Asymmetry of short-term control of spatio-temporal gait parameters during treadmill walking

0.17 MB·

by  Klaudia Kozlowska

Asymmetry of short-term control of spatio-temporal
gait parameters during treadmill walking
Klaudia Kozlowska1,+, Miroslaw Latka1,+*, and Bruce J. West2,+
1WroclawUniversityofScienceandTechnology,FacultyofFundamentalProblemsofTechnology,Departmentof
BiomedicalEngineering, Wroclaw,50-370,Poland
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2ArmyResearchOffice,InformationSciencesDirectorate,ResearchTrianglePark,27709,USA
1
*[email protected]
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2 +eachauthorcontributedequally tothiswork

n
a
ABSTRACT
J

1
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Optimizationofenergycostdeterminesaveragevaluesofspatio-temporalgaitparameterssuchasstepduration,steplength

] or step speed. However, during walking, humans need to adaptthese parametersat every step to respond to exogenous
h
and/orendogenicperturbations. Whilesome neurologicalmechanismsthat triggertheseresponsesare known, our under-
p
- standingofthefundamentalprinciplesgoverningstep-by-stepadaptationremainselusive.Wedeterminedthegaitparameters
o of 20 healthysubjectswith right-footpreferenceduringtreadmillwalking at speeds of 1.1, 1.4 and 1.7 m/s. We foundthat
bi whenthevalueofthegaitparameterwasconspicuouslygreater(smaller)thanthemeanvalue,itwaseitherfollowedimme-
. diatelybyasmaller(greater)valueofthecontralateralleg(interlegcontrol),orthedeviationfromthemeanvaluedecreased
s
duringthenextmovementofipsilateralleg(intralegcontrol). Theselectionofstepdurationandtheselectionofsteplength
c
i during such transient control events were performedin unique ways. We quantifiedthe symmetry of short-term control of
s
gaitparametersandobservedthesignificantdominanceoftherightleginshort-termcontrolofallthreeparametersathigher
y
speeds(1.4and1.7m/s).
h
p
[

1 Introduction
v
3 Ithasbeenknownforoveracenturythatthestrideintervalofhumangaitisremarkablystable. Smallfluctuationsofapproxi-
7 mately3–4%wereattributedtothecomplexityofthelocomotorsystemandtreatedasanuncorrelatedrandomprocess.1 From
1 this viewpoint,the discoveryof long-time,fractalcorrelationsin stride-intervaltime series wasunexpected.2,3 Those early
9 papersnotonlyspurredinterestintheemergingconceptoffractalphysiology,4butalsoshiftedthefocusofquantitativegait
0
analysisfromaveragevaluesoftypicalparameters(e.g. strideintervals)totheirtemporalvariability. Thatprofoundchange
.
1 ofperspectivebroughtnewinsightsintolocomotormanifestationsofHuntington’sandParkinson’sdiseases, aging,andthe
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connectionbetweengaitdynamicsandfallrisk,see5 andreferencestherein.
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From a plethora of physiologically accessible gait patterns, humans employ only walking and running. Walking feels
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: easiestatlowspeeds,andrunningfeelseasiestwhenmovingfaster. Optimizationofenergycostunderliesnotonlythechoice
v
ofgait,6–8 butalsodeterminesaveragevaluesofgaitparameters,suchassteplengthandduration.10 Duringwalking,humans
i
X need to adapt their spatio-temporalgait parameters at every step to be able to respond to exogenous(e.g. irregularities of
r walkingsurface)and/orendogenic(neuromuscularnoise)perturbations.11 Whilesomeneurologicalmechanismsthattrigger
a theseresponsesareknown,12–15 thefundamentalprinciplesgoverningstep-by-stepadaptationremainelusive.16
Treadmillwalking,especiallyathighspeeds,presentschallengesthatcanbemetonlythrougheffectiveshort-termcontrol
of spatio-temporalgait parameters. In orderto stay on a treadmill, the subject’s step durationand length must yield a step
speedwhichcanfluctuateoveranarrowrangecenteredonthetreadmillbelt’sspeed.Theresultsofpreviousexperimentswith
walkingonasplit-belttreadmillunderscoretheintricaciesofsuchaggregation.Inparticular,spatialandtemporalcontrolsof
locomotionareaccessiblethroughdistinctneuralcircuits17andneuralcontrolofintra-versusinterlimbparameters(calculated
using values from both legs, e.g., step length, double support) during walking is to a large extent independent.18 Herein,
we investigate the dynamics of time series of gait parameters (step duration, length and speed) following a sudden, large
deviationfromthemeanvalue. Inparticular,wetestthehypothesisthatwheneverthevalueofagaitparameterismarkedly
greater (smaller) than the mean value, it is either immediately followed by a smaller (greater) value of the contralateral
leg (interleg control), or the deviation from the mean value decreases during the next movementof ipsilateral leg (intraleg
control). Said differently, during treadmill walking errors are not gradually attenuated via long-term corrections, but are
correctedimmediatelybythesameoroppositeleg.Takingintoaccountdifferencesintherelativecontributionoflowerlimbs
Table1. Thestatisticsoftheoccurrenceofintraleg(columnsL-LandR-R)andinterleg(columnsR-LandL-R)control
mechanismsthatareevokedinresponsetoerrorsinstepduration,lengthandvelocity.Statisticsarepresentedforthree
valuesoftreadmillspeedv. Thenumberoferrors,definedasabruptdeviationsfromthemovingaveragevalue,ispresented
intheTotalcolumn.ThecolumnlabeledNCgivesthenumberoferrorsthatpersistedformorethantwosuccessivesteps.
TheLeftandRightcolumnsshowthenumberofcontroleventsperformedbyeachleg. Probabilitiesofoccurrenceofintra-
andinterlegcontrolmechanismsarelistedinthelastfourcolumns.
v[m/s] Total NC L-L R-L R-R L-R Left[%] Right[%] p p p p
LL RL RR LR
stepduration
1.1 180 4 42 51 32 51 53 47 0.23 0.28 0.18 0.28
1.4 121 11 29 21 18 42 45 55 0.24 0.17 0.15 0.35
1.7 103 5 22 18 26 32 41 59 0.21 0.17 0.25 0.31
steplength
1.1 380 14 125 71 122 48 54 46 0.33 0.19 0.32 0.13
1.4 257 12 73 30 93 49 42 58 0.28 0.12 0.36 0.19
1.7 186 18 42 17 87 22 35 65 0.23 0.09 0.47 0.12
stepspeed
1.1 393 5 83 115 84 106 51 49 0.21 0.29 0.21 0.27
1.4 275 14 63 61 61 76 48 52 0.23 0.22 0.22 0.28
1.7 189 8 34 45 66 36 44 56 0.18 0.24 0.35 0.19
tocontrolandpropulsion–theeffectknownasfunctionalgaitasymmetry,19wefurtherhypothesizethatinsubjectswithright
footpreferencetheshort-termcontrolofgaitspatio-temporalparametersisstrongerfortherightleg.
Results
InTable1werecordeddatawhicharecrucialfortestingthemainhypothesisofthepaper: thaterrorsingaitparametersare
ecorrectedimmediatelybythesameoroppositeleg. Letusfocusonthefirstrowofthistablewhichconcernsstepduration
controlata treadmillspeedof1.1m/s. Therewere180”errors”definedasabruptchangesinstep duration(equations2-4).
Fortheleftleg,in42cases(columnL-L),thedeviationofstepdurationfromthemeanvaluedidnottriggeracompensating
change in the step duration of the right leg. The value of the control parameter DLR, defined by equation 5, greater than
1 indicates the absence of such adjustment. However, the deviation decreased during the next left step as indicated by the
valueofintralegcontrolparameterDLL,definedbyequation7,smallerthan1(thestatisticsofbothinter-andintralegcontrol
parametersarepresentedinTable2). Inotherwords,fortheleftleg,weobservedintralegcontrolofstepdurationin42cases.
In 51 cases (columnR-L), the changein the step durationof the left leg compensatedthe deviationof step durationof the
previousrightstep(interlegcontrol). ByaddingcolumnsL-LandR-L,weobtain93controleventsperformedbytheleftleg.
Thisnumberexpressedasthepercentageofall180controleventsisgiveninthecolumnofTable1labeledasLeft. Please
notethatonlyin4(columnNC)outof180cases(2.2%),theappearanceofastepdurationerrordidnotevokeeitherofthe
controlmechanisms.
For all three gait parameters, the number of errors decreases with speed. For example, there were 180 errors in step
duration at v=1.1 m/s but only 103 at v=1.7 m/s (a 43% reduction). A comparable drop in the number of errors was
observedforstep length(51%)andvelocity(52%). Lessthanhalfofthesechangescanbeexplainedbythe22%reduction
ofthenumberofstepstakenbyallthesubjectsatv=1.7m/sincomparisonwithv=1.1m/s. Pleasenotethatthenumber
ofstepsdecreaseswithtreadmillspeedsinceateachspeed,thesubjectswereaskedtocoverthesamedistanceof400m. It
isworthemphasizingthattherewereapproximatelytwiceasmanyerrorsinsteplengthandspeedthaninstepduration. For
allthreeparameters: stepduration,length,andspeed,inatleast90%ofcases,thedeviationsfromthemeanvaluedecreased
duringthesubsequenttwostepsviaeitherintra-orinterlegcontrol.
Regardless of treadmill belt speed, the control of step length is predominatelyintraleg (Table 1). For example, for the
leftlegatthelowestspeed,theprobabilityofevokingintralegcontrol p =0.33is42%greaterthanthatofinterlegcontrol
LL
p =0.19).Inthesamecondition,fortherightlegsuchdifferenceisequalto59%( p =0.32vs p =0.13).Thereisno
RL RR LR
suchpatternfortheothertwogaitparameters.
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Table2. Thevaluesofintra-andinterlegcontrolparametersfortreadmillwalking.Dataarepresentedasmean(standard
deviation).
v[m/s] DLL DRL DRR DLR DBE DBE D DBE [%]
L R
stepduration
1.1 0.34(0.22) 0.55(0.26) 0.43(0.25) 0.58(0.21) 1.20 0.91 28
1.4 0.45(0.26) 0.55(0.25) 0.31(0.22) 0.54(0.24) 0.85 1.12 -28
1.7 0.31(0.25) 0.63(0.20) 0.36(0.28) 0.49(0.26) 0.97 1.34 -32
steplength
1.1 0.46(0.26) 0.72(0.23) 0.48(0.25) 0.62(0.30) 0.98 0.88 11
1.4 0.46(0.27) 0.79(0.16) 0.45(0.26) 0.61(0.30) 0.76 1.12 -38
1.7 0.44(0.26) 0.79(0.15) 0.44(0.27) 0.68(0.23) 0.63 1.24 -65
stepspeed
1.1 0.44(0.26) 0.65(0.26) 0.40(0.27) 0.56(0.28) 0.92 1.01 -9
1.4 0.39(0.25) 0.68(0.24) 0.33(0.23) 0.56(0.29) 0.91 1.15 -23
1.7 0.51(0.26) 0.73(0.20) 0.41(0.24) 0.58(0.27) 0.68 1.19 -55
For step duration, length, and speed, the control parameter D was independent of speed (Table 2). For all three gait
parameters, bothforthe rightandleftleg, themeanvalueofD forinterlegcontrolwasgreaterthanthatof intralegcontrol.
Forexample,forstepdurationatv=1.1m/sD =0.55andD =0.34.
RL LL
Thedifferencebetweenthevaluesofintra-andinterlegcontrolparametersforagivenlegwasstatisticallysignificantfor
allthreetreadmillspeedsforsteplength:
• at1.1m/s: p <1×10−4, p =6×10−3;
left right
• at1.4m/s: p <1×10−4, p =3×10−2;
left right
• at1.7m/s: p =1×10−4, p =2×10−3;
left right
aswellasthestepspeed:
• at1.1m/s: p <1×10−4, p =6×10−4;
left right
• at1.4m/s: p <1×10−4, p <1×10−4;
left right
• at1.7m/s: p =2×10−3, p =9×10−3.
left right
Forstepdurationsuchdifferenceswerenotsostronglypronounced:
• at1.1m/s: p =5×10−4, p =5×10−2;
left right
• at1.7m/s: p =2×10−3.
left
Withtheexceptionofstepdurationandsteplengthatthelowestspeed(v=1.1m/s),theasymmetryparameterD DBEwas
smallerthanzeroindicatingadominantroleoftherightleginshort-termcontrolofgaitparametersduringtreadmillwalking.
Table 3 shows the probabilityof compensatoryresponse to errorsin gait spatio-temporalparametersfor intra- (L-L, R-
R) and interleg(L-R, R-L) control. Such response correspondsto negativevaluesof variablesSinter (equation6) and Sintra
(equation8). Forallspeedsandparameters,theprobabilityofinterlegcompensationiscloseto1,roughlytwotimeshigher
thanthatofintralegresponse.
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Table3. Probabilityofcompensationoferrorsingaitspatio-temporalparametersforintra-(L-L,R-R)andinterleg(L-R,
R-L)control.Statisticsarepresentedforthreevaluesoftreadmillspeedv.
v[m/s] L-L R-R L-R R-L L-L R-R L-R R-L L-L R-R L-R R-L
stepduration steplength stepspeed
1.1 0.50 0.42 0.98 0.94 0.40 0.38 0.83 0.83 0.52 0.55 0.92 0.92
1.4 0.28 0.39 0.95 0.95 0.42 0.34 0.82 0.80 0.52 0.46 0.93 0.92
1.7 0.50 0.54 0.97 0.94 0.40 0.43 0.95 0.76 0.32 0.44 0.95 0.89
Discussion
In overgroundwalking with self-selected speed, fluctuations of stride interval, length, and speed exhibit persistent fractal
scaling characterized by a Hurst exponent a >0.5.2,3,20 Auditory metronomic cueing changes fractal statistics of stride
intervals from persistent to antipersistent (a <0.5).21 The super central pattern generator model, introduced by West and
Scafeta,22 elucidatesthedynamicoriginofsuchtransitions. Inparticular,thetransitionsresultfromthedrivingofa fractal
clock,whichretainsitspropertiesunderperturbation.Intreadmillwalking,fluctuationsofinterstrideintervalandstridelength
arealsopersistent.However,thetimeseriesofstridespeedisantipersistent,whichisamanifestionofincreasedcentralcontrol
ofthisgaitparameter.16,23 Terrierhasrecentlydemonstratedthatvisualcueing(alignmentofsteplengthswithmarksonthe
floor)alsoinducedanti-correlatedpatterningaitparameters.30
Toalargeextent,fluctuationsofspatio-temporalgaitparametersresultfromtheintrinsicfractalpropertiesofpatterngen-
erators. Hiddeninthesefluctuationsaresporadiccontrolevents,triggeredtoaccomplishalocomotortasksuchasremaining
ona movingtreadmillbelt. Thisiswhywe studythedynamicsoftime seriesofgaitparametersthatfollowasuddenlarge
deviationfromameanvalue. Forlackofabetterword,wedubbedsucheventserrors,butemphasizethattheymayoriginate
eitherfromthefailureofthemotorcontrolsystem,orfromthenecessaryadjustmentofthesubject’spositiononatreadmill.
Whilethedefinitionofsucheventsisarbitrary(equations2-4),itsatisfiestheresearchobjective.
We foundthatwhenthevalueofthegaitparameter(stepduration,lengthorspeed)wasconspicuouslygreater(smaller)
thanthemeanvalue,itwaseitherfollowedimmediatelybyasmaller(greater)valueofthecontralateralleg(interlegcontrol),
orthedeviationfromthemeanvaluedecreasedduringthenextmovementofipsilateralleg(intralegcontrol). Theexistence
of distinct short-term control of step frequency (the inverse of step duration) was demonstrated by Snaterse et al.24 The
timeevolutionofstepfrequencytriggeredbysuddenstepwiseincrementsintreadmillspeedwasmodeledbythesumoftwo
exponentiallydecayingterms. Thetime constantof thefirst termwas 1.44±1.14s andits amplitudewas two timeslarger
than that of the second term, whose time constant was 27.56±16.18s. For those values of time constants, step frequency
adjustmentsweretwo-thirdscompleteinlessthantwoseconds.Snaterseetal. arguedthatthefirsttermrepresentsarapidpre-
programmedresponse,whilethesloweronemodelsfine-tuningofstepfrequencydrivenbyenergyexpenditureoptimization.
Hereinweextendedthislineofreasoningbydemonstratingthatshort-termcontrolofgaitparametersmayberealizedusing
intra- andinterlegadjustments. The betterunderstandingof short-termcontrolmechanismsdoesnotbringusanycloser to
understandinghow, duringtreadmillwalking, persistentstochastic variables: step durationandstep lengthare combinedto
yieldantipersistentstepspeed. Webelievethatadifferentmechanismoperatingatalongertimescaleunderliesthiseffect.
Therearefundamentaldifferencesbetweenthecontrolofstepdurationandsteplength.Theprobabilityofevokingintraleg
controlofstep lengthatthehighesttreadmillspeed(v=1.7m/s)isapproximatelythreetimesgreaterthanthatofevoking
interlegcontrol.Thereisnosuchdistinctpatternforstepduration.Moreover,thenumberoferrorsinstepdurationishalfthat
ofsteplength,regardlessoftreadmillbeltspeed. Thisisastrongindicationthatspatialandtemporalcontrolsoflocomotion
areaccessiblethroughdistinctneuralcircuits. ThisinterpretationiscorroboratedbytheearlierstudyofMaloneandBastian,
whoinvestigatedadaptationofspatialandtemporalaspectsofwalkingtoasustainedperturbation,generatedbyasplit-belt
treadmill.18 Theydemonstratedthatconsciouscorrectionfacilitatesadaptation,whereasdistractionslowsit. Theunexpected
findingoftheirstudywasthatthosemanipulationsaffectedtheadaptationrateofthespatialelementsofwalking,butnotof
the temporalones. Inthefollow-upstudyMaloneetal25 demonstratedthattemporalandspatialcontrolsofsymmetricgait
can be adaptedindependently. Please note that continuous, consciousassessment of distance to surroundingobjects lies at
theheartofthecontrolproblemofremainingstationaryonamovingtreadmillbelt. Thus,thelargenumberoferrorsinstep
lengthascomparedtostepdurationmayreflectboththedominantroleofspatialcontrolanditssusceptibilitytodistraction.It
isworthmentioningthatincasualwalking,thecoefficientofvariationofstridetimeismuchsmallerthanthatofstridelength
andofwalkingspeed.10
Step speed may be interpreted as the output of the intricate neuromuscular control system, which integrates different
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sensory-motorprocesses. The ratio of average values of step length and frequency, or walk ratio, is constant over a broad
rangeofwalkingspeeds. Inotherwords,thereisalinearrelationbetweenthesegaitparameters(thestridelength–cadence
relationship),apre-programmedpatternwhichpresumablysimplifiesgaitcontrolinsteadystatewalking.26 Letusanalyzethe
interplayofstepdurationandsteplengthduringtransientchangesfollowingtheoccurrenceoferrors. Wepreviouslypointed
outthatthesetwoparametersarecontrolledindistinctways. Inparticular,theprobabilityofevokingtheinterlegcontrolof
steplengthisatleasttwotimessmallerthanthatofevokingtheintralegcontrol(Table1). Insharpcontrast,theprobability
ofeitherinter-orintralegcontrolofstepspeedsiscomparable.Thus,wemayhypothesizethatnegative-feedbackadjustment
ofstepdurationofthecontralaterallegunderliestheinterlegcontrolofstepspeed. Itisworthemphasizingthattheintraleg
controlofstepspeedisstrongerthantheinterlegcontrol.
TherecentworkofDingwelletal.16 providesinsightintothemaintenanceofspeedduringtreadmillwalking. Asubject
caninprinciplechooseanycombinationofstridelengthandtimethatyieldsstepspeedequaltothatofatreadmillbelt.These
pairsofvaluesforminphase-spaceadiagonallinecalledagoalequivalentmanifold(GEM).27 Dingwelletal. decomposed
deviation from this manifold into tangent and transverse components. Only the latter component was tightly controlled.
Moreover,thetimeseriesoftransversedeviationsexhibitedstatisticalantipersistencecharacteristicofstridespeed.Thisstudy
underscoresthesignificanceofinterlegcontrolofgaitparameters.WebelievethattheGEMdecompositionshouldbeapplied
totimeseriesofstepvelocitiestoquantifytheinterlegcontrolinamoresophisticatedway.
Inable-bodiedgait,asymmetryinspatio-temporalandkinematicparameters(suchasspeedprofiles,stepandstridelength,
foot placement angle, maximum knee flexion) for the left and right leg has been frequently reported.28 To the best of our
knowledge,thepresentstudyisthefirstobservationofasymmetryindynamicsofhumangaitparameters.Withtheexception
ofstepdurationcontrolatthelowestspeed,forallthreegaitparametersD DBE <0,indicatingdominanceoftherightlegin
short-termcontrol. Theoriginofthisasymmetrycanbetracedbacktodifferencesintherelativecontributionoflowerlimbs
tocontrolandpropulsion–theeffectknownasfunctionalgaitasymmetry.19 Morespecifically,thelegwithgreatermuscle
powergenerationdominatespropulsion,whilethesupportandcontrolfunctionsaremoreconspicuousforthelegwithgreater
powerabsorption.Humansaretypicallyright-footedformobilizationandleft-footedforposturalstabilization.
Specialconsiderationshouldbegiventostepdurationandsteplengthcontrolatthelowestspeedv=1.1m/s.Onlyinthis
case,theasymmetryparameterD DBEwasgreaterthanzero,indicatingthedominanceofleftlowerlimb.Notethatthelowest
asymmetry,|D DBE|,wasobservedforallthreeparametersatv=1.1m/s. Differencesinlow-speedgaithavebeenreported
before. TerrierandSchutz29 demonstratedthatduringovergroundwalking,atlowspeedsthemajorityofsubjectsadopteda
higherwalkratioandhadahighervariabilityofstridetime. However,inthisstudythelowesttreadmillspeedcoincideswith
thepreferredwalkingspeed(PWS)ofyoungsubjects.23 TherearetwopossibleexplanationsforthepositivevalueofD DBE.
It is likely that in the vicinity of PWS priority is given to balance maintenance and consequentlystride duration controlis
shiftedtotheleftleg,whichisusedforposturalstabilization. Pleasenotethatourcohortincludedonlysubjectswithclearly
pronounced right foot preference. Alternatively, reversed asymmetry for step duration and low values of |D DBE| for step
lengthandstepspeedmayindicatethatthereexistsadifferentstrategyforcontrolofgaitparametersinovergroundwalking
(treadmill walking at v=1.1m/s may not be challenging for young subjects and may resemble unconstrained overground
walking). Thisargumentisplausiblebecauseinmotorcoordinationtasks,humanscorrectonlythosedeviationsthatinterfere
withtaskgoalsandallowvariabilityinredundant(task-irrelevant)dimensions.31 Followingthelogicofthisminimuminter-
ventionprinciple,intreadmillwalking,stepspeedmustbetightlyregulated.However,inovergroundwalking,higherpriority
maybegiven,forexample,tobalancecontrol,whichwouldaffectthevalueoftheasymmetryparameterD DBE. Thesetwo
qualitatively different strategies may also reflect other fundamentaldifferences between overgroundand treadmill walking.
Therateatwhichtheenvironmentflowspasttheeyesseemstobeanimportantmechanismforregulatingwalkingspeed.32,33
Morespecifically,visionisusedcorrectivelytomaintainwalkingspeedatavaluethatisperceivedtobeoptimal.Fortreadmill
walking, a discrepancybetween observedand expected visual flow leads to a significant reduction(about20%) of PWS,34
as well as the speeds of walk-run and run-walk transitions.32 It is worth pointing out that as far as kinetic and kinematic
parametersareconcerned,treadmillandoutdoorgaitsaresimilar.35
Thediscoveryofdependenceoffunctionalasymmetryinshort-termcontrolofgaitspatio-temporalparametersontreadmill
speedwasanunexpectedoutcomeofthisresearch. Theelucidationofthetransitionfromleft-legtoright-legdominancein
short-termcontrolentailsdeterminationofthePWSforeachsubject. Furtherresearchisalsoneededtounderstandwhythe
probabilityof compensatoryresponse for interleg controlis close to 1 and is almost two times greater than that of intraleg
control (Table 3). Undoubtedly, such a strong difference indicates different roles these two mechanism play in control of
gaitduringtreadmillwalking. Onemayhypothesizethatthe primarygoalofinterlegcontrolismaintenanceofbalancevia
negative feedback from either leg while achieving specific goals such as matching the speed of the treadmill belt requires
intralegadjustments.
Duringhumanlocomotion,thelegsactastwocoupledoscillators.36 However,moststudiesdisregardbilateralcoordination
andsynchronizationdynamics37–39 andfocusonsingle-legvariability(stridetime, length,speed). Hereinwedemonstrated
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asymmetricshort-termintra-andinterlegcontrolofspatio-temporalgaitparameters. We believethatabetterunderstanding
of these effects will not only pave the way for more realistic models of gait variability and control, but also help to refine
proceduresusedinrehabilitationofgaitimpairments.
Methods
Werecruited20healthystudents(10M/10F,mean(SD):age22yr(2),height1.73m(0.1),weight71(15)kg,BMI23(4))of
theWroclawUniversityofScienceandTechnology,whoallsignedaninformedconsent.Thestudywasperformedaccording
totheDeclarationofHelsinkiandtheprotocolwasapprovedbytheEthicsCommitteeofWroclawMedicalUniversity. The
subjects were screened to exclude those with a history of orthopedicproblems, recentlower extremityinjuries, any visible
gaitanomalies,orwhoweretakingmedicationsthatmighthaveinfluencedtheirgait. Weonlyenrolledsubjectswhousedthe
rightlegto:kickatennisball,manipulateatennisballaroundacircle,makeafirststep,makeastepafterbeingpushedfrom
behind.Thesepurelybilateraltasksarefrequentlyincorporatedintofoot-preferenceinventories.40,41 Theprotocolbeganwith
a5minfamiliarizationperiodofwalkingonalevelmotor-driventreadmill.Theneachsubjectwasaskedtowalk400mthree
timesat1.1m/s,1.4m/si1.7m/s(4km/h,5km/hand6km/h).Theobjectivewastoinvestigatecontrolofgaitparametersat
treadmillspeedsequaltoorgreaterthanthePWSofyoungsubjects. Therefore,thelowestspeedwasequaltothepreferred
walkingspeedreportedbyTerrierandDeriaz23andslightlysmallerthanthevaluesdeterminedbyDaletal.34 (1.19m/s)and
Dingwell11(1.22m/s).
Thegaitparameterswereextractedfromthetrajectoriesofthe30mmopticalmarkersattachedto bothshoesbelowthe
ankle. The movementsof those markerswere recordedusing an in-house motion capture system with a frame rate of 240
Hzand720presolution. TheopticaltrackingwasimplementedinC++(VisualStudio2013)usingOpenCVlibrary. Aheel
strikewasdefinedasthepointwherethemarkeroftheforwardfootwasatitsmostforwardpointduringeachgaitcycle. A
step lengthwas the distance betweenthe ipsilateralandcontralateralheelstrikes. A step durationwas equalto the elapsed
timebetweentheipsilateralandcontralateralheelstrikes. Astepspeedwascalculatedasthequotientofsteplengthandstep
duration. Thegroupaveragednumberofstepstakenpertrialwasequalto593(23)at1.1m/s, 508(59)at1.4m/s,and456
(56)at1.7m/s.
InFig. 1wepresentatimeseriesofstepdurationfortreadmillwalkingat1.1m/s. Thecircleinthisfigureindicatesstep
durationthatwaslongerthanthemeanvalue(representedinthisfigurebyhorizontal,thick,dottedline)bymorethan3/2of
standarddeviation(theupper,horizontal,thindottedgridlinerepresentsthisthreshold). Itisapparentthatthedurationofthe
step,whichimmediatelyfollowsthe”error”,suddenlydecreases(thisshorterintervalismarkedbythefilledrectangle). This
examplehintsattheexistenceofaninterlegcontrolmechanismthatstabilizesthestrideinterval.
0.56
error
0.54
s]
n [
o 0.52
ati
r
u
d 0.5
p
e
st
0.48 left
compensation right
0.46
60 65 70 75 80 85 90
time [s]
Figure1. Timeseriesofstepdurationsfortreadmillwalkingat4km/h. Thecircleindicatesthestepdurationofleftleg
whichwaslongerthanthemeanvalue(representedinthisfigurebyhorizontal,thick,dottedline)bymorethan3⁄2of
standarddeviation(theupper,horizontal,thindottedgridlinerepresentsthisthreshold).Itisapparentthatthedurationofthe
stepwhichimmediatelyfollowsthe“error”suddenlydecreases.
LetN bethenumberofstepstakenbyeachleg. Letusintroduceanotationthatfacilitatestheanalysisofinterlegcontrol.
Wewritethetimeseriesoflength2N ofoneofthegaitparameters(stepduration,lengthorspeed){S }2N inthefollowing
n n=1
form:
{S }={I ,C ,I ,C ...I ,C }, (1)
n 1 1 2 2 N N
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where subseries(cid:8)Ij(cid:9)Nj=1 and (cid:8)Cj(cid:9)Nj=1 correspond to the ipsilateral and the contralateral leg, respectively. s I and s C are
standarddeviationsoftheseseries. Thesimplemovingaverages(theunweightedmeanofthepreviousmdata)of(cid:8)Ij(cid:9)and
(cid:8)Cj(cid:9)aredenotedbyI¯(m)andC¯(m),respectively.
WedefineaserrorsthesevaluesI whichsatisfyallofthefollowingcriteria:
i
|I −I¯(m)| > 1.5s (2)
i I
100%(cid:12)(cid:12)Ii−Ii−1(cid:12)(cid:12) > 3% (3)
(cid:12) I (cid:12)
(cid:12) i−1 (cid:12)
(cid:12)Ci−1−C¯(m)(cid:12) < 0.5s C (4)
(cid:12) (cid:12)
These undoubtedlyheuristic criteria are used to detect abruptchanges(equation (3)) which lead to conspicuousdeviations
from the movingaverage value (equation(2)) and which are not broughtaboutby a deviation in the preceding step of the
contralateralleg(equation(4)). Aspreviouslymentioned,wedubsucheventserrors,butbearinmindthattheymayoriginate
eitherfromthe motorcontrolsystemfailure, orfromthenecessaryadjustmentofthesubject’spositionona treadmill. The
rationaleforusingthemovingaverageintheabovedefinitionofanerrorstemsfromnon-stationarityofgaittimeseries. This
modification ensures that during transient linear trends the large deviationfrom the global mean value does not invoke the
detectionalgorithm.Pleasenotethatequation(3)byitselfisanothersafeguardforfalseerrordetectioncausedbythetransient
driftoflocalmeanvalue.Herein,wereportthevaluesform=10.
LetususeD todenoteadeviationofagivengaitparameterfromitsmovingaveragevalue,e.g.D I =I −I¯(m). Ininterleg
i i
controlthegaitparameterofcontralaterallegC changesinsuchawayastodecreasedeviationofI +C. Toquantifysuch
i i i
stabilization,weintroducethefollowingmetric:
|D I +D C|
Dinter = i i . (5)
i |D I +D C |
i i−1
ThestabilizationoccurswhenDinter <1. Thenumeratorintheaboveequationmaybecomesmallerthanthedenominatorin
i
twocases. InthefirstcasetheD C hastheoppositesigntoD I:
i i
Sinter=D ID C <0, (6)
i i i
in other words, the contralateral leg compensates for errors, as shown in Fig. 2a. The perfect compensation corresponds
to Dinter =0. In the alternative scenario, only the magnitudeof the deviation of the contralateralleg from the mean value
i
decreases(Sinter>0)asillustratedbyFig. 2b.
i
Figure2. IninterlegcontrolthegaitparameterofcontralaterallegC changesinsuchawayastodecreasedeviationof
i
I +C fromthemeanvalue.a)TheerrorofipsilaterallegI isimmediatelycompensatedforbythecontralateralleg. b)Inthe
i i i
alternativescenario,onlythemagnitudeofthedeviationofthecontralaterallegfromthemeanvaluedecreases.
It is possible that an error does not bring about a sudden change of gait parameter of contralateral leg. In this case,
Dinter >=1. However, stabilization may occur during the next step of ipsilateral leg. The change of I may reduce the
i i+1
deviationD I +D C. Werefertosuchascenarioasanintralegcontrolanddefineacorrespondingmetric:
i+1 i
|D I +D C|
Dintra= i+1 i . (7)
i |D I +D C|
i i
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Tobeabletodirectlycomparethepropertiesofbothtypesofcontrol(inter-andintraleg)wedistinguishwhethertheintraleg
controlwasachievedviacompensation:
Sintra=D ID I <0, (8)
i i i+1
asshowninFig. 3a,orbythereductionofthemagnitudeofthedisplacementofgaitparameteroftheispislaterallegfromthe
movingaveragevalue(Fig. 3b).
Figure3. Theerrorinagaitspatio-temporalparametermaynotevokeasuddenchangeoftheparameterofcontralateral
leg. However,thedeviationofthenextipsilateralgaitparameterI fromthemeanvalueI¯maydecreaseasaresultof:a)
i+1
compensationorb)thereductionofthemagnitudeofthedisplacementofparameteroftheispislaterallegfromthemean
value.
TheflowchartinFig. 4elucidatestheanalysisofthedynamicsofgaitparametertimeserieswhichfollowstheoccurrence
oferrors.UsingDinter andDintra,wedetecttheactivationofinter-andintralegcontrolmechanism,respectively.
Figure4. Flowchartofanalysisofdynamicsofgaitparameterwhichfollowsanerror–gait-parameter’svaluewhichsatisfy
thecriteriadescribedbyequations(2-4).
Inouranalysisofthe experimentaldata, we usea morespecificnotationforthe interlegparameterDinter. Forexample,
toindicatethatanerrorinagivengaitparameteroftheleftlegwasfollowedbyanadjustmentofthisparameterbytheright
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leg, we write DLR. In the same vein, we use DLL , DRR to denote intraleg leg control parameter for the left and right leg,
respectively.
In mostcases D values, fora givengaitparameter,speed, and controltype, were notnormallydistributed (theShapiro-
Wilktest). Foragivenspeedandgaitparameter,theLevene’stestshowedequalityofvariancesamongthecontroltypes(with
the exception of step duration at 1.1 m/s and step length at all speeds). For a given gait parameter and control type (L-L,
R-R,L-R,R-L),weinvestigatedthedependenceofDontreadmillspeed. Inthiscase,theLevene’stestshowedhomogeneity
ofvariance. Consequently,theKruskal-WallistestwithTukey’sposthoccomparisonswasusedtodetectdifferencesacross
speedandcontroltype.Thesignificancethresholdwassetto0.05.
Toquantifyfunctionalasymmetryincontrolofgaitspatio-temporalparametersweneedtotakeintoaccountthestochastic
aspectofmotorcontrolsystem. Letusemployananalogyofdetailedbalanceequationofstatisticalphysics42andcallitgait
detailedbalanceequation(DBE).Initsoriginalformulation,detailedbalancingrelatestherelativepopulationoftwostatesby
theprobabilityofatransitionbetweenthem.Theprincipleappliesequallywelltophysicalsystems,mathematicalprobability
densities,orstatisticalprocessesinavarietyofforms.
ThesmallerDthebetterstabilizationofstridegaitparameters.Consequently,theinfluenceofacontrolmechanism(inter
orintra)ongaitparametersisproportionaltoitsprobabilityofoccurrenceandtheinverseofthecorrespondingmeanvalueof
controlparameterD¯. Forexample,fortherightlowerlimb,wemaywrite:
p p
RR LR
DBE = + , (9)
R D¯RR D¯LR
inthesamevein,fortheleftlowerlimb:
p p
LL RL
DBE = + . (10)
L D¯LL D¯RL
Theperfectsymmetrycorrespondstothefollowingequality:
DBE =DBE . (11)
L R
Wequantifytheasymmetryincontrolofgaitspatio-temporalparameterswiththerelativedifferenceexpression:
DBE −DBE
D DBE=2 L R. (12)
DBE +DBE
L R
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