Heritability of Small-World Networks in the Brain: A Graph Theoretical Analysis of Resting-State EEG Functional Connectivity

Transcription

1 Human Bain Mapping 29: (2008) Heitability of Small-Wold Netwoks in the Bain: A Gaph Theoetical Analysis of Resting-State EEG Functional Connectivity Dik J. A. Smit, 1 * Conelis J. Stam, 2 Danielle Posthuma, 1 Doet I. Boomsma, 1 and Eco J. C. de Geus 1 1 Depatment of Biological Psychology, VU Univesity Amstedam, Amstedam, The Nethelands 2 Depatment of Clinical Neuophysiology, VU Univesity Medical Cente Abstact: Recent studies have shown that esting-state functional netwoks as studied with fmri, EEG, and MEG may be so-called small-wold netwoks. We investigated to what extent the chaacteistic featues of small-wold netwoks ae genetically detemined. To epesent functional connectivity between bain aeas, we measued esting EEG in 574 twins and thei siblings and calculated the synchonization likelihood between each pai of electodes. We applied a theshold to obtain a binay gaph fom which we calculated the clusteing coefficient C (descibing local inteconnectedness) and aveage path length L (descibing global inteconnectedness) fo each individual. Modeling of MZ and DZ twin and sibling esemblance indicated that acoss vaious fequency bands 46 89% of the individual diffeences in C and 37 62% of the individual diffeences in L ae heitable. It is asseted that C, L, and a smallwold oganization ae viable makes of genetic diffeences in bain oganization. Hum Bain Mapp 29: , VC 2007 Wiley-Liss, Inc. Key wods: EEG; esting state; synchonization likelihood; functional connectivity; heitability; twin study INTRODUCTION Contact gant sponso: Human Fonties of Science Pogam; Contact gant numbe: RG0154/1998-B; Contact gant sponso: Nethelands Oganization fo Scientific Reseach; Contact gant numbes: NWO/SPI , NWO/VIDI ; Contact gant sponso: Univesitai Stimuleings Fonds; Contact gant numbe: USF 96/22. *Coespondence to: D.J.A. Smit, Biologische Psychologie, van de Boechoststaat 1, 1081 BT Amstedam, The Nethelands. dja.smit@psy.vu.nl Received fo publication 4 Apil 2007; Revised 16 July 2007; Accepted 17 July 2007 DOI: /hbm Published online 6 Decembe 2007 in Wiley InteScience (www. intescience.wiley.com). Bain connectivity is likely to have evolved unde the constaints of optimized pocessing capacity while maintaining cost efficiency and esilience to loss of substate [Achad and Bullmoe, 2007; Bassett and Bullmoe, 2006]. To descibe such popeties of neual netwoks, ecent studies have applied gaph theoetical methods on data obtained with MRI, fmri, MEG, and EEG [e.g., Achad et al., 2006; Bassett and Bullmoe; 2006; Micheloyannis et al., 2006a,b; Ponten et al., 2007; Stam, 2004]. Gaph theoy descibes mathematical methods applied to epesentations of netwoks educed to thei essence: vetices (nodes) and edges (connections). In thei gound-beaking aticle, Watts and Stogatz [1998] calculated two paametes to descibe gaphs deived fom biological as well as nonbiological netwoks. The fist descibes the amount of local inteconnectedness o cliquishness called clusteing coefficient C. It takes a value between 0 and 1 indicating the popotion of neighboing vetices that ae inteconnected amongst each othe. That is, if a neighbo is defined as a vetex that is one step emoved, how many of the neighbos of one vetex ae not only connected with VC 2007 Wiley-Liss, Inc.

2 Heitability of Small-Wold Netwoks in the Bain Figue 1. (A) Aveage path length L fo X is the aveage numbe of steps fom node X to all othe nodes: L x 5 (1 þ 1 þ 1 þ 2 þ 3 þ 3)/6 $ 1.8 (B). Clusteing coefficient C fo X descibes popotional connectivity between the nodes neighboing node X: C x 5 2 out of 3 possible connections that vetex, but also with each othe. The second paamete descibes global inteconnectedness and is called the aveage path length L. It is a value simply indicating the aveage numbe of steps equied to go fom each vetex to all othes taking the shotest oute. Figue 1 shows a gaphical explanation of the calculation of C and L. Watts and Stogatz [1998] showed that C and L paametes epesent non-tivial aspects of connection pattens along the dimension anging fom highly odeed gaphs (lattices, egula netwoks) to fully andomized gaphs. Odeed gaphs ae chaacteized by high C and long L. Random gaphs have shot L and low C. By stating fom odeed gaphs and andomly econnecting single edges with a ewiing pobability P, Watts and Stogatz showed that while the aveage path length L dops quickly, clusteing coefficient C showed esilience against econnection. Figue 2 shows the development of C and L against econnection pobability fo a simulated 100-vetex gaph with degee K 5 8, whee K epesents the aveage numbe of edges pe vetex. Odeed gaphs, theefoe, equie only a few andom long distance connections to dastically shoten the path length while maintaining a high clusteing. Such efficient netwoks ae designated small-wold, efeing to the phenomenon that it takes supisingly few steps to contact any two people in the wold [Milgam, 1967] o to connect any acto to Kevin Bacon [wikipedia enty: six degee; of sepaation; Bassett and Bullmoe, 2006]. Many types of existing netwoks have been shown to possess small-wold featues, including powe gids, the wold wide web, and as indicated, human societies. In the ealm of neual netwoks, a small-wold topology has also been shown to exist in the neual netwok of C. elegans [Watts and Stogatz, 1998], in the bain anatomy in macaque and cat cotex [Hilgetag et al., 2000], and ecently also in connectivity deived fom cotical thickness measued in humans He et al., [in pess]. Besides stationay anatomical connectivity, howeve, the bain also shows nonstationay functional connectivity between bain aeas. Because of the high tempoal esolution, EEG and MEG ae paticulaly useful to study this kind of connectivity. Statistical intedependencies between EEG/MEG signals may seve as indices to these tempoay connections between bain aeas [o effective connectivity ; Aetsen et al., 1989]. Although coheence is the most widely used linea measue of connectivity of EEG/MEG pattens, nonlinea measues of coupling may be moe appopiate since bain activity is pehaps bette modeled as an ensemble of coupled nonstationay, nonlinea dynamical subsystems [Fiston, 2000; Peeda et al., 2005]. A elatively new measue that captues both the linea as well as nonlinea dependencies is synchonization likelihood (SL). With this measue, Stam and co-wokes [Montez et al., 2006; Stam and van Dijk, 2002; Stam et al., 2003] have found that both linea and nonlinea synchonization ae indeed pesent in nomal backgound EEG/MEG. In addition, they suggested clinical usefulness of SL by showing that synchonization inceased duing and slightly befoe epileptic seizues; also, alpha, beta, and gamma band SL was deceased in Alzheime s disease patients (fo eviews see: Stam, 2005, 2006). SL is based upon the concept of genealized synchonization as intoduced by Rulkov et al. [1995]. Genealized Figue 2. Development of C and L as a function of the pobability p of andomly ewiing of the edges of a 100 vetex odeed netwok with degee K 5 8. Both C p and L p ae scaled to a minimum of zeo (a fully andomized netwok) and a maximum of one (a fully odeed netwok). 1369

3 Smit et al. synchonization is said to exist between two dynamical systems X and Y if thee exists a continuous one-to-one function F such that the state of one of the systems (the esponse system) can be mapped onto the state of the othe system (the dive system): Y 5 F(X) [Ababanel et al., 1996; Kocaev and Palitz, 1996; Rulkov et al., 1995]. Intuitively, this means that genealized synchonization exists between two systems X and Y if the following holds: if X is in the same state at two diffeent times i and j, Y will also be in the same state at times i and j. By deiving the state of the system X fom one EEG/MEG signal at a cetain time point and the concuent state of system Y, we can find evidence fo connectivity between the bain aeas whose activity is eflected in the signals. In sum, SL has the advantage ove coheence as it will not show spuious connectivity between bandpass filteed white noise in which case SL will assume the fixed, pedefined value p ef, and is able to detect complex nonlinea coupling pattens. Detailed calculation pocedues ae pesented in the Methods section. The esult of calculating SL between all possible pais in a set of EEG/MEG signals can be intepeted as a geneal (linea and nonlinea) measue of connectivity stength in a functional netwok of bain aeas. By application of a theshold, a spasely connected gaph can be ceated that is suitable fo futhe gaph theoetical analysis as poposed by Watts and Stogatz. Both C and L calculated fom these gaphs can be intepeted as measues of local and global efficiency [Baahona and Pecoa, 2002; Lago- Fenandez et al., 2000; Latoa and Machioi, 2001; Masuda and Aihaa, 2004]. Shot L educes the time o effot needed to connect two vetices, allowing efficient infomation exchange between, in this case, moe distant bain egions. High C lowes the cost of building and maintaining localized netwoks and inceases eo toleance in the case of loss of connectivity [Achad and Bullmoe, 2007; Bassett and Bullmoe, 2006]. Theefoe, these paametes may eflect biologically impotant chaacteistics of the netwok. The main focus of the cuent aticle is to detemine whethe individual diffeences in the netwok popeties C and L, deived fom esting state EEG functional connectivity, have a biological basis by establishing them as heitable taits. Heitability will be assessed by compaing C and L scoes of MZ and DZ twins and thei singleton siblings, who, having vaying degees of genetic elatedness, povide infomation on the amount of vaiation that can be attibuted to genetic o envionmental souces of vaiation [Boomsma et al., 2002a; Falcone and MacKay, 1996; Fishe, 1918]. Since it has been suggested that a small-wold netwok achitectue may be optimal fo synchonizing neual activity between diffeent bain egions, it seems plausible to hypothesize that individual diffeences in netwok popeties C and L may be coelated with oveall pocessing capacity o cognitive pefomance. Indeed, using EEG measued duing a woking memoy task (the two-back task), highe educated subjects showed less of a smallwold phenomenon than subjects with less education with almost the same behavioal pefomance [Micheloyannis et al., 2006a]. We will theefoe investigate whethe the same elation can be found between small-woldness and measues of geneal cognitive ability in the context of esting state EEG functional connectivity. METHOD Subjects The sample fo this study was deived fom an ongoing twin-family study on cognition [Posthuma et al., 2001] in twins and family membes fom the Nethelands Twin Registy [Boomsma et al., 2002b]. Twins and siblings wee invited fo detailed psychophysiological study in the laboatoy. The EEG sample consisted of 760 subjects fom 309 families divided into two age cohots based on the age of the twins: a younge cohot (M yeas, SD 5 4.1) and a middle-aged cohot (M yeas, SD 5 7.2). Paticipating families consisted of one to seven siblings (including twins). On aveage, 2.50 paticipants pe family paticipated. Infomed consent was obtained in witing fo the EEG study. The study eceived appoval fom the appopiate ethical committees. Intelligence Testing IQ was measued with the Dutch adaptation of the WAISIII-R (WAIS-III, 1997). In accodance with the WAIS guidelines, the following fou dimensions wee calculated: vebal compehension (the mean pecentage coect of subtests infomation, similaities, and vocabulay), woking memoy (the mean pecentage coect of subtests aithmetic and lette-numbe sequencing), peceptual oganization (the mean pecentage coect of subtests block design, matix easoning, and pictue completion), and pocessing speed (the numbe of coect items pe 60 s of subtest digit symbol substitution). The validity of these fou dimensions was confimed by a eanalysis of the WAIS manual data by Deay [2001]. Fom these dimensions the combined full scale IQ was detemined. EEG Recoding The expeimental potocol fo backgound EEG egistation has been descibed in detail elsewhee [Posthuma et al., 2001; Smit et al., 2005], but a bief desciption will be epeated hee. EEG was measued at est. Half of egistation sessions wee duing moning hous, and half wee in the aftenoon. Subjects wee seated in a comfotable eclining chai in a dimly-lit, sound-attenuated, and electomagnetically-shielded oom. They wee instucted to close thei eyes, elax, but stay awake and minimize eye and body movement. EEG was egisteed fo 3 min with 17 Ag/AgCl electodes mounted in an electocap. Signal 1370

4 Heitability of Small-Wold Netwoks in the Bain egistation was conducted using an AD amplifie developed by Twente Medical Systems (TMS; Enschede, The Nethelands) fo 656 subjects (374 young, 282 middleaged) and NeuoScan SynAmps 5083 amplifie fo 104 subjects (24 young, 80 middle-aged). Signals wee continuously epesented online on a Nec multisync 17-in. compute sceen using Poly 5.0 softwae o Neuoscan Acquie 4.2. Standad positions wee F7, F3, Fz, F4, F8, T7, C3, Cz, C4, T8, P7, P3, Pz, P4, P8, O1, and O2. The vetical electo-oculogam (EOG) was ecoded bipolaly between two Ag/AgCl electodes, affixed 1 cm below the ight eye and 1-cm above the eyebow of the ight eye. The hoizontal EOG was ecoded bipolaly between two Ag/AgCl electodes affixed 1 cm left fom the left eye and 1 cm ight fom the ight eye. An Ag/AgCl electode placed on the foehead was used as a gound electode. Impedances of all EEG electodes wee kept below 3 kx, and impedances of the EOG electodes wee kept below 10 kx. The EEG was amplified, digitized at 250 Hz and stoed fo offline pocessing. Amplifie filte settings fo TMS wee a single ode FIR bandpass filte with cut-off fequencies of 0.05 and 30.0 Hz. NeuoScan filte settings wee a lowpass filte at 50.0 Hz. EEG Data Pocessing The signals wee ecalculated with aveaged ealobes (A1 and A2) as efeence. All EEG was automatically and visually checked fo bad channels such as absence of signal, hum, clipping, pesistent muscle tone atifacts, and extenal noise. Subjects without the full set of 17 leads wee excluded. This pocedue esulted in the exclusion of 186 subjects leaving 574 subjects. Next, the data wee cut into 16 s epochs with 8 s ovelap. Fo each subject, atifactual episodes wee identified automatically using the EEGLAB [Delome and Makeig, 2004] eject by theshold and eject by specta option. Theshold settings fo all leads was 6200 lv. The spectal analysis pocedue identified deviant epochs by compaing each epoch s powe spectum to the spectum aveaged ove all epochs. Epochs with moe than 10-dB excess powe within the fequency ange below alpha ( Hz) o above alpha ( Hz) wee maked atifactual. If less than fou nonovelapping epochs wee available, the (quite stict) citeion of 10 db was elaxed until exactly fou wee obtained. The aveage level of the db citeium was 16.8 (SD 3.4). No subject eached a citeium level ove 36 db. Visual inspection evealed that this pocedue successfully selected epochs without atifacts. Each epoch was baseline coected and filteed using theta ( Hz), lowe alpha ( Hz), uppe alpha ( ), lowe beta ( Hz), and uppe beta ( Hz) bandpass filtes. Fequencies above 25.0 Hz wee disegaded as the discepancies in hadwae filte settings between TMS and Neuoscan egisteed subjects may lead to spuious esults. SL Calculation The state of a dive system hee, an EEG signal is opeationalized with the embedded vecto X ¼fx i ; x i þ 1 l; x i þ 2 l;...; x i þðm 1Þ lg whee l is the lag and m the embedding dimension. The elements of X ae m samples taken fom the signal spaced l apat. The vecto is taken to epesent the state of the system at time i. Within the same signal ecuences ae sought at times j that eflect a simila state: a theshold distance ex is chosen such that a fixed popotion (p ef ) of compaisons ae close enough to be consideed in a simila state. Next, the same compaison is made fo a diffeent system Y at the same time points i and j and with the same value fo p ef. Now the SL S i between X and Y at time i is defined as follows: S i ¼ 1 X uðe y jy i Y j jþuðe x jx i X j jþ N j whee y is the Heaviside step function etuning 0 fo all values <0 and 1 fo values 0. N epesents the numbe of ecuences in signal X, i.e.: X uðe y jx i X j jþ j Oveall SL between X and Y is the aveage ove all possible i. To withhold the system to compae X i and X j while they epesent the same state, only values fo j ae consideed that ae at sufficient time distance. The value of this distance, W1, is the Theile coection fo autocoelation [Theile, 1986]. The value fo i 2 j is uppe bound to ceate a window (W1 < W2 < N) to shapen the time esolution of S i. Moe details on SL calculation can be found in seveal othe publications [Montez et al., 2006; Posthuma et al., 2005; Stam and van Dijk, 2002]. The paamete settings l, m, W1, and W2 wee chosen based on the filte-fequency settings. This appoach, as put fowad in Montez et al. [2006], detemines the lag l in sampling the embedded vecto on the high fequency paamete of the filte, and the embedding dimension m on the atio of the high and low fequency paametes. Fom these, windowing paametes W1 and W2 ae chosen such that embedded vectos ae not too close in time to avoid autocoelation effects, while allowing enough estimations to be made. Table I shows the paamete settings fo each fequency band. The emaining, fee paametes W2 and p ef wee fixed at fixed values of W1 þ 400 and These values eflect simila choices fom the pevious liteatue [Ponten et al., 2007; Stam, 2006]. Using data fom 51 andomly selected subjects evealed that a inceasing the value of W2 fom 430 to 830 did not change the esults in the uppe alpha band ( ). Inceasing p ef fom 0.01 to 0.05 also yielded simila esults ( ), adding to pevious obsevations that vaiation of p ef yields simila esults [Stam and van Dijk, 2002]. 1371

5 Smit et al. TABLE I. Synchonization likelihood paametes pe fequency band Band LF HF L m W1 W2 Theta Lowe alpha Uppe alpha Lowe beta Uppe beta Note. LF, low fequency filte setting; HF, high fequency filte setting; L, Lag; m, embedding dimension; W1, minimum window distance; W2, maximum window distance. LF and HF ae in Hz, all othe paametes in numbe of samples. C and L Calculation SL was computed between each pai of electodes esulting in a (17, 17) matix whee the values on the diagonal will be ignoed. To coect fo individual diffeences in oveall SL this value was subtacted fom the matix of SL connectivity. Using this matix to epesent edge stength, a binay gaph was fomed by applying a theshold y such that the aveage numbe of edges pe vetex was fixed at five diffeent levels (K [ {3,4,5,6,7}). An actual example of a gaph extacted fom a single epoch is povided in Figue 3. C and L wee calculated as explained in the intoduction and indicated in Figue 1 with the following extension. Standad C and L calculation equies that the gaph is fully connected [Latoa and Machioi, 2001; Watts and Stogatz, 1998]. Many EEG epochs, howeve, esulted in gaphs with at least one vetex unconnected. To accommodate fo eal wold applications whee unconnected nodes ae unavoidable, we followed Newman s [2003] poposal to assign the value of þ? to the path length involving unconnected nodes and use the hamonic mean: The small wold paamete was then calculated as ¼ g k. Since C an and L an ae fixed numbes fo gaphs with the same degee K, thee is no need to epeat the coelational analyses fo g and k. Theefoe, these analyses will be esticted to C, L, and. Reliability and Twin Coelations Fo all statistical modeling the feely available softwae package Mx vesion 1.65a was used [Neale et al., 2003]. A tetavaiate epeated measues stuctual equation model as depicted in Figue 4 was used, which allows the estimation of the eliability of the measuement. Vaiance of the fou occasions (epochs) is split into a coelated pat F (the tue scoes) and an uncoelated pat U which we may assume epesents measuement eo. Reliability was then defined as the popotion of noneo vaiance to the total: R epoch ¼ f 2 ðf 2 þ u 2 Þ Howeve, these single-epoch eliabilities should be coected using the total numbe of epochs actually measued to epesent eliability fo the full s duation using the following fomula: kr epoch R total ¼ 1 þðk 1ÞR epoch whee k 5 4. The coelation in Figue 4 will be estimated to have the value MZ o DZ, depending on the natue of the elation between the two individuals. Fo twins and siblings membes eithe an MZ o DZ/sibling coelation will be L ¼ N P N 1 L i i¼1 Fo each gaph we ceated 50 andomized gaphs by andomly econnecting edges, peseving the symmety of the matix. The aveage C and L values fom these gaphs wee used to calculate standadized paametes [Humphies et al., 2006]: and g ¼ k ¼ C C an L L an Figue 3. An example of a gaph deived fom the Synchonization Likelihood matix fo a single epoch. 1372

6 Heitability of Small-Wold Netwoks in the Bain genetic effects (A þ D) to the total vaiance (A þ S þ D þ E). In a twin-sibling design, howeve, no infomation is available to estimate the effects of both S and D simultaneously. The elative size of the DZ/sibling to the MZ coelation guides which is selected. If the DZ/sibling coelation is less than half the MZ coelation, then A þ D þ E ae modeled. If it is moe than half the MZ coelation, A þ S þ E ae modeled. Fo moe infomation on genetic modeling we efe to Boomsma et al. [2002a] and Posthuma et al. [2003]. WAIS Coelations Figue 4. Path model descibing the elation between any pai of siblings. V1, V2, V3, and V4 epesent the obseved vaiables C, L, o fom the fou epochs. Factos U ae uncoelated and epesent uneliable factos such as measuement eo. F is tue facto scoes epesenting the emaining vaiance. Depending on the elation between the two subjects, an MZ (MZ twins), DZ (DZ twins and siblings) is modeled. estimated. Note that the esulting twin coelations epesent the elation between tue factos F, and ae thus coected fo measuement eo. Means wee modeled with cohot and sex and the cohot by sex inteaction as covaiates. Vaiances wee tested fo heteoscedasticity between the sexes and the cohots. When significant, heteoscedasticity wee modeled by using a scala model. These models use a scala close to 1.0 to equalize the vaiances in one goup (e.g., males) to the othe (e.g., females). Significant diffeences in eo vaiances wee modeled similaly. Genetic analyses Resemblance (covaiance) in psychophysiological taits between twins and siblings deive fom genetic elatedness o shaed envionmental influences [Boomsma et al., 2002a; Falcone and Mackay, 1996]. If the coelation between DZ twins o siblings, who shae on aveage 50% of thei genetic make-up, is half the coelation between MZ twins, who ae genetically identical, this is seen as evidence fo additive genetic influences (A). If the coelation between DZ twins o siblings is less than half the coelation between MZ twins this is seen as evidence fo dominant (nonadditive) genetic influences (D). If the coelations between MZ and DZ twins/siblings ae compaable and nonzeo this is evidence fo shaed envionmental influences (S). If the coelation between MZ twins is not unity this is evidence fo envionmental effects unique to each individual (E). By compaing MZ and DZ/sibling coelations, using stuctual equation modeling as implemented in, fo example, Mx [Neale et al., 2006], we can obtain maximum likelihood estimates of the elative contibutions of each of these factos to the total tait vaiance. Heitability is defined as the popotional contibution of Coelations wee calculated between C, L, and and scoes on the fou subscales vebal compehension, woking memoy, peceptual oganization, and peceptual speed. Since dependencies exist between family membes, statistical infeence fom staightfowad coelations between taits would be incoect. We theefoe modeled the coelations while allowing fo within family coelations (i.e., MZ and DZ/sibling coelations, and cosstwin-coss-tait coelations). As with all othe modeling, the coelations wee modeled on the tue, noneo vaiance of C, L, and. RESULTS Unconnected Vetices It may have been possible that one lead esulted in an unconnected vetex in most subjects, suggesting that this lead should be emoved fom the analysis. Although T7 and T8 showed quite a high popotion of epochs which esulted in unconnected vetices, still in about 60% of the epochs they wee connected. Theefoe, we decide not to exclude these leads. Desciptives Table II shows the desciptives fo paametes C and L, and the deived vaiables g, k and Since the distibutions fo k wee in many cases ight skewed, median values ae shown to define cental tendency. Oveall, these paametes ae consistent with a small-wold oganization of the functional netwok, since k is elatively close to unity, wheeas g is lage than that (see Table II). Fo all exploed levels of K, gamma was clealy and significantly lage than unity. Howeve, inceasing K esulted in educed levels of gamma (as can be expected). The median value fo the vaiable k is only slightly lage than unity fo highe degees of K (K 5). Lowe values of K showed cleae deviation fom unity with values above 1.1 in all fequency bands. Levels of K To exploe the natue of the dependency of the gaph vaiables C and L on degee K we calculated the coelation of the 1373

7 Smit et al. TABLE II. Medians and intequatile anges fo gaph theoetical paametes of functional connectivity C L g k Band Me Quat. ange Me Quat. ange Me Quat. ange Me Quat. ange Me Quat. ange K 5 3 Theta Lowe alpha Uppe alpha Lowe beta Uppe beta K 5 4 Theta Lowe alpha Uppe alpha Lowe beta Uppe beta K 5 5 Theta Lowe alpha Uppe alpha Lowe beta Uppe beta K 5 6 Theta Lowe alpha Uppe alpha Lowe beta Uppe beta K 5 7 Theta Lowe alpha Uppe alpha Lowe beta Uppe beta gaph desciptos C and L between all levels of K. Next,we aveaged the coelations (using the Fishe tansfom) to obtain maginals epesenting the stength of coelation between a each level of K with all othe levels. Table III shows the aveage coelations fo C and L in all fequency bands. Fo C, the coelations ae somewhat smalle than those fo L, but still modeate to high (ca. 5 fo K 5 3 but 0.6 to 0.7 fo othe levels). Fo L, coelations between levels ae high. Degee K 5 3 seems to coelate the wost with all othe levels, wheeas K 5 5 seems to coelate best fo both C ( > 0.67) and L ( > 0.83) in all fequency bands. Oveall, K 5 5 seems to be the best epesentation of most of the vaiation shaed between all levels of K. In addition, C and L at degee K 5 5 seems to show small wold netwok popeties as mentioned in the pevious paagaph. To educe the amount of futhe testing, we chose to estict the genetic and phenotypic analyses to gaphs with degee K 5 5. Effects on Means and Vaiances Sex diffeences in L wee found in the lowe fequency bands such that males show shote L (theta: b sex , v 2 (1) , P ; lowe alpha: b sex , v 2 (1) , P ). No othe significant mean sex diffeences wee found. The middle-aged cohot showed significantly lowe C fo the theta band (b coh , v 2 (1) , P ), but highe C fo uppe alpha (b coh , v 2 (1) , P ). Middle-aged adults showed longe L in both these bands (theta: b coh , v 2 (1) , P ; uppe alpha: b coh , v 2 (1) , P ). In the theta band this esulted in a significantly lowe value of the small-wold vaiable sigma fo the middle-aged (b coh , v 2 (1) , P ). Vaiances diffeed between the sexes only fo L in the theta band (v 2 (1) , P ). In males tue vaiance was lage than in females. Lage diffeences in tue vaiance wee also found between the cohots (theta: v 2 (1) , P ; uppe alpha: v 2 (1) , P ), such that the olde cohot had lage vaiances in L than the younge cohot. Reliabilities Reliabilities wee calculated using the epeated measues model and coected to epesent eliability of the full measuement as specified in the methods. Table IV shows that the eliabilities wee, in geneal, modeate fo C and. This indicates that measuement eo coveed a substantial popotion of the vaiance of C (50%) and slightly moe of (50% to 65%). This popotion was lage fo the lowe fequency bands. The popotion mea- 1374

8 Heitability of Small-Wold Netwoks in the Bain TABLE III. The coelations between scoes on each level of K and scoes on all othe levels, aveaged using Fishe tansfom, vaiables C and L K 3 with 4,5,6, and 7 4 with 3,5,6, and 7 5 with 3,4,6, and 7 6 with 3,4,5, and 7 7 with 3,4,5, and 6 C Theta Lowe alpha Uppe alpha Lowe beta Uppe beta L Theta Lowe alpha Uppe alpha Lowe beta Uppe beta suement eo was makedly less fo L than fo C and. Note that in some cases heteogeneity of vaiance and/o eo vaiance wee found (as afoementioned). Since these diffeences esulted in diffeent eliabilities fo sex o age cohot goups, Fishe-z tansfomed aveages (tansfomed back) ae shown in the table in those cases. Genetic Analysis Genetic models included afoementioned significant effects on the means and vaiances. Men and women and the olde and younge cohots did not diffe in MZ and DZ/sibling coelations. The esulting twin coelations collapsed acoss cohot and sex ae shown in Table V along with 95% confidence intevals. Although the twin coelations suggested effects of shaed envionment fo theta band L, this effect was not significant. Significant dominant genetic effects wee found fo L fo the theta and uppe alpha bands (v 2 (1) 5 8.9, P , and v 2 (1) , P , espectively). Fo these vaiables boad heitabilities wee estimated, that is, including both dominant and additive genetic vaiance, TABLE IV. Reliable, noneo vaiance as a popotion of total vaiance fo gaphs with degee K 5 5 C L Theta Alpha low Alpha high Beta low Beta high Note. Fo some vaiable ICCs diffeed between cohots and/o sexes due to heteogeneity of vaiance o heteogeneity of eo vaiance. Aveage values ae shown. Reliabilities ae based on the coelations between all epochs, then coected to epesent eliability of the total of fou epochs. C 5 clusteing coefficient, L 5 aveage path length, 5 small wold vaiable g/k. and a two degee of feedom test detemining the significance of both the effects of A and D simultaneously. All heitabilities ae shown in Table VI. Highly significant heitability could be established fo L in all fequency bands (lowest significance fo theta: v 2 (2) , P ). Significant heitability fo C was found in the theta and beta fequency bands (theta: v 2 (1) 5 8.5, P ; lowe beta: v 2 (1) , P ; uppe beta: v 2 (1) , P ). Heitability estimates fo on the othe. DISCUSSION The esults fo paametes g(l/lan), k(c/can), and ¼ g k deived fom functional connectivity gaphs with degee K 5 5 ae compaable to those epoted in the existing liteatue and summaized in Table VII. Functional connectivity gaphs deived fom fmri, MEG, and EEG have shown values fo k in the ange of , but ae typically aound 1.1 which is highly compaable to ou Values fo g fom the same studies anged fom 1.58 to The values epoted hee (1.52 to 1.67) ae at TABLE V. Twin coelations fo C, L, and fo gaphs with degee K 5 5 C L MZ DZ MZ DZ MZ DZ Theta *** * Lowe alpha *** Uppe alpha *** 20.01* Lowe beta 0.66* *** Uppe beta 0.53* *** Note. All coelations wee estimated using Mx afte emoval of age and sex on the means, and significant cohot and sex effects on vaiance tems by use of a scala model. DZ coelations include all fatenal (non-identical) siblings pais, including opposite sex pais. C, Clusteing coefficient, L, Path length,, small wold index g/k* 5 P < 0.01; **P < 0.001; ***P <

9 Smit et al. TABLE VI. Heitabilities fo C, L, and C L Theta 0.50 * 0.89 * 0.37 Lowe alpha *** 0.51 Uppe alpha *** 0.00 Lowe beta 0.62 ** 0.53 *** 0.27 Uppe beta 0.54 *** 0.70 *** 0.39 Note. Heitability was modeled using additive genetic and unique envionmental effects, except fo: Dominant genetic effects wee significant fo theta L and uppe alpha L. Fo these, heitability (h 2 ) includes both additive and non-additive (dominant) genetic effects. *P < 0.01, **P < 0.001, ***P < C, clusteing coefficient; L, aveage path length;, small wold index: g/k. the lowe end of this peviously epoted ange, although this seems consistent with the esults fom the othe EEG studies. Small wold paamete in the liteatue anged fom 1.56 to Hee, too, ou esults wee at the low end of the dimension with Howeve, the small wold equiement defined as k while g 1.0 seems to be met. Modeling of MZ and DZ twin and sibling esemblance indicated that acoss vaious fequency bands 46 89% of the individual diffeences in C and 37 62% of the individual diffeences in L ae heitable. Oveall, these esults suggest that thee ae consistent individual diffeences in functional connectivity pattens that have a fim biological basis. Gaph theoetical analysis of backgound EEG connectivity eveals systematic individual diffeences in aveage path length L and clusteing coefficient C, which makes them viable biomakes fo individual diffeences in bain functioning. Howeve, uneliable vaiance constitutes a lage pat of the total vaiance of mainly C. If the uneliable vaiance can indeed be attibuted to measuement eo, that this finding indicates the impotance of measuing many epochs of sufficient length to obtain eliable estimates of individual scoes. In the pesent study, measuing fou epochs did not yield sufficient powe to establish significance of the heitability estimates fo the oveall netwok efficiency as measued with. Since it has been agued that C and L ae paametes that descibe netwok efficiency, it was easonable to hypothesize that efficient connectivity pattens would be elated to cognitive pefomance measues. Howeve, we found that WAIS was not significantly elated to eithe C, L, o small-wold paamete. Although individual diffeences in C and L may simply not eflect those in cognitive pefomance, the absence of significant coelation may also have been the esult of suboptimal measuement evidenced by the substantial amount of eo vaiance. In addition, the application of a theshold to the SL matix esulted in the loss of possibly valuable infomation egading the connectivity stength between bain aeas. To make use of this souce of infomation, altenative appoaches fo investigating netwok efficiency to the one pesently used have been poposed. Fo example, Latoa and Machiai [2001] poposed a method to descibe local and global efficiency computed fom what have been called weighted gaphs. It stands to eason that such an appoach could esult in bette estimations of oveall netwok efficiency, and hence, bette estimates of phenotypic coelations. Ultimately, though, this emains an empiical question. Anothe possible souce of bias is that volume conduction in the EEG signals could have falsely inceased TABLE VII. Gaph theoetical paametes fom psychophysiological studies of functional connectivity Study Sample Measuement Condition k g Cuent study Healthy subjects EEG theta, low alpha, high alpha, Resting state low beta, high beta, K55 Achad et al., 2006 Healthy subjects fmri Hz a Resting state Batolomei et al., 2006 Healthy subjects MEG theta, beta, gamma Resting state Bassett et al., 2006 Healthy subjects MEG Hz a Resting state 1.92 Finge tapping 1.86 Eguiluz et al., 2005 b Healthy subjects FMRI Music listening and finge tapping Micheloyannis Schizophenia EEG theta, alpha, beta, gamma WM task et al., 2006 Healthy subjects Ponten et al., 2007 Epileptic patients Depth electode EEG, Inteictal peiods to 48 Hz synchonization Salvado et al., 2005 Healthy subjects FMRI Resting state Stam et al., 2004 Healthy subjects MEG theta, gamma c Resting state Stam et al., 2006 Alzheime s MEG beta Resting state healthy subjects Results as summaized by the espective authos, except: a No aveages epoted by the authos, values aveaged acoss fequencies ae shown. b The degee K was much smalle than ln(n), which may have caused the anomalous esults (see Bassett & Bullmoe, 2006). c Stam, 2004 did not find small-wold oganization in the alpha and beta fequency bands (omitted hee). 1376

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