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Research Article: Methods/New Tools | Novel Tools and Methods

Uncovering Neuronal Networks Defined by Consistent between-NeuronSpike Timing from Neuronal Spike Recordings

Roemer van der Meij1 and Bradley Voytek

1,2,3,4

1Department of Cognitive Science, University of California, San Diego, La Jolla, CA 92093, USA

2Neurosciences Graduate Program, University of California, San Diego, La Jolla, CA 92093, USA

3Halicioglu Data Science Institute, University of California, San Diego, La Jolla, CA 92093, USA

4Kavli Institute for Brain and Mind, University of California, San Diego, La Jolla, CA 92093, USA

DOI: 10.1523/ENEURO.0379-17.2018

Received: 7 November 2017

Revised: 1 April 2018

Accepted: 21 April 2018

Published: 8 May 2018

Author contributions: R.v.d.M. and B.V. designed research; R.v.d.M. performed research; R.v.d.M.contributed unpublished reagents/analytic tools; R.v.d.M. analyzed data; R.v.d.M. and B.V. wrote the paper.

Funding: http://doi.org/10.13039/100000025HHS | NIH | National Institute of Mental Health (NIMH)MH095984-03S1

Funding: http://doi.org/10.13039/100000001National Science Foundation (NSF)1736028

Funding: http://doi.org/10.13039/100007911UC | University of California, San Diego (UCSD)

Funding: http://doi.org/10.13039/100000879Alfred P. Sloan Foundation

Conflict of Interest: Authors declare no conflicts of interest.

Correspondence to: Roemer van der Meij, Cognitive Science, University of California, San Diego, La Jolla, CA92093, USA. E-mail: roemer.van.der.meij@gmail.com

Cite as: eNeuro 2018; 10.1523/ENEURO.0379-17.2018

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Title: U o e i g eu o al et o ks defi ed o siste t et ee - eu o spike ti i g f o eu o al spike e o di gs A e iated title: U o e i g spike ti i g et o ks f o eu o al e o di gs ‘oe e a de Meij , B adle Vo tek , , , Depa t e t of Cog iti e S ie e, U i e sit of Califo ia, Sa Diego, La Jolla, CA , USA. Neu os ie es G aduate P og a , U i e sit of Califo ia, Sa Diego, La Jolla, CA , USA. Hali ioglu Data S ie e I stitute, U i e sit of Califo ia, Sa Diego, La Jolla, CA , USA. Ka li I stitute fo B ai a d Mi d, U i e sit of Califo ia, Sa Diego, La Jolla, CA , USA.

Co espo de e to: ‘oe e a de Meij, Cog iti e S ie e, U i e sit of Califo ia, Sa Diego, La Jolla, CA , USA. E- ail: oe e . a .de . eij@g ail. o Nu e of figu es: Nu e of ta les: Nu e of ulti edia: Nu e s of o ds i A st a t: Nu e s of o ds i I t odu tio : Nu e s of o ds i Dis ussio : A k o ledge e ts: We tha k S.‘. Cole, T. Do oghue, ‘. Gao, E. Pete so , a d T. T a fo thei dis ussio s a d o e ts, a d espe iall E. Ma is fo dis ussio s o the p oje t s p e ise ea s ago. Both ‘. .d.M. a d B.V. e e suppo ted the Natio al I stitute of Me tal Health MH - S a d U i e sit of Califo ia, Sa Diego, a d B.V. as additio all suppo ted a Sloa ‘esea h Fello ship a d the Natio al S ie e Fou datio . Co fli ts of i te est: The autho s de la e o o peti g fi a ial i te ests.

A stra t It is idel assu ed that dist i uted eu o al et o ks a e fu da e tal to the fu tio i g of the ai . Co siste t spike ti i g et ee eu o s is thought to e o e of the ke p i iples fo the fo atio of these et o ks. This a i ol e s h o ous spiki g o spiki g ith ti e dela s, fo i g spike se ue es he the o de of spiki g is o siste t. Fi di g et o ks defi ed thei se ue e of ti e- shifted spikes, de oted he e as spike ti i g et o ks, is a t e e dous halle ge. As eu o s a pa ti ipate i ultiple spike se ue es at ultiple et ee -spike ti e dela s, the possi le o ple it of et o ks is p ohi iti el la ge. We p ese t a o el app oa h that is apa le of e t a ti g spike ti i g et o ks ega dless of thei se ue e o ple it , a d that des i es thei spiki g se ue es

ith high te po al p e isio . We a hie e this de o posi g f e ue -t a sfo ed eu o al spiki g i to sepa ate et o ks, ha a te izi g ea h et o k s spike se ue e a ti e dela pe eu o , fo i g a spike se ue e ti eli e. These et o ks p o ide a detailed te plate fo a i estigatio of the e pe i e tal ele a e of thei spike se ue es. Usi g si ulated spike ti i g et o ks, e sho

et o k e t a tio is o ust to spiki g oise, spike ti i g jitte , a d pa tial o u e es of the i ol ed spike se ue es. Usi g at ulti- eu o e o di gs, e de o st ate the app oa h is apa le of e eali g eal spike ti i g et o ks ith su - illise o d te po al p e isio . B u o e i g spike ti i g et o ks, the p e ale e, st u tu e, a d fu tio of o ple spike se ue es a e i estigated i

g eate detail, allo i g us to gai a ette u de sta di g of thei ole i eu o al fu tio i g. Sig ifi a e state e t Spike ti i g o siste ies i eu o al et o ks a e idel thought to e o e of se e al ke p i iples

ehi d eu o al fu tio i g. The a e halle gi g to i estigate, ho e e , e ause the e is effe ti el a i fi ite u e of o i atio s of eu o s a d thei et ee - eu o ti e dela s fo a gi e e o di g. Ma te h i ues ha e ee de eloped fo thei a al sis, ut the a e still li ited the o ple it of spike ti i g patte s the a e eal. He e e p ese t a o el app oa h that a e eal

spike ti i g patte s ith a it a o i ato ial o ple it . This p o ides a e oppo tu it fo i estigati g spike ti i g et o ks, hi h is u ial to gai a ette u de sta di g of the ole the pla i eu o al fu tio i g.

I trodu tio Dist i uted et o ks of eu o s, o ell asse lies, a e idel assu ed to e fu da e tal to ai fu tio i g He , ; T eis a , ; Si ge , ; Va ela et al., ; Ha is, ; Buzsaki, . A su set of these et o ks is thought to e fo ed o siste t ti i g of a tio pote tials, o spikes,

et ee eu o s Bie e sto k, ; Si ge , ; Ai s o th et al., ; Feld a , , a featu e of spike e o di gs a oss spe ies Mai e a d Sej o ski, ; Sali as a d Sej o ski, ; Va ‘ulle et al., ; Fujisa a et al., ; ‘atté et al., . The spiki g et ee eu o s of su h et o ks a e s h o ous o i ol e ti e dela s Izhike i h, ; Fujisa a et al., ; Saku ai et al., , fo i g spike se ue es he fi i g i a o siste t o de Lee a d Wilso , ; Ikega a et al., ; Tiesi ga et al., . Spike se ue es a i ol e the sa e eu o s a d o u ithi the sa e ti e i do Mao et al., ; Ma Lea et al., ; Lu zak et al., ; Matsu oto et al., ; Mille et al., . Though the e is still u h de ate a out ho i po ta t spike ti i g is i o pa iso to alte ati es su h as ate- ased odi g s he es Ku a et al., ; ‘olls a d T e es, ; Ai s o th et al., , the i estigatio of spike ti i g et o ks a d thei spike se ue es e ai s e essa to fu the ou u de sta di g of asi eu o al ope atio s. Fi di g et o ks defi ed thei se ue es of o siste t ti e-shifted spikes et ee eu o s, de oted i the follo i g as spike ti i g etwo ks, is a t e e dous halle ge due to thei possi le o ple it , as eu o s a pa ti ipate i ultiple spike se ue es at a o ti uu of et ee -spike

ti e dela s. The past de ades ha e see the a i al of a ethods that a ha a te ize spike ti i g et o ks A eles a d Ge stei , ; Nadasd et al., ; Chapi a d Ni olelis, ; Tetko a d Villa,

; G ü et al., ; Lee a d Wilso , ; S h itze a d Meiste , ; Ikega a et al., ; Okata et al., ; S h eide et al., ; Nikoli , ; Pipa et al., ; S h ade et al., ; Be ge et al.,

; Louis et al., ; Pe a he et al., ; Elda latl et al., ; Hu ph ies, ; Lopes-dos- Sa tos et al., ; Ga sel a d Si ge , ; To e et al., . Thei appli atio has led to i po ta t i sights, et the ha e se e al li itatio s, espe iall he it o es to thei appli atio o la ge s ale

eu o al e o di gs Buzsaki, . Na el , eithe : the o ple it of the ide tified et o ks is li ited due to o i ato ial e plosio ith i easi g et o k size e.g. te plate sea hi g , the

et o ks a e des i ed o l the asso iatio of thei e e eu o s ithout des i i g spike se ue es, et ee -spike ti e dela s g eate tha a e eithe dis a ded o ot e o e ed, te po al i i g of spike ti es leads to edu ed te po al p e isio , et o ks ith o e lappi g

e e eu o s a e ot sepa ated, o , a o i atio of the a o e. Though ot i po ta t fo e e i estigatio of i te a tio s i spiki g et o ks e.g., fo highe o de i te a tio s see Nakaha a a d A a i, ; Yu et al., ; Elda latl et al., ; Staude et al., ; Balague -Balleste et al., ; Shi azaki et al., , the a e esse tial fo the e a t ide tifi atio of eu o s a d thei spike se ue es, a d i estigati g thei o u e e as a fu tio of e pe i e tal a ia les. We p ese t a o el app oa h fo e eali g spike ti i g et o ks that does ot suffe f o the a o e p o le s. The ke featu es of ou app oa h, a e that it a fi d et o ks ega dless of the o ple it of thei spiki g se ue es, a d that it des i es these se ue es ith su - illise o d

p e isio a ti e oeffi ie t pe eu o pe se ue e; see Fig . This is a hie ed appl i g a ethod de eloped fo ele t oph siologi al e o di gs i te ded fo e eali g phase- oupled os illato

et o ks, su h as t a eli g a es; a de Meij et al., , o the spe t al o a ia e o , oss spe t a o tai ed f o a spe t al a al sis of dis ete eu o al spiki g ti e se ies. I these oss spe t a, o siste t et ee - eu o spike ti e dela s a e des i ed li ea l i easi g phase diffe e es o e f e ue ies. The ethod fi ds et o ks a d thei spike se ue es thei u i ue patte s of et ee - eu o phase diffe e es o e f e ue ies a d t ials epo hs . I the follo i g, e fi st sho that ou app oa h is apa le of e o e i g si ulated spike ti i g et o ks a d thei se ue es u de a ious oise o ditio s, a d the p o ide a p oof-of- o ept sho i g et o ks

e t a ted f o at hippo a pus a d edial p ef o tal o te Fujisa a et al., , , hi h efle ted peak oss- o elog a ti e dela s ith high a u a . Togethe , this de o st ates that ou

app oa h is a o ust ethod fo e eali g a d ha a te izi g spike ti i g et o ks i eu o al e o di gs.

Materials a d Methods

. E tra ti g spiki g ti i g et orks fro euro al spike re ordi gs Spike ti i g o siste et ee eu o s is thought to e a ke featu e of eu o al spike e o di gs. I the follo i g, e des i e ho a o el appli atio of a e e t te h i ue, SPACE; a de Meij et al.,

, a e used to fi d se ue es of o siste t ti e-shifted spikes et ee eu o s, de oted i the follo i g as spike ti i g et o ks, i la ge s ale eu o al spike e o di gs, ithout a p io i i fo atio a out the i ol ed eu o s a d thei ti i g. Belo e illust ate the p o edu e fo e t a ti g spike ti i g et o ks ith details i dedi ated se tio s , a d ho to i te p et thei ha a te izatio .

We sta t ith a ki d of ulti-ele t ode eu o al e o di g o e ti e of hi h eu o s a d thei spikes ha e ee ide tified usi g e.g. ‘ossa t et al., . Suppose ou e o di g o tai s g oups of eu o s that ha e o siste t et ee - eu o spike ti i g i.e., spike ti i g et o ks, Fig

A lue/g ee , Fig B , e edded i othe spiki g a ti it . The lue et o k has a spiki g se ue e of eu o to to ith a ti eli e of s- s- s, leadi g to o siste t spike ti e dela s of s fo

pai s - a d - , a d a s dela of - . The g ee et o k has the sa e patte ut fo eu o s , , a d . If oss- o elog a s ould e o puted f o the eu o al spike e o di gs Fig B , the ould ha e peaks at s fo eu o pai s - , - , - , - , at - s fo pai - , - , - , - , at s fo pai -

, - , a d at - s fo pai - , - . Ho ofte the se ue es of the lue a d g ee et o ks o u , depe ds o t o e pe i e tal o ditio s. The lue et o ks o u o e pe t ial of o ditio A a d t i e fo those of B, Fig C , the g ee et o k i e e sa. To e t a t a pa si o ious des iptio of the a o e et o ks, e a a ge the dete ted spikes i

eu o - -ti e i a at i es = spike, = o spike; Fig C , pe t ial of the e pe i e t o a othe ea i gful te po al seg e tatio ; th oughout the te t, efe s to eu o s, to f e ue ies, to t ials . The , e o tai oss spe t a f o these t ial-spe ifi at i es. To a hie e this, e fi st o ol e the at i es ith o ple e po e tials Fig , Step at ultiple f e ue ies

, esulti g i f e ue -spe ifi a d t ial-spe ifi o ple - alued eu o - -ti e at i es . Su se ue tl , e o pute the oss p odu ts of these at i es o e ti e Fig , Step ; = o ple o jugate t a spose , esulti g i o ple - alued f e ue -spe ifi a d t ial-spe ifi eu o -

- eu o at i es : the oss spe t a Fig D . The hoi e of o ple e po e tials dete i es ke aspe ts of the spike ti i g et o ks a d thei e t a tio , a d is dis ussed i detail i Mate ials a d Methods se tio . Ha i g o tai ed the oss spe t a, e should appl a eu o - ise a d/o t ial- ise o alizatio Fig , Step ; see Mate ials a d Methods se tio . Afte o alizatio ,

e the e t a t spike ti i g et o ks usi g SPACE Fig E, Step ; hi h i ol es esti ati g the u e of et o ks to e t a t, see Mate ials a d Methods se tio .

Spike ti i g et o ks, he defi ed thei dis ete spiki g se ue es ha a te ized as ti e- shifted opies of spikes et ee eu o s, a e e t a ted f o the oss spe t a, e ause the phases of the off-diago al ele e ts of the oss spe t a o tai the o siste t ti e dela s et ee ti e- shifted opies of spikes. The u ial p i iple he e, is that the ti e diffe e e et ee t o i a spikes i the ti e do ai t a slates to phase diffe e es i the f e ue do ai , li ea l i easi g ith f e ue . That is, a s ti e dela e uals / th of a le at Hz, / th at Hz, / th at Hz, et . fo t pes of spiki g i te a tio s othe tha ti e-shifted opies of spikes, see Li de a et al., .

The e t a tio te h i ue uses this p ope t to fi d the ti e dela s et ee ti e-shifted opies of eu o al spikes that e plai s the ost a ia e i the oss spe t a.

SPACE des i es the oss spe t a ultiple et o ks, ea h et o k o sisti g of th ee pa a ete e to s Fig E : the eu o p ofile , the ti e p ofile , a d the t ial p ofile . The eu o p ofile des i es ho st o gl ea h eu o is pa t of the et o k, a si gle u e pe

eu o . The eu o p ofile of the lue et o k has high alues fo eu o s , , a d , a d lo alues

fo all othe eu o s. That is, o l eu o s to a e pa t of this et o k. Due to thei si ila eighti g, eu o s to likel ha e si ila fi i g ates, a d si ila u e of spike se ue e spikes. If a othe eu o ould ha e half the eighti g, it ill likel ha e eithe t i e as a total spikes, o o l fi es i

half of the et o k s spike se ue es e.g., a se ue e of eu o - - i half of the t ials, a d - i the est . Note, it is possi le fo the eu o p ofile to ha e o -ze o loadi gs fo o l o e eu o , see Mate ials a d Methods se tio fo a dis ussio . The ti e p ofile des i es the spike se ue e of the

et o k, a si gle ti e oeffi ie t pe eu o . Be ause the eu o p ofile of the lue et o k o l st o gl i ol es eu o s to , o l these oeffi ie ts of the ti e p ofile a e ea i gful. The ti e p ofile of eu o s to di e tl efle ts the ti eli e of the spike se ue e, s- s- s. A u ial o se atio he e, is that all the te po al elatio ships at the le el of eu o -pai s a e des i ed at the le el of i di idual eu o s the et o k s eu o p ofile elatio ship st e gth a d ti e p ofile spike se ue e ti i g . The a o e is the sa e fo the g ee et o k. I po ta tl , if the lue a d g ee spike se ue es o u at a do i te als et ee the , the lue a d g ee et o ks a e sepa ated, as the e a e o o siste t elatio ships et ee eu o s , , a d eu o s , . If the lue a d g ee spike se ue es o u ed at o siste t i te als, the ould oth e aptu ed o e et o k. This is

ot su p isi g, as the a o e ea s the e is, i fa t, o l o e spike se ue e. I the s he ati of Figu e , the e is also a diffe e e i ho ofte a se ue e of ea h et o k o u s i ea h o ditio , hi h a e efle ted the t ial p ofiles. He e, the eights fo t ials of the lue et o k efle t the atio of

spike se ue es i ea h t ial of the t o o ditio s, i.e. a t ial loadi g that is t i e as la ge fo B as it is fo A. I po ta tl , the t ial p ofile a p o ide a o e ie t a to i estigate diffe e es i spike ti e o siste at the le el of et o ks, i stead of the le el of eu o -pai s. Fo e a ple, the diffe e e et ee t o o ditio s a e i estigated o pa i g the ea s of the o ditio -spe ifi t ial

p ofile eights, o a iatio s of t ial p ofile eights a e elated to othe a ia les e.g., ea tio ti es, pa a et i sti ulus a ipulatio s, et . . Though i p i iple possi le, the t ial p ofile a e

oise se siti e see the ‘esults a d Dis ussio se tio s . Additio all , the a se e of a e t a ted et o k is ot e ide e of a et o k s a se e i the e o di g , a ious easo s a p ohi it a et o k to ot e fou d e.g., oise, see ‘esults a d Dis ussio se tio s . Fi all , see Mate ials a d

Methods se tio o ho to o pa e alues et ee a d ithi the a o e p ofiles. Whe i te p eti g the et o k p ofiles, it is u ial to keep i i d that the a e esti ated to

a i all e plai the oss spe t a see Mate ials a d Methods se tio . As su h, a thi g that affe ts the phase oupli g patte s i the oss spe t a, affe ts the p ofiles a o di gl . Fo e a ple, i ase a eu o spikes i a spike se ue e of a et o k i so e t ials, ut ot i othe s, the the phase oupli g st e gth et ee this eu o a d the othe s of the et o k ill e eake o ze o i the oss spe t a fo the latte t ials o pa ed to the fo e . Co se ue tl , the eu o p ofile of this

et o k ill ha e a lo e eight fo this eu o tha fo the othe s, a d the t ial p ofile ill ha e lo e eights fo those t ials i hi h it did t spike.

. SPACE des ri es ti e o siste i du ed phase oupli g i ross spe tra

SPACE is a de o positio te h i ue that des i es the st u tu e of phase oupli g i oss spe t a ti e dela s et ee eu o s o ele t odes/se so s/sites . The te h i ue as de eloped fo fi di g os illato phase oupli g st u tu e e.g., t a eli g a es i ele t oph siologi al e o di gs a de Meij et al., , , a t pe of data of hi h the f e ue o te t itself is of p i a i te est. This o t asts ith the appli atio e p ese t he e, fo hi h this is ot the ase. The f e ue ies of the

used spe t al t a sfo a e a tifi ial a d a e hose o l su h that the p o ide a a u ate des iptio , ia the et o ks SPACE e t a ts, of the te po al st u tu e i dis ete eu al spike ti i g ti e se ies see Mate ials a d Methods se tio . Apa t f o the a e of o st u ti g the oss spe t a, the

usage of the ethod i the u e t app oa h is ide ti al to that i the o igi al pu li atio efe ed to as SPACE-ti e the ei . The algo ith ehi d the ethod is e te si el t eated i its o igi al pu li atio

a de Meij et al., , ut see also a de Meij et al., fo a alte ate p ese tatio , a d o l ele e ts esse tial to its u e t use ill e e tio ed he e. B iefl , the te h i ue o sists of a Alte ati g Least S ua es ALS algo ith to fi d the least s ua es esti ates of its de o positio

odel. The ele e t- ise fo ulatio of this odel fo the oss spe t a a e e p essed as:

The o ple - alued oss spe t u of eu o pai a d i de ed o e all eu o s at f e ue a d t ial is des i ed as the p odu t of et o k pa a ete s, su ed o e et o ks , plus a e o te . The phase of the et o k-spe ifi p odu t is gi e the diffe e e i the ti e p ofiles of eu o a d ultiplied the f e ue i Hz, ultiplied . This phase is the eighted the p odu t of the t o eu o s eu o p ofile , the s ua ed f e ue p ofile at f e ue , a d the s ua ed t ial p ofile at t ial . As is o se ed he e, the te h i ue also p odu es a f e ue p ofile pe et o k, des i i g ho i po ta t ea h f e ue is fo a et o k. Fo the pu pose of spike ti i g et o ks, e ill ig o e this, as it does ot p o ide additio al i fo atio . It is, ho e e , a esse tial ele e t of its o igi al appli atio o ele t oph siologi al e o di gs, des i i g f e ue a d-spe ifi phase- oupled os illato et o ks, su h as t a eli g a es. Co pa ed to the efe e e pu li atio s, the a o e e uatio s ua es the t ial a d f e ue p ofile. The easo fo this, is that spike ti i g et o ks a e o e o e ie tl thought of, a al zed at, a d si ulated i , the des iptio le el of oss spe t a. This is ot the ase fo the o igi al ta get of the te h i ue, phase- oupled os illato et o ks, hi h a e o e o e ie tl thought of as ti e- a i g os illatio s o e ele t odes des i ed Fou ie oeffi ie ts of hi h the oss p odu ts o e ti e p odu e the oss spe t a . Due to this, the te h i ue p o ides t ial a d f e ue p ofiles that a e ot s ua ed, a d s ua i g e o es a e essa step p io to i estigati g the e t a ted et o ks. The e t a ted et o ks a e u i ue up to t i ial i dete i a ies ithout e ui i g o st ai ts su h as o thogo alit o statisti al i depe de e. U i ue ess is dis ussed i o e detail i the efe e e pu li atio s a de Meij et al., , . The i dete i a ies a e easil esol ed o alizatio s. He e, e iefl highlight those o alizatio s that pe tai to the u e t appli atio of

the te h i ue. The eu o a d t ial p ofiles, pe et o k, ha e u dete i ed ultipli ati e s ali g, a d a e o alized to ha e a e to L - o of . The o se ue e is that the a solute alues of eu o s a d t ials o l ha e ea i g ith espe t to the othe eu o s a d t ials of the sa e et o k. C u iall , the atios et ee eu o s a d t ials a e u affe ted this o alizatio , a d a e o pa ed f eel a oss et o ks. Additio all , thei sig is also u dete i ed, a d est i ted to ha e a positi e a e age pe et o k eu o p ofile o to e full positi e t ial p ofile . The i dete i a of the ti e p ofile is

o e o pli ated. Be ause the ti e p ofile des i es i ula phases o e ultiple f e ue ies, the ti e p ofile is i ula as ell. I sho t, e o alize it su h that the st o gest eu o of the eu o p ofile has a ti e p ofile alue of s. Due to the a o e o alizatio s, a et o k-spe ifi ultipli ati e s ali g pa a ete is also e t a ted, ut it does ot pla a ole i the i te p etatio of the i di idual

et o k pa a ete s. T o p a ti al poi ts eed to e ade fo usi g the te h i ue to e t a t et o ks. The fi st is that its algo ith is i itialized f o a do sta ti g alues. I o de to a oid u fo tu ate sta ti g alues that lead to a lo al i i u of its least s ua es loss fu tio , the algo ith eeds to e i itialized

ultiple ti es. Whe ide ti al et o ks a e fou d i those i itializatio s ith the highest e plai ed a ia e, it a e assu ed that the glo al i i u is ea hed. Ho a i itializatio s a e e ui ed

i o de to a hie e this depe ds o the pa ti ula dataset. I ou e pe ie e, it is e t e el a e to fi d a diffe e t est solutio to the loss fu tio he i easi g the u e of a do i itializatio s

e o d . The se o d p a ti al poi t is that, like elated de o positio te h i ues, the u e of

et o ks to e t a t eeds to e dete i ed. O e app oa h is to esti ate the u e of elia le et o ks. Fo this, e e t a t N et o ks f o the full e o di g, a d also f o t o splits of the e o di g, the fi st o tai i g the odd u e ed spikes of ea h eu o , the se o d the e e u e ed

spikes. If the et o ks f o the full e o di g easo a l at h those e t a ted f o of oth splits, N is i eased, a d the p o ess is epeated u til the o lo ge at h. The et o ks e t a ted f o the full data a e kept, a d those of the splits dis a ded. Othe s e a ios a e also feasi le, su h as a odd-e e t ial split, o a k-split app oa h, i hi h et o ks a e e t a ted f o k su sets of the e o di g a d o pa ed to those e t a ted f o the full e o di g. The u e of splits, a d the a e of splitti g, ill dete i e the se siti it of the app oa h. It is useful he e to ake a te h i al state e t ega di g

the splitti g of i di idual et o ks i to t o o o e s alle et o ks, hi h ould o pli ate the a o e. I o de to a oid su h splitti g, et o k i te a tio te s i the de o positio odel a e fo ed to e ze o i te a tio te s a e ot isi le i the ele e t- ise odel a o e; fo opti izatio details see a de Meij et al., , . A p a ti al o se ue e is that spike ti i g et o ks that a e

ea l pe fe tl o elated sha e spike se ue e ti i g ill likel e e t a ted as a si gle o i ed et o k see a de Meij et al., fo si ulatio s i estigati g et o k o elatio . Fi all , ega di g the a i u u e of et o ks that a e u i uel e t a ted, though o -t i ial to

esti ate Co o et al., , it likel is g eate tha the u e of eu o s. I po ta tl , a ki d of elia ilit p o edu e su h as the a o e ill p e e t e eedi g a a i u , as o -u i ue et o ks ill, defi itio , ot e elia le o e splits. I ou e pe ie e, the u e of elia le et o ks is ofte u h lo e tha the u e of e o ded eu o s.

To dete i e hethe t o et o ks a e si ila , su h as i the a o e split- elia ilit app oa h, a oeffi ie t a e o puted fo the th ee pa a ete s of the et o ks. Fo the eu o a d the t ial

p ofiles, this is si pl the i e p odu t et ee the L - o alized p ofiles of t o et o ks, a d a ges f o to ide ti al p ofiles . Fo the ti e p ofile, a oeffi ie t is the follo i g:

Ti e p ofile si ila it is o puted as the a solute alue of the i e -p odu t o e eu o s of the ti e p ofiles of t o et o ks supe s ipt a d , de otes o ple o jugate , eighted the o alized eu o p ofiles of ea h et o k de otes the ele e t- ise p odu t . He e, sta ds fo the g eatest o o di iso of the f e ue ies used to e t a t the et o ks, i Hz, hi h dete i es the le le gth of the i ula ti e p ofile a de Meij et al., . This si ila it oeffi ie t also a ges f o to ide ti al p ofiles . Allo i g fo so e diffe e es i the p ofiles due

to oise, e o side ed et o ks si ila e ough he oeffi ie ts fo the eu o , ti e, a d t ial, p ofiles a e all e ual to, o g eate tha , . . Softwa e a d ode a essi ilit The te h i ue is f eel a aila le i a pu li GitHu eposito te ed wa de o p

.githu . o / oe e a de eij/ a de o p , togethe ith tuto ials o its use. The tool o also o tai s soft a e to deal ith the p a ti al poi ts a o e. The ode is also a aila le as E te ded Data .

. O tai i g ross spe tra that are opti al for e tra ti g spike ti i g et orks

To e a le to e t a t spike ti i g et o ks e o pute oss spe t a f o i a spike t ai s, o ol i g the spike t ai s ith o ple e po e tials a elets a d o puti g thei oss p odu ts

o e ti e Fig , step - . Doi g so t a sfo s the ti e dela s et ee spikes of diffe e t eu o s, i to phase diffe e es at ultiple f e ue ies. The le gth of the o ple e po e tials, a d thei f e ue , dete i es ho se siti e the oss spe t a a e to o siste t s o - o siste t ti e dela s, a d is des i ed i the follo i g. He e, it is i po ta t to keep i i d that the oss spe t u et ee t o eu o s, is e a tl the o ple - alued su , of the phase diffe e es et ee spikes of eu o a d spikes of eu o that a e o e lappi g afte the o olutio , eighted thei a ou t of

te po al o e lap. Due to the latte , lo g ti e dela s e essa il ha e lo e eighti g i the oss spe t a tha sho t ti e dela s. O tai i g the oss spe t a a e e p essed as follo s:

That is, the oss spe t u et ee eu o eu o s a d at f e ue a d t ial is o tai ed as the oss p odu t, o e ti e, of e to s = o ple o jugate t a spose . The e to fo

eu o is o tai ed the o olutio of the i a spike t ai e to ith a o ple e po e tial u tape ed a elet , is the i agi a u it at f e ue . He e, is a e to of ti e poi ts f o

to , ith ei g the ti e do ai le gth of the o ple e po e tial, a d oth a d a e sa pled at the a i u a hie a le sa pli g ate. Note, the edges of , fo hi h the o ple e po e tial as ot full i e sed i , a e kept.

The ti e do ai le gth of the o ple e po e tials dete i es hi h et ee -spike ti e dela s a o t i ute to the oss spe t a, a d it should e hose ased o the e pe ted a ge of ti e dela s. He e, e ai to e se siti e to ti e dela s of +/- s, a a ge that aptu es o o l o u i g o siste t spike ti i g e.g., Nelso et al., ; Fujisa a et al., ; Saku ai et al., . The opti al ti e do ai le gth fo this a ge is a t ade-off. The lo ge the le gth, the lo e the se siti it ill e to the e pe ted ti e dela s, as the oss spe t a ill efle t a su of o e spike pai s. The sho te the le gth, the igge the atio et ee the eighti g sa ples o e lap of the sho test a d the lo gest e pe ted ti e dela , a d thus the st o ge the ias i se siti it to a ds the fo e . As a o p o ise et ee the t o, e hoose a ti e do ai le gth of t i e of the a i u ti e dela e ish to e se siti e fo , s. This esults i a o e lap of % of the o ple e po e tials sa ples fo

spikes at a dela of +/- s, ha i g a a epta le se siti it ias atio of s sho test; % o e lap to +/- s of : o pa ed to : fo s:+/- s at a sa pli g ate of kHz . Whe the e is o a p io i e pe tatio ega di g the le gth of the ti e dela s, the ti e do ai le gth a e ased o , e.g., a i estigati g of the peak of the et ee - eu o oss o elog a s. I ge e al, it is p efe a le to hoose a le gth that is too lo g athe tha too sho t, as the se siti it ost due to additio al spike pai s i the oss spe t u is u h less tha that of a o e ske ed se siti it ias atio a d the e pe i e tal ost of spike ti e o siste at lo ge dela s ei g i isi le. I o de to fu the edu e the ias of sho t ti e dela s to lo g ti e dela s, the o ple e po e tials should ha e o sta t

ag itude, a d ot e tape ed usi g a pa ti ula i do i g fu tio su h as a Ha i g i do . The f e ue ies of the o ple e po e tials g eatl dete i e the se siti it of the oss spe t a to o - o siste t ti e dela s. I o de to e a i all se siti e to o siste t ti e dela s, the o t i utio to the oss spe t u of all othe ti e dela s should e as s all as possi le. I the te s of

phase diffe e es i the oss spe t a, this is a hie ed he the a e age of the o ple - alued phase diffe e es of the o - o siste t ti e dela s app oa hes a ag itude of . This is the ase fo a f e ue hose le le gth is a i tege ultiple of the ti e do ai le gth hose a o e fo s,

Hz, Hz, Hz, et . , u de the assu ptio that o - o siste t spike pai s a e e uall likel at a ti e dela . To a i e he e, it is u ial to app e iate the fa t that phase diffe e es fo la ge ti e dela s a e eighted lo e tha those fo s all ti e dela s. Fo phase diffe e es o igi ati g f o ti e dela s

et ee s a d the ti e do ai le gth s a o e to ha e a a e age ag itude of , the eighti g oeffi ie ts fo phase diffe e es et ee to , the left side/ uad a t a d of the

u it i le, eed to ha e the sa e su as those fo to , the ight side/ uad a t a d of the u it i le. C u iall , fo the f e ue hose le le gth e uals the a i u ti e dela , the % s allest ti e dela s fall i uad a t , the iddle % of ti e dela s fall i uad a t a d , a d the

% la gest ti e dela s fall i uad a t . E uall u ial, the eighti g is a li ea fu tio of the ti e dela s. As fo a li ea fu tio the su of the fi st % a d the last % of a su set of its alues is e ual to the su of its iddle %, the f e ue ith a le le gth e uali g the la gest ti e dela ill

ha e phase diffe e es f o o - o siste t ti e dela s that app oa h a a e age ag itude of . This also holds fo a i tege ultiple N of this f e ue , as the a o e ill e the ase fo N e ual splits of the ti e dela a ge. Note that to o tai a a e age ag itude of , it is also e ui ed that the su of

eighti g oeffi ie ts fo uad a t a d top of u it i le is e ual to that of uad a t a d otto of u it i le . This s et is easil a hie ed ho e e , as the eighti g oeffi ie ts fo -

s to s p og ess alo g the u it i le i opposite di e tio tha those fo + s to s. As a last ote, the f e ue ies of the o ple e po e tials also dete i e the o ust ess to jitte a ou d a o siste t ti e dela . The lo e the f e ue , the lose the phases of jitte ed ut o siste t ti e

dela s, the highe thei a e age ag itude, a d thus, the o e o ust to jitte . Co ludi g, e o pute oss spe t a o ol i g spike t ai s ith o ple e po e tials of

s le gth, o sta t ag itude, a d at f e ue ies f o Hz to Hz i steps of Hz. Whe i estigati g lo ge ti e-s ale eu o al d a i s, a a alogous set ould e e.g. s le gth, at Hz to

Hz i steps of Hz. The u e of f e ue ies to use is so e hat a it a . I itial si ulatio s sho ed o oti ea le diffe e e e o d f e ue ies a d Hz is al ead e se siti e to jitte , a d si ulated et o ks e e elia l e o e ed f o si ulatio s as little as f e ue ies.

. Nor alizatio s of the ross spe tra The eu o p ofiles of spike ti i g et o ks des i e the off-diago al ele e ts of ea h oss spe t u , efle ti g et ee - eu o spike pai s, a d the diago al ele e ts, i.e. po e , efle ti g the total u e of spikes of eu o s. I ealisti data, the fi i g ates of eu o s a diffe g eatl , esulti g i

la ge diffe e es i po e . Be ause the po e of ea h oss spe t u is t pi all u h la ge tha its off-diago al ele e ts, this a lead to spike ti i g et o ks hose eu o p ofiles a e d i e o e fi i g ates of i di idual eu o s, athe tha o siste t spike ti i g et ee eu o s. A e t e e e a ple is a eu o p ofile ith a o -ze o eighti g fo o l a si gle eu o . Su h a et o k o l des i es the diago al ele e t i.e. fi i g ate of the oss spe t a of the espe ti e eu o , a d should

e o side ed as a a tefa tual et o k. To i ease se siti it to o siste t spike ti i g, i o de to a oid the a o e, po e diffe e es et ee eu o s a e o alized Fig , step . No alizi g po e su h that it is e ual to a Nth oot, su ed o e f e ue ies a d t ials, is o e su h

o alizatio :

He e, is a diago al at i o tai i g o l the diago al ele e ts of . B i easi g N, the po e diffe e es et ee eu o s de ease. Ideall , the po e of e e eu o e o es e ual, i.e. the oss spe t a e o e ohe e spe t a, as this ill ha e the highest se siti it to spike ti e o siste . Ho e e , this a ha e the u i te ded o se ue e of i te fe i g ith the split elia ilit p o edu e fo esti ati g ho a et o ks to e t a t. B iefl , he e t a ti g fe e tha the total u e of

et o ks, hi h et o ks a e e t a ted st o gl depe ds o thei e plai ed a ia e; those ith the ost, te d to e e t a ted fi st as the et o ks a e fou d a a do l i itialized least s ua es

algo ith . Whe the diffe e es i e plai ed a ia e et ee et o ks de ease, the o de i hi h the a e e t a ted e o es o e a ia le, hi h a p e atu el stop the split elia ilit p o edu e. I easi g eu o - ise o alizatio st e gth a esult i de easi g diffe e es i e plai ed

a ia e. As su h, hile eu o - ise o alizatio i eases the useful ess of the et o ks, it a also esult i fi di g less split- elia le et o ks. P a ti all , a opti al o alizatio st e gth a e fou d

as follo s. Fi st, a split elia ilit p o edu e is u ith o alizatio st e gth N = o o alizatio . If this esults i , split- elia le et o ks, a d, et o ks that a e u likel to efle t spike ti e o siste eu o p ofiles that ha e st o g eighti g fo o l o e eu o , the o alizatio

st e gth N is dou led a d the split elia ilit p o edu e epeated. This is epeated u til suffi ie t elia le et o ks a e fou d that efle t spike o siste . A o e ie t ua tifi atio of he et o ks a e u likel to efle t spike ti e o siste is to o pute the atio of the st o gest a d

se o d st o gest eights of the eu o p ofile; the highe this atio, the o e likel the se o d st o gest eight efle ts oise a d that the et o k does ot efle t spike ti e o siste . The , i the a o e p o edu e, a ut-off atio of -to- a e used as a o se ati e ite io see also ‘esults se tio . C u iall , the a o e o l affe ts the p o a ilit of u o e i g those spike ti i g et o ks that al ead e ist i the e o di g, the phase oupli g st u tu e i the oss spe t a i du ed the

et o ks e ai s u affe ted. The fi i g ate of eu o s a also diffe g eatl et ee t ials. Be ause the t ial p ofile efle ts

a iatio s i the oss spe t a o e t ials i its eights, it efle ts oth the t ial-spe ifi fi i g ate of the i ol ed eu o s as ell as thei t ial-spe ifi a ou t of spike ti i g o siste . Si ila to the a o e,

o alizi g the oss spe t a o e t ials a edu e the i pa t of fi i g ate o the t ial p ofile. No alizi g oss spe t al po e su h that it is e ual a oss t ials is o e su h o alizatio :

As a o e, is a diago al at i o tai i g o l the diago al ele e ts of . I po ta tl , i the o o ase of a eu o ot spiki g i a pa ti ula t ial, its ele e ts of the oss spe t a ill e

ze o, leadi g to di isio - -ze o e o s du i g the a o e o alizatio . This is a oided addi g a do oise of t i ial st e gth lose to the used u e i al p e isio to the espe ti e ele e ts of the oss spe t a p io to o alizatio . T ial- ise o alizatio is a hie ed fi st o alizi g f e ue -

a d t ial-spe ifi oss spe t al po e to di isio itself , a d the ultipl i g it ith the f e ue -spe ifi su ed po e o e t ials ; o puted p io to o alizatio . T ial- ise

o alizatio is i depe de t of the a o e eu o - ise o alizatio , oth o alizatio s a e applied joi tl . The o alizatio a o e is e t e e, as it e o es all oss spe t al po e a ia ilit o e t ials. Ho e e , e ause o alizatio o u s ia a diago al at i as is the ase ith eu o -

ise o alizatio , the off-diago al ele e ts of the oss spe t a o l u de go a s ali g p opo tio al to thei diago al ele e ts; thei ag itudes still efle t the elati e a ou t of spike ti i g o siste et ee thei i ol ed eu o s. As su h, the e ai i g t ial- -t ial a iatio s i oss

spe t al ag itudes a i all efle ts t ial- -t ial a iatio s i the a ou t of spike ti i g o siste . Si ila to the eu o - ise o alizatio , the t ial- ise o alizatio a ake it ha de to u o e

et o ks that a e p ese t i the e o di g. This a e dealt ith o e o e ie tl ho e e . Fi st, et o ks a e e t a ted usi g a split elia ilit p o edu e as dis ussed a o e, ithout t ial- o alizi g oss spe t a. O e elia le et o ks a e o tai ed, the t ial p ofiles a e e-esti ated i o e fi al

de o positio usi g oss spe t a that a e additio all t ial- o alized, i hi h the eu o a d ti e p ofiles a e kept o sta t. Although the a o e -step app oa h is ad ised, i the ase of ou si ulatio s the diffe e es e e egligi le, a d fo si pli it the esults that a e p ese ted e e e t a ted f o t ial- ise o alized oss spe t a i o e step.

. Si ulati g a d e tra ti g ois spike ti i g et orks To i estigate the effe ts of a ious ki ds of oise o spike ti i g et o k e t a tio , e si ulated spike e o di gs of eu o s at t ials of s o tai i g spike ti i g et o ks. Net o k spiki g se ue es had a fi ed te po al st u tu e that as epeated et ee - p edete i ed pe t ial

. Hz a e age spike se ue e ate fo ea h et o k . Withi ea h t ial, ea h spike se ue e ould o u a he e ith u ifo p o a ilit , ith a s offset f o t ial ou da ies. O t ials he e spike se ue es of ultiple et o ks e e p ese t thei o de as a do ized, a d ith a i i u of

s et ee se ue es. Th ee ki ds of oise e e si ulated. Fi st, all eu o s of a si gle si ulatio had a oise spiki g ate of Hz, Hz, Hz, Hz, o Hz, as Poisso spiki g supe i posed o the

et o k spike se ue es. Se o d, ea h spike of ea h spike se ue e ould ha e a i di idual a do jitte u ifo l dist i uted at a a i u of s, . s, . s, s, o s. Thi d, ea h spike i ea h spike se ue e o u e e had a i di idual deletio p o a ilit of %, %, %, %, %, esulti g i pa tial spike se ue es. C oss spe t a of ea h si ulatio u e e o tai ed as des i ed

a o e, usi g a ti e- i do le gth of s a d f e ue ies f o Hz to Hz i steps of Hz. The et o ks e e e t a ted usi g a do i itializatio s of the e t a tio algo ith . Note that the

pu pose of these si ulatio s is to sho ho ell spike se ue es a e e t a ted u de ois o ditio s, a d ot ho su h a patte a e ge e ated ph siologi all , o hethe su h a patte is

ph siologi all ea i gful. As su h, e si ulated data f o the pe spe ti e of spikes, i stead of odel eu o s ge e ati g spikes, hi h also p o ides a o e ie t g ou d t uth fo al ulati g e o e .

. Qua tif i g re o er of si ulated spike ti i g et orks

To ua tif the e o e of the e t a ted eu o p ofile, ti e p ofile, a d the t ial p ofile, the e e o pa ed to thei si ulated e ui ale ts. The si ulated eu o p ofiles e e o st u ted as a i a

e to pe et o k, its alues i di ati g et o k e e ship of ea h eu o . Si ila l , si ulated t ial p ofiles e e o st u ted as e to , its alues efle ti g the u e of se ue e epeats li ea odulatio of et o k a ti it o e t ials . Fi all , si ulated ti e p ofiles e e o st u ted as a

e to , its alues des i i g the te po al se ue e of spikes i se o ds o - e e eu o s set a it a il to . Fo displa pu poses these si ulated pa a ete s e e o alized i the sa e a e as the e t a ted et o k pa a ete s. I o de to o pute e o e , e t a ted et o ks e e pai ed to the si ulated et o ks usi g the si ila it oeffi ie ts des i ed a o e, fi st dete i i g the ost si ila pai , the the e t ost si ila i the e ai de , et . ‘e o e of eu o a d t ial p ofiles as dete i ed usi g a Pea so o elatio oeffi ie t. Ti e p ofile e o e as judged the follo i g oeffi ie t:

Ti e p ofile e o e is o puted as the a solute alue of the eighted su o e eu o s of the o ple - alued diffe e e of the i ula ti e p ofiles of the e t a ted a d si ulated et o ks supe s ipt e a d s esp.; de otes o ple o jugate , eighted the si ulated eu o p ofile

de otes the ele e t- ise p odu t . Si ila to the si ila it oeffi ie t des i ed a o e, is the g eatest o o di iso of the f e ue ies used fo e t a tio i.e., Hz , a d is used to deal ith the i ula it of the ti e p ofile. This oeffi ie t a ges f o to pe fe t e o e .

. E tra ti g spike ti i g et orks fro re ordi gs of rat edial prefro tal orte a d hippo a pus

As a p oof of p i iple, e e t a ted spike ti i g et o ks f o eal spiki g e o di gs, o tai ed f o a dataset pu li all a aila le at C‘CNS.o g Fujisa a et al., , . This dataset o tai s ide tified

eu o s a d thei s spikes f o e o di gs o tai ed f o at edial p ef o tal o te a d a ea CA of the hippo a pus, hile the at pe fo ed a odo - ased dela ed at hi g-to-sa ple task, e ui i g it to u th ough eithe the left o ight a of a aze to o tai its e a d. A i al e o di g p oto ols

e e app o ed the I stitutio al A i al Ca e a d Use Co ittee of ‘utge s U i e sit , Ne a k, NJ, USA. The e o di g used at GG. a e f o a d ele t ode sha ks μ sha k sepa atio i

edial p ef o tal o te a d CA espe ti el , ea h sha k o tai i g o ta ts μ o ta t sepa atio ; μ o ta t su fa e . The e o di g as sa pled at kHz, a d offli e spike so ti g as pe fo ed afte a d-passi g et ee . - kHz usi g KlustaK ik fo spike so ti g details see Fujisa a et al., , . eu o s e e ide tified o sha ks. The dataset o tai ed left a d

ight t ials, ha i g a a e age du atio of . s SD = . s . O l eu o s ith a e age spiki g ates

of Hz a d a o e e e sele ted. I o de to e t a t et o ks, e fi st o tai ed oss spe t a as des i ed i Mate ials a d Methods se tio , usi g a ti e i do of s a d f e ue ies f o Hz to Hz i steps of Hz, di idi g ea h oss spe t a its t ials du atio . Su se ue tl , oss spe t a

e e eu o - ise o alized as des i ed a o e. A o alizatio su h that oss spe t al po e as e ual to its d oot o alizatio as hose as a opti al o alizatio , e ause th oot

o alizatio esulted i o split elia le et o ks likel due to a i g et o k o de e tio ed i i Mate ials a d Methods se tio , a d th oot o alizatio esulted i a et o ks ostl o sisti g of si gle eu o s i.e. the ias to a ds po e diffe e es et ee eu o s as too st o g to

o e o e . The u e of et o ks to e t a t as dete i ed usi g odd-e e spike split elia ilit p o edu e des i ed i Mate ials a d Methods se tio ith a si ila it oeffi ie t ut-off of . , a do i itializatio s e e used at ea h step. This esulted i et o ks ei g e t a ted. Co ti uous oss- o elog a s e e o tai ed at ti e lags of s at . s steps su atio of the lagged

i a spike t ai s afte the e e o ol ed ith a Gaussia ith full- idth at half- a i u of . s a i u = .

Results Spike ti i g et o ks o sist of ultiple eu o s that ha e o siste t ti e dela s et ee thei spikes, fo i g a spike se ue e. He e, e alidate a o el app oa h fo fi di g a d ha a te izi g these et o ks i eu o al spike e o di gs. Fi st, e e aluate its o ust ess to a ious oise o ditio s. We sho ho the e o e of si ulated spike ti i g et o ks is affe ted spike jitte i

the et o k spike se ue es a d a ia ilit of eu o pa ti ipatio i the et o k, u de i easi g spiki g oise of all si ulated eu o s. The , e sho ho a ia le fi i g ates of eu o s affe ts e o e , a d hat a tio s a e take to edu e egati e effe ts. Fi all , e p o ide a p oof-of-

p i iple sho i g et o ks e t a ted f o at hippo a pus a d edial p ef o tal o te Fujisa a et al., , , a d o pa e the e t a ted spike ti i g elatio s to oss- o elog a s of the i ol ed eu o s.

. Si ulated spike re ordi gs fro spike ti i g et orks To i estigate the o ust ess of spike-ti i g et o k e t a tio to a ious ki ds of oise e si ulated spike e o di gs of eu o s o e t ials of s o tai i g et o ks fo si ulatio details see Mate ials a d Methods se tio . A et o k as defi ed as a g oup of eu o s that spike i se ue e,

ith et ee -spike ti e dela s a gi g f o s h o ous to . s. The spike se ue e ti eli es e e - - - . - . - - . - . s fo et o k , - - - - s fo et o k , - - - s fo et o k all

s h o ous , a d - . - . s fo et o k . Ea h et o k s spike se ue e as epeated - ti es i g oups of t ials to si ulate li ea odulatio s of et o k a ti it a oss the task. So e of the et o ks had eu o s that e e i ol ed i othe et o ks spike se ue es: all of the eu o s of the spike se ue e of et o k e e also pa t of et o k , a d o e eu o as sha ed et ee et o k a d

, a d et o k a d . I Figu e A-C e sho the et o ks eu o p ofiles Fig A , ti e p ofiles Fig B , a d t ial p ofiles Fig C . The si ulated e o di gs of the eu o s esult i a pai - ise spike

ti e elatio ships et ee eu o s, hi h e sho s he ati all i Figu e D. These pai - ise elatio ships a also e isualized as oss- o elog a s fo all eu o s, hi h e sho i Figu e E.

The p ofiles i Figu e A-C a e di e tl o pa a le to the th ee p ofiles of spike ti i g et o ks e t a ted usi g ou app oa h see Mate ials a d Methods se tio fo ho to i te p et the p ofiles , a d a e used i ‘esults se tio s - fo judgi g e o e of the si ulated et o ks the e t a ted spike ti i g et o ks. Note, the a solute alues of the eu o a d t ial p ofiles a e ot ea i gful, o l the withi - etwo k atios a e see Mate ials a d Methods se tio . As su h, it is ot the spike se ue e epeats pe t ial that is des i ed the t ial p ofile i.e. , , , ; Fig C , ut athe the atio

et ee the e.g. a t ial ith se ue es ha i g a eight that of a se ue e t ial . To i estigate he the e o e of the si ulated et o ks fails, e a ied the st e gth of th ee ki ds of oise see Mate ials a d Methods se tio . These e e: spiki g oise, o o -

et o k spikes, supe i posed o the spike se ue es see Fig F , jitte of ea h spike i a spike se ue e o u e e, a d, pa tialit of et o k spike se ue es a do spikes issi g f o the se ue e . The a ge of ea h of the oise le els as hose to p o ide a i tuitio fo he a e pe ted et o k a still e e o e ed, a d to p og essi el esult i failu e to e o e the si ulated

et o ks. As su h, the highe le els a e ot e essa il ph siologi all easo a le. The si ulated et o ks e e also diffe e t i size, spike se ue e ti i g, spatial o e lap, a d t ial o e lap, to i ease

the likelihood that a elated eak esses of the te h i ue ould e e ealed.

. Re o er of si ulated spike ti i g et orks ith spiki g jitter he surrou ded spiki g oise Neu o s a e ois , a d a spike se ue e of a spike ti i g et o k is likel e edded i othe spikes of the sa e eu o s. Fu the o e, p e ise spike ti es depe d o the flu tuati g e a e pote tial of the eu o a d othe fa to s, pote tiall addi g te po al jitte . To i estigate ho these

t o fa to s i flue e et o k ha a te izatio , e si ulated spike ti i g et o ks ith diffe e t le els of a kg ou d spiki g oise a d ith diffe e t le els of jitte of ea h spike i the spike se ue es. Net o ks e e si ulated ti es fo ea h o i atio of the oise fa to s. We o puted e o e of si ulated et o ks a d sho the esult i Figu e . ‘e o e of the eu o a d t ial p ofiles as o puted as the Pea so o elatio et ee the e o e ed a d si ulated p ofiles. Fo the t ial

p ofiles, the o elatio also di e tl efle ts the e o e of the li ea odulatio of et o k a ti it a oss the task pe fe t e o e is . ‘e o e of the ti e p ofiles as o puted usi g a e o e oeffi ie t that a ges f o to pe fe t e o e ; see Mate ials a d Methods se tio .

Fi stl , e o se e that ith easo a le jitte i.e. . s o pa ed to - . s spike se ue e dela s a d spiki g oise e.g. Hz s . Hz a e age et o k spiki g ate the eu o a d ti e p ofiles e e e o e ed ith easo a le a u a , ith the t ial p ofile ei g the ost affe ted. At

Hz spiki g oise a d . s jitte the li ea odulatio of et o k a ti it as still isi le ut eake ed Fig , otto : ea SEM o e si ulatio s of Pea so s o elatio s fo et o k - : . . , . . , . . , . . . Sho i the e a ples Fig , otto , the effe t of oise o

the t ial p ofile a e o se ed as a sh i ki g of the atios et ee loadi gs of t ials ith a diffe e t u e of si ulated et o k se ue es a d a i ease i the t ial p ofiles aseli e ; the loadi gs of

those t ials hi h had et o k se ue es. The latte is i po ta t i p a ti e, e ause u de the assu ptio that a et o k is ot a ti e i all t ials, the lo est t ial loadi gs ith espe t to the highe t ial loadi gs a e used as a i di atio of the elia ilit of et o k pa a ete s. Se o dl , e o se e that, e ept f o the la gest jitte ase s , et o k spike jitte had a si ila effe t o e o e of

et o k pa a ete s as spiki g oise, as e ide ed the si ila it et ee the Hz/ s a d the Hz/ . s ases, a d the Hz/ s a d the Hz/ . s ases. Thi dl , e o se e that, u de

st o g oise o ditio s > Hz spiki g oise a d > s jitte , the li ea odulatio of et o k a ti it e a e e eak to la gel i isi le a i u ea Pea so s o elatio o e si ulatio s of . , . , . , . fo et o k - . ‘ega di g et o k spe ifi e o e , though the e as so e a iatio

i e o e , apa t f o the a o e, the diffe e es e e i i al a d did ot highlight a se siti it to a pa ti ula aspe t of the si ulated et o ks.

. Re o er of si ulated spike ti i g et orks ith partial spiki g he surrou ded spiki g oise

To i estigate ho pa tial spiki g i spike ti i g et o ks, i.e. ot all e e eu o s joi i g i ea h spike se ue e, affe ts ha a te izatio of the full spike se ue es, e si ulated et o ks he e ea h spike of a se ue e had a ha e to e deleted. Si ila to the a o e, e did so ti es fo ea h le el of spike deletio p o a ilit , a d of spiki g oise. The esults a e sho i Figu e . We o se e that,

as the ha e of spike deletio i eased, e o e a u a as de eased, as ith spiki g jitte / oise, the t ial p ofile as o e affe ted oise tha the eu o p ofile, as ith spiki g jitte / oise, the effe ts of spike deletio o e o e e e si ila to those of spiki g oise, the full spike se ue es i the ti e p ofile e e a u atel e t a ted u de easo a le oise Hz ith % p o a ilit of spike deletio , e e though the ajo it of i di idual spike se ue es e e i o plete, a d, u de the sa e oise o ditio s the li ea odulatio of et o k a ti it as eak ut dete ta le fo et o ks a d , a d ea l i isi le fo et o ks a d ea SEM Pea so s o elatio o e si ulatio s of . . , . . , . . , . . fo et o ks - . These

diffe e es possi l ste f o et o k o e lap et o k sha es all its eu o s ith et o k a d et o k size et o k is the s allest .

. Cross spe tra or alizatio di i ishes effe ts of differe tial firi g rates of u its a d trials

The spike ti i g et o ks si ulated a o e e e e t a ted u de oise elated fi i g ates that e e ide ti al o e eu o s a d o e t ials. This as hose to sho the o e all effe t of spiki g oise, ut is

at pi al fo eal e o di gs. He e, e sho the effe t o et o k e o e of fi i g ate diffe e es et ee eu o s a d t ials, hile keepi g the u e of spike se ue es o sta t.

We fi st sho the e o e of si ulated et o ks he the fi i g ate diffe s o e eu o s Fig . We si ulated et o ks ti es, ith eu o a e e i et o k a d a d eu o e e i et o k a d ha i g Hz spiki g oise, the othe eu o s Hz Fig A . Net o k

spiki g jitte as . s. The e o e of the et o ks Fig B; o pa e to Fig as disto ted: the eu o p ofiles of the et o ks fo eu o s a d e e st o gl i eased/de eased, the oise

of eu o led to a st o g loadi g fo et o k , of hi h it as ot a e e , et o k as do i ated eu o , , the t ial p ofiles sho ed de eased e o e o pa ed to Fig , a d although the ti e p ofile of et o ks , , a d e e ot oti ea l disto ted, et o k s is. O e all, the diffe e tial fi i g ate a e said to ha e pulled the esti ated et o k pa a ete s to a ds those

eu o s ith o e spiki g. This effe t ho e e , a e su sta tiall edu ed o alizi g oss spe t a p io to et o k e t a tio . He e, e sho the effe t of o alizi g oss spe t a su h thei po e is e ual to thei Nth oot see Mate ials a d Methods se tio , edu i g diffe e es i fi i g ates. We sho its effe ts p og essi el usi g N = , , , , Fig C , a d e di g ith N = Fig D . We o se e that the e o e of the eu o p ofiles as i p o ed, ith et o k sho i g the ost e ai i g disto tio at eu o , the t ial p ofiles e e si ila to the ase ith Hz spiki g

oise fo all eu o s see Fig , a d, the e o e of the ti e p ofile of et o k as i p o ed su h that the disto tio is i i al. To i estigate the effe t of diffe e tial spiki g ate o e t ials e si ulated et o ks ith Hz spiki g oise, e ept fo t ials to , hi h had Hz spiki g oise Fig A . Net o k spiki g jitte as set at . s spiki g jitte , a d et o ks e e si ulated ti es. The t ials ith additio al spiki g

oise e e hose su h that the oth i ol ed % of t ials of se ue e epeats a d fo et o k , fo et o k , fo et o k a d a pa tial set of se ue e epeats % of a d

fo et o k , % of fo et o k . The e o e ithout o alizatio is sho i Figu e B. We o se e that the e o e of the eu o p ofiles a d ti e p ofiles as si ila to the ase of Hz

oise a d . s spiki g jitte see Fig a d, as su h, the e e i i all affe ted the diffe e tial oise o e t ials, , the li ea odulatio of et o k a ti it as e o e a le, ut eake ed, fo all et o ks espe iall et o k ; ea SEM Pea so s o elatio o e si ulatio s of . < . , . . , . < . , . . fo et o k - o pa ed to ithout t ial a iatio s of fi i g ate .

< . , . < . , . < . , . . ; Fig , a d, the t ial p ofile loadi gs fo those t ials affe ted i eased spiki g oise e e disto ted su h that the atios of loadi gs o lo ge efle ted the o e t o de of the u e of se ue e epeats i.e. > t ials fo et o k a d . Though the li ea odulatio as ode atel e o e a le, the latte ea s a i estigatio of the et o k a ti ities i spe ifi t ials of et o k a d suppo ted e.g., a i depe de t sa ples t-test ould ha e esulted i the i o e t o lusio of o e et o k a ti it ei g p ese t i o pa ed to t ials. As as the ase fo diffe e tial oise o e eu o s, o alizatio of the affe ted di e sio a i p o e e o e . He e, e o alized the oss spe t a su h that thei po e fo e e t ial is e ual to thei po e su ed o e t ials see Mate ials a d Methods se tio . C u iall , this does ot affe t the atio of the off-diago al ele e ts to the diago al po e . As su h, t ials that ha e a spike

se ue es st o g off-diago al ele e ts o pa ed to po e a e still disti guisha le f o t ials ith fe spike se ue es eak off-diago al ele e ts o pa ed to po e . Note, as ell, that this t ial- ise

o alizatio is u elated to the eu o - ise o alizatio i the a o e, a d the a e applied joi tl . We sho the esult of the t ial- ise o alizatio i Figu e C. We o se e that, the t ial p ofile e o e as i p o ed su h that the o de of thei loadi gs agai efle ted the o de of the

u e of se ue e epeats, , e o e of the li ea odulatio of et o k a ti it as g eatl i p o ed ea SEM Pea so s o elatio o e si ulatio s of . < . , . < . , . < . ,

. < . , a d, though i p o ed, the t ial p ofiles e o e as poo e tha those at e ual oise

le els fo all t ials see Fig . We additio all o se e that the o alizatio also affe ted the t ial p ofile loadi gs of t ials that did ot ha e i eased oise. This as ost oti ea le i the loadi gs fo those t ials of et o k that had a d se ue e epeats t ials to a d to : the atio of the loadi g of epeats to that of epeats as u h highe ithout o alizatio Fig B , tha ith

o alizatio Fig C . As t ials ith epeats should ideall ha e a loadi g of , the highe this atio the ette . I te esti gl , e e though the t ial p ofile sho ed o se e o e o e all, the e o e of the

li ea odulatio of et o k a ti it afte t ial- ise o alizatio as ette tha the e o e at e ual oise le els a oss t ials ithout o alizatio , espe iall fo et o k ea SEM o elatio of . < . a d . . esp. . This as likel aused the t ial p ofiles of the fo e sho i g less a ia ilit tha those of the latte i.e., the oeffi ie t of a iatio of the t ial p ofile of et o k , a e aged o e si ulatio s, as . % a d . % esp. .

. Spike ti i g et orks e tra ted fro real re ordi gs refle t et ee - euro spike ti i g relatio ships

To p o ide a p oof-of-p i iple e e t a ted spike ti i g et o ks e t a ted f o spike e o di gs f o edial p ef o tal o te a d hippo a pus of a at pe fo i g a odo - ased dela ed at hi g-to-

sa ple task Fig ; see Mate ials a d Methods se tio . Afte odo p ese tatio , the at had to u th ough the left o ight a of a figu e-eight T- aze to o tai its e a d. Net o ks e e e t a ted si ila l to the si ulatio s a o e, usi g a eu o - ise d oot po e o alizatio , a d a split-half elia ilit app oa h to dete i e the u e of et o ks see Mate ials a d Methods se tio . This esulted i e t a ted et o ks.

We sho eu o p ofiles, ti e p ofiles, a d t ial p ofiles fo ea h e t a ted spike ti i g et o k i Figu e . To p o ide a g ou d-t uth esti ate of hethe the et ee - eu o spike ti es

f o the et o ks efle t eal spike ti i g elatio ships i the e o di gs, e also sho fo ea h et o k o ti uous oss- o elog a s o puted post-ho ; see Mate ials a d Methods se tio of

the eu o s ostl st o gl o t i uti g to ea h et o k. I po ta tl , i ea h of these oss- o elog a s e i di ate he the oss- o elatio is e pe ted to e highest, ased o the ti e p ofile

of the et o ks. Fo et o k , eu o pai s - , - , a d - had peaks i thei oss- o elog a s that

at hed the ti e p ofile s spike ti i g elatio ships ithi . s, . s, a d . s espe ti el . Neu o does ot appea to ha e o siste t spike ti i g elatio ship ith the fi st th ee, hi h is u su p isi g gi e that its eight i the eu o p ofile is u h eake suggesti g its eight efle ts, at least ostl , oise . Although the e appea s to e a diffe e e i et o k a ti it et ee left a d ight t ials, this likel due to fi i g ate diffe e es et ee o ditio s, as t ial p ofiles al ulated o t ial- ise

o alized oss spe t a sho ed o statisti all sig ifi a t diffe e e see Mate ials a d Methods se tio ; this should e i te p eted ith autio ho e e , as the p ofile s lo est eights suggested the e ai ed ois . Fo et o k , eu o pai s - , - , a d - had oss- o elog a peaks that

at hed the ti e p ofile ithi . s, . s, a d . s espe ti el . Neu o pai - a d pai - did ot ha e si gle oss- o elog a peak though thei e te peaks at hed ithi . s a d

. s esp. , a d eu o pai - appea s i hi ito . These o se atio s ould i di ate that the spike se ue e did ot i ol e all eu o s i a su set of t ials. It is useful to eite ate he e, that the e t a ted spike se ue e should e o side ed o l at the le el of the full e o di g i.e. oss spe t a of all t ials . That is, the e t a ted spike se ue e should e o side ed as a des iptio of the N- a elatio ship et ee N eu o s, i.e. the la gest possi le spike se ue e fo the et o k, a d se e as a

sta ti g poi t fo ta geted a al ses. Net o k sho a si ila patte as et o k a d a o di g the oss- o elog a s, ith the peaks of eu o pai s - , - , - , a d - , at hi g the ti e p ofile ithi . s, . s, . s, a d . s esp. Net o k likel efle ts o siste t spike ti i g o l

et ee the st o gest t o eu o s at hi g ithi . s , as the eu o p ofile has fe eu o s ith st o g loadi gs.

Dis ussio Ide tif i g a d i estigati g ell-asse lies ith spike ti i g o siste et ee eu o s is ke to gai a fu the u de sta di g of thei ole i eu o al odi g Bie e sto k, ; Si ge , ; Tiesi ga et al., ; Pa ze i et al., , ut fi di g the is a t e e dous halle ge due to the possi le o ple it of patte s of et ee - eu o spike ti e dela s. He e, e i t odu ed a d alidated, i

si ulated a d eal data, a o el app oa h fo e t a ti g et o ks defi ed thei et ee - eu o spike ti i g o siste , he fo i g se ue es of ti e-shifted spikes, f o eu o al spike e o di gs fo othe t pes of i te a tio s, see e.g. Li de a et al., . The ke featu es of this

app oa h a e that et o ks a d thei spike se ue es a e e t a ted ega dless of thei o ple it i size a d spike ti i g patte s, a d the spike se ue es of the et o ks a e spe ified ith high te po al p e isio . Net o ks o sist of th ee p ofiles, des i i g hi h eu o s a e

i ol ed i hi h et o ks, ith hi h spike ti i g patte , i hi h t ials o o ditio s. The latte a i p i iple e used as a i de fo et o k a ti it . Togethe , these p ofiles fo a pa si o ious des iptio of the spike ti i g patte s i the e o di g, a d a used as a asis fo su se ue t spike t ai a al ses of e pe i e tall ele a t a iatio s i et o k su sets. Usi g si ulatio s, e sho ed ho the e t a ted et o ks e e affe ted spiki g jitte , a ia ilit i

et o k pa ti ipatio its e e eu o s, a d o - et o k elated spiki g a ti it . Net o ks e e e o e a le u de easo a le oise o ditio s, ith the ti e p ofile ei g espe iall o ust to the

si ulated oise. Though the t ial p ofiles e e st o gl i flue ed oise, the still t a ked si ulated et o k a ti it to a deg ee. Usi g eu o al spike e o di gs f o ats, e sho ed e e e a le to

e t a t et o ks f o eal e o di gs, of hi h the ti e p ofile efle ted et ee - eu o spike ti i g o siste that at hed oss- o elog a s ith high a u a . Togethe , this sho s that ou

app oa h a e useful fo the i estigatio of spike ti i g et o ks. The e t a ted et o ks a e of a it a o ple it i size a d ti e dela s. This is a o se ue e of the fa t that the u de l i g ethod fi ds et o ks ot i the eu o - -ti e ti e

se ies, ut i the eu o - - eu o oss spe t a. These oss spe t a o tai all of the spike ti i g o siste ies of the spike se ue es, o de sed i to et ee - eu o phase oupli g. Net o ks a e

sepa ated he thei spike se ue es ha e diffe e t et ee - eu o phase oupli g patte s, a d diffe e es i phase oupli g patte s o e t ials o epo hs i eases thei sepa a ilit . Net o ks a e e t a ted fi di g those eu o , ti e, a d t ial p ofiles hose phase oupli g patte s e plai the

ost a ia e i the oss spe t a. Be ause the esti ated p ofiles ha e the sa e size fo ea h et o k, la ge et o ks o l diffe f o s alle et o ks thei diffe e t dist i utio of eight ag itudes. As a la ge et o k does ot i ol e esti ati g a la ge u e of eights, the e is o o i ato ial e plosio ith i easi g et o k size. I fa t, highe o ple it et o ks a e likel easie to fi d tha lo e o ple it et o ks, as the ill t pi all e plai o e a ia e i the oss spe t a. The a o e is diffe e t f o te h i ues that sea h fo te plate spike se ue es i thei o igi al eu o - -ti e ep ese tatio s A eles a d Ge stei , ; Nadasd et al., ; Tetko a d Villa, ; Lee a d Wilso ,

; S h itze a d Meiste , ; Ikega a et al., ; Ga sel a d Si ge , . As these sea h fo e a t spiki g te plates, the ha e to do so ithi so e est i ted spa e to a oid a o i ato ial e plosio . Although fi di g high o ple it et o ks is i p a ti al ith su h app oa hes, the ha e the ad a tage of ei g a le to fi d spike se ue es that epeat e fe ti es i the ou se of a e o di g. Be ause ou app oa h is ost se siti e to those et o ks that e plai the ost a ia e i the oss spe t a, it is ot ell suited fo fi di g se ue es ith e fe epeats, as the t pi all e plai e little a ia e i the oss spe t a. As su h, ou app oa h t ades se siti it to su h se ue es fo se siti it to se ue es ith a it a il high o ple it , ut that a e o e p o i e t. A i po ta t aspe t of the ethod ehi d ou app oa h is that it is a de o positio of

et ee - eu o oss spe t a o e f e ue ies a d t ials i to sets of et o k p ofiles. Be ause this

de o positio atte pts to fi d p ofiles that pa si o iousl e plai all of the a ia e i the oss spe t a, its p ofiles eed to ot o l des i e et ee - eu o spike pai s, ut also thei total u e of spikes. I po ta tl , the latte t pi all out u e the fo e to a st o g deg ee e.g., Go hi et al.,

; Nelso et al., ; K eite a d Si ge , ; B os h a d S h ei e , . This i pa ts the i te p etatio of the eu o p ofile eights. Fo a t o eu o s, thei eights i the eu o p ofile

eed to des i e fou ag itudes of the oss spe t a: thei total u e of spikes i the ag itude of thei oss spe t al po e , a d thei spike ti i g o siste i the ag itude of thei off-diago al ele e ts. I the ase these ag itudes diffe , the eu o p ofile eights e o e a o p o ise, a d a e d a to those ag itudes that e plai the ost a ia e. These eights should the efo e e i te p eted ith autio , a d should e o side ed o e as a i di atio of et o k e e ship he suffi ie tl a a f o , tha as a st aightfo a d i de i to the st e gth of thei spike ti i g o siste . This is also the easo h a eu o - ise o alizatio of the oss spe t a is ad isa le, as

it ill edu e the effe t of fi i g ate diffe e es. I fa t, i ou e pe ie e, he the oss spe t a a e ot eu o - ise o alized, fe e t a ted et o ks ill o sist of o e tha o e eu o i.e.,

a tefa tual et o ks that a e ot ased o spike ti i g . If it is also the ase that the total u e of spikes of eu o s diffe s o e o e t ials tha the u e of thei spike pai s do, the the t ial p ofile

eights ill e d a to a ds the fo e , as the ill e plai o e a ia e i the oss spe t a. This as likel the ase fo the et o ks e e t a ted f o at hippo a pus a d edial p ef o tal o te

see e.g. et o k i Fig , a d is also likel the easo h the t ial p ofile as st o gl i pa ted si ulated spiki g oise. A t ial- ise o alizatio fo this as i t odu ed, that i the spe ifi ase of ou si ulatio s, i p o ed e o e of the li ea odulatio of et o k a ti it . No etheless, the t ial p ofile e ai ed se siti e to oise a d, as su h, should e used ith autio , ideall ith o ple e ta a al ses su h as a ta geted sea h, see elo . The a o e o t asts ith a p e ious

appli atio o hu a ele t oph siologi al e o di gs, he e the t ial p ofile as less se siti e to oise likel aused o e spatiall e te ded et o ks; a de Meij et al., , .

Ou app oa h des i es the st u tu e of spike ti i g o siste ies i the oss spe t a of the e ti e e o di g. This ea s that the spike se ue es ep ese ted ea h et o k s ti e p ofile des i e the spike ti i g o siste ies of the i ol ed eu o s o e the e ti e e o di g. As su h, the ti e p ofile efle ts a agg egate spike se ue e, o e that does ot e essa il e a tl epeat i ea h of the i ol ed t ials. Fo e a ple, so e t ials ight o l o tai a pa t of the se ue e. This p ope t a also e o side ed e efi ial, a d it is so ethi g e e pli itl tested i ou si ulatio s ith pa tial spike se ue es. I the ase of st o g a ia ilit i the e a t spike se ue e of e e t ial, the ai se ue e ould still e ide tified. The agg egate atu e of ou spike se ue es o t asts ith those of app oa hes that sea h fo e a tl epeati g spike se ue es A eles a d Ge stei , ; Nadasd et al., ; Tetko a d Villa,

; Lee a d Wilso , ; S h itze a d Meiste , ; Ikega a et al., ; Ga sel a d Si ge , . These app oa hes t pi all also i o po ate so e fo of statisti al testi g of the ide tified spike se ue es, hi h is e essa to o tai o e e tai t that the fou d se ue es a e ot a a ide tal o se ue e of statisti al p ope ties of the fi i g ates see e.g. G ü , , fo a dis ussio of su ogate

data fo this pu pose . I po ta tl , e o side ou app oa h ot as a alte ati e to the a o e, ut as o ple e ta . That is, ou spike ti i g et o k p ofiles a e used to o st u t a et o k-spe ifi

spiki g te plate ith et ee -spike ti e dela s at high te po al esolutio , that a e used i app oa hes like the a o e to lo ate dis ete o u e es of the et o k s spike se ue es. This ould allo fo su se ue t i estigatio s i to, e.g., spike ti e a ia ilit ithi se ue es, a ia le o u e e of se ue es o e o ditio s, a d spike se ue e o plete ess, of et o k spike se ue es ith a o ple it that ould othe ise e p ohi iti e. A gua l the app oa hes losest to ou s a e those that also depe d o eu o - - eu o ep ese tatio s to i estigate spike ti i g o siste . Of these app oa hes, so e sta t f o a

P i ipal Co po e t A al sis PCA o the et ee - eu o oss- o elatio s Chapi a d Ni olelis, ; Pe a he et al., ; Lopes-dos-Sa tos et al., . These app oa hes esult i a eu o p ofile

pe o po e t, des i i g o elated a d a ti- o elated eu o s, a d a te po al p ofile, p o idi g a o po e t a ti it ti e ou se of so e fo that a e at hed to the o igi al eu o al spiki g ti e

se ies. The iggest diffe e e to ou app oa h, is that, i those ethods, et ee - eu o ti i g i fo atio is lost he t a sfo i g the eu o al spiki g ti e se ies to oss- o elatio at i es. Apa t f o losi g the spe ifi atio of the o de a d ti i g of the et o k spiki g se ue e, this also

akes it o e diffi ult to disti guish et ee those et o ks that i ol e the sa e eu o s, ut at diffe e t et ee - eu o spike ti es. This adds u to the otatio al a iguit of PCA that i flue es

et o k ide tifi atio a d sepa atio , although Lopes-dos-Sa tos et al., ade sig ifi a t ad a es ith espe t to the latte . I o pa iso , the ethod ehi d ou app oa h a de Meij et al., , a d elated ethods Ha sh a a d Lu d , ; B o, ; Kie s et al., ; Sidi opoulos et al., ; Mo up et al., , e t a ts et o ks that a e u i ue ithout otatio al a iguit , a d sepa ates the o the asis of thei diffe e t st u tu e a oss eu o s, f e ue ies, a d t ials. Se e al othe app oa hes use oss- o elatio at i es i a that did allo fo a i estigatio of et ee -

eu o spiki g at ti e dela s S h eide et al., ; Nikoli , ; Hu ph ies, , ut these app oa hes e e ot ta geted at ide tif i g a d sepa ati g et o ks a d thei spiki g se ue es. I su a , e ha e p ese ted a app oa h that a e t a t et o ks defi ed thei

et ee - eu o spike ti i g o siste , ith a it a et o k size a d high te po al p e isio of the ide tified spike se ue es. Espe iall the latte is i po ta t o side i g the g o i g u e of

eu o s that a e e o ded si ulta eousl , a d the o ple it of spike se ue es that a thus e easu ed. Ulti atel , the useful ess of ou app oa h a d those elated to it, lies i hethe spike

ti i g pla s a u ial ole i la ge, dist i uted, eu o al et o ks. Bei g a le to sea h fo these et o ks ith i eased se siti it is esse tial to the i estigatio of thei e iste e a d fu tio .

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Figure aptio s Figu e . S he ati of e tra ti g spike ti i g et orks. Neu o al spiki g ti e se ies a o tai o siste t spike ti i g et ee eu o s, fo i g spike se ue es. A, s he ati of t o spike ti i g et o ks, ith thei eu o s i les a d se ue e spikes e ti al li es olo ed lue a d g ee . The

dashed li es efle t the et ee - eu o o siste t spike ti e elatio ships esulti g f o the spike se ue es. The lue et o k s se ue e goes f o eu o to to da k dashed li es , ith s ti e dela s, esulti g i a s dela f o to light dashed li e . The g ee et o k s se ue e is the sa e ut f o eu o to to . B, the spike ti e o siste ies i A a also e isualized as oss- o elog a s et ee all eu o pai s, at lags a gi g f o - s to s ith s i s. C, the et o ks i A a d B ut sho as spike t ai s pe t ial of t o e pe i e tal o ditio s. The lue et o k t ials ha e o e se ue e i o ditio A a d t o se ue es i o ditio B, i e e sa fo the

g ee et o k. To e t a t these t o et o ks, e a a ge spike t ai s of all eu o s i a eu o - - ti e i a at i . These spike t ai s a e the o ol ed ith o ple e po e tials o a elets of e ual le gth at diffe e t f e ue ies, esulti g i a o ple - alued eu o - -ti e at i pe f e ue pe t ial. D, the oss p odu ts a e the o puted alo g the ti e di e sio , esulti g i a

eu o - - eu o oss-p odu t at i pe f e ue pe t ial: the oss spe t u . The et ee - eu o phase diffe e es of the oss spe t a o e f e ue ies, efle t the o siste t et ee - eu o

spike ti e dela s. E, usi g a e e t te h i ue de oted as SPACE, the st u tu e i the oss spe t a o e f e ue ies a e e t a ted, a d des i ed as sepa ate spike ti i g et o ks. The lue a d g ee

et o ks a e ea h des i ed a eu o p ofile, des i i g et o k e e ship a si gle eight pe eu o , a ti e p ofile, des i i g the spike se ue e a ti e oeffi ie t pe eu o , a d a t ial p ofile,

ha i g a si gle eight pe t ial, i di ati g ho st o gl the et o k as p ese t. Fo details see Mate ials a d Methods se tio , , . Figu e . Si ulated spike ti i g et orks. To i estigate the o ust ess of et o k e t a tio to

a ious ki ds of oise e si ulated spike e o di gs f o eu o s o tai i g spike ti i g et o ks a oss t ials. A-C, des iptio of si ulated et o ks i sa e fo at as e t a ted et o ks. A, the eu o p ofile of ea h et o k des i es o - e e eu o s s a d e e eu o s s. Be ause a solute ag itudes of the eu o a d t ial p ofiles of et o ks a e ot ea i gful, the a e L - o alized o e tio leadi g to the isi le a it a et ee - et o k

a plitude diffe e es ; the et ee - eu o /t ial atios a e ea i gful. B, the spiki g se ue e of ea h et o k, sho as thei ti e p ofiles o l e e eu o s a e sho . C, the t ial p ofile. Ea h et o k spiki g se ue e as epeated - ti es i ea h t ial, sho pe t ial i the fi st o . The

se o d o of C sho s a alte ati e isualizatio of the t ial p ofile, hi h is o e ie t fo isualizi g e o e see Fig - . He e, o alized t ial p ofile eights -a is a e sho as thei ea SD , pe

si ulated u e of spike se ue e epeats -a is . D, s he ati of all o siste t spike ti i g elatio ships esulti g f o the si ulated spike se ue es. Ea h i le is a eu o , ea h dashed li e efle ts a spike ti i g elatio ship. Fo isi ilit , the fi st-o de elatio ships a e da k olo ed, all othe s

a e light olo ed. Nu e s i di ate the fi st-o de ithi -se ue e spike ti e dela s. E, oss- o elog a s o puted fo all eu o -pai s f o a si ulatio u ith Hz spiki g oise, at lags a gi g f o - s to s ith s i s. F, aste plots of e a ple spike t ai s as a fu tio of

spiki g oise le els used i the si ulatio s. Ea h e ti al dash is a si gle spike. Ea h o o sists of o ate ated t ials, sepa ated a e ti al li e. Net o k spiki g se ue es a e sho thei olo as

i A-D. See Mate ials a d Methods se tio . Figu e . Re o er of si ulated spike ti i g et orks ith spiki g jitter a d spiki g oise. Net o ks

e e si ulated ti es at le els of spiki g jitte a d le els of spiki g oise. ‘e o e of the eu o

a d t ial p ofiles a e sho as Pea so o elatio s et ee the e t a ted a d si ulated et o ks a ged f o - to , isualized f o to ; a e aged o e si ulatio s; shadi g = SEM . The e o e of

the ti e p ofile is sho a e o e oeffi ie t a gi g f o to pe fe t e o e ; a e aged o e si ulatio s; shadi g = SEM . Net o ks a e olo ed as i Figu e . Botto pa els isualize e t a ted

et o ks as i Figu e at se e al e a ple jitte a d oise le els. Neu o p ofiles a e sho as ea s o e si ulatio s shadi g = SD , ith that of i di idual si ulatio s as thi li es. Ti e p ofiles a e displa ed as a e age o e si ulatio s e o a = SD; alig ed usi g a e age diffe e e et ee si ulated a d e o e ed et o ks . T ial p ofiles sho ea s o e t ial eights pe # of si ulated se ue e epeats, a e aged o e si ulatio s e o a = SD . The si ulated t ial p ofiles a d ti e p ofiles a e i di ated i g a fo efe e e. Note that, he spiki g oise a d jitte i eased, the t ial p ofiles aseli e eights of o - o t i uti g t ials that should e g aduall i eased, spiki g oise had a st o ge effe t o the t ial p ofiles of et o ks ith fe e eu o s, a d, the ti e p ofiles e e o e o ust to oise tha the eu o p ofiles a d t ial p ofiles, ith a u ate e o e e e he spike jitte as a ultiple of the et ee - eu o ti e dela s. Also ote i the e a ples that as oise i eased, at hi g of si ulated et o ks to e t a ted et o ks e a e t ou leso e, leadi g to diffe e es et ee et o k-spe ifi e o e e o i g less ea i gful. See Mate ials a d Methods se tio a d . Figu e . Re o er of si ulated spike ti i g et orks ith partial et ork spiki g a d spiki g oise. Net o ks e e si ulated ti es at p o a ilit le els of spike deletio , a d le els of spiki g oise. P o a ilit is the ha e fo ea h i di idual o - oise spike to e deleted. ‘e o e a d e a ples a e displa ed ide ti all to Figu e . Note that, the effe t of spike deletio affe ted the eu o p ofiles, ti e p ofiles, a d t ial p ofiles si ila l to that of spike jitte a d spiki g oise, a d, e e he the spiki g se ue es of the et o ks e e highl a ia le % ha e of ea h spike s a se e the

et o ks ould still e ide tified i the e a ples at Hz spiki g oise. See Mate ials a d Methods se tio a d . Figu e . Cross spe tra or alizatio di i ishes effe ts of differe tial euro firi g rates. I ealisti

eu o e o di gs, the fi i g ate of eu o s t pi all diffe . To sho the effe t of diffe e tial fi i g ates o et o k e o e , e si ulated spike ti i g et o ks spiki g jitte = ± . s; spike deletio

= % ti es ith t o eu o s ha i g Hz spiki g oise, the othe eu o s Hz. To i p o e et o k e o e , the oss spe t a a e o alized. O e ethod is to o alize the su h that the

po e of the oss spe t a e o es e ual to thei Nth oot. A, spiki g oise as a fu tio of eu o s, ith the si ulated eu o p ofiles i the a kg ou d. B, et o k e o e ithout o alizatio .

Though the et o ks a e e og iza le, e o e as lea l affe ted. Net o ks a e displa ed ide ti all to e a ples i Figu e . C, the effe t of s ua e, th, th, th, d oot po e o alizatio o e o e ed

et o ks, ul i ati g i : D, e o e ed et o ks afte th- oot po e o alizatio . See Mate ials a d Methods se tio a d . Figu e . Trial- ise ross spe tra or alizatio di i ishes effe ts of differe tial trial firi g rates. I ealisti e o di gs, the fi i g ate of eu o s a diffe o e t ials. To sho its effe t o et o k e o e , e si ulated spike ti i g et o ks spiki g jitte = ± . s; spike deletio = % ti es ith t ials ha i g Hz spiki g oise fo all eu o s , the othe t ials Hz. To i p o e et o k

e o e , the oss spe t a a e o alized i a si ila a as fo diffe e tial eu o fi i g ates. He e, e o alize the oss spe t a of ea h t ial su h that thei po e is e ual to that su ed o e t ials. A, spiki g oise as a fu tio of t ials, ith the si ulated t ial p ofiles i the a kg ou d. B, e o e ed et o ks ith t ial- ise o alizatio . Net o ks a e displa ed ide ti all to e a ples i

Figu e . The t ial p ofiles of the e o e ed et o ks e e st o gl affe ted. C, like B ut fo et o ks

e o e ed afte t ial- ise o alizatio . While the t ial p ofiles still de iated f o the si ulated et o ks, the atios of thei eights . .t. the # of si ulated se ue e epeats e e pa tiall esto ed.

See Mate ials a d Methods se tio a d . Figu e . E a ple spike ti i g et orks e tra ted fro rat edial prefro tal orte a d hippo a pus. We e t a ted spike ti i g et o ks f o e o di gs i hi h a at eithe had to take the left o ight a of a figu e-eight T- aze. The u e of et o ks to e t a t as esti ated usi g a split-half app oa h. The fi st o of ea h et o k sho s the eu o p ofile, the ti e p ofile, a d the t ial p ofile. The ti e p ofile o l sho s the st o gest eu o s of the t ial p ofile as gi e the

eu o p ofile . Se e al of the st o gest eu o s a e highlighted i ea h eu o p ofile. To sho that the et o ks efle t spike ti i g o siste ies i the data, e also sho oss- o elog a s i the se o d a d thi d o . The oss- o elog a s of ea h pai of the highlighted eu o s a e sho as spike de sities, the -a is li it oughl efle ts spike ou ts. The dashed g a li e is the ti e dela et ee the eu o s as gi e the ti e p ofile of the et o k. We o se e the follo i g. Fo et o k ,

eu o pai s - , - , a d - the e t a ted ti e dela s a e lose to the oss- o elog a . Though the th eu o has a highe eight tha the o -highlighted eu o s i the eu o p ofile, the oss- o elog a s a e ot as st o gl peaked as fo the othe pai s. Fo et o k , the e t a ted ti e dela s

of pai s - , - , - a e losest to thei oss- o elog a s. Though fo et o k the oss- o elog a s sho eake spike ti i g o siste highe aseli e spike de sit , the e t a ted ti e

dela s of pai - , - , - , a d - a e lose to thei peaks. Net o k i ol es fe st o g eu o s, as i di ated the eu o p ofile; o l the eu o pai - is lose to its peak. See Mate ials a d Methods se tio .