Coupling Functions in Neuroscience
CCoupling functions in neuroscience
Tomislav Stankovski
Abstract
The interactions play one of the central roles in the brain mediat-ing various processes and functions. They are particularly important for thebrain as a complex system that has many different functions from the samestructural connectivity. When studying such neural interactions the couplingfunctions are very suitable, as inherently they can reveal the underlayingfunctional mechanism. This chapter overviews some recent and widely usedaspects of coupling functions for studying neural interactions. Coupling func-tions are discussed in connection to two different levels of brain interactions– that of neuron interactions and brainwave cross-frequency interactions. As-pects relevant to this from both, theory and methods, are presented. Althoughthe discussion is based on neuroscience, there are strong implications from,and to, other fields as well.
Many systems in nature are found to interact, between each other or with theenvironment. The interaction can cause gradual or sudden changes in theirqualitative dynamics, leading to their grouping, self-organizing, clustering,mutual coordinated synchronization, even to some extremes when their veryexistence is suppressed [1, 2, 3, 4, 5, 6]. An important class of such dynamicalsystems are oscillators, which also often interact resulting in a quite intricatedynamics.
Tomislav StankovskiFaculty of Medicine, Ss. Cyril and Methodius University in Skopje, Skopje, N. Mace-donia; Department of Physics, Lancaster University, Lancaster, United Kingdom,e-mail: [email protected]
Preprint chapter to the book ”
The Physics of Biological Oscillators ” 1 a r X i v : . [ n li n . AO ] A ug Tomislav Stankovski
On the quest to untangle and better understand interactions, one couldstudy several complementary aspects [7]. One is structural connectivity,where physical actual connection is studied. Often this is not directly ob-servable, or it exist but it is not active and dynamic all the time. Further on,one could study functional connectivity i.e. if a functional dependence (likecorrelation, coherence or mutual information) exist between the observeddata. Finally, one could study the causal relations between dynamical mod-els and observe the effective connectivity. In this way, the interactions canbe reconstructed in terms of coupling functions which define the underlayinginteraction mechanism.With their ability to describe the interactions in detail, coupling functionshave received a significant attention in the scientific community recently [8, 9].Three crucial aspects of coupling functions were studied: the theory, meth-ods and applications. Various methods have been designed for reconstructionof coupling functions from data [10, 11, 12, 13, 14, 15]. These have enabledapplications in different scientific fields including chemistry [16], climate [17],secure communications [18, 19], mechanics [20], social sciences [21], and os-cillatory interaction in physiology for cardiorespiratory and cardiovascularinteractions [10, 22, 23, 24, 25].Arguably, the greatest current interest for coupling functions is comingfrom neuroscience. This is probably because the brain is a highly-connectedcomplex system [27], with connections on different levels and dimensions,many of them carrying important implications for characteristic neural statesand diseases. Coupling functions are particularly appealing here because theycan characterize the particular neural mechanisms behind these connections.Recent works have encompassed the theory and inference of a diversity ofneural phenomena, levels, physical regions, and physiological conditions [28,29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].The chapter gives an overview of the topic of coupling function, with par-ticular focus on their use and suitability to neuroscience. This will be ex-plained through observations on two levels of brain connectivity – the neuronsand the brainwaves level. The relationship between the appropriate theoryand methods will be also given. On systemic level, the focus will be on neu-ronal oscillations, thus positioning around and complementing the main topicof the book – biological oscillators. The chapter will finish by outlook andsome thoughts on the future developments and uses of coupling function inneuroscience. However, before going into greater detail, first the basics ofwhat coupling functions are discussed briefly bellow.
The system setup to be studied is one of an interacting dynamical systems,with the focus of coupled oscillators. Then, coupling functions describe the oupling functions in neuroscience 3 physical rule specifying how the interactions occur and manifest . Because theyare directly connected with the functional dependencies, coupling functionsfocus not only on if the interactions exist, but more on how they appear anddevelop. For example, when studying phase dynamics of coupled oscillatorsthe magnitude of the phase coupling function affects directly the oscillatoryfrequency and will describe how the oscillations are being accelerated or de-celerated by the influence of the other oscillator. Similarly, if one considersthe amplitude dynamics of interacting dynamical systems, the magnitude ofcoupling function will prescribe how the amplitude is increased or decreaseddue to the interaction.First we consider two coupled dynamical systems given in the followinggeneral form: ˙ x = f ( x ) + g ( x, y )˙ y = f ( y ) + g ( x, y ) , (1)where the functions f ( x ) and f ( y ) describe the inner dynamics, while g ( x, y ) and g ( x, y ) describe the coupling functions in the state space. Then,given that the two dynamical systems are oscillators, and under the assump-tion that they are weakly nonlinear and weakly coupled, one can apply thephase reduction theory [3, 55, 54]. This yields simplified approximative sys-tems where the full (at least two dimensional) state space domain is reducedto a one dimensional phase dynamics domain:˙ φ = ω + q ( φ , φ )˙ φ = ω + q ( φ , φ ) , (2)where φ , φ are the phase variables of the oscillators, ω , ω are their nat-ural frequencies, and q ( φ , φ ) and q ( φ , φ ) are the coupling functions inphase dynamics domain. For example, in the Kuramoto model [3] they wereprescribed to be sine functions from the phase differences:˙ φ = ω + ε sin( φ − φ )˙ φ = ω + ε sin( φ − φ ) , (3)where ε , ε are the coupling strength parameters. Apart from this exampleof sinusoidal form, the coupling functions q ( φ , φ ) and q ( φ , φ ) can havevery different and more general functional form, including a decompositionon a Fourier series. Given in the phase dynamics like this Eqs. 2, the couplingfunctions q ( φ , φ ) and q ( φ , φ ) are additive to the frequency parameters ω , ω , meaning that their higher or lower values will lead to acceleration ordeceleration of the affected oscillations, respectively.Coupling function can be described in terms of its strength and form .The coupling strength is a relatively well-studied quantity, and there aremany statistical methods which detect measures proportional to it (e.g. the Tomislav Stankovski mutual-information based measures, transfer entropy and Granger causality).It is the functional form of the coupling function, however, that has provideda new dimension and perspective probing directly the mechanisms of theinteractions. Where, the mechanism is defined by the functional form thatgives the rule and process through which the input values are translated intooutput values i.e. for the interactions it prescribes how the input influencefrom one system is translated into the output effect on the affected or thecoupled system.In this way a coupling function can describe the qualitative transitionsbetween distinct states of the systems e.g. routes into and out of synchro-nization, oscillation death or network clustering. Moreover, depending on theknown form of the coupling function and the detected quantitative inputs,one can even predict transitions to synchronization. Decomposition of a cou-pling function provides a description of the functional contributions fromeach separate subsystem within the coupling relationship. Hence, by describ-ing the mechanisms, coupling functions reveal more than just investigatingcorrelations and statistical effects.
The human brain is an intriguing organ, considered to be one of the mostcomplex systems in the universe. The adult human brain is estimated tocontain 86 ± ±
10 billion)of non-neuronal cells [40]. Out of these neurons, 16 billion (19%) are locatedin the cerebral cortex, and 69 billion (80%) are in the cerebellum. One of themain features of the brain is how the neurons are connected, and when andhow they are active in order to process information and to produce variousfunctionalities.In neuroscience, the brain connectivity is classified in three different typesof connectivity. That is, the brain connectivity refers to a pattern of links(”structural, or anatomical, connectivity”), of statistical dependencies (”func-tional connectivity”) or of causal model interactions (”effective connectivity”)between distinct units within a nervous system [41, 42, 7]. In terms of graphtheory of the brain, the units correspond to nodes, while the connectivitylinks to edges [43]. The connectivity pattern between the units is formed bystructural links such as synapses or fiber pathways, or it represents statisticalor causal relationships measured as cross-correlations, coherence, informationflow or the all-important coupling function . In this way, therefore, the brainconnectivity is crucial to understand how neurons and neural networks pro-cess information.The units can correspond to individual neurons, neuronal populations, oranatomically segregated brain regions. Taking aside the anatomically struc-tural brain regions, the other two – the neurons and their populations – are oupling functions in neuroscience 5 of particular interest from a system neuroscience point of view. Moreover,for certain conditions these systems may operate in the oscillatory regimefor some time. When having an oscillatory nature their dynamics and con-nectivity can be modeled as coupled oscillators (see for example Fig. 1). Inthis constellation, a coupling function with its functional form can be verysuitable effective connectivity measure through which much can be learnedabout the mechanisms and functionality of the brain.
Fig. 1
A schematic ex-ample of the brain, anelectroencephalography(EEG) signal recording asa measure of the neuralpopulation electrical ac-tivity, and the schematicmodel of two oscillatorsand their coupling func-tions which can be usedto model a particularbrainwave activity. Thefive distinct brainwave( δ, θ, α, β, γ ) frequencyintervals are also given onthe right of the figure.
As the two connectivity units, the neurons and the neuronal populations,are of particular interest to the focus of coupling functions and oscillatorydynamics, bellow they will be discussed separately in light of the utility ofcoupling functions.
The neurons are archetypical cells which act as basic units from which thestructure of the brain is realized. Existing in great numbers, they are inter-connected in various network configurations giving rise to different functionsof the brain. One should note that besides neurons other cell types may alsocontribute to the brain overall function [26]. As such the brain is a complexsystem which can perform large number of neural functions from relativelystatic structure [27]. For comparison, in terms of functions the brain is muchmore complex than for example the heart, which performs generally only onefunction – pumping blood to other parts of the body. Importantly for thebrain, the neurons are electrically excitable cells, which are active only in theact of performing certain function.Based on their function, neurons are typically classified into three types:sensory neurons, motor neurons and interneurons. Number of neuron models
Tomislav Stankovski exist which describe various features, including but not limited to the Hodgk-inHuxley, the Integrate-and-fire, the FitzHughNagumo, the MorrisLecar andthe Izhikevich neuronal model [44, 45, 46, 47, 48]. These models describethe relationship between neuronal membrane electrical currents at the inputstage, and membrane voltage at the output stage. Notably, the most extensiveexperimental description in this category of models was made by Hodgkin-Huxley [45], which received the 1963 Nobel Prize in Physiology or Medicine.The mathematical description of the neuronal models is usually representedby a set of ordinary or stochastic differential equations, describing dynamicalsystems which under specific conditions exhibit nonlinear oscillatory dynam-ics.Importantly, the neurons are highly interconnected forming a complexbrain network. Their interactions give rise to different neural states and func-tions. In terms of system interactions, such brain interactions could lead toqualitative transitions like synchronization and clustering, on the whole orpart of the brain network. When observing the neuronal models as dynami-cal systems, the mechanisms of the interactions are defined by the neuronalcoupling functions. On this level, coupling functions have been studied exten-sively, although more in an indirect way through the neuronal phase responsecurve (PRC) [49, 50]. Namely, coupling function is a convolution between twofunctions, the phase response curve and the perturbation function [3] i.e. onefunction of how an oscillator responds to perturbations and the second func-tion defining the perturbations from the second oscillator, respectively. Thereare generally two types of such response curves, type I with all positive, andtype II with positive and negative values. Different types of phase responsecurves were studied (especially theoretically) forming different types of neu-ronal models [51, 52, 53]. The phase response curves are typically defined forweakly coupled units [54, 55].An important feature of the neuronal oscillations are that they are ex-citable and have non-smooth spike-like trajectories. Such dynamics of theneuronal oscillations are highly nonlinear. For many applications, the neu-ronal activity is studied completely through the timing of the spike events[56]. In general, such spike-like oscillations act similar as a delta function,hence the phase response curves will have a similar delta function-like form[57]. This can have direct effect when observing the coupling function whichcan be a convolution between the a delta-like functions.In terms of methods for neuronal coupling functions, a number of methodsexist for reconstructing the neuronal phase response curves and the associatecoupling functions [58, 30]. However, there are many open problems on thistask and many applications on different types of signals from interactingneurons are yet to be resolved. oupling functions in neuroscience 7
Studying some kind of property of a large number of neurons at once, as awhole or region of the brain, scales up the observation on higher level. In thisway the resultant measurement of the brain, or region of the brain, is in a waysome kind of mean field, a sum of all the functional activities of the individualneurons in a group, ensemble or network. For example such measurementsinclude the neural EEG, iEEG, NIRS, MRI, CT and PET, which measuredifferent characteristics like the electrical activity, the hemodynamic activity,the perfusion etc. of the whole brain or on specific spatially localized brainregions.Arguably, the most used high level observable is the EEG. Electroen-cephalography (EEG) is a noninvasive electrophysiological monitoring methodto record electrical activity of the brain. EEG measures voltage fluctuationsresulting from ionic current within the neurons of the brain [59]. EEG mea-sures electrical activity over a period of time, usually recorded from multipleelectrodes placed on the scalp according to some widely accepted protocols,like the International 1020 system [60] (internationally recognized protocolto describe and apply the location of scalp electrodes).At first sight the EEG signal looks random-like and complex (see e.g. Fig.1), however, a detail spectral analysis reveals that there are number of dis-tinct oscillating intervals – called brainwaves . The most commonly studiedbrainwaves include the delta δ , theta θ , alpha α , beta β and gamma γ neuraloscillation [61]. The frequancy intervals of these brainwaves are also given inFig. 1. Apart from these, there are also other brainwaves, including the mu µ ,faster gamma1 γ and gamma2 γ brainwaves, and other more characteristicoscillations like the sleep spindles, thalamocortical oscillations, subthresh-old membrane potential oscillations, cardiac cycle etc. The brainwaves areoften linked to specific brain functions and mechanisms, though not all ofthem are known and they are still very active field of research. The existenceand strength of the brainwave oscillations are usually determined by spectralFourier or Wavelet analysis.The brainwave oscillations emanate from the dynamics of large-scale cellensembles which oscillate synchronously within characteristic frequency in-tervals. The different ensembles communicate with each other to integratetheir local information flows into a common brain network. One of the mostappropriate ways of describing communication of that kind is through cross-frequency coupling , and there has been a large number of such studies inrecent years to elucidate the functional activity of the brain underlying e.g.,cognition, attention, learning and working memory [62, 63, 64, 65, 66]. Thedifferent types of cross-frequency coupling depend on the dynamical proper-ties of the oscillating systems that are coupled, e.g., phase, amplitude/powerand frequency, and different combinations of brainwaves have been investi-gated, including often the δ - α , θ - γ and α - γ cross-frequency coupling relation.These types of investigation are usually based on the statistics of the cross- Tomislav Stankovski frequency relationship e.g., in terms of correlation or phase-locking, or on aquantification of the coupling amplitude.Recently, a new type of measure for brain interactions was introducedcalled neural cross-frequency coupling functions [33]. This measure is one ofthe central aspects in this chapter. The neural cross-frequency coupling func-tions describe interactions which are cross-frequency coupling i.e. betweenbrainwaves but now describing not only the coupling existence and strengthbut also the form of coupling function. This functional form acts as anotherdimension of the coupling with the ability to describe the mechanisms, orthe functional law, of the underlaying coupling connection in question [8]. Insimple words, not only if , but also how the neural coupling takes place.When studying brainwave interactions the neural cross-frequency couplingfunctions are very suitable. Namely, the fact that the brainwaves are de-scribed by oscillations can be used to model the interacting dynamics withthe coupled phase oscillator model [3]. In this way one can have a direct 1:1correspondence between the number of observables and the dimensions of themeasured signals – having a 1D signal and 1D model for the phase dynam-ics for each system i.e. there will be no hidden dimensions. To illustrate thesteps of the analysis an example of δ -to- α phase neural coupling function isconsidered: • First one needs to extract the δ and α oscillation signals – this is donewith standard filtering of the EEG signals. • After this, one needs to detect the instantaneous phase signals from theoscillations, which can be done by Hilbert transform, and further trans-forming this with protophase-to-phase transformation [20]. • Such phases φ δ ( t ) and φ α ( t ) are then inputs to a method for dynamicalinference which can infer a model of two coupled phase oscillators wherethe base functions are represented by Fourier series (set of sine and cosinefunctions of the φ δ ( t ) and φ α ( t ) arguments). In our calculations we usedthe method for dynamical Bayesian inference [14] and Fourier series asbase function up to the second order. • The resulting inferred model explicitly gives the desired neural couplingfunctions. • After reconstructing the neural coupling functions of interest, one can usethem to perform coupling function analysis in order to extract and quantifyunique characteristics.The phase coupling functions give the precise mechanism of how oneoscillation is accelerated or decelerated as an effect of another oscillation.For example, lets consider the δ -to- α phase neural coupling function. Fig.2 presents such δ -to- α coupling function q α ( φ δ ( t ) , φ α ( t )) from three studiesinvolving resting state and anaesthesia, from single electrode or from spa-tially distributed electrodes [33, 37, 67]. Fig. 2 (a) shows the coupling exis-tence, strength and significance in respect of surrogates, while the Fig. 2 (b)shows the all-important neural coupling function q α ( φ δ ( t ) , φ α ( t )). Observ- oupling functions in neuroscience 9 Fig. 2
Examples of δ - α neural coupling functions. (a) The coupling strength spatialdistribution and significance in respect of surrogates. (b) The δ - α phase couplingfunctions of resting state – with 3D and 2D plots and a polar index of couplingfunction similarity. (c) The effect of anaesthesia on the δ - α coupling functions – thethree group functions are for the awake, anesthetised with propofol and anesthetizedwith sevoflurane states. (d) and (e) depict the spatial distribution and the averageresting state δ - α coupling function, respectively. (a) and (b) are from [33], (c) is from[37], (d) and (e) are from [67]. ing closely the 3D plot in Fig. 2 describes that the q α ( φ δ ( t ) , φ α ( t )) couplingfunction which is evaluated in the φ α ( t ) dynamics changes mostly along the φ δ ( t ) axis, meaning it is a predominantly direct coupling from δ oscillations.Detailed description of the direct form of coupling function, which is notanalytical for non-parametric functional form, are presented elsewhere [67].The specific form of the coupling function describes the coupling mechanismthat when the δ oscillations are between 0 and π the coupling function isnegative and the α oscillations are decelerated, while when the δ oscillationsare between π and 2 π the coupling function is positive and the α oscillations are accelerated. The rest of the figures tell similar story – Fig. 2 (c) presentthree cases of q α ( φ δ ( t ) , φ α ( t )) coupling functions for awake and anaesthetizedsubjects (with propofol nd sevoflurane anaesthetics, respectively), while Fig.2 (d) and (e) present the q α ( φ δ ( t ) , φ α ( t )) in spatial distribution on the cor-tex and its average value. The 3D plots present the qualitative description,while for quantitative analysis one can extract two measures – the couplingstrength and the similarity of form of coupling function [8]. The theory and methods for studying coupling functions of brain interactionsare developed unsymmetrically. Namely, it seems that theoretical studies aremore developed for the neuronal level, while the methods are largely de-veloped for studying the large-scale (brainwaves) systems. Of course, this isnot a black-and-white division, however the predominance of the two aspectscertainly seems to be like this.The large populations of interacting neurons, in form of ensembles andnetworks, have been studied extensively in theory. The celebrated Kuramotomodel [68, 3] has been exploited in particular. It is a model of large popu-lation of phase oscillators, one which has an exact analytic solution for thesynchronization state of the whole ensemble. The coupling functions is asimple sine function of the phase difference. Kuramoto discussed that thiscoupling function is not very physical, however his interest was in finding ananalytically solvable model. The Kuramoto model has been particularly pop-ular in neuroscience with its ability to describe analytically the synchronousstates of large populations of neurons [69, 70, 71]. Other two recently intro-duced approaches, known as the OttAntonsen [72] and WatanabeStrogatz [73]reductions, provide reduced model equations that exactly describe the col-lective dynamics for each subpopulation in the neural oscillator network viafew collective variables only. A recent review provides a comprehensive andupdated overview on the topic [39]. The theoretical studies on the large-scalebrainwave interactions are often performed through the common frameworkof two or few coupled oscillatory systems [4].To infer coupling functions from data one needs to employ methods basedon dynamical inference. These are class of methods which can reconstruct amodel of ordinary or stochastic differential equations from data. The couplingfunctions are integral part of such models. In this chapter example wereshown from the use of specific method based on dynamical Bayesian inference[14, 74, 75], however any other method based on dynamical inference (oftenreferred to also as dynamic modelling or dynamic filtering) can also be used[10, 11, 12, 13, 15]. The differences between the results of these methodsin terms of the coupling functions are minor and not qualitatively different. oupling functions in neuroscience 11
Often, there is a need for coupling functions to be inferred from networksof interacting systems, and several methods have been applied in this way[76, 67, 13]. In neuroscience, such methods have been used mainly on twoto several brainwave oscillation systems, and it has been argued that theprecision and feasibility are exponentially reduced as the number of systemsincreases and it is recommended not to go beyond
N >
10 [77]. For thisreason and due to the exponentially increasing demand for larger numberof systems, there are not many effective methods for inference of couplingfunctions in low-level large populations of neuronal interactions.In terms of methodology and analysis, few other aspects are importantwhen analysing coupling functions. One is that once coupling functions areinferred they give the qualitative mechanisms but for any quantitative eval-uations and comparisons (for example in a multisubject neuroscience study)one can conduct coupling function analysis i.e. it can calculate the couplingstrength and the similarity of the form of coupling function [8, 10, 25]. Also,of paramount importance is to validate if the inferred coupling functions arestatistically significant in respect of surrogate time series [78, 79]. Usually onetest the if the coupling strength of coupling functions is significantly higherthan the coupling strength from large number of randomized surrogate timeseries which have similar statistical properties as the original data. Also oneshould be careful when analysing neural coupling functions as it has beenshown that they can be time-varying [33, 80, 81], hence this should be takeninto account in the analysis.
In summary, this chapter gives an overview of how coupling functions arerelevant and useful in neuroscience. They bring an additional dimension –the form of coupling function – which revels the mechanism of the neuralinteractions. This is relevant in neuroscience, as it can describe and be linkedto the many different brain functions.Two largely studied levels of neural interactions were discussed, the low-level individual neurons and the high-level systemic processes like the brain-wave oscillations. Of course, these two levels are not excluding but they areclosely related, i.e. the brainwaves are like a mean-field averages of activitiesfrom billions of neurons. In fact studies exist where the brainwave oscillationsare modeled as Kuramoto ensembles but the large-scale cross-frequency cou-plings for the modelling are inferred from data [82, 69]. Needless to say,coupling functions have implications for other levels and depths of the brainother than the two discussed here.The focus was on phase coupling functions, though the interactions can bein amplitude, or combine phase-amplitude based domains [62, 64, 83]. Manymodeling methods used in neuroscience actually inferred dynamical systems where coupling functions were an integral part [84, 15]. In such cases cou-pling functions were implicit, and they were not treated as separate entities,nor were they assesses and analysed separately. These tasks are yet to bedeveloped properly for the amplitude and the phase-amplitude domains.As an outlook, with all their advantages one could expect that couplingfunctions will continue to play an important role in future neuroscience stud-ies, maybe even to extend their current use. The ever demanding compu-tational power for calculations on large populations of neuron interactionswill be more accessible in future, as new improved and faster methods willbe developed. The artificial neural networks take on increasing importancerecently, with many application across different disciplines and industries[85, 86]. The coupling function theory and the different findings in manyneuroscience studies could play an important role in establishing improvedand more efficient artificial neural networks. Also, the models could be ex-tended and generalized further for easier applications on amplitude andphase-amplitude domains. The theory needs to follow closer the new dis-coveries from neural coupling functions analysis. The coupling function de-velopments in other fields, especially in physics, could play an important rolefor neuroscience tasks, and vice versa . References
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