Sta$onary versus Bifurca$on Regime for Standing Wave Central Pa

Sta$onaryversusBifurca$onRegimeforStandingWaveCentralPa:ernGenerator
RobertoMar*n-del-Campo1andEdmondA.Jonckheere2
1.ResearchGoals
Thegoalsofthepresentstudyarethree-fold.
Reproducibility
Ø Thecontrolsubjectsofthetwocasestudiesarehealthy
individuals,whosignedtheIRB-approvedinformedconsent.
First(“CaseStudyI”),weshowthatthe
resultsuponwhichtheCentralPaBern
Generator(CPG)hypothesisrestsare
reproducible[3].
Ø Weplaced4surfaceElectromyography(sEMG)reducednoisetripolarelectrodesontheskinoftheindividuals,
alongtheirspine,atcervical(C2-C3),thoracic(T4-T6),
lumbar(L3),andsacral(S2-S4)posi6ons.
Bifurca*on
Second(“CaseStudyII”),weaddanother
aBributeassociatedwithaCPG:theability
ofthespinalwavetoundergo“bifurca*on,”
confirmedbysta*s*calsignalprocessing.
Lessthanidealsignals
Third,anotherpurposeofCaseStudyIIisto
showhowtodealwithsignalsthatareless
thanideal.WeshowthattheshiKin
correla*onpaLerntypicalofbifurca6on
survivesthenoisecorrup6onbyperforming
atestofsignificancewithbootstrapping.
2.Introduc*on
Ø Theso-calledspinalwaveisaphenomenon
duringwhichthespinegoesthrougha
rhythmicoscilla6onelicitedbylightfinger
pressureatsomesensi6zedareasofthe
spine,typically,theneckandsacrum.
Ø Thecrucialfeaturesthatthemovementis
rhythmic,thata?ersomeini6als6mulusit
becomesself-sustainedandhencehasno
sensoryinput,alreadypointtoaCentral
PaBernGenerator.
Ø ThestandingwaveaspectoftheCPGis
confirmedbyobservingthecorrela6onnode
paBernamongthecervical,thoracic,lumbar
andsacralsurfaceelectromyographic(sEMG)
signals,whichappearsmoreclearlyonthe
andsubbandsoftheDaubechiesDB3
waveletdecomposi6on.
Ø Sincethespinalstandingwavehasits
coherenceextendingfromthenecktothe
sacrum,itisconsideredaphenomenonof
coherenceatadistance,presentinnormal
subjectsbutdepletedinquadriplegicsubjects.
4.Results
3.Methods
DuringCaseStudyI,we
recorded720,000
samplesatarateof4
kHzandcenteredthe
analysisaroundasec6on
of100,000samples,
whereaphenomenonof
synchronicityofsignalsis
visuallyevident.
Ø TherandomnessisanalyzedintermsofthenormalizedcPSD
distribu6ons,andwedefinethe“Before”and“A?er”probability
densityfunc6ons,withmeans,respec6vely.
Ø Thefigures
depictthe
crosscorrela6on
betweenthe
fourdifferent
sEMGsignals
obtainedfrom
subband.
Ø Wehavetoshow,sta6s6cally,thatthereis95%confidencein
asser6ngthatathighfrequencies,andatlow
frequencies.
Ø Ourinves6ga6onshaveshownthatdonotfollow
theGaussdistribu6on.Thus,togoaroundthelackofGaussian
property,weneedtoperformbootstrappingofthecPSDvalues.
Ø Thezerocorrela6onnodesaremarkedwithblack
circlesandtheydevelopwithhigherlevelofaccuracy
thanthenodes,markedwithdoBedcircles.
DuringCaseStudyII,we
recordedthreeseconds
ofdataatarateof4kHz.
Thissec6oncomprisesa
setof12,000samples,in
whichthebifurca6onis
visuallypresentbetween
samples4,300and5,300.
Ø ThefundamentalprocessinthesEMGsignalanalysisis
theDaubechiesDB3decomposi6on,whichischosen
becauseitsmotherfunc6onmimicsthesinglemotorunit
ac6onpoten6al.
Ø Letbeeithertheorthe
subbandofthecervical,thoracic,lumbarandsacralsEMG
signalsrespec6vely.Wedefinethecorrela6onsasfollows,
Ø Themovementhasacoherentstandingwaveifthere
existsomedelayssuchthat
whereThepoints
InCaseStudyII,thespecificdatasec6onthat
comprisesthebifurca6onisdeliberatelyavoided,
becausethestandingwavemomentarilydisappears
andthesignalsarenolongersta6onary;thus,the
analysisisbrokendownintwoparts:“before”and
“a?er”thebifurca6on.
BeforetheBifurca*on
AKertheBifurca*on
Ø Thesubband
Ø Themostaccurate
“zerocorrela6onnodes” wasnolongerthebest
areonthesubband. torevealcoherence.
Ø Thesubband
showedlessaccurate
“zerocorrela6on
nodes.”
5.Problem
Ø Thesubband
exhibitsbeBer“zero
correla6onnodes.”
Toconfirmthevisibleshi?ofcoherencefromthe
“slow”tothe“fast”,wedevelopedasta6s6cal
testofsignificancewhereweexpectthecrossPower
SpectralDensity
(cPSD),between
twosignals,tobe
richerinhigh
frequencyaKer
thebifurca*on,
andricherinlow
frequencybefore
thebifurca*on.
7.ResultsoftheStatedProblem
Thebootstrappingwas
implementedonMatLabby
MathWorksandonJMP
Pro11bytheSASIns6tute
usingandboth
returnedsimilarp-values.
Electrode
posi$ons
6.Analy*calMethodology
Ø BootstrappingisaMonteCarlomethodthatemploys
repeatedsampleswithreplacementfromtheoriginaldata.
Ø Usingbootstrappingweconductedasta*s*calhypothesis
testofdifferencebetweenmeanstoes6matetheAchieved
SignificanceLevel(ASL),alsoknownasp-value.
Ø Wecalculatethevalueoftheteststa6s6cforthesamplesas
follows:
Wehighlightthosecasesthatconcur,with95%
confidencelevel,withtheexpectedqualita6vebehavior.
Where,arethesamplemeansand,thesample
standarddevia6onsofthecPSDvaluesbeforeanda?erthe
bifurca6onrespec6vely;isthesamplesize.
8.Conclusions
Ø Wetransformthevaluesfromthe“Before”sampleas
andthosefromthe“A?er”sampleas
Weillustratedthereproducibilityoftheresultsof[3]
,whereisthemeanofthe
indica6ngthatthespinalwaveisacoherentmovement
combinedsamples.
elicitedbyaCentralPaBernGenerator,openingthe
Ø A?errandomlysamplingandwithreplacement,
roadforthepoten6alofthiscoherenceanalysisto
werepeatthepreviousstepatotalof6mes.
becomepartoftheneurologicalsuite[4].
Ø Foreachbootstrapsample,wecomputetheteststa6s6c:
Thestandingwavecanundergoabifurca6onwitha
shiKofcoherencefromto,meaningthatthe
mo6onspeedsbyafactorof2,inaprocessthat
appearstobethereverseofthewell-knownperiod
doublingindynamicalsystemstheory.
Ø WedefinetheNullHypothesisasfollows:
​"↓$ =​& ↑( −​& ↑) /√⁠​(​
-↑( )↑2 +​(​-↑) )↑2 // .
Ø WedefinetheAlterna$veHypothesisandthebootstrap
es6matedp-valueas:
Ø Thesta6s6caltestconfirmstheshi?ofthe
coherencewiththerestric6onthatthishappenstobe
prevalentamongthelowerspinesignals,whenthe
sacraloscillatorisbeBerengagedthanthecervicalone.
Ø Itishopedthatthetechniquedevelopedinthis
researchwillbeapplicabletoconfirmotherbifurca6ons.
havebeencalledzerocorrela$onnodesandare
Acknowledgement:Dr.BartKosko’shelpindevelopingthesta6s6caltestisgratefullyacknowledged.
Ins*tu*onalReviewBoard(IRB):ThisresearchwasapprovedbytheIRBoftheUniversityofSouthernCalifornia,UP-06-00152-CR010.
rdIEEEGlobalConferenceSIP.(2015)
References:[1]E-mail:mart737@usc.edu[2]URL:hBp://eudoxus2.edu.usc[3]E.Jonckheere,P.Lohsoonthorn,S.Musuvathy,V.Mahajan,M.Stefanovic.OnastandingwavecentralpaBerngeneratorandthecoherenceproblem.ElsevierBSPC(2010)336-347.[4]R.Mar6n-del-Campo,E.Jonckheere.Sta6onaryregimeforStandingWaveCentralPaBernGenerator.3
manifesta*onsofacoherentstandingwave.
accepted.
Sta$onaryversusBifurca$onRegimeforStandingWaveCentralPa:ernGenerator
RobertoMar*n-del-Campo1andEdmondA.Jonckheere2
Spinalwave
Ø Thepresentstudyappliestoaspinalwavecharacterizedbytwomodeshape
standingwave,awavethatnormalsubjectareabletodevelopwithminimum
entrainment.
Ø Amorecomplicatedwavewiththreemodeshapean6nodescanbeelicited,butwith
moredifficul6esthough.
MoredetailsonCaseStudyIIsignals
Herewedisplaythecompletesetof4sEMGsignalsofCase
StudyII.ThesignalfromtheLumbarspine(red)ishighly
corruptedwithnoise,whichmakesthebifurca6ondifficulttosee
withthenakedeye.
Acknowledgement:Dr.BartKosko’shelpindevelopingthesta6s6caltestisgratefullyacknowledged.
Ins*tu*onalReviewBoard(IRB):ThisresearchwasapprovedbytheIRBoftheUniversityofSouthernCalifornia,UP-06-00152-CR010.
References:[1]E-mail:mart737@usc.edu[2]URL:hBp://eudoxus2.edu.usc[3]E.Jonckheere,P.Lohsoonthorn,S.Musuvathy,V.Mahajan,M.Stefanovic.OnastandingwavecentralpaBerngeneratorandthecoherenceproblem.ElsevierBSPC(2010)336-347.[4]R.Mar6n-del-Campo,E.Jonckheere.Sta6onaryregimeforStandingWaveCentralPaBernGenerator.3rdIEEEGlobalConferenceSIP.(2015)
accepted.
Sta$onaryversusBifurca$onRegimeforStandingWaveCentralPa:ernGenerator
RobertoMar*n-del-Campo1andEdmondA.Jonckheere2
EnlargedFiguresoftheResultsofCaseStudyI
Acknowledgement:Dr.BartKosko’shelpindevelopingthesta6s6caltestisgratefullyacknowledged.
Ins*tu*onalReviewBoard(IRB):ThisresearchwasapprovedbytheIRBoftheUniversityofSouthernCalifornia,UP-06-00152-CR010.
References:[1]E-mail:mart737@usc.edu[2]URL:hBp://eudoxus2.edu.usc[3]E.Jonckheere,P.Lohsoonthorn,S.Musuvathy,V.Mahajan,M.Stefanovic.OnastandingwavecentralpaBerngeneratorandthecoherenceproblem.ElsevierBSPC(2010)336-347.[4]R.Mar6n-del-Campo,E.Jonckheere.Sta6onaryregimeforStandingWaveCentralPaBernGenerator.3rdIEEEGlobalConferenceSIP.(2015)
accepted.
Sta$onaryversusBifurca$onRegimeforStandingWaveCentralPa:ernGenerator
RobertoMar*n-del-Campo1andEdmondA.Jonckheere2
BeforetheBifurca*on
AKertheBifurca*on
Acknowledgement:Dr.BartKosko’shelpindevelopingthesta6s6caltestisgratefullyacknowledged.
Ins*tu*onalReviewBoard(IRB):ThisresearchwasapprovedbytheIRBoftheUniversityofSouthernCalifornia,UP-06-00152-CR010.
References:[1]E-mail:mart737@usc.edu[2]URL:hBp://eudoxus2.edu.usc[3]E.Jonckheere,P.Lohsoonthorn,S.Musuvathy,V.Mahajan,M.Stefanovic.OnastandingwavecentralpaBerngeneratorandthecoherenceproblem.ElsevierBSPC(2010)336-347.[4]R.Mar6n-del-Campo,E.Jonckheere.Sta6onaryregimeforStandingWaveCentralPaBernGenerator.3rdIEEEGlobalConferenceSIP.(2015)
accepted.
Sta$onaryversusBifurca$onRegimeforStandingWaveCentralPa:ernGenerator
RobertoMar*n-del-Campo1andEdmondA.Jonckheere2
Acknowledgement:Dr.BartKosko’shelpindevelopingthesta6s6caltestisgratefullyacknowledged.
Ins*tu*onalReviewBoard(IRB):ThisresearchwasapprovedbytheIRBoftheUniversityofSouthernCalifornia,UP-06-00152-CR010.
References:[1]E-mail:mart737@usc.edu[2]URL:hBp://eudoxus2.edu.usc[3]E.Jonckheere,P.Lohsoonthorn,S.Musuvathy,V.Mahajan,M.Stefanovic.OnastandingwavecentralpaBerngeneratorandthecoherenceproblem.ElsevierBSPC(2010)336-347.[4]R.Mar6n-del-Campo,E.Jonckheere.Sta6onaryregimeforStandingWaveCentralPaBernGenerator.3rdIEEEGlobalConferenceSIP.(2015)
accepted.