-A receptors.The interregional coupling is considered excitatory-to-excitatory only, and a feedback inhibition controlin the excitatory current equation is included [115]. The output variable of the model is the firing rate ofthe excitatory population that is then included in a nonlinear hemodynamical model [121] to simulate theregional BOLD signals.The key idea behind the mean-field approximation is to reduce the high-dimensional randomlyinteracting elements to a system of elements treated as independent. Then, an average external fieldeffectively replaces the interaction with all other elements. Thus, this approach represents the averageactivity of an homogeneous population of neurons by the activity of a single unit of this class, reducing inthis way the dimensionality of the system. In spite of these approximations, the dynamic mean field modelincorporates a detailed biophysical description of the local dynamics, which increases the interpretabilityof the model parameters.4.2.2. Stuart-Landau non-linear oscillator modelThis approach builds on the idea that neural activity can exhibit – under suitable conditions –self-sustained oscillations at the population level [102,109,110,119,122]. In this model, the dynamicalbehaviour is represented by a non-linear oscillator with the addition of Gaussian noise at the proximityof a Hopf bifurcation [123]. By changing a single model parameter (i.e. bifurcation parameter) acrossa critical value, the model gives rise to three qualitatively different asymptotic behaviours: harmonicoscillations, fixed point dynamics governed by noise, and intermittent complex oscillations when thebifurcation parameter is close to the bifurcation (i.e. at dynamical criticality). Correspondingly, the modelis determined by two parameters: the bifurcation parameter of the Hopf bifurcation in the local dynamics,and the coupling strength factor that scales the anatomical connectivity matrix. In contrast to the DMFmodel, coupled Stuart-Landau non-linear oscillators constitute a phenomenological model, i.e. the modelparameter does not map into any biophysically relevant variable. In this case, the model is attractive due toits conceptual simplicity, which is given by its capacity to produce three qualitatively different behavioursof interest by changing a single parameter.
Whole-brain models are tuned to reproduce specific features of brain activity. The way in which thisis ensured is via optimisation of the free parameters in the local dynamics plus the coupling strength.Parameter values are usually selected such that the model matches a certain statistical observable computedfrom the experimental data.For example, the DMF whole-brain model introduces one parameter to scale the strength of theconnectivity matrix, usually known as the global coupling parameter . During model training, an exhaustiveexploration of this parameter is conducted over a wide range of values. The parameter value is chosento maximise the similarity between the observable computed from simulated and experimental data.For instance, the parameter can be chosen to minimise the Kolmogorov-Smirnov distance between thefunctional connectivity dynamics (FCD) distributions of the simulated and real data [115].This kind of brute-force optimisation is employed when the number of free parameters is low (i.e. twoor three). However, it is also possible to separately optimise the parameters governing the local dynamicsof each node, which dramatically increases the dimensionality of the search space, and thus requiresmore elaborated optimisation techniques, such as gradient descent [124] or genetic algorithms [119]. Theadvantage of considering a small set of global parameters resides in its simplicity and scalability, butunfortunately it misses the dynamical heterogeneity among brain regions. These heterogeneities can bemodelled at the expense of increasing the parameter space. Essentially, the choice of model complexity (i.e.the number of free parameters) depends on the scientific question and its associated hypotheses.Since adding more free parameters increases the computational cost of the optimisation procedure, itbecomes critical to choose parameters reflecting variables that are considered relevant, either from a generalneurobiological perspective or in the specific context of the altered state under investigation. Dependingon the latter, the parameters could be divided into groups that are allowed to change independently basedon different criteria, including structural lesion maps, receptor densities, local gene expression profiles,and parcellations that reflect the neural substrate of certain cognitive functions, among others.After choosing the parcellation, the equations governing the local dynamics and their interactionterms, the interregional coupling given by the structural connectivity matrix, and selecting a criteria toconstrain the dimensionality of the parameter space, the last critical step is to define the observable whichwill be used to construct the target function for the optimisation procedure. As mentioned above, onepossibility is to optimise the model to reproduce the statistics of functional connectivity dynamics (FCD).Perhaps a more straightforward option is to optimise the "static" functional connectivity matrix computedover the duration of the complete experiment, an approach followed by Refs. [119] and [110], among others.Other observables related to the collective dynamics can be obtained from the synchrony and metastability,as defined in the context of the Kuramoto model [110,125]. In general, any meaningful computationsummarising the spatiotemporal structure of a neuroimaging dataset constitutes a valid observable, withthe adequate choice depending on the scientific question and the altered state of consciousness understudy.Since different observables can be defined, reflecting both stationary and dynamic aspects of brainactivity, a natural question arises: is a given whole-brain model capable of simultaneously reproducingmultiple observables within reasonable accuracy? We consider this question to be very relevant, yet atthe same time it has been comparatively understudied. For instance, a review of articles using coupledStuart-Landau oscillators shows that dynamical observables are reproduced when the oscillators operateat dynamical criticality (i.e. near the Hopf bifurcation), yet stationary observables (such as the "static"functional connectivity") are best reproduced for other parameter combinations [110,119,124]. This suggeststhat exploring bifurcations with higher co-dimensions or even chaotic dynamics unfolding in the proximity of strange attractors could enable the simultaneous optimisation of several observables, a possibility thatis discussed later in this article.Finally, some natural candidates for observables to be fitted by whole-brain models are preciselythe high-level signatures of consciousness put forward by theoretical predictions, such as the differentmeasures of information integration, complexity and entropy that were reviewed in the previous section.The objective is to fit whole-brain models using these signatures as target functions and then assess thebiological plausibility of the optimal model parameters, which allows to test the consistency of thesesignatures from a bottom-up perspective. Alternatively, signatures of consciousness can be computedfrom the model –initially fitted to other observables– and compared to the empirical results. Again, thishighlights the need to understand which kind of local dynamics allow the simultaneous reproduction ofmultiple observables derived from experimental data.
The available evidence suggests that states of consciousness are not determined by activity inindividual brain areas, but emerge as a global property of the brain, which in turn is shaped by itslarge-scale structural and functional organisation [48,126,127]. According to this view, whole-brainmodels provide a fertile ground to explore how global signatures of different states of consciousnessemerge from local dynamics. This promise is already being met, as shown by several recent articles[33,102,109,110,118,119,122,128].For example, transitions from wakefulness into other states, such as the different stages of humansleep or the state induced by general anaesthetics, have been interpreted as phase transitions in neuralmass models and in terms of the collective dynamics of coupled Stuart-Landau oscillators [109,110,119].Noise-driven systems at dynamical criticality result in dynamics compatible with neuroimaging recordingsobtained during conscious wakefulness, and departures from these dynamics better reflect different statesof unconsciousness [33,102,122,128–130]. As will be discussed in the following section, the stochasticswitching between different attractors results in the kind of metastable behaviour that is characteristic ofconscious wakefulness [131]. These results are consistent with the hypothesis of statistical criticality (e.g.proximity to a second order phase transition) as a fundamental principle of brain organization [132]. Eventhough parallels can be drawn between statistical and dynamical criticality, we limit our discussion to theformer since the relationship between both concepts is complicated and beyond the scope of this article.Following the example of the PCI index (which is obtained by perturbing the cortex with TMSand measuring the complexity of the elicited response) [77], whole-brain models can be systematically"perturbed" by incorporating changes into the dynamical equations. The in silico rehearsal of perturbationsis useful to test hypotheses concerning which parts of the model are essential to produce differentsignatures of consciousness. A prominent example of this perturbational analysis applied to whole-brainmodels can be found in a recent article [118] where a whole-brain model based on coupled Stuart-Landauoscillators was fitted to empirical fMRI data acquired from subjects during deep sleep. The model wasthen modified by changing local bifurcation parameters with a greedy optimization algorithm, whichunveiled the optimal perturbation profile to increase the similarity to a target brain state (in this case,conscious wakefulness). Another relevant example of this perturbational approach is found in Ref. [116],where a transition was shaped by the effects of neuromodulation. The authors investigated the transitionfrom resting state activity acquired under a placebo condition towards the altered state of consciousnessinduced by the serotonin 2A receptor agonist lysergic acid diethylamide (LSD). A dynamical mean-fieldmodel was fitted to minimize the difference between FCD of the simulated activity and the empirical dataof subjects in the placebo condition, which allowed to determine the optimal value of the global couplingparameter. Then, an empirical map of 5- HT A receptor density was used to modulate the synaptic gain, effectively simulating the heterogeneous effects of LSD across the whole brain. As a control, the authorsshowed that using maps for the density of other serotonin receptor sub-types decreased the goodness offit, thus corroborating the well-known association between LSD and the 5- HT A receptor.Another interesting possibility is to assess the consequences of stimulation protocols that areimpossible to apply in vivo . An example is the Perturbative Integration Latency Index (PILI) [122], whichmeasures the latency of the return to baseline after a strong perturbation that generates dynamical changesdetectable over long temporal scales (on the order of tens of seconds). This in silico perturbative approachallows to systematically investigate how the response of brain activity upon external perturbations isindicative of the state of consciousness, providing new mechanistic insights into the capacity of the humanbrain to integrate and segregate information over different time scales.In Ref. [119], the authors used a model of coupled Stuart-Landau oscillators to model the regionalchanges in dynamical stability that occur during the wake-sleep cycle. Brain regions belonging to differentresting state networks (RSN) [133] were considered as independent sources of variation for the localmodel parameters. Using a stochastic optimisation algorithm, the authors represented the transition fromwakefulness into deep sleep as a sequence of changes in the stability of brain activity within canonical RSN.A follow-up paper extended this analysis to other states of reduced consciousness (including anaesthesiaand patients suffering from disorders of consciousness) and investigated the possibility of inducingtransitions to conscious wakefulness by means of simulated periodic stimulation at the resonant frequencyof each node in the model [134].
5. Proposed research agenda
Consciousness research is in need of mechanistic accounts to explain why brain signals recordedduring different states of consciousness can be consistently characterised by the presence of certain globalsignatures. Our motivation is not the replacement of the explanations of these signatures provided bytheories such as GNW or IIT. Instead, we aim to put forward a framework for their investigation froma bottom-up perspective. Eventually, we expect to converge on the high-level explanations furnishedby some of these theories. Our inspiration is partially drawn from statistical thermodynamics, whichprovides a clear example of how the bottom-up and top-down perspectives can converge into a consistentpicture of physical reality. Importantly, in this case the resulting theory remained useful both as a setof phenomenological principles and computational rules (i.e. classical thermodynamics), but also as aframework to establish connections between those principles and the rules governing the microscopicproperties of matter.Following this concept, we strive to use our current knowledge about neural dynamics to producemodels whose behaviour agrees with the constraints of some theories formulated from a top-downperspective, while weakening the support for others as a result of inconsistent predictions. Here it becomesimportant to clarify our intended meaning of the word "prediction". When it comes to complex systemssuch as the brain, predictions are considered possible only in a statistical sense [132]. Accordingly, we donot expect that the time series generated by computational models directly correspond to their empiricalcounterparts; however, we can expect a match for statistical observables.This motivates our study of altered states of consciousness, since their extended temporal durationguarantees the possibility of extracting robust statistical characterisations from multivariate neuroimagingrecordings. An example of this characterisation is the matrix derived from computing all pairwisecorrelations between regional time series, which is considered a marker of inter-areal functionalconnectivity (sometimes referred to as the "functional connectome") [135]. We consider that whole-brain computational models have been developed to a point where they contain sufficient empirical ingredientsto predict the second-order statistics of brain activity. Thus, the field is ripe to welcome a framework whichmay provide solid ground to investigate signatures of consciousness from a mechanistic perspective.The following example is aimed to motivate the proposal we put forward in the next section. Weknow that activity within a network of brain regions including the fronto-parietal cortex is correlatedwith conscious experience [8,62,136–138]. On the other hand, conscious experience is also characterised bysignatures such as information integration, entropy and neural complexity. Is it possible to determine thecausal role that these anatomical regions play in the generation of these signatures of consciousness bymeans of computational models?
The principal idea behind our proposal is that whole-brain models can be used to test hypothesesconcerning the mechanistic and causal underpinnings of different states of consciousness. We do notexpect that whole-brain models are sufficiently advanced to identify those precise mechanisms; however,we propose that they can contribute to narrow the space of possible mechanistic explanations, thereforecomplementing current theories of consciousness from a bottom-up perspective.The fundamental objective of this research program is to foster the development of this novelapproach to study altered states of consciousness. Our framework rests upon the complementary natureof three key ingredients: experimental data obtained through neuroimaging experiments, theoreticalapproaches to characterise signatures of consciousness, and bottom-up whole-brain computational models.The application of modern neuroimaging techniques to the study of signatures of consciousness hasprovided very effective tools to predict the brain activity patterns that are associated with different statesof consciousness. However, as René Thom famously stated "to predict is not to explain" [139]. Hence, wenow turn to the discussion of how models could bridge the gap between prediction and explanation.The proposed framework to model altered states of consciousness is based on adjusting threeindependent variables (see Figure 2):
A. Connectome:
Is the state of consciousness implicated with local or diffuse structural abnormalities?This is frequently the case for neurological conditions such as coma and post-comatose disorders ofconsciousness (e.g. unresponsive wakefulness syndrome, minimally conscious state) [140]. Also,subtler structural modifications can be implicated in certain psychiatric conditions presentingepisodes of altered consciousness, such as different forms of schizophrenia [141].
B. Modulation:
Is the state of consciousness a consequence of neuromodulatory changes, eitherendogenous or induced externally by means of pharmacological manipulation? Two typical examplesare the altered states of consciousness induced by psychedelics/dissociatives, which are linked toagonism/antagonism at serotonin/glutamate receptors [142]. Certain psychiatric conditions arebelieved to arise as a consequence of neuromodulatory imbalances, e.g. dopaminergic imbalances arebelieved to play an important role in the pathophysiology of schizophrenia [143]. Most anaestheticdrugs reduce the complexity of the brain activity by targeting specific neuromodulatory sites, suchas those activated by gamma-aminobutyric acid (GABA) [144]. Finally, sleep is a state of reducedconsciousness triggered by activity in monoaminergic neurons with diffuse projections throughoutthe brain [145].
C. Dynamics:
Is the altered state of consciousness captured by well-understood dynamical mechanisms?Does the model include parametrically controlled external perturbations? While changes in thelocal excitation/inhibition balance are ultimately caused by neurochemical processes, they are bestunderstood in terms of their dynamical consequences. States such as epilepsy, deep sleep and generalanaesthesia are believed to involve unbalanced excitation/inhibition [146]. In some cases, dynamics R e c e p t o r D e n s i t y Local Dynamics and PerturbationsStructuralConnectivity NeuromodulationNeurostimulation PharmacologySleep AnesthesiaConsciousWakefulnessDOCComa Dynamics S c h i z o p h r e n i a E p il e p s y ModulationConnectome
Figure 2.
Representation of the three key variables that can be modified to construct whole-brain modelsof different altered states of consciousness. These variables correspond to local dynamics, anatomicalconnectivity, and priors related to neuromodulatory systems necessary to accommodate physiological,pathological and pharmacologically-induced altered states of consciousness. Certain states may requirethe modification of multiple variables; for instance, focal seizures and propofol-induced anaesthesia areboth associated with low complexity patterns of brain activity, yet in the first case these dynamics reflectstructural abnormalities, while in the second case they reflect the activation of certain inhibitory pathways. may be sufficiently idiosyncratic to be captured by low dimensional phenomenological models, asin the case of certain forms of epileptic activity [147]. Finally, local dynamics could be modified tosimulate the effects of external neurostimulation [118,134].Depending on the answers to these questions, the whole-brain model should incorporate changes toanatomical connectivity, local dynamics, or include empirical receptor density maps to add a new layer ofneurobiological detail.
The dynamics of whole-brain models can be perturbed arbitrarily. This is significant since it allowsto explore different mechanisms leading to the observed empirical dynamics (as described in a previousparagraph) and to explore how external stimulation can force transitions between states of consciousness,including the clinically relevant case of displacing whole-brain models from unconscious states towardswakefulness [118,134]. Therapeutic alternatives to accelerate the recovery of DOC patients are scarce,and while some studies support the therapeutic role of external electrical stimulation [148], very littleis known about the optimal choice of stimulation sites and parameters. Whole-brain models could be useful for the optimization of stimulation protocols, as well as for assisting in clinical decision making.Localized stimulation and/or resection of neural tissue are surgical alternatives to treat certain severeforms of epilepsy, and whole-brain models have been explored with success to predict the outcomeof these interventions [149]. The same concept could apply to the development and in silico testingof new pharmaceuticals to treat psychiatric conditions, where whole-brain models could be used toreverse-engineer the optimal receptor affinity profiles required to restore statistical signatures of healthybrain dynamics. Finally, the combination of data produced by whole-brain models and machine learningclassifiers could be useful for data augmentation in the context of automated diagnosis of rare neurologicaldiseases [150], and to generate input for deep learning architectures (e.g. variational autoencoders) capableof representing altered states of consciousness as trajectories within a low dimensionality latent space.[151].
To further highlight what we can learn from whole-brain models, we discuss an illustrative example ofa bottom-up model that successfully matches a global signature of altered conscious [152]. Using the DMFmodel optimised to fit the FCD of placebo and LSD conditions [116], a significant entropy increase of brainsignals was found in LSD vs. placebo as a consequence of simulated 5- HT A receptor activation. Thus, themodel was capable of identifying a low-level (i.e. molecular scale) mechanism leading to increased neuralentropy, which is a robust signature of the psychedelic state [33,63].Since activation of the 5- HT A receptor is causally implicated with the conscious state induced byserotonergic psychedelics [142,153,154], the effect of the drug was modelled as a local change in thenon-linearity of the regional firing rate. This change was proportional to the local density of 5- HT A receptors as determined by PET imaging. Brain entropy increases during the psychedelic state were theresult of heterogeneous changes in the entropy of the regional firing rates (i.e. some regions increasedwhile others decreased their entropy). These changes in firing rate entropy depended both on the localanatomical connectivity and the 5- HT A receptor density.Thus, starting from local dynamics describing the behaviour of coupled excitatory and inhibitory poolsof neurons, and introducing a perturbation which reflects serotonergic activation, the model provideda bottom-up confirmation of 5- HT A activation as the source of increased neural entropy during thepsychedelic state. In the context of Fig. 2, the model adopted changes in local dynamics (bottom left)informed by empirical maps of 5- HT A receptor density (bottom right).
6. Future directions
The question of the ultimate substrate of consciousness is part of a long-standing philosophical debate,with positions including functionalism (the substrate is irrelevant insofar it instantiates the adequate set ofcausal relationships) [38], biological naturalism (the view that consciousness arises as a consequence ofbiochemical processes in the brain) [155], and proposals of consciousness as a manifestation of quantummechanics [156]. Even though we choose to sidestep this complicated discussion, our modest aim ofbuilding bottom-up models of brain activity still requires the specification of some physical or biologicalsubstrate, which in turn determines the level of realism displayed by the equations that govern localdynamics.Many signatures of consciousness are directly related to the global complexity of brain dynamics,reflecting the widespread hypothesis that consciousness plays an integrative role in the brain [127].According to this hypothesis, consciousness could be considered a dynamical process "gluing" together the output of specialised neural circuits. While tampering with these circuits could modify some specificcontents of consciousness, only the disruption of large-scale neural communication would result ina state of altered or reduced consciousness. Since this view disregards the contribution of specificcomputations that are implemented in local neural circuitry, we could expect that bottom-up modelscapable of reproducing an adequate set of canonical dynamics will suffice to span the spectrum ofsignatures of altered consciousness. Conversely, it could be that the large-scale dynamics that supportinter-areal communication at the same time interact and shape local information processing, and vice-versa.In this case, we expect that increasingly complex and biologically realistic models will be needed to advancewith our proposal.This crucial point results in a ramification within our proposal to investigate altered states ofconsciousness using whole-brain models. On one hand, models could be enriched by increasingly detailedand sophisticated sources of empirical information with the purpose of linking signatures of consciousnessto the biophysical details of neural activity. This direction is already suggested by studies modelling theeffects of 5- HT A activation using receptor density maps produced by PET imaging [116,152]. Followingthis direction, future models could be expanded to include fine-grained details of local wiring patterns,different cell types and their projections, as well as their interaction with diffuse neuromodulatory systems.However, as complexity is increased, the conceptual interpretation of models becomes less clear. On theother hand, it is known that dynamical systems may exhibit canonical behaviours when their solutionsundergo changes in their qualitative behaviour (i.e. bifurcations) [157]. Recent work fitting whole-brainmodels to the results of fMRI experiments suggests that bifurcations play a key role in the reproduction ofthe second-order statistics of empirical data [102,109,110,119,122]. This occurs because noisy dynamicsclose to a bifurcation point switches between different attractors, producing rich and complex dynamicstypical of brain signals. This observation raises the question of whether more complex models reproducethe statistics of empirical observables by virtue of their universal behaviour near bifurcation points, or as aconsequence of their stationary solutions away from dynamical criticality. Contrary to the dictum by Norbert Wiener ( "The best material model of a cat is another, or preferably the same,cat" ) we propose that even if vast sources of biological information can be incorporated into whole-brainmodels, striving for such level of detail defeats the purpose of unveiling concrete and interpretablemechanisms underlying signatures of consciousness. Thus, we suggest that models could be classified bythe kind of large-scale activity patterns they are capable of generating. In other words, we propose thatthe "bottom" of bottom-up models should not be related to the scale of the biological substrate, but to theminimal set of simple dynamical behaviours necessary to reproduce a certain signature of consciousness.Paralleling the definition of NCC given by Crick and Koch [4,5], we could introduce the "dynamicalcorrelates of consciousness" (DCC); but we opt to not introduce yet another acronym in an already crowdedfield.Interestingly, Batterman has suggested that multiple realizability, the "metaphysical mystery" thattroubled Jerry Fodor, among other great philosophers of the mind, is as mysterious as the observation thatphysical matter behaves in ways which are entirely independent from the vast majority of its details [158].For a typical example consider a pendulum, whose behaviour is described by the same differential equationregardless of the colour of the swinging bob. Furthermore, in the small amplitude regime all systems with Here, canonical dynamics refers to dynamics in the proximity of a class of topologically equivalent attractors. The reader shouldthink of the result of simplifying the equations into the normal forms corresponding to the bifurcations present in the system[157].7 of 28 an U-shaped energy landscape can be approximated by an harmonic solution, with examples rangingfrom electrical circuits to orbital mechanics. Northoff and colleagues have argued that the spatiotemporaldynamics constitutes the fundamental substrate underlying human consciousness [159], which resonateswith Batterman’s proposal, as well as with our suggestion that the "bottom" (i.e. the maximum necessarylevel of detail) is best understood as a comprehensive list of the dynamical behaviours that the system candisplay. We postpone taking a stance towards these metaphysical speculations, and proceed to developthese ideas in the context of building useful bottom-up models in the future.A set of qualitatively different dynamics is provided in Fig. 3, illustrating a Takens-Bogdanovbifurcation diagram [160]. Whole-brain models can be constructed by coupling the dynamics given as anequation in the inset (left panel) either by variables x , y , or both. The equation and its solutions depend ontwo parameters, α and β . Under the weak coupling assumption, modifying these two parameters willresult in qualitative changes in the local dynamics (where these changes occur in the diagram could bemodified by the coupling strength). For uncoupled dynamics, parameter combinations at points a , c , e result in a stable constant level of activity (i.e. fixed point dynamics). Parameter combinations at points b , d , f give rise to oscillations of different spectral content (i.e. limit cycles).In the right panel of Fig. 3, the solutions can be visualised either as time series or as two dimensionaldiagrams known as phase portraits, where each axis corresponds to a variable (in this case, x and y ) andthe arrows stand for the vector field (in this case, ˙ x and ˙ y ). Insofar the bifurcations in the left panel of Fig.3 are not crossed, changes in the parameters α and β only result in deformations of the phase portrait,representing solutions that are equivalent in a qualitative sense (more formally, the phase portraits aretopologically equivalent). Crossing a bifurcation results in an abrupt change that cannot be understood asa small deformation of the phase portrait, implying a qualitatively different behaviour of the system.The richness of coupling this kind of simple dynamical models stems from the possibility of inducingstochastic transitions across bifurcations by incorporating an additive noise term. In this way, dynamicsswitch intermittently between two qualitatively different solutions. In the case of the Hopf bifurcation,for instance, noise-driven dynamics at the bifurcation point are neither stable nor oscillatory, but presentcomplex amplitude fluctuations [124]. The noise-driven exploration of a system’s attractor space is amainstay of computational neuroscience [161] and could represent an useful methodological resource tobuild whole-brain models to explore altered states of consciousness.Following the pioneering work of Deco and colleagues [124], the most frequently explored transitionis between stable noise-driven dynamics and self-sustained harmonic oscillations, corresponding to theHopf bifurcation (vertical red line in Fig. 3), which appears in Stuart-Landau nonlinear oscillators. Atthe bifurcation point, dynamics show the kind of complexity that is compatible with certain signaturesof consciousness, with departures from this point being reported for states of reduced consciousnesssuch as sleep and anaesthesia [110,118,119,122] (as it is clear from Fig. 3, however, this bifurcation is onlyone among multiple possibilities). The upper panel of Fig. 4 illustrates this situation by presenting thephase space and temporal evolution of a noise-driven Stuart-Landau nonlinear oscillator near dynamicalcriticality. The signal evolves with complex amplitude fluctuations as noise drives the dynamics acrossthe bifurcation. Also, at dynamical criticality small fluctuations tend to be amplified [110,124], thuswhole-brain models far from criticality reproduce the lack of sustained and complex responses to externalperturbations seen in states of reduced consciousness [77].The inclusion of noise in whole-brain models raises questions concerning the mechanisms thatendow biological systems with stochastic dynamics [161]. Again, we postpone these difficult questions in lieu of more practical considerations, and propose that noise-driven equilibrium dynamics increaseinterpretability at the expense of two main shortcomings. First, parameter fine-tuning is required to posedynamics near dynamical criticality. As discussed above, optimisation procedures can be applied to obtainthe parameters which best reproduce certain empirical observables. However, the biological variables Figure 3.
Left panel:
Takens-Bogdanov bifurcation diagram, which is obtained by changing parameters α and β in the normal form equations (included as an inset). Depending on the combination of parameters,this simple dynamical system can present qualitatively different solutions. The green line stands for asaddle-node bifurcation, where two equilibrium points collide and disappear. Crossing the red line resultsin a Hopf bifurcation, where dynamics switch from a fixed point to stable harmonic oscillations. The dashedline represents a homoclinic bifurcation, where the limit cycle collides with a saddle point resulting againin steady dynamics. Right panel:
The phase portraits a-f illustrate the dynamics at different regions of thebifurcation diagram, with individual trajectories highlighted in red and presented both as curves in phasespace and as time series. a) Stable fixed point, b) Self-sustained harmonic oscillation after the appearanceof a stable limit cycle, c) Three fixed points appear due to a saddle-node bifurcation, resulting in a stablefixed point, d) One of the stable fixed points loses its stability and dynamics undergo a Hopf bifurcation, e)The limit cycle undergoes a homoclinic bifurcation, f) A saddle-node on a limit cycle (SNIC) bifurcationoccurs, resulting in periodic dynamics with complex spectral content. For a detailed description of theTakens-Bogdanov bifurcation see Ref. [160]. Left panel adapted from Ref. [162]. captured by the optimal combination of parameters could change upon small perturbations, leading tomodels that always predict intrinsically unstable states of consciousness. The second problem is thatonce parameters are optimised to reproduce a certain observable, other different observables could bepoorly captured by the model, thus questioning the extent to which the model is adequately describingthe empirical data. We propose that both problems could be simultaneously addressed by exploringnon-stochastic models of chaotic coupled oscillators, such as Rossler oscillators. In this model, dynamicsunfold near a strange attractor with positive Lyapunov exponent for a comparatively ample range ofparameters [163]. Thus, complex dynamics do not depend on a bifurcation parameter taking a precisevalue, but instead arise over an extended range of parameter values. This kind of phenomenologicalmodels of whole-brain activity is comparatively understudied, and could represent a valuable target forfuture developments.
7. Final remarks
The history of science shows an intensive ongoing debate about the position of scientific inquires withrespect to the study of consciousness. As a matter of fact, until recently the largest part of the scientificcommunity did not consider consciousness as a suitable topic for investigation. While the ultimate natureof consciousness is still full of mysteries, it is evident that deepening our knowledge of the mechanistic,statistical, and dynamical relationships within the brain in its different possible states of consciousness canonly increase our understanding of the relationship between mind and body.
Figure 4.
Upper panel:
Phase space of a single Stuart-Landau nonlinear oscillator near dynamical criticality(Hopf bifurcation) with an additive noise term. The radius of the limit cycle fluctuates unpredictably,resulting in complex signal amplitude modulations.
Bottom panel:
Phase space of a chaotic Rossler oscillationin a regime with positive Lyapunov exponent, without the addition of noise. Dynamics unfold in theproximity of a strange attractor, which results in complex but deterministic dynamics.
A key factor supporting the modern discipline of consciousness research is the extraordinarydevelopment of neuroimaging technologies that occurred over the last decades, which plays a similarfundamental role than the one played by telescopes in the discovery of the nature of the solar system.However, making progress in the problem of consciousness not only depends on technological advances,but also on our capacity to explore and chart the contents and boundaries of consciousness itself.Consciousness research needs neuroimaging as much as any other branch of human neuroscience, but alsoneeds to devise and explore new methods to induce altered states of consciousness, and to break througharbitrary regulatory restrictions preventing the exploration of certain older but very powerful researchtools [164,165].These technological advances, matched with increases in computational capability, and a renewedappreciation of the role that altered states of consciousness play in scientific research, have prepared afertile ground for whole-brain models to open a new window of research possibilities. In effect, whilemuch progress has been made during the last decades in the problem of identifying top-down signaturesof consciousness, most of these tools have not yet reached a stage of maturity to allow clinical applications.We expect that pursuing the problem from a different perspective will be invigorating for the field as awhole, increasing the appreciation for the role that low-level biological mechanisms play in the emergenceof high-level signatures of consciousness.Consciousness research is not alone in its need for low-level mechanistic explanations. The projectof formulating psychiatric diagnosis in biological terms [166] will require a systematic exploration ofthe low-level mechanisms giving rise to the behavioural manifestations of mental disorders [167,168].We expect that many of the ideas and methods here proposed will seamlessly translate into the field of computational psychiatry, even for the study of disorders which do not include altered consciousness as adefining feature (e.g. depression).In the same way that scientific inquiry has eventually succeeded explaining seemingly mysteriousphenomena such as heat (in terms of kinetic considerations), combustion (in terms of chemical reactions)and genes (in terms of molecular replication), it is reasonable to expect that consciousness will also beexplainable someday in mechanistic terms. If this is to happen, the perspective of bottom-up modelling islikely to play a crucial role, as it was the case for the three aforementioned examples. It is our hope that thepresent proposal will serve both as an encouragement and as a roadmap to invest future research efforts inthe computational modelling of altered states of consciousness.
Author Contributions:
Conceptualization, R.C., R.H., P.A.M.M, F.E.R, Y.S.P and E.T; methodology, R.C., R.H., P.A.M.M,F.E.R, Y.S.P and E.T; writing–original draft preparation, R.C., R.H., P.A.M.M, F.E.R, J.P, Y.S.P and E.T; writing–reviewand editing R.C., R.H., P.A.M.M, F.E.R, J.P, Y.S.P, and E.T
Funding:
R.C. was supported by Fondecyt Iniciación 2018 Proyecto 11181072. R.H. was funded by CONICYTscholarship CONICYT-PFCHA/Doctorado Nacional/2018-21180428. P.M. was funded by the Wellcome Trust (grantno. 210920/Z/18/Z). F.R. was supported by the Ad Astra Chandaria Foundation. E.T. and Y.S.P. were supported byANPCyT (Argentina), grant PICT-2018-03103
Conflicts of Interest:
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:NCC Neural correlates of consciousnessDMF Dynamic mean fieldfMRI Functional magnetic resonance imagingBOLD Blood oxygen level–dependentPET Positron emission tomographyDTI Diffusion tensor imagingEEG ElectroencephalographyMEG MagnetoencephalographyIIT Integrated Information TheoryGNW Global neuronal workspaceEBH Entropic brain hypothesisLZc Lempel-Ziv complexityFCD Functional connectivity dynamicsPCI Perturbational complexity indexTMS Transcranial magnetic stimulationPILI Perturbative Integration Latency IndexLSD Lysergic acid diethylamideAAL Automated anatomical labellingDOC Disorder of consciousnessGABA Gamma-aminobutyric acidRSN Resting-state networks
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