Accepted in Physical Review Research 3 April, 2020. Click title to verify. Published under CC-BY 4.0.

General anaesthesia reduces complexity and temporal asymmetry of the informational structuresderived from neural recordings in Drosophila

Roberto N. Muñoz,1, ∗ Angus Leung,2, † Aidan Zecevik,1, ‡ Felix A. Pollock,1, §

Dror Cohen,3, 4, ¶ Bruno van Swinderen,5, ∗∗ Naotsugu Tsuchiya,4, 6, 3, †† and Kavan Modi1, ‡‡

1School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia2School of Psychological Sciences, Monash University, Clayton, Victoria 3800, Australia

3Center for Information and Neural Networks (CiNet),National Institute of Information and Communications Technology (NICT), Suita, Osaka 565-0871, Japan

4School of Psychological Sciences and Turner Institute for Brain and Mental Health,Monash University, Melbourne, Victoria 3800, Australia

5Queensland Brain Institute, The University of Queensland, St Lucia, Queensland 4072, Australia6Advanced Telecommunications Research Computational Neuroscience Laboratories,

2-2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan(Dated: June 4, 2020)

We apply techniques from the field of computational mechanics to evaluate the statistical complexity ofneural recording data from fruit flies. First, we connect statistical complexity to the flies’ level of consciousarousal, which is manipulated by general anaesthesia (isoflurane). We show that the complexity of even singlechannel time series data decreases under anaesthesia. The observed difference in complexity between the twostates of conscious arousal increases as higher orders of temporal correlations are taken into account. We thengo on to show that, in addition to reducing complexity, anaesthesia also modulates the informational structurebetween the forward and reverse-time neural signals. Specifically, using three distinct notions of temporalasymmetry we show that anaesthesia reduces temporal asymmetry on information-theoretic and information-geometric grounds. In contrast to prior work, our results show that: (1) Complexity differences can emergeat very short time scales and across broad regions of the fly brain, thus heralding the macroscopic state ofanaesthesia in a previously unforeseen manner, and (2) that general anaesthesia also modulates the temporalasymmetry of neural signals. Together, our results demonstrate that anaesthetised brains become both lessstructured and more reversible.

I. INTRODUCTION

Complex phenomena are everywhere in the physical world.Typically, these emerge from simple interactions among ele-ments in a network, such as atoms making up molecules ororganisms in a society. Despite their diversity, it is possibleto approach these subjects with a common set of tools, usingnumerical and statistical techniques to relate microscopic de-tails to emergent macroscopic properties [1]. There has longbeen a trend of applying these tools to the brain, the archety-pal complex system, and much of neuroscience is concernedwith relating electrical activity in networks of neurons to psy-chological and cognitive phenomena [2]. In particular, thereis a growing body of experimental evidence [3], that neuralfiring patterns can be strongly related to the level of consciousarousal in animals.

In humans, level of consciousness varies from very lowin coma and under deep general anaesthesia, to very highin fully wakeful states of conscious arousal [4]. With the

∗ roberto.munoz@monash.edu† angus.leung1@monash.edu‡ aidanzecevik@gmail.com§ felix.pollock@monash.edu¶ dror.cohen@nict.go.jp∗∗ b.vanswinderen@uq.edu.au†† naotsugu.tsuchiya@monash.edu‡‡ kavan.modi@monash.edu

current technology, precise discrimination between uncon-scious vegetative states and minimally conscious states areparticularly challenging and remains a clinical challenge [5].Therefore, substantial improvement in accuracy of determin-ing such conscious states using neural recording data willhave significant societal impacts. Towards such a goal, neuraldata has been analysed using various techniques and notionsof complexity to try to find the most reliable measure of con-sciousness [6, 7].

One of the most successful techniques to date in distin-guishing levels of conscious arousal is the perturbationalcomplexity index [8–10], which measures the neural activitypatterns that follows a perturbation of the brain through mag-netic stimulation. The evoked patterns are processed througha pipeline then finally summarised using Lempel-Ziv com-plexity [9]. This method is inspired by a theory of conscious-ness, called integrated information theory (IIT) [11, 12],which proposes that a high level of conscious arousal shouldbe correlated with the amount of so-called integrated infor-mation, or the degree of differentiated integration in a neu-ral system (see Ref. [13] for details). While there are vari-ous ways to capture this essential concept [14, 15], one wayto interpret integrated information is as the amount of lossof information a system has on its own future or past statesbased on its current state, when the system is minimally dis-connected [16–18].

These complexity measures, inspired by IIT, are motivatedby the fundamental properties of conscious phenomenology,such as informativeness and integratedness of any experience

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[11]. While there are ongoing efforts to accurately translatethese phenomenological properties into mathematical postu-lates [13], such translation often contains assumptions aboutthe underlying process which are not necessarily borne outin reality. For example, the derived mathematical postulatesin IIT assume Markovian dynamics, i.e., that the future evo-lution of a neural system is determined statistically by itspresent state [15]. Moreover, IIT requires computing the cor-relations across all possible partitions between subsystems,which is computationally heavy [16] in relation to methodswhich do not require such partitioning to work. Assumingthat the hierarchical causal influences in the brain would man-ifest as oscillations across a range of frequencies and spa-tial regions [19], non-Markovian temporal correlations likelyplay a significant role in explaining any experimentally mea-surable behaviours, including the level of conscious arousal.There is therefore, scope for applying more general notionsof complexity to meaningfully distinguish macroscopic brainstates that support consciousness.

A conceptually simple approach to quantifying the com-plexity of time series data, such as the fluctuating potentialin a neuron, is to construct the minimal model which statisti-cally reproduces it. Remarkably, this minimal model, knownas an epsilon machine (ε-machine), can be found via a sys-tematic procedure which has been developed within the fieldof computational mechanics [20–22]. Crucially, ε-machinesaccount for multiple temporal correlations contained in thedata and can be used to quantify the statistical complexity ofa process – the minimal amount of information required tospecify its state. As such they have been applied over vari-ous fields, ranging from neuroscience [23, 24] and psychol-ogy [25] to crystallography [26] and ecology [27], to the stockmarket [28]. Lastly, unlike IIT the ε-machine analysis can beperformed for data coming from a single channel.

In this paper, we use the statistical complexity derived froman ε-machine analysis of neural activity to distinguish statesof conscious arousal in fruit flies (D. melanogaster). We anal-yse neural data collected from flies under different concen-trations of isoflurane [29, 30]. By analysing signals fromindividual electrodes and disregarding spatial correlations,we find that statistical complexity distinguishes between thetwo states of conscious arousal through temporal correlationsalone. In particular, as the degree of temporal correlationsincreases, the difference in complexity between the wakefuland anaesthetised states becomes larger. In addition to mea-suring complexity, the ε-machine framework also allows us toassess the temporal irreversibility of a process- the differencein the statistical structure of the process when read forwardsvs. backwards in time. This may be particularly importantfor wakeful brains which are thought to be sensitive to thestatistical structure of the environment which runs forward intime [30–32]. Using the nuanced characterisation of temporalinformation flow offered by the ε-machine framework [33],we then analyse the time irreversibility and crypticity of theneural signals to further distinguish the conscious states. Wefind that the asymmetry in information structure between for-ward and reverse-time neural signals is reduced under anaes-thesia.

The present approach singularly differentiates between

highly random and highly complex information structure; ac-counts for temporal correlations beyond the Markov assump-tion; and quantifies temporal asymmetry of the process. Noneof the standard methods possesses all of these features withina single unified framework. Before presenting these results indetail in Sec. III and discussing their implications in Sec. IV,we begin with a brief overview of the ε-machine frameworkwe will use for our analysis.

II. THEORY: ε-MACHINES AND STATISTICALCOMPLEXITY

To uncover the underlying statistical structure of neural ac-tivity that characterises a given conscious state, we treat themeasured neural data, given by voltage fluctuations in time,as discrete time series. To analyse these time series, we usethe mathematical tools of computational mechanics, whichwe outline in this section. We start with a general discus-sion on the ways to use time series data to infer a model ofa system while placing ε-machines in this context. Next, weexplain how we construct ε-machines in practice. Finally, weshow how this can be used to extract a meaningful notion ofstatistical complexity of a process.

A. From time series to ε-Machines

In abstract terms, a discrete-time series is a sequence ofsymbols r = (r0, . . . , rk, . . .) that appear over time, one af-ter the other [34]. Each element of r corresponds to a sym-bol from a finite alphabet A observed at the discrete timestep labelled by the subscript k. The occurrence of a sym-bol, at a given time step, is random in general and thus theprocess, which produces the time series, is stochastic [35].However, the symbols may not appear in a completely in-dependent manner, i.e., the probability of seeing a particularsymbol may strongly depend on symbols observed in the past.These temporal correlations are often referred to as memory,and they play an important role in constructing models thatare able to predict the future behaviour of a given stochasticprocess [36].

Relative to an arbitrary time k, let us denote the future andthe past partitions of the complete sequence as r = ( ~r, ~r),where the past and the future are ~r = (. . . , rk−2, rk−1) and~r = (rk, rk+1, . . .) respectively. In general, for the predictionof the immediate future symbol rk, knowledge of the past` symbols ~r` := (rk−`, . . . , rk−2, rk−1), may be necessary.The number of past symbols we need to account for in orderto optimally predict the future sequence is called the Markovorder [37].

In general, the difficulty of modelling a time series in-creases exponentially with its Markov order. However, notall distinct pasts lead to unique future probability distribu-tions, leaving room for compression in the model. In a sem-inal work, Crutchfield and Young showed the existence of aclass of models, which they called ε-machines, that are prov-ably the optimal predictive models for a non-Markovian pro-cess under the assumption of statistical stationarity [20, 21].

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Constructing the ε-machine is achieved by partitioning setsof partial past observations ~r` into causal states. That is, twodistinct sequences of partial past observations ~r` and ~r′` be-long to the same causal state Si ∈ S, if the probability ofobserving a specific ~r given ~r` or ~r′` is the same; that is

~r` ∼ε ~r′` if P (~r | ~r`) = P (~r | ~r′`), (1)

where ∼ε indicates that two histories correspond to the samecausal state. The conditional probability distributions inEq. (1) may always be estimated from a finite set of statis-tically stationary data via the naive maximum likelihood es-timate, given by P (rk| ~r`) = ν(rk, ~r`)/ν( ~r`), where ν(X) isthe frequency of occurrence of sub-sequence X in the data.For the case of non-stationary data, the probabilities obtainedby this method will produce a non-minimal model that cor-responds to a time-averaged representation of the time series.We now discuss how to practically construct an ε-machine fora given time series.

B. Constructing ε-machines with the CSSR algorithm

Several algorithms have been developed to constructε-machines from time series data [20, 38, 39]. Here, webriefly explain the Causal State Splitting Reconstruction(CSSR) algorithm [25], which we use in this work to inferε-machines predicting the statistics of neural data we provideas input.

The CSSR algorithm proceeds to iteratively constructsets of causal states accounting for longer and longer sub-sequences of symbols. In each iteration, the algorithm first es-timates the probabilities P (rk| ~r`) of observing a symbol con-ditional on each length ` prior sequence and compares themwith the distribution P (rk|S = Si) it would expect fromthe causal states it has so far reconstructed. If P (rk| ~r`) =P (rk|S = Si) for some causal state, then ~r` is identified withit. If the probability is found to be different for all existing Si,then a new causal state is created to accommodate the sub-sequence. By constructing new causal states only as neces-sary, the algorithm guarantees a minimal model that describesthe non-Markovian behaviour of the data (up to a given mem-ory length), and hence the corresponding ε-machine of theprocess.

The CSSR algorithm compares probability distributionsvia the Kolmogorov-Smirnov (KS) test [40, 41]. The hypothe-sis that P (rk| ~r`) and P (rk|S = Si) are identical up to statis-tical fluctuations is rejected by the KS test at the significancelevel σ when a distance DKS [42] is greater than tabulatedcritical values of σ [43]. In other words, σ sets a limit on theaccuracy of the history grouping by parametrising the proba-bility that an observed history ~r` belonging to a causal stateSi, is mistakenly split off and placed in a new causal stateSj . Our analysis, in agreement with Ref. [25], found thatthe choice of this value does not affect the outcome of CSSRwithin the tested range of 0.001 < σ < 0.01. As a result, weset σ = 0.005.

As it progresses, the CSSR algorithm compares futureprobabilities for longer sub-sequences, up to a maximum pasthistory length of λ, which is the only important parameter that

must be selected prior to running CSSR in addition to σ. If theconsidered time series is generated by a stochastic process ofMarkov order `, choosing λ < ` results in poor prediction be-cause the inferred ε-machine cannot capture the long-memorystructures present in the data. Despite this, the CSSR algo-rithm will still produce an ε-machine that is consistent withthe approximate future statistics of the process up to order-λ correlations [25]. Given sufficient data, choosing λ ≥ `guarantees convergence on the true ε-machine. One impor-tant caveat to note is that the time complexity of the algorithmscales asymptotically as O(|A|2λ+1), putting an upper limitto the longest history length that is computationally feasibleto use. Furthermore, the finite length of the time series dataimplies an upper limit on an ‘acceptable’ value of λ. Estimat-ing P (rk| ~rλ) requires sampling strings of length λ from thefinite data sequence. Since the number of such strings growsexponentially with λ, a value of λ that is too long relative tothe size N of the data, will result in a severely under-sampledestimation of the distribution. A distribution P (rk| ~rλ) thathas been estimated from an under-sampled space is almost al-ways never equal to P (rk|S = Si), resulting in the algorithmcreating a new causal state for every string of length λ it en-counters. A bound for the largest permissible history lengthis L(N) ≥ log2N/ log2 |A|, where L(N) denotes maximumlength for a given data size of N [44, 45]. Once these con-siderations have been taken into account, the ε-machine pro-duced by the algorithm provides us with a meaningful quan-tifier of the complexity of the process generating the time se-ries, as we now discuss.

C. Measuring the complexity and asymmetry of a process

The output of the CSSR algorithm is the set of causal statesand rules for transitioning from one state to another. That is,CSSR gives a Markov chain represented by a digraph [20, 37]G(V,E) consisting of a set of vertices vi ∈ V and directededges {i, j} ∈ E, e.g. Figs. 1(c) and (d). Using these rules,one can find P (Si), which represents the probability that theε-machine is in the causal state Si at a any time. The Shannonentropy of this distribution quantifies the minimal number ofbits of information required to optimally predict the futureprocess; this measure, first introduced in Ref. [20], is calledthe statistical complexity:

Cµ := H [S] = −∑i

P (Si) logP (Si). (2)

Formally, the causal states of a time series depend uponthe direction in which the data is read [33]. The main conse-quence of this result is that the set of causal states obtainedby reading the time series in the forward direction S+, are notnecessarily the same as those obtained by reading the time se-ries in the reverse direction S−. Naturally, this correspondsto potential differences in forward and reverse-time processesand the associated complexities, which is known as causal ir-reversibility

Ξ := C+µ − C−µ , (3)

capturing the time-asymmetry of the process.

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Another (stronger) measure of time-asymmetry is cryptic-ity:

d := 2C±µ − C+µ − C−µ . (4)

This quantity measures the amount of information hidden inthe forwards and reverse ε-machines that is not revealed in thefuture or past time series, respectively. Specifically, it com-bines the information that must be supplemented to determinethe forwards ε-machine given the reverse ε-machines and theinformation to determine reverse ε-machines given the for-wards ε-machine. In each case, this is equivalent to the dif-ference between the complexity of a bidirectional ε-machine,denotedC±µ [33], and that of the corresponding unidirectionalmachine. Throughout this manuscript, we implicitly refer tothe usual forward-time statistical complexity C+

µ when writ-ing Cµ, unless otherwise stated.

Finally, an operational measure for time-asymmetry isdefined by the microscopic irreversibility, which quanti-fies how statistically distinguishable the forwards and re-verse ε-machines are, in terms of the sequences of sym-bols they produce. If the forward-time ε-machine producesthe same sequences with similar probabilities to the reverse-time ε-machine, then the process is reversible. Should a se-quence available to M+ be impossible for M− to produce,then the process is strictly irreversible. Here, we assess thedistinguishability between two ε-machines by estimating theasymptotic rate of (symmetric) Kullback-Leibler (KL) diver-gence DKLS between long output sequences; this measure iscommonly applied to stochastic models [46]. Specifically

DKLS = DKL(M+‖M−) +DKL(M−‖M+), (5)

where DKL is the regular, non-symmetric estimated KL di-vergence rate [47]. The KL divergence can be proved to bea unique measure that satisfies all of the theoretical require-ments of information-geometry [17, 48, 49].

A few remarks are in order: in general, any one of theabove measures vanishing does not imply that the other mea-sures must also vanish. For instance, consider the case wherethe structures of the forward (M+) and reverse-time (M−)ε-machines are different but they happen to have the samecomplexities, i.e., C+

µ = C−µ . Then, clearly we have Ξ = 0but d 6= 0 and DKLS 6= 0. On the other hand, considerthe case when M+ and M− are the same; here, we haveΞ = DKLS = 0, yet may not have d = 0. This means thatvanishing DKLS implies that Ξ = 0 (but not the converse,and not d = 0). This turns out be an interesting extremal casebecause, while the forward and reverse processes are iden-tical, the non-vanishing crypticity accounts for the informa-tion required to synchronise the corresponding ε-machines,i.e., producing the joint statistics of the paired ε-machines.Moreover, we can conclude that microscopic irreversibility isa stronger measure than causal irreversibility; this comes atthe expense of computational cost, i.e., the former is harderto compute than the latter. In essence, each measure aboverepresents a different notion of temporal asymmetry, with itsown operational significance. Causal irrversibility and cryp-ticity are information-theoretic constructs, while microscopicirrversibility is a information-geometric construct.

In the next section, we describe the experimental and an-alytical methods, as well as the results: that the statisticalcomplexity and temporal asymmetry of the neural time se-ries, taken from fruit flies, significantly differ between statesof conscious arousal.

III. EXPERIMENTAL RESULTS AND ANALYSIS

A. Methods

We analysed local field potential (LFP) data fromthe brains of awake and isoflurane-anaesthetised D.melanogaster (Canton S wild type) flies. Here, we brieflyprovide the essential experimental outline that is necessary tounderstand this paper. The full details of the experiment arepresented in Refs. [29, 30]. LFPs were recorded by inserting alinear silicon probe (Neuronexus 3mm-25-177) with 16 elec-trodes separated by 25 µm. The probe covered approximatelyhalf of the fly brain and recorded neural activity as illustratedin Fig. 1(a). A tungsten wire inserted into the thorax actedas the reference. The LFPs at each electrode were recordedfor 18s while the fly was awake and 18s more after the flywas anaesthetised (isoflurane, 0.6% by volume, through anevaporator). Flies’ unresponsiveness during anaesthesia wasconfirmed by the absence of behavioural responses to a seriesof air puffs, and recovery was also confirmed after isofluranegas was turned off [29].

We used data sampled at 1kHz for the analysis [29], and toobtain an estimate of local neural activity, the 16 electrodeswere re-referenced by subtracting adjacent signals giving 15channels which we parametrise as c ∈ [1, 15]. Line noise wasremoved from the recordings, followed by linear de-trendingand removing the mean. The resulting data is a fluctuatingvoltage signal, which is time-binned (1ms bins) and bina-rised by splitting over the median, leading to a time series,see Fig. 1(b).

For each of the 13 flies in our data set, we considered 30time series of length N = 18, 000. These correspond tothe 15 channels, labelled numerically from the central to pe-ripheral region as depicted in Fig. 1(a), and the two statesof conscious arousal. Using the CSSR algorithm [25], weconstructed ε-machines for each of these time series as afunction of maximum memory length within the range λ ∈[2, 11], measured in milliseconds. This is below the mem-ory length L(N) ∼ 14 beyond which we would be unableto reliably determine transition probabilities for a sequenceof length N (see Sec. II B) [51]. For a given time directionξ ∈ {+ : forward,− : reverse}, we recorded the resulting3, 900 ε-machine structures and their corresponding statisti-cal complexities C(ξ,ψ)

µ , and grouped them according to theirrespective level of conscious arousal, ψ ∈ {w, a} for awakeand anaesthesia, channel location, c, and maximum memorylength, λ. Thus, the statistical complexity we computed ina given time direction is a function of the set of parameters{ψ, c, λ} for each fly, f . We also determined the irreversibil-ity Ξ, crypticity d, and symmetric KL divergence rate DKLSfor each fly and again grouped them over the same set of pa-rameters {ψ, c, λ}. While we found that not all the data is

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Figure 1. Evolution of experimental data from neural signals to ε-machines. (a) Representative schematic of D. melanogaster brain (modifiedfrom Ref. [50]) depicted with probe and approximate channel locations. Each channel c ∈ [1, 15] samples around a localised region in thebrain, with numerical labels ordered from the central (c = 1) to peripheral (c = 15) regions. (b) Example reading of a processed local fieldpotential (LFP) for a single channel. Points along the x-axis represent LFP measurements at each sampling time step. The median LFPmeasurement of the sample is shown as the grey line bisecting data. LFP binarisation is determined via splitting over the median with theencoding scheme 0 : LFP ≤ Median, and 1 : otherwise. The ε-machines are inferred by using the binary string as the input to the CSSRalgorithm. (c) Digraph representation of the CSSR-inferred ε-machine for channel 1 readings of fly 1 under anaesthesia (0.6 vol.% isoflurane)with σ = 0.005 and λ = 3. Graph vertices correspond to causal states. Vertex labels distinguishing causal states are assigned arbitrarily anddo not imply state equivalence across multiple graphs. Directed edges correspond to transitions between causal states. Edge labels denote theprobability (2 significant figures) of a transition occurring, and edge colour encodes the emitted symbol upon making the transition (1: Red,0: Blue). The histories stored in the causal states for this ε-machine are visualised in Fig. 6. (d) Digraph representation of ε-machine for thewakeful (0 vol.% isoflurane) level of conscious arousal for the same channel, fly, σ, and λ as in (c). We report the forward-time statisticalcomplexities Caµ = 1.88 and Cwµ = 2.96 for (c) and (d) respectively.

strictly stationary, in that the moving means of the LFP sig-nals were not normally distributed, the conclusions we drawfrom them are still broadly valid. As mentioned in Sec. II A,ε-machines reconstructed from approximately stationary dataare time-averaged models, and are likely to underestimate thetrue statistical complexity of the corresponding neural pro-cesses.

We are principally interested in the differences the infor-mational quantities Qψ ∈ {C(ξ,ψ)

µ , Ξψ, dψ, DψKLS} haveover states of conscious arousal and thus consider

∆Q := Qw −Qa, (6)

for fixed values of {f, c, λ}. Positive values of ∆Q indi-cate higher complexities observed in the wakeful state rela-tive to the anaesthetised one. Finally, we use the notation〈Qψ〉x to denote taking an average of any information quan-tity Qψ , over a specific parameter x ∈ {f, c, λ}. For ex-ample 〈∆C+

µ 〉f means taking the fly-averaged difference inforward-time statistical complexity.

To assess the significance of each of the parameters ψ, c, λ,and ξ, or some combination of them, have on the response of

the elements in the set Q across flies, we conducted a statisti-cal analysis using linear mixed effects modelling [52] (LME).The LME analysis describes the response of a given quantityQ by modelling it as a multidimensional linear regression ofthe form

Q = Fβ + Rb + E . (7)

The resulting model in Eq. (7) consists of a family of equa-tions where Q is the vector allowing for different responsesof a quantity Q for each specific fly, channel location, levelof conscious arousal, and time direction where applicable.Memory length λ, channel location c, state of consciousarousal ψ, and time direction ξ (again, where applicable)are the parameters that Q responds to. To account for vari-ations in the response caused by interactions between param-eters (e.g. between memory length and channel location), weincluded them in the model. Letting X = {λ, c, ψ, ξ} bethe set of the parameters which may induce responses, wecan write all the non-empty k-combinations between themas F = {λ, c, ψ, ξ, λc, λψ, ..., λcψξ} =

(Xk

)\∅. The ele-

ments in F are known as the fixed effects of Eq. (7), and are

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contained as elements within the matrix F. The vector β,contains the regression coefficients describing the strength ofeach of the fixed effects F .

In addition to fixed effects affecting the response of an el-ement of Q in our experiment, we also took into account anyvariation in response caused by known random effects. In par-ticular, we expected stronger response variations to be causedby correlations occurring between the channels within a sin-gle fly, compared to between channels across flies. These ran-dom effects are contained as elements of the matrix R, andthe vector b encodes the regression coefficients describingtheir strengths. Finally, the vector E describes the normally-distributed unknown random effects in the model. The re-gression coefficients contained in the vectors β and b, wereobtained via maximum likelihood estimation such that E areminimised. The explicit form of Eq. (7) used in this analysisis detailed in the Appendix IV A.

With the full linear mixed effects model given by Eq. (7),we tested the statistical significance of a fixed effect in F .This was accomplished by comparing the log-likelihood ofthe full model with all fixed effects, to the log-likelihood ofa reduced model with the effect of interest removed [53] (re-gression coefficients associated with the effect are removed).This comparison between the likelihood models is given byΛ = 2(hfull − hreduced), where Λ is the likelihood ratio, hfullis the log-likelihood of full model, and hreduced is the log-likelihood of the model with the effect of interest removed.

Under the null hypothesis, when a fixed effect does nothave any influence on an informational quantity Q, i.e., theregression coefficients for the effect are vanishing, the likeli-hood ratio Λ is χ2 distributed with degrees of freedom equalto the difference in the number of coefficients between themodels. Therefore, we considered any fixed effect in the setF to have a statistically significant effect on a quantity if theprobability of obtaining the likelihood ratio given the relevantχ2 distribution was less than 5% (p < 0.05). Thus, for eachsignificant effect we report the fixed effect being tested, i.e.,an element of F , the obtained likelihood ratio χ2(n−1) withn associated degrees of freedom, and the associated probabil-ity p of obtaining the statistic under the null hypothesis.

In addition to all the quantities in the set Qψ , the LME andlikelihood ratio test was also performed for ∆Q, in order tofind the significant interaction effects of the parameters. Here,we also modelled ∆Q as dependent on a fixed effect in F asin Eq. (7), but excluding the parameter ψ as it was alreadyimplicitly considered. Once the significant effects of memorylength, level of conscious arousal, and channel location werecharacterised with our statistical analysis, we followed withpost-hoc, paired t-tests for elements in Qψ given by

t =〈∆Q〉fsf/√|f |, (8)

where sf is the standard deviation of 〈∆Q〉f , and |f | = 13is the sample size. The paired t-tests examine the nature ofinteractions between the parameters on a given quantity overthe two states of conscious arousal. Positive t-scores indi-cate a quantity is larger for the wakeful state. We presentthe results of these analyses in the following sections, sortedcategorically by whether time-direction is considered.

Anaesthetised (0.6% isoflurane)Wakeful (0% isoflurane)

Figure 2. Colour map of statistical complexity response averagedover (n = 13) flies 〈C(+,ψ)

µ 〉f , during wakefulness (left) and anaes-thesia (right), over channel location and memory length λ, measuredin milliseconds. Hatched cells on the right sub-figure, show regionswhere Cµ did not decrease under anaesthesia.

B. Results

1. Forward-time complexity results

In order to observe the effects of isoflurane on neuralcomplexity, we began by visually inspecting the structure ofthe reconstructed ε-machines for the two levels of consciousarousal for the forward-time direction. We took special inter-est in observing the differences in the characteristics of thetwo groups of ε-machines heralding the two levels of con-scious arousal. Here, memory length λ plays an importantrole. At a given λ, the maximum number of causal statesthat may be generated scales according to |A|λ [25]. In ourcase, the alphabet is binary,A = {0, 1}. This greatly restrictsthe space of ε-machine configurations available for short his-tory lengths [54]. For λ = 2 we can observe up to four dis-tinct configurations, which is unlikely to reveal the differencebased on conscious states. Given the previous findings [30],we generally expected that the data from the wakeful statewould present more complexity than those from the anaes-thetised state.

Visual inspection of the directed graphs indeed suggestedhigher ε-machine complexity during the wakeful state com-pared to the anaesthetised state, at a given set of parameters{f, c, λ}. In particular, the data from the anaesthetised statetended to result in fewer causal states and overall reducedgraph connectivity. Panels (c) and (d) of Fig. 1 are examplesof ε-machines (channel 1 data recorded from fly 1, at maxi-mum memory length λ = 3), where a simpler ε-machine isderived from the data under the anaesthetised condition. Dif-ferentiating between two conscious arousal states by visualinspection quickly becomes impractical because of the largenumber of ε-machines. Moreover, for large values of λ thenumber of causal states is exponentially large and it becomesdifficult to see the difference in two graphs. To overcomethese challenges, we looked at a simpler index, the statisti-cal complexityCµ, to differentiate between conscious arousalstates. To systematically determine the relationships betweenCµ and the set of variables {c, f, ψ} we employed the LMEanalysis outlined in Sec. III A. We first tested whether λ sig-nificantly affects Cµ. We found λ to indeed have a significant

7

effect on Cµ (λ, χ2(1) = 443.64, p < 10−16). Fig. 2 showsthat independent of the conscious arousal condition or chan-nel location, Cµ increases with larger λ. This indicates thatthe Markov order of the neural data is much larger than thelargest memory length (λ = 11) we consider. Nevertheless,we have enough information to work with.

We then sought to confirm if the complexity of ε-machinesduring anaesthesia are reduced, as suggested from visual in-spection. Our statistical analysis indicates that Cµ is notinvariably reduced during anaesthesia (ψ, χ2(1) = 0.212,p = 0.645) at all levels of λ and all channel locations.This means that Cµ cannot simply indicate the causal arousalstate without some additional information about time (λ) orspace (c). In addition, we found that neither c alone nor cψstrongly affects Cµ. However, we found significant reduc-tions in complexity when either the level of conscious arousalor the channel location, interacted with memory length (λψ,χ2(1) = 14.63, p = 1.31×10−4) and (cλ, χ2(14) = 42.876,p = 8.97 × 10−5) respectively. Moreover, the three-wayinteraction also had a strong effect (λψc, χ2(14) = 24.00,p = 0.0458).

As the three-way interaction between λ, ψ, and c compli-cates interpretation of their effects, we performed a secondLME analysis where we modelled ∆Cµ instead of Cµ, thusaccounting for ψ implicitly. In doing so, we investigatedwhether the change in statistical complexity due to anaes-thesia is affected by memory length λ or channel locationc. Using this model, we found a non-significant effect of con ∆Cµ, while a significant effect of λ on ∆Cµ was seen(λ, χ2(1) = 20.97, p = 4.65 × 10−6), indicating that ∆Cµoverall changes with λ. Specifically, ∆Cµ tended to increasewith larger λ when ignoring channel location, as is evident inFig. 3 (top). Further, explaining our previous interaction be-tween λ and ψ, ∆Cµ was not clearly larger than 0 for smallmemory length (λ = 2; the top panel of Fig. 3). This sug-gests that the information to differentiate between states ofconscious arousal is contained in higher order correlations.We also found that the interaction between λ and channel lo-cation has a significant effect on ∆Cµ (λc, χ2(14) = 37.19,p = 6.90 × 10−4), indicating that the effect of λ is not con-stant across channels. Given that ∆Cµ overall increases withλ, we considered that that the largest ∆Cµ should occur atthe largest λ. Fig. 3 (bottom) examines ∆Cµ across channelsat λ = 11.

To further break down the interaction between λ and c,we performed a one sample t-test at each value of memorylength and channel location to find regions in the parame-ter space (λ, c) where Cµ reliably differentiates wakefulnessfrom anaesthesia across flies. We plot the t-statistic at eachparameter combination in the top-left panel of Fig. 4, out-lining regions in the parameter space where ∆Cµ is signif-icantly greater than 0 (with p < 0.05, uncorrected, two-tailed), finding that the majority of the significance map isdirected towards positive values of the t-statistic. However,only a subset of (λ, c) cells contain values which are signif-icantly different from 0. Interestingly, we observed that forλ = 2, ∆Cµ is actually significantly negative, correspondingto greater complexity during anaesthesia, not during wakeful-ness. This marks λ = 2 as anomalous relative to other levels

Figure 3. Statistical complexity differences ∆Cµ = Cwµ − Caµ ofε-machines between states of conscious arousal for: (Top) increas-ing memory length λ. Grey lines indicate complexity averages overchannels per fly (n = 13), 〈∆Cµ〉c, while the blue line denotesthe average over both channels and flies 〈∆Cµ〉c,f . Error bars are95% confidence intervals of the population. (Bottom) maximummemory length λ = 11 (in milliseconds), mapped throughout thefly brain (channels). Grey and red lines indicate the result per flyand the average over (n = 13) flies, 〈∆Cµ〉f , respectively. Errorbars corresponding to the 95% confidence intervals over the sampleof files.

of λ, and this reversal of the direction of the effect of anaes-thesia likely contributed to the interaction between λ and ψ.

Disregarding λ = 2, we find ∆Cµ to be significantlygreater than 0 for channels 1, 3, 5-7, 9, 10, and 13, at vary-ing levels of λ. As expected from our reported interactionbetween λ and c, we observe ∆Cµ to already be significantlygreater than 0 at small λ for channels 5-7, while ∆Cµ onlybecame significantly greater at larger λ for channels 1, 3, 10and 13. Further, other channels such as the most peripheralchannel (c = 15) did not have ∆Cµ significantly greater than0 at any λ. All significance results, due to LME tests, arereported in Table I.

The above results suggest that the measured difference incomplexity is present across various brain regions (top-leftpanel of Fig. 4), and that it grows as longer temporal correla-tions are taken into account (up to the largest value λ = 11tested). While Fig. 3 shows a continued increase in the differ-ence of statistical complexity, ∆Cµ, as a function of λ, we didnot analyse longer history lengths, due to limitations in theamount of the data and stability of the estimation of Cµ. Inaddition to this general observation of increasing ∆Cµ withλ, we observe that, remarkably, some brain regions discrimi-nate the conscious arousal states with a history length of only3. One trivial explanation for this effect is that under anaes-thesia, the required memory length is indeed λ = 2, while theoptimal λ for awake is much larger. However, a quick obser-vation of Fig. 2 rules out this simple possibility; under both

8

Figure 4. Colour map of two-tailed paired t-scores over channel location and memory length λ for statistical complexity differences〈∆C+

µ 〉f = 〈C(+,w)µ − C

(+,a)µ 〉f (top left); causal irreversibility differences 〈∆Ξ〉f = 〈Ξw − Ξa〉f (top right); crypticity differences

〈∆d〉f = 〈dw − da〉f (bottom left); and the differences in KL divergence rate 〈∆DKLS〉f = 〈DwKLS −DaKLS〉f (bottom right). The dottedlines indicate the memory length and channel locations that exceed p < 0.05 (uncorrected). The colour scale is consistent across all subplots.

wakeful and anaesthetised states, Cµ continues to increase.

It is likely, however, that the tested range for λ remainsbelow the Markov order of the neural data; this is clearlyindicated by the lack of a plateau in statistical complexityin Fig. 3. This suggests that we are far from saturating theMarkov order of the process, and with more data we wouldbe able to further distinguish between the two states. Fu-ture analyses with longer time series would also contributeto our understanding of the Markov order (maximum mem-ory length) differences between the two states of consciousarousal. Nevertheless, our results, in Figs. 3 and 4, demon-strate that saturation of Markov order is not required fordiscrimination between conscious arousal states. This find-ing has a practical implication about the empirical utility ofε-machines; even if the history length is too low, the inferredε-machine and its statistical complexity can be useful. Wenow discuss the temporal asymmetry of neural processes.

2. Temporal asymmetry

Unlike other complexity measures, we obtain a distinctε-machine from each given time series, and for each directionwe read the time series, i.e., forward or backward in time.Based on the notion that wakeful brains should be better atpredicting the next sensory input [30], we expect that anaes-thesia should alter the information structures depending onthe time direction. Our expectation translates to the follow-ing three hypotheses:

1. Causal irreversibility (Ξ := C+µ −C−µ ), which is purely

based on the summary measure of statistical complex-ity, should be higher for awake but lower for anaes-thetised brains;

2. Crypticity (d := 2C±µ − C+µ − C−µ ) should be higher

for wakeful than anaesthetised brains;

3. Symmetric KL divergence rate (DKLS) should behavesimilarly.

On visual inspection of the variation in Ξ for the wake-ful and anaesthetised conditions, both appeared close to zero,

9

suggesting that Ξ would not have significant dependenceon the condition. This impression was confirmed statisti-cally with two-tailed t-tests against zero with corrections formultiple comparisons, as shown in the top-right panel ofFig. 4. Thus, Hypothesis 1 above, that irreversibility shouldbe higher for wakeful over anaesthetised brains, is not sup-ported by the data. However, as mentioned earlier, vanishingΞ does not imply that either d = 0 or DKLS = 0. To rule outthe possibility that the information structure of ε-machinesare different when read forwards, as opposed to backwards,depending on the condition, we also tested the latter two hy-potheses.

With respect to crypticity, first, visual inspection of thetwo-tailed t-score map, which compares crypticity for thewakeful dw and anaesthetised da conditions (bottom-leftpanel of Fig. 4) strongly implies that crypticity is larger inthe former compared to the latter. This difference is largestover channels 5-7 and 9. To systematically evaluate thisimpression, we used LME statistical analysis (described inSec. III A) to determine the relationships between cryptic-ity, d, and the set of variables {c, λ, ψ} we employ. Asexpected, we found that both memory length (λ) and levelof conscious arousal (ψ) significantly affected crypticity (λ,χ2(1) = 470.5, p < 10−16) and (ψ, χ2(1) = 5.896,p = 0.0152) respectively. Crypticity also depended on asignificant interaction between memory length and condition(λψ, χ2(1) = 6.119, p = 0.0134). Specific increases incrypticity around the middle brain region (bottom-left panelof Fig. 4) were also evident, with a strong interaction betweenchannel location and memory length (λc, χ2(14) = 35.86,p = 1.09× 10−3), which is similar to the result obtained for∆Cµ. This LME analysis, together with the direction of ef-fects in Fig. 4 (bottom-left) strongly confirms our Hypothesis2.

Furthermore, as a more direct measure of microscopicstructure, we also analysed the symmetric KL divergencerate, DKLS . Again, the two-tailed t-score map (Fig. 4,bottom-right panel) showed support for our hypothesis. Ourformal statistical analysis with LME confirmed a criti-cal interaction between memory length and condition (λψ,χ2(1) = 15.37, p < 10−16), meaning that time-asymmetricinformation structure is lost due to anaesthesia, especiallywhen a long memory length is taken into account. (We alsonote other significant effects: mainly the effect of memorylength (λ, χ2(1) = 127.4, p < 10−16) and interaction be-tween memory length and channel location (λc, χ2(14) =85.81, p < 10−16). Again, all significant results, due to LMEtests, are reported in Table I.

Taken together, these results show that the relative com-plexity of the forward versus reverse direction, as measuredby causal irreversibility, does not distinguish between thewakeful and anaesthetised states. However, our crypticity re-sults demonstrate that, under anaesthesia, the structures of theforward and reverse processes are relatively similar, whereasduring wakefulness their structures differ. Fig. 5 demon-strates this effect with exemplar ε-machines reconstructedfrom a representative channel, from which we derived six dis-tinct ε-machines: three for wakeful (a, c, d), and three foranaesthetised (e-g) flies. Panel (b) shows how the time series

and the transitions in the causal states of forward, reversed,and bidirectional ε-machines are related.

Our finding, that causal irreversibilities were not abovezero for wakeful brains, corresponds to the fact that com-plexities of forward and reverse ε-machines were not signif-icantly different. However, the bidirectional ε-machines forthe wakeful condition were substantially more complex thanthose for the anaesthetised condition. The statistical complex-ity of bidirectional ε-machines should equal that of forward orreverse ε-machines if the process is completely time symmet-ric and deterministic [33], resulting in zero crypticity. How-ever, for non-deterministic processes, additional informationfor synchronising the forward and reversed process may beneeded, which would mean d > 0. For instance, in Fig. 5(e-f), if we are told that the forward machine is in causal stateA, we need extra information to determine whether the re-versed machine is in causal state X or Y . Yet, the detailedstructure of the forward and reverse machines are the samein this example. Our analysis is supplemented with a studyof the symmetric KL divergence rate between forwards andbackwards processes, which measures the distance betweenthe reconstructed ε-machines. In other words, crypticity andsymmetric KL divergence rate quantify two different notionsof temporal asymmetry; the former is information theoretic,and the latter is information-geometric. Indeed, in generalwe find that the processes in the two directions are differentin both ways and, further, their difference varies significantlybetween conditions, as shown in the bottom two panels ofFig. 4.

IV. DISCUSSION

Discovering a reliable measure of conscious arousal in an-imals and humans remains one of the major outstanding chal-lenges of neuroscience. The present study addresses thischallenge by connecting a complexity measure to the degreeof conscious arousal, taking a step forward to strengthen-ing the link between physics, complexity science, and neu-roscience. Here, we have taken tools from the former andhave applied them to a problem in the latter. Namely, wehave studied the statistical complexity and time asymmetryof neural recordings in the brains of flies over two states ofconscious arousal: awake and anaesthetised. We have demon-strated that differences between these macroscopic states canbe revealed by both the statistical complexity of local elec-trical fluctuations in various brain regions, and various mea-sures of temporal asymmetry of hidden models that explaintheir behaviour. Specifically, we have analysed the single-channel signals from electrodes embedded in the brain usingthe ε-machine formalism, and quantified the statistical com-plexity Cµ, causal and microscopic reversibility Ξ & DKLS ,and crypticity d of the recorded data for 15 channels in 13flies over two states of conscious arousal. We find the statis-tical complexity is larger on average when a fly is awake thanwhen the same fly is anaesthetised (∆Cµ > 0; Figs. 3 and 4),and that the structural complexity of information and its timereversibility captured by crypticity and KL rate are also re-duced under anaesthesia (∆d > 0 and ∆DKLS > 0; Fig. 4).

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Figure 5. Exemplary digraph representations of ε-machines for wakeful (a-d) and anaesthetised (e-g) conditions for forward-time (a, e),reverse-time (c, f), and bidirectional (d, g) analyses, all constructed from channel 5 in fly 7, at memory length λ = 3. Panel (b) gives anexample emission sequence and causal state sequence for forward and reverse-time ε-machine pair (a) and (c). The vertex labelling denotingcausal states in (a-d) is consistent to show composition of forward and reverse-time ε-machines in the bidirectional ε-machine. The ε-machinesfor the wakeful condition have statistical complexity of C(+,w)

µ = 1.76, C(−,w)µ = 1.50, and C(±,w)

µ = 3.25. In this example the processis irreversible for all three quantities. The ε-machines for the anaesthetised condition have statistical complexity of C(+,a)

µ = C(−,a)µ = 1.0

and C(±,a)µ = 1.9989. The process is causally and microscopically reversible, but has finite crypticity.

As we have demonstrated in this study, the local informa-tion contained within a single channel can contain informa-tion about global conscious states, which are believed to arisefrom interactions among many neurons. Theoretically, singlechannels can reflect the complexity of the multiple channelsdue to the concept of Sugihara causality [55]. This arises dueto any one region of the brain causally interacting with therest of the brain, making the temporal correlation in a singlechannel time series contain information about the spatial cor-relations, i.e., information that would be contained in multiplechannels. With this logic, Ref. [56] infers the complexity ofthe multi-channel interactions from a single channel temporalstructure of the time series. This is often known as the back-flow of information in non-Markovian dynamics [57]. Theperiodic structure of statistical complexity observed acrosschannels in Fig. 2, demonstrates an unexpected example ofspatial effects present in our study – one that was not observedwith conventional LFP analyses. This observation may pro-vide a motivation for multi-channel analyses.

While we already find differences between conscious statesin the single channel based ε-machine analysis, it would bebeneficial to extend the present analysis to the multi-channelscenario, in which ε-machine can be contrasted with themethods of IIT [9–18]. Formal comparison of the distinguish-ing power of conscious states among proposed methods (suchas those in Ref. [6, 7]) will contribute to refining models andtheories of consciousness.

Our results can be informally compared with a previousstudy, where the power spectra of the same data in the fre-quency domain [30] was analysed. There, a principal obser-vation was the power in low-frequency signals in central andperipheral regions, which was more pronounced in the cen-

tral region (corresponding to channel 1-6 in this study). Ourε-machine analysis here reveals that the region between pe-riphery and centre (channels 5-7) shows most consistent dif-ference in Cµ across history length λ > 2. Ultimately, thereason for this difference is due to our distinct approach, in-sofar as ε-machines are provably the optimal predictive mod-els of a large class of time series that take into account higherorder correlations memory structure [20, 21]. Thus, our ap-plication of ε-machines contrasts with the power spectra anal-ysis, by considering these higher order correlations for veryhigh-frequency signals, instead of only two-point correlationsin both high- and low-frequency signals. Finally, the top-leftpanel of Fig. 4 shows that in regions corresponding to chan-nels 1 and 13, the differences in the conditions are only seenat high values of λ.

Our multi-time analysis further reveals an interesting ef-fect when we look more closely at, e.g., the anaesthetisedε-machine example shown in Fig. 1(c). When we examine thebinary strings belonging to each causal state, we find a clearsplit between active (consecutive strings of ones) and inactive(consecutive strings of zeros) neural behaviour correspondingto the left and right hand sides of Fig. 6 respectively. Previ-ous studies have demonstrated an increase in low-frequencyLFP and EEG power for mammals and birds during sleep andanaesthesia, mediated by similar neural states of activity andinactivity known as ‘up’ and ‘down’ states [58, 59]. A simi-lar phenomenon has recently been observed in sleep deprivedflies [60]. Consistent with other studies, our study, using gen-eral anaesthesia, does not observe this slow oscillations. Fu-ture studies with more formal comparisons between up anddown states and ε-machines, in both theory and computersimulations, may be a fruitful avenue for further research in

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Figure 6. ε-machine for same channel, fly, conscious state asFig. 1(c), but with histories stored in each causal state explicitlystated. The sequences after the asterisk ∗ represent the sequenceof symbol observations with the most recent observed symbol onthe far right. Sequences collected within a causal state (grey circle)warrant significantly different future statistics to observed sequencesin other causal states. The red lines emit a “1" upon transition, andblue lines emit “0"s.

this regard.An analysis in terms of ε-machines has also allowed us to

discriminate between levels of conscious arousal by examin-ing causal structures found in both forward and reverse timedirections. Based on our previous finding [30] as well as re-lated concepts in temporal predictive [31, 61–63] and causalmatching [32], we hypothesised that the wakeful brain maybe tuned to causal structures of the world, which run forwardin time, and thus ε-machines would be more complex for for-ward than reverse readings. Further, we hypothesised thatsuch temporally tuned structural matching will be lost underanaesthesia. Our results (Sec. III B 2) are highly intriguingin three ways. First, near-zero causal irreversibility indicatesthat reducing the structural complexity to a simple index isnot enough to capture effects on the information structure thatare sensitive to the direction of time. This is the case regard-less of the level of consciousness (at least at the timescales ofthis study). Second, nonzero crypticity indicates that the un-derlying information structure is not symmetric in time. Moreprecisely, the signals themselves encode different amountsof information when run forwards as opposed to backwards.Third, the KL divergence rate analysis definitively demon-strated the existence of greater temporal irreversibility in thewakeful as opposed to the anaesthetised state. Having saidthis, we are limited in drawing strong conclusions due, inpart, to the relatively small observed effect size of Ξ, likelya consequence of our relatively small data set. Despite this,even at millisecond time scales, our study successfully iden-tifies significant differences in the time direction of the neuralrecordings.

Identifying the decrease in temporal-reversibility due toanaesthesia in tandem with complexity is of broad interestin neuroscience. While some physicists and neuroscientistshave conjectured links among physics, the brain, and evenconsciousness through the lens of the direction of the time,their accounts have remained rather speculative, and not builton any solid theoretical foundations (for related and alter-

native theoretical foundations, see the work by Cofré andcolleagues [64, 65]). For example, using reversely playedmovies, the sensitivity to the direction of time is shown todiffer across brain regions in humans [66]. In animal stud-ies, some populations of neurons (in the hippocampus) areknown to become activated in a particular sequential orderwhile the animal experiences a particular event. For example,in anticipation of the event, the neurons activate in a forwarddirection, but in retrospection, the neurons activate in reverseorder [67]. While direct links between these empirical find-ings and the ε-machine framework remains elusive, we fore-see that our unified theoretical and analytical framework canpotentially bridge this gap in the future.

Our study is not the first to apply complexity measure inconsciousness research. Indeed, many definitions and mea-sures of complexity have been proposed in the literature (seeRef. [68] for a list). Moreover, there is a flow of ideas goingthe other way as well [69–71]. However, many, if not most,of these measures cannot account for temporal correlations(memory), temporal asymmetry, or differentiate between ran-dom and structured processes. Our interdisciplinary study,based on ε-machines, opens up new possibilities; physics canimprove its theoretical constructs through the application oftools to empirical data, while neuroscience can benefit fromrigorous quantitative tools that have proven their physicalbasis across different spatio-temporal scales. Among thosecomplexity measures, Cµ can be easily interpreted in termsof temporal structure [72], as it has a direct relation to pro-cess predictability and memory requirements. We empha-sise that statistical complexity Cµ derived from ε-machines,drastically differentiates itself from other scalar complex-ity indices such as Lempel-Ziv complexity [73]. For oneLempel-Ziv complexity is maximal for a random noise pro-cess whereas statistical complexity for the same process iszero (see Eq. (2)). In addition, the notion of temporal re-versibility available in the ε-machine framework has no coun-terpart in Lempel-Ziv complexity. This is a critical differ-ence since it is known that a low-complexity forward-timeε-machine consisting of only two causal states can have avery high-complexity reverse-time ε-machine with countablyinfinite states [74]. Thus, explicitly considering the influenceof time is critical for addressing questions about complexity.When coupled with our results, we can conclude that anaes-thetised brains become less structured, more random, morereversible, and approach a stochastic process with a smallermemory capacity compared to the wakeful brains.

Overall, our results suggest that measures of complexityextracted from ε-machines might be able to identify furtherstructures that are affected by anaesthesia at different spatialand temporal scales. It is also likely that applying a similaranalysis to other data sets, in particular, human EEG data willlead to new discoveries regarding the relationship betweenconsciousness and complexity that can be retrieved simply atthe single channel level.

12

ACKNOWLEDGMENTS

RNM, FAP, NT, KM acknowledge support from MonashUniversity’s Network of Excellence scheme and the Foun-dational Questions Institute (FQXi) grant on Agency in thePhysical World. AZ was supported through Monash Uni-versity’s Science-Medicine Interdisciplinary Research grant.DC was funded by an Overseas JSPS Postdoctoral Fellow-ship. NT was funded by Australian Research Council Discov-ery Project grants (DP180104128, DP180100396). NT andCD were supported by a grant (TWCF0199) from Temple-ton World Charity Foundation, Inc. We thank Felix Binder,Alec Boyd, Mile Gu, Rhiannon Jeans, and Jayne Thompsonfor valuable comments. KM is supported through AustralianResearch Council Future Fellowship FT160100073.

APPENDIX

A. Linear mixed-effects model

In this section, we demonstrate an example of an LMEanalysis for the case of statistical complexity Cµ in the for-ward time direction. For the case when time direction ξ is in-cluded as an effect, the only change this makes to the processis increasing the dimensions of the effects matrix F. Perform-ing an LME analysis on other quantities used in this study likecrypticity or KL rate follow the same procedure outlined here.

The main goal of the LME analysis we perform in thisstudy is to determine the degree of contributions each andcombinations of memory length (λ), channel location (c),and level of conscious arousal (ψ) have on statistical com-plexity Cµ. LME accomplishes this by modelling statisticalcomplexity as a general linear regression equation (Eq. (7)),whose response is predicted by the aforementioned param-eters λ, c, and ψ. In this Appendix, we show the exactform of the linear regression equation used in this analysis,while referring to the terminology introduced in the methods(Sec. III A).

We begin by restating Eq. (7) for the case of statistical com-plexity, C = Fβ + Rb + E , which has the form of a generalmultidimensional linear equation. We will set aside the righthand side of the equality for now. On the left hand side, sta-tistical complexity takes the form of a column vector C. Eachrow corresponds to the unique response of Cµ, at a specificselection of parameters. There is a general freedom of choice

associated with the number of parameters one would like toassign to the elements C. We index the rows with fly numberf , channel location c, and the conscious arousal state ψ. Thatis, the (i, j, k)th element is

[C](i,j,k) = C(i,j,k)µ . (9)

In other words, it is the ith fly’s jth channel in kth condition.Thus, C has length of |f | × |c| × |ψ| = 390. Each Cµ in thisvector is a function of λ.

The matrix F introducing the set of fixed effects F ={λ, c, ψ, λc, λψ, cψ, λcψ} into the model (known in the con-text of general linear models as the design matrix) can thenbe represented as F = (F1, . . . ,F13)T , with each elementcorresponding to the design matrix of a specific fly. Theseindividual fly response matrices can be explicitly expressedas

Ff =

(~λ D ~ΨW λD λ~ΨW DΨW

λDΨW

~λ D ~ΨA λD λ~ΨA DΨAλDΨA

), (10)

where ~λ = (λ, . . . , λ)T and ~ΨX = (ΨX , . . . ,ΨX)T are col-umn vectors of length 15 containing the predictor variables ofmemory length and level of conscious arousal respectively, Dis the 15× 15 identity matrix which “selects out" the channelof interest, DΨX

= diag(ΨX , . . . ,ΨX) is the 15× 15 matrixwhich “selects out" the condition of interest correlated withthe level of conscious arousal, where

ΨW (A) =

{1 if ψ = wakeful (anaesthetised)0 otherwise.

(11)

In a similar fashion, the expression for the matrix contain-ing the random effects R can be determined. For the case ofour study, we only consider random effects arising due to cor-relations between channels within a specific fly. The result ofthis is an adjustment to the intercept of the linear model foreach fly and channel combination. Therefore, the random ef-fects matrix R is simply an identity matrix of dimension 390.The accompanying elements of the random effects vector bconsist of regression coefficients bfc describing the strengthof each intercept adjustment.

The execution of the LME analysis which included coeffi-cient fitting, and log-likelihood estimations was facilitated byrunning fitlme.m in MATLAB R2108b.

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Q 1st Order 2nd Order 3rd OrderC+µ λ : χ2(1) = 443.64 p < 10−16 λc : χ2(14) = 42.876 p = 8.97× 10−5 λcψ : χ2(14) = 24.00 p = 0.0458

λψ : χ2(1) = 14.63 p = 1.31× 10−4

∆C+µ λ : χ2(1) = 20.97 p = 4.65× 10−6 λc : χ2(14) = 37.19 p = 6.90× 10−4

Ξ ψ : χ2(1) = 4.870 p = 0.0273 λψ : χ2(1) = 5.565 p = 0.0183 λcψ : χ2(14) = 31.79 p = 4.29× 10−3

λ : χ2(1) = 6.725 p = 9.51× 10−3

d ψ : χ2(1) = 5.896 p = 0.0152 λψ : χ2(1) = 6.119 p = 0.0134

λ : χ2(1) = 460.5 p < 10−16 λc : χ2(14) = 35.86 p = 1.09× 10−3

DKLS λ : χ2(1) = 127.4 p < 10−16 λc : χ2(14) = 85.81 p < 10−16

λψ : χ2(1) = 127.4 p < 10−16

Table I. Significant χ2 and p values of effects of channel c, memory length λ, and condition ψ, for informational quantities Q obtained viaLME analysis. First order effects correspond to significant channel, memory, or condition responses on an informational quantity, whilesecond and third-order effects correspond to interactions between these effects. χ2 values are reported with n− 1 degrees of freedom in theparentheses, corresponding to the number of effects removed under the null model, described in Sec. III A

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