SELF-ORGANIZED CRITICALITY AND THE DEVELOPMENT OF EEG PHASE RESET

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1 This is a preprin of an aricle published in Human Brain Mapping, Jan 2008 Copyrigh 2007 Wiley-Liss, Inc SELF-ORGANIZED CRITICALITY AND THE DEVELOPMENT OF EEG PHASE RESET Thacher, R.W. 1,2, Norh, D.M. 2, and Biver, C. J. 2 Deparmen of Neurology, Universiy of Souh Florida College of Medicine, Tampa, Fl. 1 and EEG and NeuroImaging Laboraory, Applied Neuroscience, Inc., S. Peersburg, Fl 2 Send Reprin Requess To: Rober W. Thacher, Ph.D. NeuroImaging Laboraory Applied Neuroscience, Inc. S. Peersburg, Florida (727) , rwhacher@yahoo.com

2 Thacher e al 2 ABSTRACT Objecives: The purpose of his sudy was o explore human developmen of self-organized criicaliy as measured by EEG phase rese from infancy o 16 years of age. Mehods: The elecroencephalogram (EEG) was recorded from 19 scalp locaions from 458 subjecs ranging in age from 2 monhs o years. Complex demodulaion was used o compue insananeous phase differences beween pairs of elecrodes and he 1 s & 2nd derivaives were used o deec he sudden onse and offse imes of a phase shif followed by an exended period of phase locking. Mean phase shif duraion and phase locking inervals were compued for wo symmerical elecrode arrays in he poserior-o-anerior locaions and he anerior-o-poserior direcions in he alpha frequency band (8 13 Hz). Resuls: Log-log specral plos demonsraed 1/f α disribuions (α 1) wih longer slopes during periods of phase shifing han during periods of phase locking. The mean duraion of phase locking ( msec) and phase shif (45 67 msec) generally increased as a funcion of age. The mean duraion of phase shif declined over age in he local fronal regions bu increased in disan elecrode pairs. Oscillaions and growh spurs from mean age 0.4 years o 16 years were consisenly presen. Conclusions: The developmen of increased phase locking in local sysems is paralleled by lenghened periods of unsable phase in disan connecions. Developmen of he number and/or densiy of synapic connecions is a likely order parameer o explain oscillaions and growh spurs in self-organized criicaliy during human brain mauraion. Key Words: Developmen of EEG phase rese, phase locking, chaos, self-organizaional criicaliy.

3 Thacher e al Inroducion Undersanding he mechanisms of neural synchronizaion and desynchronizaion is imporan in undersanding human brain dynamics. Recen sudies have shown ha sudden ransiions in he ampliude of he human elecroencephalogram (EEG) are represened by power laws and scale invariance and long-range emporal correlaions (Freeman 2003; Nikulin and Brismar, 2005; Linkenkaer-Hansen e al, 2001; Parish e al, 2004). These sudies are imporan because long-range emporal correlaions are a reliable mehod o ransfer informaion in neuronal populaions as well as providing a linkage o general laws of physics of complex sysems (Bak e al, 1987; 1988; Chialvo and Bak, 1999; Beggs and Plenz, 2003; Rios and Zang, 1999). The rapid creaion and desrucion of mulisable spaial-emporal paerns have been evaluaed in evoked, ransien and sponaneous EEG sudies (Breakspear and Terry, 2002a, 2002b; Rudrauf e al, 2006; Le Van Quyen, 2003). The paerns of sponaneously occurring synchronous aciviy involve he creaion of differeniaed and coheren neural assemblies a local, regional and large scales (Breakspear and Terry, 2002a; 2002b; Rudrauf e al, 2006; Sam and de Bruin, 2004; Varela, 1995; Freeman and Rogers, 2002). The dynamic balance beween synchronizaion and desynchronizaion is considered essenial for normal brain funcion and abnormal balance is ofen associaed wih pahological condiions such as epilepsy (Lopes da Silva and Pihn, 1995; LeVan Quyen e al, 2001b; Chevez e al, 2003; Neoff and Schiff, 2002), schizophrenia (Lere e al., 2002) and demenia (Sam e al., 2002a; 2002b). Synchronizaion is commonly defined as an adjusmen of rhyhms of oscillaing objecs due o heir weak ineracion and nearly always involves a period of sable phase relaions or phase locking (Pikovsky e al, 2003). De-synchronizaion is he opposie of synchronizaion and is defined as a shif in he phase difference of synchronized oscillaors and eliminaion of phase locking. Noice ha boh synchronizaion and de-synchronizaion sar wih a phase shif or adjusmen bu differ in he absence or exen of phase locking. Phase locking is a ell ale sign of synchronizaion and his is why Freeman and colleaques (Freeman, 2003; Freeman and Rogers, 2002) and Blacksone and Williams (2004) and ohers (Lachaux e al, 2000; LeVan Quyen e al, 2001b) use phase locking as a measure of EEG synchronizaion. The sudy of rapid changes in phase difference followed by periods of phase locking is called Phase Rese (PR). Phase rese occurs in coupled nonlinear oscillaors when here is a sudden shif of he phase relaionship of oscillaors o a new value followed by a period of phase locking or phase sabiliy also called phase synchronizaion (Pikovsky e al, 2003). The erm phase synchrony is

4 Thacher e al 4 synonymous wih phase locking and is someimes preferred in order o emphasize he saisical naure of phase sabiliy (Rudrauf e al, 2006). However, he erm phase synchrony is ofen used in reference o EEG coherence whereas phase locking is more specific o phase rese. Wheher one refers o phase locking or phase synchrony wha is imporan in he measuremen of phase rese is ha here is a prolonged period of phase sabiliy following a phase shif. This is imporan because random phase shifs wihou sabiliy exhibi whie noise disribuions (Pikovsky e al, 2003; Tass, 1997). Phase rese is also imporan because i resuls in increased EEG ampliudes due o increased phase locking of synapic generaors (Cooper e al, 1965; Nunez, 1994; Lopes da Silva, 1994). The inegraed rapid sequencing of phase shifs followed by phase locking ( i.e., he wo fundamenal componens of phase rese) have been correlaed o he alpha frequency band during cogniive asks (Kahana; 2006; Kirschfeld, 2005; Tesche and Karhu, 2000; Jensen and Lisman, 1998),working memory (John, 1968; Rizzuo e al, 2003; Damasio, 1989; Tallon-Baudry e al, 2001), sensory-moor ineracions (Vaadia e al, 1995; Roelfsema e al, 1997), hippocampal long-erm poeniaion (McCarney e al, 2004) and consciousness (Cosmelli e al, 2004; Varela e al, 2001; John, 2002; 2005). The presen sudy builds on hese previous sudies by paramerically analyzing he mauraion of he wo fundamenal componens of phase rese: 1- phase shif followed by, 2- phase sabiliy in a large populaion of subjecs from infancy o adolescence. There are wo general and equivalen mehods for sudying phase rese: 1- narrow band decomposiion and, 2- broad band decomposiion (Rudrauf e al, 2006; Le Van Quyen e al, 2001a; Bruns, 2004). Boh mehods use analyic ransforms such as he Fourier ransform, Wavele ransform and Hilber ransform. Which mehod is used depends on he frequency resoluion desired and he naure of he ransien signals ha are o be deeced (Rudrauf e al, 2006;Tass, 1997; Le Van Quyen e al, 2001a; Lachaux e al, 2000; Freeman e al, 2003; 2006; Freeman and Rogers, 2002). In he presen paper we used complex demodulaion as an analyic signal processing mehod similar o Lachaux e al (2000) and Blacksone and Williams (2004) which is mahemaically he same as he Hilber ransform (Pikovsky e al, 2003; Oppenheim and Schafer, 1975). All mehods measure he phase difference of pairs of signals evaluaed over successive inervals of ime where phase sabiliy is when he firs derivaive approximaes zero or dφ i,j /d 0. In general, he magniude of phase shif is defined as he difference beween he pre-phase shif value minus he pos-phase shif value and if a sudden and significan phase difference occurs followed by an exended period of phase sabiliy hen he poin in ime when he phase shif sared is he ime when he firs derivaive exceeded some hreshold value

5 Thacher e al 5 (Rudrauf e al, 2006; Tass, 1997; Tass e al, 1998; Le Van Quyen e al, 2001a; Blacksone and Williams, 2004). This poin in ime marks he onse of phase rese. EEG phase shif offse is defined in a reverse manner and he onse and offse imes define he phase shif duraion. The phase shif duraion is ypically in he range of 20 msec o 80 msec (Buzaski, 2006; Freeman, 2003; Freeman and Rogers, 2002). Phase locking or phase sabiliy ha follows a phase shif is ofen 200 msec o 600 msec in duraion in single cell analyses (Gray e al, 1989) and 100 msec o 1 second in surface recordings (Freeman and Baird; 1987; Freeman and Rogers, 2002; Freeman e al, 2006; Blacksone and Williams, 2004). Recenly, Linkenkaer-Hansen e al (2001); Sam and de Bruin (2004); Blacksone and Williams, 2004; Freeman e al (2003; 2006) and Buzaski (2006) have shown ha he EEG specrum is bes fi by a power funcion wih a 1/f α disribuion. The one over f disribuion is shared by a very wide range of observaions in he universe and was one of he inriguing myseries in physics unil Bak e al (1987) creaed a mahemaical and compuer model of he process of 1/f. Bak e al (1987) referred o heir model as self-organized criicaliy (SOC) because of heir discovery of he sponaneous emergence of minimal sabiliy wih spaial scaling ha leads o a 1/f power law for emporal flucuaions. Many sudies have subsequenly replicaed and exended he mahemaics and physics of SOC models (Bak, 1996). An invarian feaure of SOC is he presence of acivaion-deacivaion processes and dissipaion of energy which are fundamenal o he 1/f disribuion (Riuos and Zhang, 1999; Davidson and Schuser, 2000). The inerplay of he dissipaion and supply of energy from a source deermines he ampliude and phase of all self-susained nonlinear oscillaors and acivaion-deacivaion and dissipaion are fundamenal characerisics of neurons as well as mos biological oscillaors (Winfree, 1980; Pikovsky e al (2003).The link of SOC models o EEG requires a minimum of hree facors: 1- measuremen of an acivaion-deacivaion process (e.g., rapid phase shif followed by sabiliy), 2- an approximae 1/f disribuion in a log-log plo and, 3- spaial scaling and emporal flucuaions where here are long-range spaial correlaions (Bak e al, 1987; 1988; Bak, 1996; Davidson and Schuser, 2000). The analyses of Freeman e al (2003; 2006) and Blacksone and Williams (2004) demonsraed ha he fine emporal srucure of he EEG is characerized by 1/f disribuions wih periods of unsable phase dynamics followed by periods of phase sabiliy. These sudies and ohers (Breakspear and Terry, 2002a; 2002b) indicae ha he human EEG fundamenally exhibis characerisics of self-organizaional criicaliy, where sudden phase shifs are unsable phase dynamics and phase locking or phase synchrony is phase sabiliy. Furhermore, hese sudies

6 Thacher e al 6 demonsraed ha he erms phase rese and SOC when coupled wih he 1/f disribuion and acivaion-deacivaion of self-susaining nonlinear oscillaors are synonymous erms in complex sysems (Bak e al, 1987; 1988; Bak, 1996; Rios and Zang, 1999). Currenly here are no sudies of he early childhood developmen of EEG phase rese. Nikulin and Brismar (2005) sudied EEG age dependence of 1/f α disribuions in 96 adul subjecs. Ineresing age and gender correlaions were presened, however, phase shif duraion and phase locking were no separaely analyzed. Developmenal changes in he number of phase reses per second and he duraion of phase shifs and he lengh of phase locking are currenly unknown and such informaion may be imporan in undersanding he developmen of human neural dynamics. Therefore, he purpose of he presen sudy is o invesigae he developmen of human EEG phase rese from infancy o 16 years of age. The null hypoheses o be esed are: 1- here are no 1/f disribuions of EEG phase rese, 2- here are no lef and righ hemispheric differences in he developmen of phase rese; 3- here are no differences in phase rese as a funcion of anerior-o-poserior vs. poserior-o-anerior direcion, 4- here are no differences in phase rese as a funcion of iner-elecrode disance, 5- here are no changes in phase rese as a funcion of age. 2.0 Mehods 2.1 Subjecs A oal of 458 subjecs ranging in age from 2 monhs o years (males = 257) were included in his sudy. The subjecs in he sudy were recruied using newspaper adverisemens in rural and urban Maryland (Thacher e al, 1987; 2003; 2007). The inclusion/exclusion crieria were no hisory of neurological disorders such as epilepsy, head injuries and repored normal developmen and successful school performance. None of he subjecs had aken medicaion of any kind a leas 24 hours before esing. All of he subjecs were wihin he normal range of inelligence as measured by he WISC-R and were performing a grade level in reading, spelling and arihmeic as measured by he WRAT and none were classified as learning disabled nor were any of he school aged children in special educaion classes. All subjecs 2 years of age were given an eigh-iem laeraliy es consising of hree asks o deermine eye dominance, wo asks o deermine foo dominance, and hree asks o deermine hand dominance. Scores ranged from 8 (represening srong sinisral preference or lef handedness), o +8 (represening srong dexral preference or righ handedness). Dexral dominan children were defined as having a laeraliy score of 2 and sinisral dominan children were defined as having a laeraliy score of

7 Thacher e al 7-2. Only 9% of he subjecs had laeraliy scores - 2 and 87% of he subjecs had laeraliy scores 2 and hus he majoriy of subjecs were righ side dominan. 2.2 EEG Recording Power specral analyses were performed on 58 seconds o 2 minue 17 second segmens of EEG recorded during resing eyes closed condiion. The EEG was recorded from 19 scalp locaions based on he Inernaional 10/20 sysem of elecrode placemen, using linked ears as a reference. The average reference and a Laplacian reference were no used because hese reference mehods involve mixing he ampliude and phase from differen scalp locaions resuling in phase and coherence disorions as shown by Rappelsberger (1989), Kamiński e al (1997) and Essl and Rappelsberger (1998). Eye movemen elecrodes were applied o monior arifac and all EEG records were visually inspeced and manually edied o remove any visible arifac. Each EEG record was ploed and visually examined and spli-half reliabiliy and es re-es reliabiliy measures of he arifaced daa were compued using he Neuroguide sofware program (NeuroGuide, v2.3.8). Spli-half reliabiliy ess were conduced on he edied EEG segmens and only records wih > 90% reliabiliy were enered ino he specral analyses. The amplifier bandwidhs were nominally 1.0 o 30 Hz, he oupus being 3 db down a hese frequencies. The EEG was digiized a 100 Hz and up-sampled o 128 Hz and hen specral analyzed using complex demodulaion (Granger and Haanaka, 1964; Ones and Enochson, 1978) (see secion 2.3). EEG phase differences and phase rese merics was compued in he alpha frequency band ( Hz) for anerior-o-poserior elecrodes Fp1/2-F3/4; Fp1/2-C3/4; Fp1/2-P3/4 and Fp1/2-O1/2 and for poserior-o-anerior elecrodes O1/2-P3/4; O1/2-C3/4; O1/2-F3/4 and O1/2- Fp1/2. This recording arrangemen provides a sable marix and es of spaial homogeneiy by using five equally spaced elecrodes per hemisphere wih increasing disance beween elecrodes in he anerior-o-poserior and poserior-o-anerior direcions. According o volume conducion here is a homogeneous decline of volage as a funcion of disance from any source a near zero phase delay. The five equally spaced elecrode locaions is a direc es of volume conducion versus corical conneciviy (Thacher e al, 1986; 1998). Facors used in he mulivariae analysis of variance were: 1- Hemisphere, 2- Direcion, 3- Iner-elecrode disance and 4- Age. The analyses of differen frequency bands demonsraed ha he alpha frequency band exhibis he sronges developmenal rends and herefore his sudy will focus exclusively on he 8 13 Hz alpha frequency band (Niedermeyer and Lopes da Silva, 1994).

8 Thacher e al 8 Fig. 1 Experimenal design. Lef head diagram shows he locaion of elecrodes for he compuaion of coherence and phase differences in he anerior-o-poserior direcion. Righ head diagram shows he locaion of elecrodes in he poserior-o-anerior direcion. Local disances (6 cm) are in adjacen elecrode combinaions (O1/2-P3/4 and Fp1/2-F3/4). Longes disance (24 cm) elecrode combinaions are Fp1/2-O1/ Complex Demodulaion and Join-Time-Frequency-Analysis Complex demodulaion was used in a join-ime-frequency-analysis (JTFA) o compue insananeous coherence and phase-differences (Granger and Haanaka, 1964; Ones and Enochson, 1978; Bloomfield, 2000). This mehod is an analyic linear shif-invarian ransform ha firs muliplies a ime series by he complex funcion of a sine and cosine a a specific cener frequency (Cener frequency = 10.0 Hz) followed by a low pass filer (6 h order low-pass Buerworh, bandwidh = 2.0 Hz) which removes all bu very low frequencies (shifs frequency o 0) and ransforms he ime series ino insananeous ampliude and phase and an insananeous specrum (Bloomfield, 2000). We place quoaions around he erm insananeous o emphasize ha, as wih he Hilber ransform, here is always a rade-off beween ime resoluion and frequency resoluion. The broader he band widh he higher he

9 Thacher e al 9 ime resoluion bu he lower he frequency resoluion and vice versa. Mahemaically, complex demodulaion is defined as an analyic ransform (Z ransform) ha involves he muliplicaion of a discree ime series {x, = 1,..., n} by sine ω 0 and cos ω 0 giving ' = x sinω (1) x 0 and x '' = x cosω (2) 0 and hen apply a low pass filer F o produce he insananeous ime series, Z and Z where he sine and cosine ime series are defined as: Z = F( x sinω 0) (3) Z = F( x cosω 0) (4) and 2 2 [( Z ) + ( Z ) ] 1 / 2 (5) 2 is an esimae of he insananeous ampliude of he frequency ω 0 a ime and Z Z 1 an (6) is an esimae of he insananeous phase a ime. A his sep he complex demodulaion ransform is he same as he Hilber ransform (Pikovsky e al, 2003, p. 362; Oppenheim and Schaefer, 1975). The insananeous cross-specrum is compued when here are wo ime series {y, = 1,.

10 Thacher e al 10.., n} and {y, = 1,..., n} and if F [ ] is a filer passing only frequencies near zero, hen, as R = F y sinω + F y cosω = F y e is he esimae of he iω above [ ] [ ] [ ] 2 0 ampliude of frequency ω 0 a ime and phase of frequency ω 0 a ime and herefore, F iω0 iϕ [ y e ] = R e 0 0 [ y ] sinω 0 [ y cosω ] 1 F ϕ = an is an esimae of he F 0, (7) and likewise, F iω iϕ [ y e ] = R e 0 (8) The insananeous cross-specrum is i [ y e ] F[ y e ] ω 0 i ω 0 V = F [ ϕ ϕ ] i = R Re (9) and he insananeous coherence is R V 2 2 R 1 (10) The insananeous phase-difference is ϕ ϕ. Tha is, he insananeous phase difference is compued by esimaing he insananeous phase for each ime series separaely and hen aking he difference. Insananeous phase difference is also he arcangen of he imaginary par of V divided by he real par (or he insananeous quadspecrum divided by he insananeous cospecrum) a each ime poin. 2.4 Phase Sraighening We used he phase sraighening mehod of Ones and Enochson (1978) o remove he phase angle disconinuiy, i.e., where 0 and 360 are a opposie ends while in he circular

11 Thacher e al 11 disribuion 0 0 = The Ones and Enochson (1978) procedure involves idenifying he poins in ime when phase jumps from o and hen adding or subracing depending on he direcion of sign change. For example, θ = (180 ε) 0 + (180 ε) 0 = ε which is he same as 2ε since -(180 ε) 0 = ε. This procedure resuls in phase being a smooh funcion of ime and removes he disconinuiies due o he arcangen funcion. We found ha absolue phase differences wihou phase sraighening gave similar resuls o he sraighened phase differences. This is because he vas majoriy of EEG phase relaionships are less han ± However, phase sraighening is imporan when compuing he firs and second derivaives of he ime series of phase differences because he disconinuiy beween o can produce arifacs. Accordingly, all of he derivaives and phase rese measures in his paper were compued afer phase sraighening Compuaion of he 1 s and 2nd Derivaives of he Time Series of Phase Differences The firs derivaive of he ime series of phase-differences beween all pair wise combinaions of wo channels was compued in order o deec advancemens and reducions of phase-differences. The Savizgy-Golay procedure was used o compue he firs derivaives of he ime series of insananeous phase differences using a window lengh of 3 ime poins and he polynomial degree of 2 (Savizgy-Golay, 1964; Press e al, 1994). The unis of he 1 s derivaive are in degrees/poin which was normalized o degrees per cenisecond (i.e., degrees/cs = degrees/100 msec). The second derivaive was compued using a window lengh of 5 ime poins and a polynomial degree of 3 and he unis are degrees per ceniseconds squared (i.e., degrees/cs 2 = degrees/100 msec. 2 ). 2.6 Calculaion of Phase Rese The ime series of 1 s derivaives of he phase difference from any pair of elecrodes was firs recified o he absolue value of he 1 s derivaive (see fig. 2). The sign or direcion of a phase shif is arbirary since wo oscillaing evens may sponaneously adjus phase wih no saring poin (Pikovsky e al, 2003: Tass, 2007). The onse of a phase shif was defined as a significan absolue firs derivaive of he ime series of phase differences beween wo channels, i.e., d( ϕ ϕ ) /d > 0, crierion bounds = 5 0. Phase sabiliy or phase locking is defined as ha period of ime afer a phase shif where here is a sable near zero firs derivaive of he insananeous phase differences or d( ϕ ϕ )/d 0. The crieria for a significan 1 s derivaive is imporan and in he presen sudy a hreshold crieria of 5 0 was seleced because i was > 3

12 Thacher e al 12 sandard deviaions where he mean phase shif ranged from 25 deg/cs o 45 deg/cs. Changing he hreshold o higher values was no significan, however, eliminaing he hreshold resuled in greaer noise and herefore he crieria of 5 0 is an adequae crieria. As poined ou by Blacksone and Williams (2004) visual inspecion of he daa is he bes mehod for selecing an arbirary hreshold value and he hreshold value iself is less imporan han keeping he hreshold consan for all subjecs and all condiions. Figure wo illusraes he concep of phase rese. Phase differences over ime on he uni circle are measured by he lengh of he uni vecor r. Coherence is a measure of phase consisency or phase clusering on he uni circle as measured by he lengh of he uni vecor r. The illusraion in figure 2 shows ha he resulan vecor r 1 = r 2 and herefore coherence when averaged over ime 1.0 even hough here is a brief phase shif. As he number of phase shifs per uni ime increases hen coherence declines because coherence is direcly relaed o he average amoun of phase locking or phase synchrony (Benda and Piersol, 1980). Fig. 2 Illusraions of phase rese. Lef is he uni circle in which here is a clusering of phase angles and hus high coherence as measured by he lengh of he uni vecor r. The op row is an example of phase reducion and he op righ is a ime series of he

13 Thacher e al 13 approximaed 1 s derivaive of he insananeous phase differences for he ime series 1, 2, 3, 4 a mean phase angle = 45 0 and 5, 6, 7, 8 a mean phase angle = The vecor r1 = 45 0 occurs firs in ime and he vecor r2 = 10 0 and (see boom lef) occurs laer in ime. Phase rese is defined by a sudden change in phase difference followed by a period of phase locking. The onse of Phase Rese is beween ime poin 4 and 5 where he 1 s derivaive is a maximum. The 1 s derivaive near zero is when here is phase locking or phase locking and lile change in phase difference over ime. The boom row is an example of phase advancemen and he boom righ is he 1 s derivaive ime series. The sign or direcion of phase rese in a pair of EEG elecrodes is arbirary since here is no absolue saring poin and phase shifs are ofen sponaneous and no driven by exernal evens, i.e., self-organizing criicaliy. When he absolue 1 s derivaive 0 hen wo oscillaing evens are in phase locking and represen a sable sae independen of he direcion of phase shif. Figure 3 shows he ime markers and definiions used in his sudy. As menioned above he peak of he absolue 1 s derivaive was used in he deecion of he onse and offse of a phase shif and he second derivaive was used o deec he inflecion poin which defines he fullwidh-half-maximum (FWHM) and phase shif duraion. As seen in Figure 3, Phase Rese (PR) is composed of wo evens: 1- a phase shif of a finie duraion (SD) and 2, followed by an exended period of phase locking as measured by he phase locking inerval (LI) and PR = SD + SI. Phase Shif duraion (SD) is he inerval of ime from he onse of phase shif o he erminaion of phase shif where he erminaion is defined by wo condiions: 1- a peak in he 1 s derivaive (i.e., 1 s derivaive changes sign from posiive o zero o negaive) and, 2- a peak in he 2 nd derivaive or inflecion on he declining side of he ime series of firs derivaives. The peak of he 2 nd derivaive marked he end of he phase shif period. Phase shif duraion is he difference in ime beween phase shif onse and phase shif offse or SD() = S() onse S() offse. Phase locking inerval (LI) was defined as he inerval of ime beween he end of a significan phase shif (i.e., peak of he 2 nd derivaive) and he beginning of a subsequen significan phase shif, i.e., marked by he peak of he 2 nd derivaive and he presence of a peak in he 1 s derivaive or SI() = S() offse S() onse See figure 3 is a diagram of phase shif duraion and phase locking inervals. In summary, wo measures of phase dynamics were compued: 1- Phase shif duraion (msec) (SD) and, 2- Phase locking inerval (msec) (LI). Figure hree illusraes he phase rese merics and figure four shows an example of he compuaion of phase rese merics in a single subjec.

14 Thacher e al 14 Fig. 3- Diagram of phase rese merics. Phase shif (PS) onse was defined a he ime poin when a significan 1 s derivaive occurred ( 5 0 /cenisecond) followed by a peak in he 1 s derivaive, phase shif duraion (SD) was defined as he ime from onse of he phase shif defined by he posiive peak of he 2 nd derivaive o he offse of he phase shif defined by he negaive peak of he 2 nd derivaive. The phase locking inerval (LI) was defined as he inerval of ime beween he onse of a phase shif and he onse of a subsequen phase shif. Phase rese inerval (PRI) is composed of wo evens: 1- a phase shif and 2- a period of locking following he phase shif where he 1 s derivaive 0 or PRI = SD + LI.

15 Thacher e al 15 Fig. 4- Example from one subjec. Top are he EEG phase differences beween Fp1-F3, Fp1-C3, Fp1-P3 and Fp1-O1 in degrees. Boom are he 1 s derivaives of he phase differences in he op races in degrees/ceniseconds. A 1 s derivaive 5 0 /cs marked he onse of a phase shif and an inerval of ime following he phase shif where he 1 s derivaive 0 defined he phase locking inerval as described in figure Sliding Averages In order o increase emporal resoluion one year sliding averages of EEG phase differences were compued. The procedure involved compuing means and sandard deviaions for phase locking inervals and phase shif inervals over a one year period, e.g., birh o 1 year, hen compuing means and sandard deviaions from 0.25 years o 1.25 years, hen a mean and sandard deviaion for he ages from 0.5 years o 1.5 years, ec. This resuled in a 75% overlap of subjecs per mean wih oally unique subjecs on a one year inerval. The sliding average procedure produced 64 equally spaced mean values wih a 0.25 year resoluion. Table I shows he age range per bin, he mean ages per bin and he number of subjecs per age group from mean age of 0.4 years o 16.2 years. Relaive small Ns were presen from 0.4 o 1.4 years of age and larger sample sizes (max N = 50) were presen beyond one year of age. In spie of he relaively

16 Thacher e al 16 small sample sizes a 0.4 years o 1.4 years he mean values of phase shif and phase rese were well behaved (see figs. 7 & 8). The overall average number of subjecs per age bin = Table I AGE BINs MEAN AGE N-SIZE AGE BINs MEAN AGE N-SIZE Table I Age ranges, mean ages and he number of subjecs per age bin using sliding average compuaion. 2.8 Specral Analyses of 1/f disribuion To evaluae possible 1/f specral disribuions he fas Fourier ransform (FFT) of he ime series of he 1 s derivaive of phase differences were compued for individual subjecs using

17 Thacher e al 17 edied EEG daa. The FFT epoch lengh was 2 seconds and he sample rae was 128 Hz. Compuaions were firs conduced on he enire edied EEG record (58 sec o 2 min 17 sec) and hen separae selecions of periods of phase shif and phase locking were subjeced o separae FFT analyses in order o deermine if he specra were differen beween he phase locking vs. he phase shif periods in he EEG record. Tess of he 1/f disribuion involved ploing he log 10 ransforms of frequency and magniude and hen a linear regression was used o deermine he slope (α coefficien) and inercep of he linear fi. 2.9 Specral Analyses of Developmenal Ulraslow Oscillaions During he Lifespan As explained previously, he sliding averages produced 64 equally spaced mean values of phase shif duraion and phase locking inervals in each elecrode pairing (a 3 monh or 0.25 year resoluion) from 0.4 years o 16.2 years. This resuled in a developmenal ime series of equally spaced mean ages for phase shif duraion and phase locking inervals which were hen specrally analyzed. The Fas Fourier Transform (FFT) was used o analyze he frequency specrum of he developmenal rajecories of phase shif and phase locking. Derending was used prior o he FFT o remove he low frequency developmenal rends in order o analyze he frequency and power of rhyhmic changes during he developmenal period. The number of ime poins = 64, and he epoch lengh or Lifespan = 16.2 years. This produced a frequency resoluion of 6 monhs and a maximum frequency of 32 cycles per epoch. The unis of frequency were cycles per lifespan (cpl) and wavelengh (λ) = 16.2/cpl. The unis of frequency are cpl. The magniude of he specrum ploed on he y-axis are milliseconds/cycle/lifespan or msec/cpl. 3.0 Resuls 3.1 1/f Phase Rese Disribuions As discussed in he inroducion, he human EEG is ofen characerized by 1/f disribuions which are revealed by log-log plos of he power specrum (Freeman e al, 2003; 2006; Buzaki, 2006). All of he subjecs exhibied 1/f disribuions of he 1 s derivaive of phase differences. Figure 5 shows examples of linear regression fis of he log-log plos of he

18 Thacher e al 18 Fig. 5- Examples of he log-log plos of he FFT of he 1 s derivaives of EEG phase differences in he anerior-o-poserior direcion. Solid line is he FFT specral values and he doed line is he linear regression fis o he specral values. Table I shows he slope or α exponens in he 1/f α equaion. power specrum of he 1 s derivaive of phase differences in four subjecs in he anerior-oposerior direcion. Very similar specra were obained independen of direcion of hemisphere. Table II shows he slope or α values, he inercep and he regression correlaion coefficiens which all yielded 1/f α specral disribuions wih a range from 0.86 o 0.54 and he average α = To furher invesigae he naure of he 1/f disribuion, sub-componen analyses were conduced by selecing he 1 s derivaive of phase difference during phase rese by separaely selecing he phase shif periods and phase locking periods and hen specrally analyzing he wo differen daa selecions. Frequency and magniude were log 10 ransformed and linear regression fis were conduced in order o deermine he slopes of he specra. The resuls of he sub-componen analyses are shown in figure 6 and Table III. In all insances and in all subjecs

19 Thacher e al 19 Table II The resuls of he linear regression fi o he log-log plo of he power specra in Figure 6. The sign of he correlaion and α is in accordance wih 1/f α. he slope of he linear fi was > 1.0 for phase locking and < 1.0 for phase shif periods. The average slope or α coefficien was close o 1.0 wih a mean = As seen in Table III, here were saisically significan differences in slope (i.e., α) beween periods of phase shif vs. periods of phase locking where he slope was always seeper for phase locking han phase shif. Alhough all of he regression fis were saisically significan, noneheless, he linear fi for phase locking accouned for less variance han for phase shif duraion which is likely due o he exponenial shape of he phase locking specral disribuion.

20 Thacher e al 20 Fig. 6- Log-log plos of he Fourier analyses of he 1 s derivaive of phase differences during periods of phase locking versus periods of phase shif in he four differen subjecs in figure 6. Solid line is he FFT specral values and he doed line is he linear regression fis o he specral values. A 1/f α disribuion is presen in all insances in which he slope coefficiens were higher for he phase locking periods in comparison o he phase shif periods. Table II shows he differences in slopes and he1/f alpha coefficiens for phase shifing vs. phase locking as well as he average α 1. Table III shows he alpha slope values and age regression correlaions and -ess beween he specral disribuions for phase shif vs. phase locking inervals for he four sample subjecs in figure 6. The α values for phase shif were saisically significanly smaller han he α values for phase locking and he average α value was close o 1.0 which indicaes ha decomposiion of he log-log specral disribuion ino sub-componens of phase shif vs. phase locking is useful in order o reveal more of he underlying EEG dynamics.

21 Thacher e al 21 Table III- Summary of he log-log regression fis and esimaes of he slope (i.e., alpha) for he same subjecs as in figs. 6 & 7. The alpha values in he 1/f α disribuions are shown for he phase locking vs. phase shif periods. The average α values were near o 1.0. The sub-componen analyses of phase shif vs. phase locking reveals ineresing differences in he slope or alpha of he 1/f disribuion for phase shif vs. phase locking. 3.2 Developmen of Phase Shif Duraion Figure 7 shows he mean duraion of phase shif in he alpha frequency band from 0.4 o 16.2 years of age. The op row are mean phase shif duraion values in he anerior-o-poserior direcion (see fig. 1) and he boom row are he poserior-o-anerior elecrode combinaions. The lef column are he mean phase rese duraions for he lef hemisphere and

22 Thacher e al 22 Fig. 7- Mean EEG phase shif duraion from 0.44 years of age o years of age. Top row are from he anerior-o-poserior elecrode combinaions and boom row are from he poserior-o-anerior elecrode combinaions (see Fig. 1). The lef column is from he lef hemisphere and he righ column is from he righ hemisphere. I can be seen ha phase shif duraion increases in mos elecrode combinaions bu decreases in he shor iner-elecrode disance (6 cm) in he anerior-o-poserior direcion. he righ column are he righ hemisphere values. I can be seen ha here were oscillaions and sudden changes in he mean duraion of phase shif and here was a seady increase in he mean duraion of phase shif as a funcion of age in he long iner-elecrode disances (18 cm & 24 cm) and reduced phase shif duraion in he shor iner-elecrode disance (6 cm) in anerior-poserior direcions. In he poserior-anerior direcion, an increase in phase shif duraion was presen as a funcion of age in all iner-elecrode disances, alhough he long iner-elecrode disances (18 cm & 24 cm) exhibied a more pronounced increase in phase duraion wih age han he shor inerelecrode disances (6 cm). Table IV shows he resuls of a linear fi of he mean duraion of phase shif as a funcion of age for all elecrode pairings. I can be seen in Table IV ha here were saisically significan negaive

23 Thacher e al 23 slopes in he shor iner-elecrode disance (6 cm) in he anerior-poserior direcion and posiive slopes in all oher insances. Saisically significan age regressions were presen for all of he developmenal rajecories. Table IV Age Regression of Phase Shif Duraion LEFT Anerior - Poserior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001 RIGHT Anerior - Poserior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001 LEFT Poserior - Anerior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001 RIGHT Poserior - Anerior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001

24 Thacher e al 24 Table IV Linear regression saisics for mean phase shif duraion from 0.44 years o years of age. The op wo rows are from he anerior-o-poserior direcion and he boom wo rows are from he poserior-o-anerior direcion. Shor Iner-elecrode disance (6 cm) in he anerior-o-poserior direcion exhibied a negaive slope as a funcion of age while all oher iner-elecrode disances and direcion exhibied posiive slopes as a funcion of age. The y-inercep, regression correlaion and saisical significance are also shown. Mulivariae analyses of variance (MANOVA) were conduced wih he facors being direcion (anerior-o-poserior vs. poserior-o-anerior), lef hemisphere vs. righ hemisphere and disance (6 cm, 12 cm, 18 cm & 24 cm). No significan lef vs. righ hemisphere affec was presen (F= , P < 0.628). However, here was a saisically significan direcion affec (F = , P <.0001) wih a saisically significan Bonferroni pos hoc es (P <.0001). There was also a saically significan disance affec (F = , P <.0001) wih saisically significan Bonferroni pos hoc ess (P >,0001) for all iner-elecrode disances excep for 24 cm 18 cm (P < 0.68). 3.3 Developmen of he Phase Locking Inerval Figure 8 shows he mean phase locking inerval in he alpha frequency band from 0.4 o 16.2 years of age. The op row are mean phase locking inerval values in he anerior-oposerior direcion (see fig. 1) and he boom row are he poserior-o-anerior elecrode combinaions. The lef column are he mean phase locking inervals for he lef hemisphere and

25 Thacher e al 25 Fig. 8- Mean EEG phase locking inervals from 0.44 years of age o years of age. Top row are from he anerior-o-poserior elecrode combinaions and boom row are from he poserior-o-anerior elecrode combinaions (see Fig. 1). The lef column is from he lef hemisphere and he righ column is from he righ hemisphere. Growh spurs and oscillaions during developmen are seen. Also, i can be seen ha phase locking inervals increase as a funcion of age in all elecrode combinaions. he righ column are he righ hemisphere values. I can be seen ha here were oscillaions and sudden changes in he mean phase locking inervals and here was a seady increase in he mean phase locking inerval as a funcion of age in he shor iner-elecrode disance (6 cm) wih less increased phase locking inervals in he long iner-elecrode disances. Sudden increases in he mean phase locking inerval were presen n all elecrode combinaions a ages 9 and 14 years, especially in he shor iner-elecrode disances (6 cm) and in he poserior-o-anerior direcion. Table V shows he resuls of a linear fi of he mean phase locking inerval as a funcion of age for all elecrode pairings. I can be seen in Table V ha here were saisically significan posiive slopes in all insances. The shor iner-elecrode disance (6 cm) exhibied a seeper developmenal slope in he poserior-o-anerior direcion han in he anerior-o-poserior direcion.

26 Thacher e al 26 Table V Age Regression of Phase Locking Inerval LEFT Anerior - Poserior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001 RIGHT Anerior - Poserior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001 LEFT Poserior - Anerior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001 RIGHT Poserior - Anerior 6cm 12cm 18cm 24cm SLOPE INTERCEPT CORRELATION SIGNIFICANT P<.0001 P<.0001 P<.0001 P<.0001 Table V Linear regression saisics for mean phase locking inervals from 0.44 years o years of age. The op wo rows are from he anerior-o-poserior direcion and he boom wo rows are from he Poserior-o-anerior direcion. The y-inercep, regression correlaion and saisical significance are also shown. Mulivariae analyses of variance (MANOVA) were conduced wih he facors being

27 Thacher e al 27 direcion (anerior-o-poserior vs. poserior-o-anerior), lef hemisphere vs. righ hemisphere and disance (6 cm, 12 cm, 18 cm & 24 cm). No significan lef vs. righ hemisphere affec was presen (F= 3.033, P < 0.082). However, here was a saisically significan direcion affec (F = 82.02, P <.0001) wih a saisically significan Bonferroni pos hoc es (P <.0001). There was also a saically significan disance affec (F = , P <.0001) wih saisically significan Bonferroni pos hoc ess ( P >,0001) for all iner-elecrode disances excep for 24 cm 12 cm (P < 0.271) and 24 cm 18 cm (P < 0.783) Relaions Beween EEG Phase Rese and Coherence Coherence is a measure of phase sabiliy and one would expec a posiive correlaion beween he duraion of phase locking and coherence. We esed his hypohesis using a Pearson produc correlaion coefficien of he developmenal ime series of coherence and phase shif duraion and phase locking inerval in he shor iner-elecrode disances (6 cm). Table VI shows he average correlaion of he shor iner-elecrode disance measures in which here was a negaive correlaion beween coherence and phase shif duraion (i.e., inverse relaionship o unsable phase dynamics ) and a posiive correlaion beween coherence and phase locking inervals (i.e., direc relaionship o Sabiliy ). The hypohesis of a posiive relaionship beween coherence and phase locking was confirmed. As expeced here was an inverse relaionship beween phase shif duraion ( unsable phase dynamics ) and he phase locking inerval ( sabiliy ), however, he correlaions were relaively small indicaing ha he majoriy of variance is unaccouned for when correlaing phase shif duraions wih phase locking inervals. Table VI Correlaions beween Coherence and Phase Shif Duraion and Phase Locking Inerval Phase Shif Phase Locking Coherence Phase Shif Phase Locking Table VI- Average correlaions beween coherence and phase shif duraion ( Unsable phase dynamics ) and phase locking ( Sabiliy ) a he shor iner-elecrode disance (6 cm). There was a negaive correlaion beween coherence and Unsable phase dynamics and a posiive correlaion beween coherence and sabiliy.

28 Thacher e al 28 To furher explore he naure of phase shif and phase duraion he emporal boundaries and frequency disribuions were sudied. Figure 9 shows he frequency disribuion of phase shif duraions (Top) and he phase locking inervals (Boom) for shor and long disance connecions (average of lef and righ hemisphere) in 215 subjecs beween 10 and years of age. Phase shif duraion exhibied emporal boundaries or window lenghs wih no duraions

29 Thacher e al 29 Fig. 9 Frequency hisograms of phase shif duraion (Top) and phase locking inervals (Boom) from 215 subjecs beween 10 and years of age. 5 cm anerior-o-poserior (AP) iner-elecrode disance, 6 cm iner-elecrode disance for poserior-o-anerior direcion (PA) and he long (24 cm) iner-elecrode disance which is he same for AP and PA (see fig. 1). Lef and righ hemispheres were averaged ogeher. The y-axis is he number of subjecs and he x-axis is msec.

30 Thacher e al 30 less han 25 msec and no duraions greaer han 100 msec. Local (6 cm) fronal disances exhibied he mos peaked disribuion a 45 msec duraion and he long disance (24 cm) connecions were shifed in peak duraion by approx msec. The phase locking inerval exhibied emporal boundaries or window lenghs wih no duraions less han 150 msec and 99% of he duraions less han 900 msec. The frequency disribuions as a funcion of disance were similar alhough he long disance (24 cm) connecions were mos peaked a 200 msec and exhibied an approximae 50 msec. shif in peak duraion in comparison o he local connecions. Phase locking (sable dynamic) was on he average shorer and phase shif duraion (unsable dynamic) was longer in he long disan connecion sysem. 3.5 Developmenal Oscillaions Examinaion of figures 7 and 8 shows ulra-slow oscillaions wih iner-peak inervals of approximaely 2 o 3 years. Specral analyses of he developmenal ime series of mean phase shif duraion from 0.4 years o 16.2 years for he 6 cm and 24 cm iner-elecrode disances are shown in figure 10. The op row are he fronal-o-poserior elecrode combinaions and he boom row are he occipial-o-anerior combinaions. The lef column are he lef hemisphere mean FFT values and he righ column are he righ hemisphere values (see Fig. 1). In general here was greaer developmenal specral energy in he shor iner-elecrode disance (6 cm) in comparison o he long iner-elecrode disance (24 cm). Mos of he developmenal specral energy was in he ulraslow frequency range of 1 cycle per lifespan (i.e., a wavelengh of 16 years) o approximaely 12 cycles per lifespan (i.e., a wavelengh of 1.3 years). The highes peak frequency was 31 cycles per lifespan (i.e., a wavelengh of 0.5 years or 6 monhs).

31 Thacher e al 31 Fig. 10 Fourier specral analyses of he developmenal rajecories of phase shif duraion from 0.44 years o years of age in shor (6 cm) (dashed line) and long (24 cm) (solid line) iner-elecrode disances in he anerior-o-poserior and poserior-o-anerior direcions. Magniude is on he y-axis and frequency on he x-axis. Disan iner-elecrodes exhibied greaer power in he anerior-o-poserior direcion while local connecions exhibied he greaer power in he poserior-o-anerior direcion. Specral analyses of he developmenal ime series of mean phase locking inervals from 0.4 years o 16.2 years for he 6 cm and 24 cm iner-elecrode disances are shown in figure 11 The op row of figure 11are he anerior-o-poserior elecrode combinaions and he boom row are he poserior-o-anerior combinaions. The lef column are he lef hemisphere FFT values and he righ column are he righ hemisphere values (see Fig. 1). Similar o phase shif duraion, phase locking exhibied mos of he specral energy in he ulraslow frequency range of 1 cycle per lifespan (i.e., a wavelengh of 16 years) o approximaely 20 cycles per lifespan (i.e., a wavelengh of 0.8 years). Similar o phase shif duraion, phase locking in he shor inerelecrode disance (6 cm) was greaer han he long disance (24 cm) in he poserior-o-anerior direcion.

32 Thacher e al 32 Fig. 11 Fourier specral analyses of he developmen of phase locking from 0.44 years o years of age in shor (6 cm) (dashed line) and long (24 cm) (solid line) iner-elecrode disances in he anerior-oposerior and poserior-o-anerior direcions. Magniude is on he y-axis and frequency on he x-axis. The greaes specral energy was in he shor disance iner-elecrodes (6 cm) in he poserior-o-anerior direcion and repeiive 8 year and 4 year and 2 year cycles. See Table VI for he lamda values in years. Tables VII and VIII are summaries of he cycles per lifespan (cpl) and he wavelengh (16 yrs/cpl) of he specral peaks in he FFT analyses of he mean duraion of phase shif and mean phase locking developmenal rajecories from 0.4 years o 16.2 years shown in figures 9 & 10 respecively. Table VII shows he FFT peak values for phase shif duraion over he lifespan in he fronal-o-poserior direcion (Top) and for he poserior-o-anerior direcion (Boom). I can be seen ha shor (6 cm) and disan (24 cm) exhibied differen specra for phase shif duraion and differen specra as a funcion of direcion. In general here are more specral peaks and greaer power in he long disan iner-elecrode connecions (24 cm) in he anerior-oposerior direcion while here are more specral peaks in he 6 cm disance in he poserior-oanerior direcion.

33 Thacher e al 33 Table VI Summary of specral peaks of he phase shif duraion in he anerior-o-poserior direcion (op) and he poserior-o-anerior direcion (boom) for 6 cm and 24 cm iner-elecrode disances. Lifespan = 16 years and cpl = cycles per lifespan and λ = wavelengh in years or lifespan/cpl. Table VIII are summaries of he cycles per lifespan (cpl) and he wavelengh (16 yrs/cpl) for he mean phase locking developmenal rajecories from 0.4 years o 16.2 years. The larges number of specral peaks was in he poserior-o-anerior direcion in he shor iner-elecrode disance (6 cm).

34 Thacher e al 34 Table VII Summary of specral peaks of phase locking inervals in he anerior-o-poserior direcion (op) and he poserior-o-anerior direcion (boom) for 6 cm and 24 cm iner-elecrode disances. Lifespan = 16 years and cpl = cycles per lifespan, and λ = wavelengh in years or lifespan/cpl. 4.0 Discussion This sudy exends he invesigaion of he spaial and emporal properies of EEG phase rese o a large populaion of subjecs from infancy o adolescence and focuses on brain developmen of phase rese in he alpha frequency band. An imporan finding is ha he average phase shif duraion (unsable dynamics) and he average phase locking inerval (sabiliy) were significanly correlaed wih age and exhibied differen mauraional rajecories, oscillaions and growh spurs in he anerior-oposerior vs. poserior-o-anerior direcions (see figs. 7 & 8). A second significan finding was ha he fronal phase shif duraion (unsable dynamics) increased as a funcion of age in disan connecions in conras o poserior local connecions (see fig. 7 & Table IV). The hird significan finding was ha phase sabiliy increased as a funcion of age in boh local and disan connecions (see fig. 8 and Table V). A fourh significan finding was he presence of 1/f α log-log disribuions near o 1 hus showing scale invarian flucuaions linked o he developmen of minimally sable saes such as human EEG