ENTROPIES OF THE EEG: THE EFFECTS OF GENERAL ANAESTHESIA.

Transcription

1 NTROPIS OF TH G: TH FFCTS OF GNRAL ANASTHSIA. J. W. Sligh 1,. Olofsn 2, A. Dahan 2, J d God 3 & A Styn-Ross 4. Dpartmnt of Anasthsia 1, Waikato Hospital, Dpartmnts of 2 Anasthsiology and 3 Physiology, Lidn Univrsity Mdical Cntr, Lidn, Th Nthrlands, and Dpartmnt of Physics 4, Univrsity of Waikato, Hamilton, Nw Zaland slighj@hwl.co.nz Th aim of this papr was to compar th prformanc of diffrnt ntropy stimators whn applid to G data takn from patints during routin induction of gnral ansthsia. Th qustion thn aros as to how and why diffrnt G pattrns could affct th diffrnt stimators. Thrfor w also compard how th diffrnt ntropy stimators rspondd to artificially gnratd signals with prdtrmind, known, charactristics. This was don by applying th ntropy algorithms to psudog data: (1) computr-gnratd using a scond-ordr autorgrssiv (AR2) modl, (2) computr-gnratd whit nois addd to stp signals simulating blink and ymovmnt artifacts and, (3) sing th ffct of xognous (computr-gnratd) sin-wav oscillations addd to th actual clinically-drivd G data st from patints undrgoing induction of ansthsia. BACKGROUND AR G NTROPY STIMATORS PURLY AN LGANT MTHOD OF SIGNAL PROCSSING? What is ntropy? Claud Shannon dvlopd th modrn concpt of 'information' or 'logical' ntropy as part of information thory in th lat 194s(Shannon C 1948). Information thory dalt with th nascnt scinc of data communications. Shannon ntropy (H) is givn by th following quation: H = -Σ p k log p k, whr p k ar th probabilitis of a datum bing in bin k. It is a masur of th sprad of th data. Data with a broad, flat probability distribution will hav a high ntropy. Data with a narrow, pakd, distribution will hav a low ntropy. As applid to th G - is ntropy mrly just anothr statistical dscriptor of th variability within th G signal (comparabl to othr dscriptors, such as th spctral dg, of th shift to low frquncis that occurs on induction of gnral ansthsia)? Part of th answr to this qustion lis back in th original dfinition of thrmodynamic ntropy in th nintnth cntury by Clausius and othrs. Th chang in thrmodynamic

2 2 ntropy (ds) of a closd systm is dfind as a quantity that rlats tmpratur (T) to th nrgy (= hat, dq) transfrrd to th molculs via th quation: ds = dq/t Bcaus ntropy changs with chang of stat (.g. solid to liquid), and bcaus ntropy tnds to incras with tim, it can b considrd to b a masur of th dgr of 'disordr' of th systm. Howvr, 'disordr' is a loosly dfind trm. Boltzman showd that thrmodynamic ntropy could b dfind prcisly as (Boltzman's) proportionality constant (k) multiplid by th logarithm of th numbr of indpndnt microstats (W) availabl for th systm: S = k log (W) Bcaus h was abl to xplain th changs in macroscopic obsrvabls (such as tmpratur), from th changs in kintic nrgy of a collction of individual molculs - h thus pionrd th scinc of statistical mchanics. Thrmodynamic ntropy has a wll-stablishd physical basis. It is possibl to driv th Shannon/'information' ntropy (H) from th thrmodynamic Boltzman formula (S). Howvr it must b clarly statd that, bcaus thr xists a formal analogy btwn H and S, it dos not imply that thr ncssarily is a matrial basis for quating H and S in rgards to th cortical function. Nvrthlss thr dos xist tantalizing nurophysiological vidnc that th utility of information ntropy stimators as masurs of cortical function is bcaus - as th cortx bcoms unconscious - a tru dcras in (th logarithm of) th numbr of accssibl microstats (S) occurs at th nuronal lvl (Styn-Ross t al. Towards a thory of th gnral ansthtic-inducd phas transition of th crbral cortx: 1. A thrmodynamics analogy, and 2. Numrical simulations, spctral ntropy, and corrlation tims. Phys Rv, in prss),(john, aston t al. 1997; Michloyannis, Arvanitis t al. 1998; Styn-Ross 1999.; Quiroga, Arnhold t al. 2). If this is tru, it mans that th chang in information ntropy within th G may window a ral chang in cortical functional organization. Thus th trm 'ntropy' may b mor than mrly a statistical masur of G pattrn, but may in som way truly rflct th intra-cortical information flow. ntropy is th logarithm of th numbr of ways that th microstat can rarrang itslf and still produc th sam macrostat. In thrmodynamics, th microstat is th momntum and position of ach molcul. Th diffrnc btwn tru thrmodynamic ntropy and othr information ntropis is that th distribution of th kintic nrgy of individual molculs is not ncssarily involvd in information ntropy stimators. Th nrgy has bn abstractd away from hat nrgy and thrmodynamics, and can man any chang in th activity of th particls that mak up th systm undr obsrvation. In this papr th 'nrgy' is th changs in cortical pyramidal mmbran potntial - which produc fluctuations in th local fild potntial of th cortx which, in turn, ar thn physiologically and statistically avragd to produc th scalp-masurd G signal. What is th significanc for th G?

3 3 If th G is to som dgr a window on cortical procsss, th changs in ntropy of th G may b xpctd to indirctly coarsly masur changs in th ntropy occurring within th crbral cortx itslf. Assuming that th main function of th conscious cortx is th procssing and manufactur of information, it would not b unrasonabl that som sort of 'information masur' may b usful. Th problm is that (lik th word 'disordr'), th word 'information' carris too many othr manings and connotations, and has to b carfully, and scintifically dfind to b rally usful in this contxt. Prhaps th simplst, practical, working dscription of th ntropis of th G would b - masurs of th xtnt that constraints (in our cas, gnral ansthsia), limit th numbr of accssibl stats availabl to th cortx. This statmnt is prsntd without proof, but is consistnt with th xprimntally-obsrvd changs prsntd latr in this papr. It is clar from this that, whilst it could b xpctd that incrasd numbr of microstats is in gnral associatd with a mor 'complx' systm, ntropy dos not itslf ncssarily dirctly masur th 'complxity' of a systm - which has othr implications of variability in rspons to inputs tc. If ntropy is dfind as th logarithm of th numbr of commonly accssibl cortical microstats, th qustion ariss - what ar cortical microstats? Thr is growing nurophysiological vidnc that cognitiv activity involvs th transint formation and dissolution of intrconncting cortical nuronal assmblis ( activation and quiscnc )(John, aston t al. 1997). It would not b unrasonabl to claim that ths cohrnt assmblis ar ffctivly th functional cortical microstats. This activity is manifst in th scalp G as a broadr-band, 'whit-nois' spctrum(stam, Tavy t al. 1993; Thomr, Stam t al. 1994). If, (1) th stat of conscious awarnss rquirs th fficint formation of many cortical microstats, and (2) if th microstats of th G signal rflct in som way th cortical microstats, thn th dcras in G ntropy (as is sn with gnral ansthsia) is an indicator that thr ar fwr availabl cortical microstats(wiss 1992). Th Cortx-Consciousnss Paradox: If you call ntropy disordr, th highr valus of ntropy found in th G from th awak cortx imply that th cortx in th conscious stat is mor disordrd than th unconscious stat. This paradox highlights th problms with quating ntropy with disordr. At prsnt w lack th mans to discrn th xquisit high dimnsional pattrns gnratd by th conscious cortx during cognition, and call it nois! This is th rason why w prfr to dfin ntropy in trms of availabl microstats rathr than ordr. Prhaps a bttr mtaphor would b to dscrib ntropy as 'frdom of choic'. Th conscious cortx is fr to mov to many availabl microstats. A VISIT TO TH NTROPY ZOO

4 4 Th dcras in complxity of th G signal during induction of gnral anasthsia is manifst primarily as a chang in th undrlying slop of th G's powr spctrum. Th paramtr commonly usd to dscrib th slop is th Hurst xponnt. This paramtr is most convnintly stimatd using dtrndd fluctuation analysis (DFA) (Hnghan C and Mcdarby G 2). Although it is not a tru ntropy stimator, w hav includd th DFA as on of th G paramtrs in our mthods bcaus it is an asily calculatd, robust masur of th autocorrlation structur of th G. Rcntly a numbr of diffrnt ntropy stimators hav bn applid to G data attmpting to quantify complxity and/or dpth-of-ansthsia. Ths tchniqus do not masur th shap of th distribution of th G voltags pr s, but instad dscrib how th G signal changs with tim ithr in frquncy-spac or phas-spac. Thy may b thrfor loosly classifid into two groups. 1) Spctral ntropis Thr ar various ways of stimating th changs in th amplitud componnt of th powr spctrum of th G. Ths us th amplitud componnts of th powr spctrum as th probabilitis in th ntropy calculations. Intrstingly, by using frquncy-spac w ar dfining th microstats in trms of rats-of-chang. A wid rang of accssd frquncis (a flat spctrum) implis many possibl diffrnt rats-of-chang of summd pyramidal cll mmbran potntials. Th prototyp of this group is th Spctral ntropy (SN) (Inouy, Shinosaki t al. 1991; Fll, Roschk t al. 1996). Th SN is th Shannon ntropy formula suitably normalisd and applid to th powr spctral dnsity of th G signal. SN = Σ p k log p k / log(n), whr p k ar th spctral amplituds of frquncy bin k. Σ p k =1, and N = numbr of frquncis. In our papr w hav usd k from 1 to 47, rprsnting a frquncy histogram with 1Hz bins ovr th rang 1-47Hz inclusiv. Th SN can tak valus from zro (if th spctrum contains purly a singl oscillatory pak) to on (if th spctrum is that of uncorrlatd whit nois i. p k = 1/N). Th SN can b shown to b a spcial cas of a sris of ntropis trmd Rnyi ntropis (R(α)) (Amari S 1985; Grassbrgr, Schribr t al. 1991; Gonzalz Andino, Grav d Pralta Mnndz t al. 2). Thir formula is: R(α) = α/(1 α) Σ log p k α. (α 1) Takn togthr a spctrum of ths Rnyi ntropis can b usd to dfin a givn probability distribution in a mannr similar to th convntional us of statistical momnts for th sam purpos. In this papr w us only th valu of α = -1, which w hav trmd th Gnralisd Spctral ntropy (GS). Bcaus it is a rciprocal transformation, this transformation diffrs from th SN in that th sum is wightd towards frquncis with rlativly smallr amplituds. Ths tnd to b thos in th highr nd of th

5 5 frquncy band. Thus huristically, th rsults from th GS may not b dissimilar to th SN calculatd ovr a highr frquncy band (g. 2-45Hz). Th dissimilarity btwn two probability dnsity functions can b quantifid using th Kullback-Liblr ntropy (K-L) (Torrs M and Gamro LG 2) also somtims calld th 'Rlativ ntropy', or 'Cross-ntropy'(Qian 21). Th K-L has bn shown to b ssntially quivalnt to yt anothr ntropy, th Rnormalizd ntropy (Quiroga, Arnhold t al. 2). W will only considr th K-L. It is possibl to stablish a baslin spctrum (q k ) of th G whn th patint is awak and thn us th K-L to stimat how much it changs whn th patint is givn anasthsia and loss consciousnss. K-L is simply dfind as: K-L(p q) = Σ p k log (p k /q k ) Obviously this tchniqu cannot b institutd halfway through an opration! It dos hav th possibl advantag of individualising th G changs to b spcific for ach patint. For compltnss thr ar a larg numbr of possibilitis of using an ntropy stimator to stimat th sprad of any masur of G pattrns. Spcifically, w hav lookd at: 1) Th Shannon ntropy of th bispctrum, (SNhos), 2) Th Shannon ntropy of th wavlt spctrum (SNwv)(Rosso, Blanco t al. 21), and 3) A variant of th spctral ntropy, that uss th spctrum of th scond drivativ of th tim sris calld th acclration spctrum ntropy(stam, Tavy t al. 1993). 4) A quantity calld th Fishr Information (I) (Fridn BR 2). I = ( k -1 ) Σ {[p(k n+1 ) - p(k n )] 2 /p(k n )}, This masur has som intrsting proprtis, and has bn trmd a mothr ntropy. Firstly, it is thortically, complmntary to th Shannon ntropy (which is a global masur) bcaus I has th proprty of locality. (i. if you shuffl p k s, (I) will b diffrnt.) Also I is rlatd to K-L by th following rlation: I = (2/ k 2 ) K-L [p(k),p(k+ k)], whr k is th width of th frquncy bin. Thus I may b considrd as bing proportional to th cross ntropy (K-L) btwn th PDF p(k) and its shiftd vrsion p(k+ k). Huristically, I is a masur of th absolut gradints within th spctrum. I is also proportional to th Rnyi ntropis R(α) = ( k 2 /2) α/(1 α) Ι. (α 1)

6 6 In practical trms non of ths thr last-mntiond tchniqus appard to contribut significant nw information to that gaind from th othr ntropy stimators, and thus thir rsults ar not rportd furthr. 2) mbdding ntropis Th scond clustr of tchniqus ar thos which dirctly us th G tim sris. Th ntropis that incorporat this as part of thir calculation ar th Approximat ntropy (Apn), and th ntropy of th Singular-Valu Dcomposition (SVDn). Information about how th G signal fluctuats with tim is obtaind by comparing th tim sris with itslf, but laggd by a spcifid tim intrval. This practic is usually tchnically trmd as mbdding th on-dimnsional signal in a phas-spac. Intuitivly it sms snsibl that, if an G signal is irrgular, th position of a particular point will not b asily prdictd using knowldg of its prvious points; whras in a rgular signal th position of th point will b mor rliably prdictd. Th numbr of prvious (laggd) points usd in making th prdiction is th mbdding dimnsion (m). For a procss whos undrlying dimnsion is n (i. Which can b dscribd uniquly in trms of n paramtrs), th rquird mbdding dimnsion is; m 2n+1, and th rquird minimum dat siz to xtract ths n paramtrs is 1 m. Suppos that th G (and by assumption th cortx) was oprating undr xtrm constraints and had a dimnsion (n) of only 1 (i 1 paramtrs could dscrib th signal), thn th minimum rquird data lngth is 1 21 points! This xponntial dpndnc of data lngth on th complxity of th undrlying procss is calld th curs of dimnsionality in nonlinar analysis. It is impractical to proprly mbd th G signal. Thus ths tchniqus ar NOT abl to fulfil thir thortical promis and xtract high-dimnsional information from th univariat G data-stram. Using ths mbddings, th thortical masur of th rat of "information" gnration by a systm is th Kolmogorov-Sinai ntropy(grassbrgr, Schribr t al. 1991). Howvr this masur divrgs to a valu of infinity whn th signal is contaminatd by th slightst nois! A practical solution to ths problms has bn put forward using a family of statistics calld th Apn (Pincus, Gladston t al. 1991), and Sampn (Richman and Moorman 2). Th Apn, as applid to G signals in patints undr gnral ansthsia, has bn vry wll dscribd in dtail in an articl by Bruhn(Bruhn, Ropck t al. 2). In short th Apn looks at squncs of lngth m, and thn stablishs th ngativ logarithmic probability that ths squncs prdict a nw squnc of m+1 points to within an rror rang of r typically st at.2 SD. In a rgular signal most squncs will succssfully prdict th nxt data points, and th Apn will b low. In an irrgular signal, thr will b fw succssful prdictions and th Apn will b corrspondingly high. Th xact valu of th Apn will dpnd on th valus chosn for th thr paramtrs of th statistic: N = numbr of sampls, m = mbdding dimnsion, and r = nois thrshold. Bruhn suggstd that for a data lngth of N = 124 points, r =.2 SD, and m = 2; th maximum valu of th Apn should b about 1.7. Almost all publishd paprs us low valus of m = 2 or 3 in th calculation of th Apn(Rzk and Robrts 1998). Bcaus th Apn(m=2) statistic is ffctivly only

7 7 xtrapolating using a coupl of prvious data points, it may b mrly applying a linar prdiction. This may not b using any information byond that mor asily obtaind from th SN. In this papr w compard th Apn obtaind using valus of m = 2 with thos whn using m = 5 and m = 1. It is also possibl to calculat an Apn to stimat th similarity of two diffrnt signals - this is, confusingly somtim calld th 'cross-ntropy'. It should not b confusd with th tru Kullback-Liblr cross-ntropy. An altrnativ mthod for xtracting information from th mbddd tim sris data is to do a Singular-Valu Dcomposition (Broomhad DS and King GP 1986; Grassbrgr, Schribr t al. 1991). It is possibl to dcompos a matrix U into thr matrics: U = CΣV T Th 'singular valus' ar th positiv squar roots of th ignvalus of a matrix (C) multiplid by its transpos (V T ). Th m diagonal lmnts of th diagonal matrix Σ ar th singular valus. Thus thy can b thought of as 'psudo-ignvalus' of non-squar matrics. In our G mbddings (m=4, lag=2) th matrix (U) consists of 64 4 lmnts. ach of th 4 column vctors is th G signal laggd by 2 1/sampling frquncy (msc). If ach column vctor is indpndnt of ach othr, thn ach singular valu will b larg. This is th cas in th awak stat, whr w will hav as many significant singular valus as thr ar vctors. Whn th patint bcoms ansthtizd th G tim sris vctors bcom lss indpndnt - bcaus th G dvlops slow oscillations (longtrm tmporal corrlations). Thus th vctors that mak up th matrix (U) ar mor dpndnt, and thr a fwr significant singular valus. Th Singular-valu Dcomposition ntropy (SVDn) was dfind by Sabatini (Sabatini 2) using th Shannon formula applid to th lmnts of Σ as follows: SVDn = Σσ i ln σ i, whr is normalizd as σ i = σ i /Σσ j ar th diagonal lmnts. In ssnc, similar to th Spctral ntropy, th SDVn masur stimats th dviation of th singular valus away from a uniform distribution. Mthods. XPRIMNTAL COMPARISON OF NTROPY STIMATORS

8 8 Calculation of ntropis Th various ntropy stimators wr calculatd according to standard algorithms (Inouy, Shinosaki t al. 1991; Fll, Roschk t al. 1996; Bruhn, Ropck t al. 2; Hnghan C 2; Quiroga, Arnhold t al. 2; Sabatini 2). Th SN, GS, and K-L wr calculatd ovr th frquncy band 1 to 47Hz. Th baslin for th K-L ntropy was obtaind from th singl 5sc poch of G data at th start of rcording. A pictur of how th ntropy stimators transform th powr spctral dnsitis for a typical G poch is shown in figur 1. W hav plottd th transformd ntropy valu at ach frquncy (k) (i. at th stp bfor summing across th frquncis to produc th final ntropy stimator). In th lft uppr graph w can s that th SN k is vry clos to th raw spctral dnsity at ach frquncy bin in th awak (flat) spctrum. In contrast th right uppr graph shows th pakd distribution charactristic of an ansthtizd patint's G. Th p log(p) transformation has th ffct of xaggrating th 'spctral dnsity' around th spctral paks. Convrsly th rciprocal GS transformation has th ffct of incrasing th 'spctral dnsity' in th troughs of th spctrum. Awak Anasthtisd SN k Raw k GS.1 k SN = SN = PSD ch 1 PSD ch 2 K-L Dist k Awak.2.15 Anasthtisd.1.1 K-L = K-L = Frquncy (Hz) Frquncy (Hz) Figur 1. Th ffct of th ntropy transformations in th frquncy domain. xampls of th valus of diffrnt spctral ntropy for ach frquncy bin. Th horizontal axis of frquncy(hz), and vrtical axis is spctral powr. In this graph, th labls "RAW k ", SN k ", "GS k " and "K-L k " rfr to th valus calculatd for ach frquncy bin bfor summation. In th rst of th txt th labls SN, GS and K-L rfr to th totals of all th frquncis (k). Th valus of th GS hav bn scald to fit thm on th sam graph. Th top row illustrat how th SN

9 9 transformation xaggrats th paks, and how th rciprocal (GS) transformation smooths th dips. Th graphs in th bottom row ar illustrativ of th typs of ffcts sn whn using th K-L ntropy. For graphical clarity w hav compard th K-L btwn 2 contmporanous G channls - not as in th txt whr w hav compard th G spctrum at tim (t) vs that at th start of th rcording (t=5sc). Thy dmonstrat pictorially th fact that ngativ distancs ar possibl, and that th ffctiv distanc may b incrasd if th absolut valu of th dnominator is nar zro. For th Apn w usd r =.2 SD, n = 128, lag = 2 data points, m = 2, 5 and 1. In th initial calculations of th SVDn it bcam apparnt that th us of mbdding dimnsions gratr than four did not contribut significant additional information, so a valu of m = 4 was usd for all calculations rportd in this papr. Th dtrndd fluctuation analysis (DFA) is an fficint robust mthod of calculating th Hurst xponnt of th data. In ssnc th signal is intgratd and dividd into pochs. ach poch is dtrndd and th root man squar of th rsultant fluctuations in ach poch obtaind. As th siz of th pochs incras, th root man squar of th fluctuations incras. If ths incras in a linar bilogarithmic fashion, th slop of th lin is th Hurst xponnt (DFA). W calculatd th slop using pochs ranging from 2 to 25 data points (~1 to 128Hz). (A) Patints Aftr obtaining rgional thical committ approval and writtn informd consnt, G data wr obtaind from 6 adult patints during induction of anasthsia with propofol (1-2.5 mg/kg iv). Som of this data hav bn prviously rportd. Th xact tim at which th patints bcam unconscious (dfind as bcoming unrsponsiv to vrbal command) was notd. Th G signal was obtaind from a bifrontal montag (F7: F8), via th Aspct A-1 G monitor (Aspct Mdical Systms, Nwton, MA), using a sampling frquncy of 256/sc, band-width1:47hz, and 5sc pochs. Th raw G data wr thn downloadd to a computr for offlin analysis. Th various ntropy stimators wr calculatd from G data at th start (START = bfor induction of anasthsia), 15sc prior to th point of loss-of-consciousnss (LOC-15), th point of loss-ofconsciousnss (LOC), 15sc aftr th point of loss-of-consciousnss(loc+15), and 3sc aftr th point of loss-of-consciousnss(loc+3). Th xact poch chosn for th start poch varid slightly bcaus th xact pochs wr slctd manually as thos containing minimal y-blink and frontalis MG artifact. Thr was no smoothing of any of th paramtrs. (B) Th AR2 modl This modl was usd bcaus it gnrats sris with known charactristics/oscillations, and it provids a 'tst-bd' to compar th prformanc of th ntropy stimators in a simpl wll-dscribd systm, without complication of unknown ffcts of non-linaritis in th signal. Historically, highr ordr autorgrssiv modls hav bn usd to modl th ral G(Wright, Kydd t al. 199). Data pochs of 128 sampls lngth wr gnratd. Th paramtrs of th AR2 modl wr chosn to giv spctra similar to thos ncountrd with ral G s in patints undrgoing gnral ansthsia (s figur 2).

10 1 u l a V r t m a r a P A2 SVDn SN DFA A5 #1 #2 #3 #4 #5 Sris numbr r b m u n s i r S r b m u n s i r S #5 #4 #3 #2 #1 #5 #4 #3 #2 #1 PsudoGs Tim (sc) Spctra Frquncy (Hz) Figur 2. Th AR2 modl data. Tim sris, powr spctra and accompanying changs in ntropy stimators. Th ρ2 valu for sris #1 to #5 wr -.9 to -.99 in stps of.2. ρ2 was kpt constant at.99. Th paramtr valus ar xprssd as th man (± on SD) of 3 sris. SN=spctral ntropy, DFA=dtrndd fluctuation analysis, SVDn=singular valu dcomposition ntropy, A2=approximat ntropy (m=2), and A5=approximat ntropy (m=5). S txt for dtails of th calculation of ths paramtrs. (C) Addition of Artifacts to Whit nois 'psudog's. W cratd Gaussian whit nois sris (128 sampls lngth, quivalnt to 5sc of ral G data (Fs=256/sc), man=, SD=1) to simulat th G in th alrt (pr-induction) stat. W thn addd: (i) low frquncy (.8Hz) sin wav oscillations to simulat y movmnts (ii) a stp chang half-way through th signal (iii) a stp chang followd by an xponntial rturn to th baslin Ths produc wavforms similar to thos sn with ral blink artifacts (s figur 5). W progrssivly incrasd th magnitud of th artifact componnt of th signal from 1 to 6 units. W thn calculatd th various ntropy stimators at ach lvl of artifact magnitud, in ordr to valuat how robust ach stimator is to non-g nois.

11 11 (D) Addd oscillations to patint data. Bcaus it is clar that th valus of th various ntropy stimators may b rducd by spctral paks, w tstd th ffcts of adding an artificial sin-wav to th patint G data st to produc a st of composit data (trmd G+oscillations ). W addd a computr-gnratd sin wav at thr diffrnt frquncis (6.4Hz, 19Hz, and 32Hz). Th amplitud was calculatd such that th standard dviation (SD) of th sin-wav qualld th SD of th raw G signal. W trialld a numbr of diffrnt magnituds and found that th ffcts on th ntropy stimators wr almost linarly rlatd to th rlativ magnitud. Thrfor w dcidd on th valu of th SD bcaus it appard to produc a physiologically ralistic magnitud of artifactual signal, but still showd an apprciabl ffct on th G paramtr. xampls of th raw wavforms and th composit (G+oscillations) wavforms and spctral dnsitis ar shown in figur 3. Not that th y axis of th powr spctra is a logarithmic scal. G w a R z H 6 + G z H G 2-2 On Sc G z 2 H G Tim (s) V u 2 g o L Powr Spctra Frquncy (hz) Figur 3a. xampls of th ffcts of addd oscillations to Gs of a patint whil awak.

12 12 G w a R z H 6 + G z H G 2 On Sc G z 2 H G Tim (s) V u 2 g o L Powr Spctra Frquncy (hz) Figur 3b. xampls of th ffcts of addd oscillations to Gs of a patint whil aslp. Statistical Analysis Th rlativ fficacy of ach paramtr was compard using th ara undr th rcivr oprating curv (ROC) - using th valus obtaind at th START poch (= awak stat) with thos obtaind 3 sc aftr loss of consciousnss (LOC+3) (= unconscious stat). Th ROC of ach ntropy stimator was compard pair-wis using a t-tst. Rsults. (A) Th Patint Data. Th changs in th diffrnt G paramtrs during induction wr similar (tabl 1); and wr comparabl in diffrntiating th awak(start) from anasthtisd (LOC+3 sc) stats. Intrstingly Apn(m=5, and m=1) incrasd significantly during induction of ansthsia. This is an opposit dirction to that of Apn(m=2).

13 13 Tabl 1. Th changs in G ntropy stimators (man(sd)) at diffrnt points during induction of gnral ansthsia in th group of 6 patints. (LOC = point of loss-of-consciousnss dfind as no rspons to vrbal command. ROC = ara undr rcivr oprating charactristic curv for awak(start) vs LOC+3sc tim points.) Tim Paramtr Start LOC-15sc LOC LOC+3sc ROC DFA.63(.22).81(.26) 1.11(.21) 1.24(.19).97 SN.9(.6).89(.7).82(.7).76(.7).93 GS.9(.5).87(.4).81(.8).74(.4).95 SNhos 1.32(.15) 1.26(.14) 1.21(.19) 1.12(.16).83 K-L.8(.4).13(.14).23(.16).29(.17).88 Apn (m=2) 1.56(.21) 1.53(.18) 1.35(.23) 1.18(.28).89 Apn (m=5).29(.16).35(.17).49(.17).57(.28).83 Apn (m=1).9(.3).3(.1).14(.3).3(.5).83 SVDn.98(.2).92(.8).82(.1).74(.11).97 All paramtrs changd significantly with tim (p<.5, paird t-tst). Th changs in ach paramtr wr broadly comparabl. Thr wr no significant diffrncs in th ROC curvs for th diffrnt paramtrs xcpt that Apn (m=5, and m=1), and SNhos wr lss than th DFA and SVDn (p<.5, t-tst). Th Parson linar corrlation cofficints (R) btwn th diffrnt paramtrs for th combind patint data at all tim points ar shown in tabl 2. (Bcaus Apn(m=1) was vry similar to Apn(m=5) w hav not includd it for simplicity in prsntation.) Tabl 2. A matrix of th corrlations btwn ach paramtr. (Masurd using th Parson linar corrlation cofficint (R).) SN GS Apn(m=2) Apn(m=5) K-L DFA(>1Hz) SN 1. GS Apn (m=2) Apn (m=5) K-L DFA SVDn

14 14 (B) ffcts of oscillations on ntropy masurs Th AR2 simulatd Gs (s fig 2) show that both oscillatory spctral paks (sris #5), and a shift to low frquncis (sris #1) rduc th SN, GS, and Apn similarly compard to th psudo 'awak' spctrum (sris #3). In this linar, Gaussian, data st, th SN is highly corrlatd with Apn (r=.91) and GS (r=.97). (C) Th ffct of othr 'simulatd artifact' signals on th ntropy stimators Figur 4 shows how incrasing th magnitud of th addd artifact progrssivly rducs th ntropy stimators from thir valu of on for pur whit nois, (or 1.5 for th Apn). In this simulation th mbdding stimators sm to b mor robust to th ffcts of nois than th spctral stimators. It can b sn that th ffcts on SN ar gratr than thos on GS which in turn ar lss than SVDn and Apn. Th xcption is th stp-chang whr th stimators ar similarly affctd. 5 PsudoG tim sris Chang in ntropy vs magnitud 1.5 SVDn GS SN Apn Magnitud of "y Movmnt" 5 y p o r t n Magnitud of "Blink" 5 y p o r t n Tim (sc) Magnitud of "Stp" Figur 5. Th ffct of adding - (a) a low frquncy (.8Hz) sin wav, (b) a stp and xponntial dcras, and (c) a stp - to a whit nois 'psudog' signal. On th lft ar xampls of th raw tim sris (with maximum amplitud addd artifact). On th right ar graphs showing how th valus of th various ntropy stimators ar progrssivly rducd as th amplitud of th addd artifact incrass rlativ to th original whit nois signal.

15 15 (D) Th ffct of addd oscillations to ral patint G data Figur 6 is an xampl of th changs causd by adding artificial oscillations to ral G data in on patint during induction of ansthsia. N S N P A n D V S A F D Tim (5sc pochs) Raw 6.4Hz 32Hz N δ S n p δ A n D V δ S SN Apn 6.4Hz 32Hz SVDn A F D δ DFA Figur 6. An xampl of th changs in ntropy stimators during induction of ansthsia in patint #18 (raw). Th ffcts of addition to th raw G of artificial oscillations at 6.4Hz(upward triangls) and 32Hz(downward triangls) on ach ntropy paramtr ar shown in th ntropy stimator timsris' on th lft. Th plots on th right show oscillation-inducd diffrnc in ach paramtr (as indicatd by th dlta symbol) vs th actual valu of th paramtr as applid to th raw G data st. Apn = approximat ntropy(m=2), SN = spctral ntropy, SVDn = ntropy of th singular valu dcomposition(m=4), and DFA = dtrndd fluctuation analysis. A typical xampl of th ffcts of th oscillation on th major ntropy stimators is shown in figur 6. It suggsts that th SN, whn th patint is awak, is last robust to th oscillation - as compard to th Apn and th SVDn. Also it is apparnt that th oscillations hav diffring influnc dpnding on whthr th patint is awak (high SN >.75) or comatos (corrsponding to a low SN<.75). Th oscillations tnd to dcras th SN whn th patint is awak, but hav lss ffct whn th patint is comatos. Also th ffcts ar rlativly insnsitiv to th frquncy of th imposd oscillation. In contrast, whn th patint is awak, th Apn and SVDn ar rlativly insnsitiv to addd oscillations. Howvr both ths masurs ar artificially incrasd maximally if th patint is comatos and th frquncy is abov 3Hz (figur 5 and tabl 3). Th SN is lss affctd, and vn furthr dcrasd in th unconscious patint. Th DFA is rsistant to th ffcts of oscillation in both th awak and unconscious stats.

16 16 Tabl 3. Combind data from all patints. Cofficint of variation (R 2 ) Tim Start (Awak) 3sc post loss-of-consciousnss Frquncy of addd Oscillation (Hz) SN Apn SVDn DFA Th dgr of agrmnt btwn th raw G and th oscillation+g data was quantifid using th cofficint of variation (R 2 ). Clarly, th SN in th awak patints was most snsitiv to th addition of xognous oscillatory activity, whn compard to th othr masurs n p A δ SN Apn.6.4 Awak Aslp.6.4 n D V S δ.2 A F D δ SVDn DFA Figur 7. Diffrnc btwn raw G and G+32Hz (dlta) vs actual raw G valus. Data ar combind from all 6 patints. (Awak=START, and aslp=loc+3sc pochs.)

17 17 Th slop of all curvs is positiv This indicats that th addition of 32Hz oscillations tnds to dcras th valus (th dlta is positiv) of ach paramtr whn ths paramtrs hav high valus, and incras th paramtr valus (dlta is ngativ) whn th paramtrs tak on low valus. Th absolut diffrncs ar last for th SVDn. Th approximatly linar natur of th curvs (with th possibl xcption of th Apn) suggsts that th ffct of addd oscillations is not constant but highly dpndnt on th proprtis of th pr-xisting G signal. Discussion. Although th diffrnt ntropy stimators wr closly corrlatd with ach othr whn applid to th AR2 modl data, thy wr lss wll intr-corrlatd whn compard using ral G data. It was difficult to stablish whthr th diffrncs btwn th diffrnt ntropy stimators wr du to inhrnt diffrncs in th typ of information that thy obtaind from th G signal, or whthr th diffrncs could b mrly attributd to diffrnt dgrs of numrical robustnss of th signal procssing and algorithms. Thortically both Apn and SN masur th dynamic changs of th G. Th SN masurs in th frquncy domain, and th Apn in th tim domain (phas-spac, or altrnativly, Markov-spac). Th Apn and SN producd vry similar rsults from AR2-gnratd psudog data. This suggsts that th two ntropy stimators ar probably quivalnt whn applid to data drivd from a linar systm - both achiving maximal valus with uncorrlatd whit nois, and dcrasing whn th signal bcoms mor corrlatd. Whn analyzing th ral G data, it smd that mbdding-drivd ntropy stimators (Apn and SVDn) wr lss affctd by th addition of xognous oscillations than spctrum-drivd ntropy stimators (SN, GS, K-L). Th rason why this should b so is not ntirly clar. Intuitivly, vn a high-frquncy sin-wav oscillation (32Hz) should b xtrmly rgular, and thrfor th addition of such a wav to th (irrgular) raw wavform in from an awak patint should hav th ffct of making th signal mor rgular on avrag, and lowring th Apn in all cass. W obsrvd that whn patints wr ansthtizd, th addition of such an oscillation had th opposit ffct lvating th Apn in a frquncy dpndnt mannr. This may b xplaind as th addition of th high frquncy oscillation to a signal with an xisting larg low-frquncy pak, it has th ffct of making th spctrum flattr and broadr (s fig 3b). Hnc th incrasd Apn. This is vidnc that, in practic, Apn dos not purly stimat rgularity as has bn claimd(pincus, Gladston t al. 1991), but actually stimats th ffctiv narrownss of th G powr spctrum. Most patints rciving modst doss of midazolam or propofol will xhibit a pronouncd spctral pak in th bta frquncyband. Our data would prdict that th ffct of this phnomnon on th various ntropy stimators will thrfor diffr (ithr lvat or dprss th stimator); dpnding on whthr thr is xisting background of dlta activity in th G. Th sam argumnt could b applid to th ffct of spindls (transint 1-14Hz G oscillations), on th valus of th various ntropy stimators.

18 18 Th SN and GS quantify th dgr to which th G spctrum dviats from whit nois (markd by a uniformly distributd spctrum). During incrasing dpth of ansthsia (with GABArgic agnts), th G spctrum dvlops an incrasingly stpr slop (th DFA jumps from ~.5 to ~1.5), and th SN and GS dcras from thir maximal valus of on. Th K-L ntropy masurs th distanc that th spctrum lis from a rfrnc spctrum. In our cas th spctrum was obtaind from th awak patint just prior to th start of induction of ansthsia. Bcaus th G spctrum in th awak (and nrvous) patint is oftn comparabl to th uniform whit-nois spctrum, th K-L ntropy of th spctrum uss much th sam information as th SN. All spctraldrivd masurs ar rlativly snsitiv to artifacts such as frontalis MG and yblinks and y-movmnts - which caus pisodic normous incrass in low frquncy powr. It must b notd that th dltrious ffcts on th SN, causd by adding oscillations to th tru G, may hav bn lss if w had usd mor aggrssiv prprocssing and filtring of th G data to rduc th ffct of artifacts. Our data rprsnts 'worst-cas', ral-lif data takn during busy surgical lists. This most closly rsmbls th day-to-day us of G monitoring in ansthsia. Th approximat ntropy is drivd in a substantially diffrnt fashion bing a practical approximation to th tru Kolmogorov-Sinai ntropy. With m = infinity, and r =, th Apn = Kolmogorov-Sinai ntropy. It is said to stimat th inhrnt prdictability of th signal. Howvr, crucially, its proprtis will dpnd on th dimnsion of th mbdding spac (m). It is a rcurring problm in using nonlinar tchniqus to quantify th G, that thr is not sufficint stationary data to nabl to rliabl high dimnsional mbdding. In our calculation of Apn, m = 2 implis that w ar using th prvious G data point (sinc w usd lag of 2 data points (= 1/128sc), about 8msc) to prdict th nxt data point. Th irrgularity masur will thrfor b prdicting th data point 8msc into th futur thus, ffctivly th gnral ansthtic-inducd dcras in Apn(m=2) is an stimat of th loss of high frquncis. With m = 1 th Apn is using information from data squncs up to ~1msc into th past. Thrfor th Apn(m=1) is wightd towards th influnc of lowr frquncis. Prsumably this is th xplanation as to why Apn(m=1) incrass with incrasing dpth of ansthsia - compard with Apn(m=2) which dcrass. In simpl trms, th Apn(m=2) stimats th ansthtic-inducd dcras in gamma wavs, whras th Apn(m=1) stimats th ansthtic-inducd incras in thta and dlta wavs. This opposit dirction of chang for Apn(m=5 or 1) vs Apn(m=2) was also sn using th linar AR2 modl data. This would imply that this paradoxical phnomnon is causd by intrinsic proprtis of th Apn algorithm, and is not du to non-linaritis in th G signal. This study shows that induction of gnral anasthsia with propofol causs similar changs in magnitud in all G ntropy stimators, mdiatd prdominantly by a dcras in rlativ high frquncy componnts of th G signal. Furthrmor a dcras in th valu of th ntropy stimator of th G dos not diffrntiat btwn ithr th apparanc of an oscillation, or an incras in slop of th spctrum. Convrsly, th addition of a high-frquncy oscillation on top of an alrady stply slopd spctrum, causs a paradoxical incras in Apn and SVDn, but lss prdictabl

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