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Feasibility theory
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Content
by
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
Copyright
Acknowledgements
Iwouldliketoexpressmystrongestgratitudetomyadvisor,FranciscoJ.Valero-Cuevas,whobothledme
towardapathofadventure,andinstilledinmeanewsenseofscientificcuriosity. WithhissupportIsought
newanglesinresearch,inventednewtoolstosolveproblems,andenjoyedapathofdiscovery. Tomycom-
mittee members, Heather Culbertson, Stefanos Nikolaidis, Lori Michener, and Gaurav S Sukhatme, I am
deeply thankful for their support in shaping my perspectives from the view-points of clinical medicine,
bio-robotics, and theoretical computer science; these perspectives shaped the story this dissertation tells.
MyappreciationextendstomyfellowmembersoftheBrain-BodyDynamicsLaboratory–—myunwaver-
ingsupportnetworkthatkepttheshipafloat. Andfinally,Iamextremelygratefultomyfriendsandfamily,
whoaremygreatestinspirationandmystrongestsourceofjoy.
ii
TableofContents
Acknowledgements ii
ListOfTables v
ListOfFigures vi
Abstract xii
Chapter1: Exploringthehigh-dimensionalstructureofmuscleredundancyviasubject-specific
andgenericmusculoskeletalmodels 1
1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.0.1 AuthorContribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Cathindlimbmodel: Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Vectormappingofthefeasibleforceset . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Vectormappingofthefeasibleactivationset . . . . . . . . . . . . . . . . . . . . 6
1.4 Cathindlimbmodel: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Intra-speciesdifferencesinthefeasibleforceset . . . . . . . . . . . . . . . . . . 7
1.4.2 Structureoffeasibleactivationsets . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Humanarmmodel: Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.1 Kinematicsofthrowingaflyingdiscandresultingmusclefibervelocities . . . . . 9
1.6 Humanarmmodel: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.1 Musclefibervelocitiesforflyingdiscthrow . . . . . . . . . . . . . . . . . . . . . 11
1.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7.1 Structureofthefeasibleactivationandfeasibleforcesetsofthecathindlimb . . . 13
1.8 Muscleactivationforfasteverydayrecreationalandsportstasks . . . . . . . . . . . . . . 15
1.9 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter2: Structureofthesetoffeasibleneuralcommandsforcomplexmotortasks 26
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.0.1 AuthorContribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Hit-and-Runalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Realisticindexfingermodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
iii
Chapter 3: Feasibility Theory reconciles and informsalternative approaches to neuromuscular
control 36
3.0.0.1 Affiliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.0.1 AuthorContribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 Exampleofatendon-drivensystem . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.2 Analysisoffeasibleactivationspaces . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 Parallelcoordinatevisualizationnaturallyreveals
thestructureofthefeasibleactivationspace . . . . . . . . . . . . . . . . . . . . . 49
3.4.2 Low-dimensionalapproximationstothefeasibleactivationspace . . . . . . . . . . 51
3.4.3 Changesintheprobabilisticstructureofthefeasibleactivationspacewithincreas-
ingtaskintensity,orhowmuscleredundancyislost. . . . . . . . . . . . . . . . . 52
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.2 Thevalueofacostfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.3 Freedomunderconstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5.4 HowtoapplyFeasibilityTheoryinanexperiment . . . . . . . . . . . . . . . . . . 56
3.5.5 Extensiontodynamicalforceproductionormovement . . . . . . . . . . . . . . . 57
3.5.6 Structure,correlation,andsynergies . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.7 Towardprobabilisticneuromuscularcontrol . . . . . . . . . . . . . . . . . . . . . 60
3.5.8 FeasibilityTheoryasatheoryofmotorcontrol . . . . . . . . . . . . . . . . . . . 61
3.6 DataAvailabilityStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter4: Spatiotemporaltunnelsconstrainneuromuscularcontrol 72
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.0.1 AuthorContribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.0.1 HitandRunsamplingofthefeasibleactivationspace . . . . . . . . . . 77
4.3.0.2 Definingthetemporalconstraints . . . . . . . . . . . . . . . . . . . . . 78
4.3.1 Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.1.1 Analyzingunseededvsseededactivationtrajectorydistributions . . . . 79
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 SupplementaryInformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 Conclusionandongoingwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
iv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ListOfTables
3.1 ApplicabilityandcompatibilityofFeasibilityTheorywithdominanttheoriesofneuromus-
cularcontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
v
ListOfFigures
1.1 Bone lengths, joint axes of rotation, and moment arm matrix for the species average cat
hindlimb model, in cm. Positive values are shown in red and negative values in blue, as
pertheright-hand-rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Left: The polygon of the 2-D feasible force set in the sagittal plane. The color-coded
vectormapping of radial lines indicate the magnitude of the maximal feasible force along
thatdirection,thenvectormappedontotheperimeterofthecirclesurroundingtheFFS.The
verythinlinesemanatingfromtheoriginarethelinesofactionofeachofthe31muscles.
Center: thepolyhedronofthe3-DFFS,againwiththevectormappingofforcemagnitude
values onto a circle in the sagittal plane. Right: The color-coded vectormapping onto the
surface of a sphere indicating the maximal feasible force in every direction in 3-D. Note
the FFS is rather flat on the sagittal plane, but elongated towards the posterior direction.
AlldataareforthecatcalledBirdyin[113]. . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Top: Vectormapoftheaverageofmaximalfeasibleforceacrossallsampledoutputvectors
in three feline hindlimbs. Bottom: A vectormap displaying regions of the feasible force
space that have higher standard deviation across three cat hindlimbs. Force in Newtons
representedbycolorscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Structure of the feasible activation set for three muscles. The large vectormaps on the far
Right show their unique activation level for maximal force output in every 3-D direction.
Because multiple activation levels can produce submaximal forces, the small vectormaps
to the Left show the lower and upper bounds of those feasible activation levels for force
magnitudes(a)graduallyincreasingfrom50%ofmaximalinevery3-Ddirection. . . . . . 22
1.5 Moment arm values for human arm model. The moment arms from the 17 muscles con-
sidered in this model and their associations with the five DoFs are illustrated, in cm. The
moment arms are grouped by DoF and are shown below the associated joint. Positive
valuesareshowninredandnegativevaluesinblue. . . . . . . . . . . . . . . . . . . . . . 23
1.6 Top view of the 3-D human arm model. This figure illustrates the initiation of forward
motionthroughfollow-throughoftheflyingdiscthrow. Thereferencepostureisshownin
black and the release point in the throw is shown in red. The interpolated joint angles for
the45posturesdescribingthismotion,obtainedfrom[58],areshowninthebottompanel. 24
vi
1.7 Normalized instantaneous fiber velocities during the throw for the nominal model. Top:
The muscles are listed on the y-axis and the 45 postures making up the throw are shown
onthex-axis. Excessivemusclevelocitiesareshowninred(shortening)andblue(length-
ening). Bottom: The same data are illustrated with individual traces for each muscle that
show the fiber velocity. Muscles controlling the shoulder, elbow, and wrist are illustrated
in blue, red, and green, respectively. Instantaneous fiber velocity is given on the y-axis
and the postures during the throw are on the x-axis. Regions of the traces outside of the
horizontal dashed lines indicate excessive muscle velocities. In both figures, the release
pointofthethrowisindicatedwithaverticaldashedline. . . . . . . . . . . . . . . . . . . 25
2.1 The feasible activation set for a three-muscle system meeting one functional constraint is
apolygoninR
3
. Notethatmuscleactivationsareassumedtobeboundedbetween0and1. 30
2.2 GraphicaldescriptionoftheHit-and-Runalgorithm. . . . . . . . . . . . . . . . . . . . . . 31
2.3 The index finger model simulated 50% of maximal force production in the palmar direc-
tion. Adaptedfrom[144]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 We show one histogram for each muscle of the index finger to illustrate how the muscle
is used across all feasible solutions. For this set of distributions, the task was 50% of
maximal force output in the palmar direction. Muscles are FDP, FDS,EIP,EDC, LUM,
DI, and PI are shown in that order from top to bottom. The orange dotted lines are the
lowerandupperboundsofactivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Emergence and interpretation of feasible activation spaces for a particular motor
task. The descending motor command for a given task is issued by the motor cortex (a),
which projects onto inter-neurons and alpha-motor neuron pools in the spinal cord (b).
The combined drive to all alpha-motor neurons of a muscle can be considered its total
muscle activation level (a value between 0 and 1). If we consider that muscles can, to
a large extent, be controlled independently and in different ways, then the overall motor
command can be conceptualized as a multi-dimensional muscle activation pattern (i.e. a
point) in a high-dimensional muscle activation space[23, 116, 70, 144, 129] (c). For that
muscleactivationpatterntobevalid,ithastoelicitmuscleforces(d)capableofsatisfying
the mechanical constraints of the task—in this case defining a well-directed sub-maximal
fingertip force (e). Given the large number of muscles in vertebrates, there can be muscle
redundancy: whereagiventaskcanbeaccomplishedwithalargenumberofvalidmuscle
activationpatterns. Weproposethatournovelabilitytocharacterizethehigh-dimensional
structure of feasible activation spaces (i) allows to us to compare, contrast, and reconcile
today’sthreedominantapproachestomuscleredundancyinsensorimotorcontrol(f,g,h). 38
3.2 Parallelcoordinatescharacterizethehigh-dimensionalstructureofafeasibleactiva-
tionspaces. Considerfourpoints(i.e. muscleactivationpatterns)fromthepolygonthatis
afeasibleactivationspace(a). Theactivationlevelforeachmuscle(i.e. thecoordinatesof
eachpoint)aresewnacrossthreeverticalparallelaxes(b). Asiscommonwhenevaluating
musclecoordinationpatterns,eachpointcanalsobeassignedacostasperanassumedcost
function. The associated cost for each muscle activation pattern can also be shown as an
additional dimension. We show three representative cost functions (c). Activation levels
areboundbetween0and1,andcostsarenormalizedtotheirrespectiveobservedranges. . 65
vii
3.3 Activationpatternsofthesevenmusclesoftheindexfingeracrosssixintensities(mag-
nitudes)ofafingertipforcevectorinthedistaldirection. The connectivity across par-
allel coordinates visualizes the correlations among muscle activation patterns at different
task intensities. At the extremes of 0% and 100% we have, respectively, the coordina-
tion patterns that produce pure co-contraction and no fingertip force, and the one unique
solution for maximal fingertip force [144]. In between, we see how the structure of the
feasible activation spaces changes, and that much redundancy is lost rather late (at inten-
sities greater than 80%, in agreement with [110]). In blue are the activation values, and in
red are normalized costs for four common cost functions in the literature. For each task
intensity, we produced 1,000 points that are uniformly distributed in the polytope via the
Hit-and-Runmethod. The musclesare FDP: flexor digitorum profundus, FDS: flexor dig-
itorum superficialis, EIP: extensor indicis proprius, EDC: extensor digitorum communis,
LUM:lumbrical,DI:dorsalinterosseous,PI:palmarinterosseous. Colorisusedsolelyto
differentiatemuscleactivations(blue)fromcostvalues(red). . . . . . . . . . . . . . . . . 66
3.4 Exploration of the feasible activation space for task intensity of 80%. Here we show
three informative examples of constraints applied to the points sampled from the feasible
activation space (n=1,000; axes match those of Fig. 3.3). With this interactive visual-
ization, we can easily see how the size (i.e. number of solutions) and characteristics of
the family of valid muscle activation patterns change. For example, in the event of (Top)
weaknessofagroupofmuscles(54%reduction),(Middle)selectionofthelowest5%ofa
givencostfunction(95%reduction),and(Bottom)enforcingthelowest10%ofcostrange
across multiple cost functions (99.6% reduction). In all cases, the family of valid muscle
activation patterns retains a wide range of activation levels for some muscles. While it is
challengingtounderstandthestructureofthefeasibleactivationspacewithastaticplotof
the parallel coordinates, interactively manipulating the muscle ranges on one or multiple
axesmakesitveryeasytoviewanddescribehowmuscleactivationschangeinthefaceof
differentconstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Approximating the structure of feasible activation spaces via principal components
analysis (PCA) is sensitive to both the task intensity and the amount of input data
used. Rowsshowthevarianceexplainedbythefirst(top)throughthird(bottom)principal
components with increasing data points for a given replicate (left to right). Hit-and-Run
sampling provides the ground truth for the high-dimensional structure of the feasible ac-
tivation set at each task intensity. Each box plot, across all subplots, is formed from 100
metrics (replicates), where each metric is the PC variance explained for a replicate ‘sub-
ject’whichperformedthetaskntimes(wherenisoneof10,100,or1000taskrepetitions).
WefindthatPCAapproximationstothisstructuredonotgeneralizeacrosstasksintensities
(i.e. the polytope changes shape as redundancy is lost), and numbers of points. That is,
> 100 muscle activation patterns should be collected from a given subject to confidently
estimate the real changes in variance explained as a function of task intensity. Compare
pointslabeleda,b,c,correspondingto11,66,and88%oftaskintensity,respectively. . . . 68
viii
3.6 PCAloadingschangewithtaskintensityForeachof1,000taskintensities,wecollected
1,000 muscle activation patterns from the feasible activation space and performed PCA.
ThefacetrowsshowthechangesinPCloadings,whichdeterminethedirectionofallPCs
in 7-dimensional space. Note that the signs of the loadings depend on the numerics of
the PCA algorithm, and are subject to arbitrary flips in sign [27]—thus for clarity we plot
them such that FDP’s loadings in PC1 are positive at all task intensities. Dotted vertical
linesconnectloadingsofPC2andPC3inspiteofflipsinsign. Adiscontinuityhereisnot
indicative of a major change to the feasible activation space. It instead, is a result of how
PCA selects loadings. The shape of the activation space has tilted at these points, thereby
flippingthesign. Notethatthevaluesarethesamebeforeandafterthejump,lessthesign.
These loadings (i.e. synergies) change systematically, as noted for representative task
intensities a, b, c in Fig. 3.5, and more so after b. This reflects changes in the geometric
structureofthefeasibleactivationspaceasredundancyislost. . . . . . . . . . . . . . . . 69
3.7 The within-muscle probabilistic structure of feasible muscle activation across 1,000
levels of fingertip force intensity. The cross-section of each density plot is the 50-bin
histogram of activation for each muscle, at that task intensity. The changes in the breadth
andheightforeachmuscle’shistogramrevealmuscle-specificchangesintheirprobability
distributionswithtaskintensity. Heightrepresentsthepercentageofsolutionsforthattask.
The axis going into the page indicates increasing fingertip force intensity up to 100% of
maximal. Color is used to provide perspective. It is interesting to note that, for example,
both extensor and flexor muscles are used to produce this ‘precision pinch’ force. This is
to be expected as the activity in the extensors is necessary to properly direct the fingertip
forcevector[141]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8 SpatiotemporalTunneling. A dynamical movement can be decomposed into a sequence
ofslicesintime,whereeachslicehasacorrespondingfeasibleactivationspace. Strungto-
gether,thesequenceoffeasibleactivationspacesformthe‘spatiotemporaltunnel’through
which the neuromuscular system must operate. In this 3-dimensional schematic example,
theblacklinerepresentsonevalidtime-varyingsequenceofactivationsforthreemuscles.
Becausethissequenceexistswithineachfeasibleactivationspace,itnecessarilymeetsthe
constraintsofthedynamicaltaskateachinstant. . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 Overview of the primary objective of this work. Our objective is to computationally
surveytheFeasibleTrajectorySpaceinthecontextofactivation-contractionconstraint,to
betterinformourperspectivesofdescendingneuromuscularcontrolparadigms. . . . . . . 75
4.2 Consequencesofselectingaspecificinitialmuscleactivationpatternforamaxactivation-
contractionspeedof0.25Hereweshowthedistributionsofthreetrajectoryseedsselected
across a uniform sample of the unseeded H (Eq. 4.3). As in 4.10, lines are drawn by
connecting the midpoints of 100 histogram bins. We observe strong hysteresis in the po-
sitioning of muscle activation when the seed trajectory locks the activation high or low
on a given muscle, and that selecting a seed point implies that you cannot easily return to
anotherseedpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
ix
4.3 Method for generating unseeded and seeded trajectories Unseeded trajectories can
originate in any valid solution at t = 0 show their evolution across the subsequent poly-
topes (i.e., solution spaces) subject to the temporal constraints of activation-contraction
dynamics of muscle. A seeded trajectory, on the other hand, is pulled from the same con-
straintmatrix,butwithanadditionalconstraint: allofthepointsselectedfromaseedstart
at a same seed point (i.e., valid solution at t = 0). A seed point can be extracted from
the unseeded trajectories. Seeded points can only evolve in time into subregions of the
subsequent solution spaces that are reachable given the starting point and the temporal
constraints of activation-contraction dynamics of muscle. Importantly, unseeded trajecto-
riesallmeetactivation-contractionconstraintsaswell . . . . . . . . . . . . . . . . . . . . 81
4.4 Quantifyingtheevolutionovertimeofthedistributionofsolutionsforunseededand
seededtrajectories Here we detail our method for analyzing and visualizing the effect of
selectingasolutionseededint =0. Webeganbyextractingonehundredthousandactiva-
tion trajectories from H as in Eq. 4.3. With 10 of those trajectories, we extracted only the
first value, then ran a further sampling paradigm on a modified constraint equation where
the first activation pattern (of 7 muscle activations) had to match the seed’s activations at
t=0. As we want to visualize the effect of selecting a seed point, but cannot easily plot a
4D structure embedded in 7D, we applied principal component analysis to each of the 7
moments of time across the unseeded distribution. We then projected both the unseeded,
andseededactivationtrajectoriesacrossthefirsttwoPCs,highlightingwhereinthelower-
dimensionalspacethosesolutionsweremostprobable. . . . . . . . . . . . . . . . . . . . 82
4.5 Ten example trajectories with three levels of activation-contraction constraints For
each level, we show a) ten example trajectories, where each color is a different trajectory.
b) Those trajectories, differentiated to show how quickly the activations were changing
withtheupperandloweractivation-contractionconstraintsshownasdottedlines,andc)a
distributionofthetrajectory‘activation-contractionspeeds’,groupedbymuscle. Notethat
colors on part c) do not relate to a) and b). Outliers are not shown on c. Note that unlike
Figure 4.6 which shows the max(|˙ a
i
|), this figure shows the raw differentiated muscle
activationsas ˙ aandthusissignedfrom ±1. . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6 Theeffectofdifferingactivation-contractionconstraintsonthedistributionofmax(|˙ a
i
|),
comparedacrossmusclesWhenwesampletrajectories,wegetabunchofn-dimensional
trajectories, where n=7 muscles. From each of those trajectories, we differentiate them
(e.g. ˙ a
i
=a
LUM
i+1
a
LUM
i
), and we show here the distributions of e.g. ˙ a
LUM
. These speeds
are grouped by the applied activation-contraction constraint. The case with no activation-
contraction constraints is a 1.0; a 0.1 means a muscle is spatiotemporally constrained so
thatitcannotchangebymorethan10%within50ms. . . . . . . . . . . . . . . . . . . . . 85
4.7 Spatiotemporal tunnels for each of 10 seed points The ‘seed’ activation you choose
in the first moment highly constrains where your muscle activations can go across the
following six tasks. Shown for a activation-contraction constraint of 0.12 (in that no
muscle can change more than 12% in tension from slice to slice). Each slice of the tunnel
is a task, where the points have been projected onto the un-seeded PCs (PC1 and PC2),
whichwerecomputedseparatedforeachslice,providingpolytope-relevantchangesinthe
distributionsofseededdistributionswithrespecttotheunseededtrajectorydistribution. . . 86
x
4.8 Activation distributions under differing activation-contraction constraints Taking all
of the points collected, we group them by muscle, task, and by the spatiotemporal con-
straint under which they were collected. You can see each color represents a different
spatiotemporal constraint, and the boxplots represent the way each muscle was used, at
thattaskindex. Alltrajectoriessampledareunseeded. . . . . . . . . . . . . . . . . . . . . 91
4.9 Supplemental Figure: Variance ofa across trajectories (within a given muscle) does
notnecessarilygodownasthefeasibleactivationspaceisundermorestrictactivation-
contraction constraint Given the velocity constraints, we extract a long series of activa-
tions for each muscle, at each task index. Per muscle, we computed the variance of each
series, creating a visualization of the feasible activation space as the task is performed,
and across differing activation-contraction constraint (and the degenerate case). dimen-
sionsarebarelyaffectedbyeitherthechangeinthetask,norbytheactivation-contraction
constraint. variance, and also had a bigger effect in their variance shrinking under more
activation-contraction constraint. Temporal constraint led to reduction in variance across
those muscles, indicating that the distribution across the muscle may become more uni-
formlydistributed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.10 Interactive seed trajectory explorer In comparing unseeded distributions with seeded
distributions, we designed an interactive data exploration supplement to highlight how
different the seeded trajectories could be, and how they were often highly constrained by
their activation in t=0. Bottom: we provided a slider so the user could change the seeds,
and see how the distributions compared with the unseeded distribution (which remained
constant across all seeds, for this given redirection task). Lines are drawn by connecting
themidpointsof100histogrambins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.11 Spreadofdifferentseedpointsundervaryingactivationcontractionconstraints . . . 93
xi
Abstract
Feasibility Theory is a conceptual and computational approach to understand the dimensionality of how
tendon-driven limbs are controlled. How do the brains of animals control their bodies? This remains one
of the deepest mysteries in biology, a concept with an enormous consequence upon how we understand,
diagnose, and treat diseases or injuries that rob animals of manipulation and locomotion. Engineers and
scientists have tackled this problem from rigorous mathematical and scientific perspectives and much
progresshasbeenmade—primarilydescriptiveapproachesthatattempttopredicthowthenervoussystem
solves a motor task (e.g. modeling observed behavior based on recordings of muscle activity). However,
this rigor has a downside: the efficient ways we know how to solve problems mathematically (via formal
optimization)cansurreptitiouslybiasthescientificcommunityintohypothesizingthatthebrainalsosolves
problemsthisway. Thesemethodsdescribehowthemusclesfunction,buttheydonotdescribewhycertain
patternsofcontrolareevolved(overmillennia),learned(overalifetime),orchosen(withinjustonemotor
task). The work presented herein delves into the problem from a full-dimensional perspective of motor
control—requiring a truly Big Data exploration into, first, a view of the feasible options for control, and
second,asetofconstraintswhichfaithfullydescribethewaysinwhichtendon-drivenlimbstruly must be
controlled. I present new mathematicallyformal ways to show thatmany muscles are indeedneeded even
for simple tasks, and indicate how even the most ‘optimal’ solutions are highly prone to disruption, even
withminordisabilityofjustonemuscle. Iproposenewcomputationalmethodsforreconcilingalternative
approaches to motor control, including techniques in dimensionality reduction, Bayesian representation,
and optimization. Ultimately, this work now enables a new perspective towards exploring how motor
controlaffectshealthandqualityoflife.
xii
Chapter1
Exploringthehigh-dimensionalstructureofmuscleredundancyvia
subject-specificandgenericmusculoskeletalmodels
FranciscoJ.Valero-Cuevas,
1,2
,*BrianA.Cohn,
1
H¨ ordurF.Ingvason,
3
andEmilyL.Lawrence
1
1
DepartmentofBiomedicalEngineering,UniversityofSouthernCalifornia,LosAngeles,CA,USA
2
Division of Biokinesiology and Physical Therapy, University of Southern California, Los Angeles, CA,
USA
3
SwissFederalInstituteofTechnology-Zurich,Zurich,Switzerland
1.1 Abstract
Subject-specific and generic musculoskeletal models are the computational instantiation of hypotheses,
and stochastic techniques help explore their validity. We present two such examples to explore the hy-
pothesis of muscle redundancy. The first addresses the effect of anatomical variability on static force
capabilities for three individual cat hindlimbs, each with seven kinematic degrees of freedom (DoFs) and
31 muscles. We present novel methods to characterize the structure of the 31-dimensional set of feasi-
ble muscle activations for static force production in every 3-D direction. We find that task requirements
strongly define the set of feasible muscle activations and limb forces, with few differences comparing
individual vs. species-average results. Moreover, muscle activity is not smoothly distributed across 3-D
1
directions. The second example explores parameter uncertainty during a flying disc throwing motion, by
usingagenerichumanarmwithfiveDoFsand17musclestopredictmusclefibervelocities. Weshowthat
themeasuredjointkinematicsfullyconstraintheeccentricandconcentricfibervelocitiesofallmusclesvia
their moment arms. Thus muscle activation for limb movements is likely not redundant: there is little, if
any,latitudeinsynchronizingalpha-gammamotoneuronexcitation-inhibitionformusclestoadheretothe
time-critical fiber velocities dictated by joint kinematics. Importantly, several muscles inevitably exhibit
fiber velocities higher than thought tenable, even for conservative throwing speeds. These techniques and
results,respectively,enableandcompelustocontinuetorevisetheclassicalnotionofmuscleredundancy
forincreasinglymorerealisticmodelsandtasks.
1.1.0.1 AuthorContribution
My primary contribution to this work was the extension of [113], invention of the methods for generating
FAS and FFS species averages, and the invention of vector mapping for tendon-driven feasible activation
sets. Further,Icontributedtotheanalysisandvisualizationoffrisbee-throwingbyEL.
1.2 Introduction
This invited paper has the dual purpose of being didactic about computational methods to test neurome-
chanicalhypothesesinthecontextofhigh-dimensionalsubject-specificandgenericmodels;andapplying
these methods to explore the classical notion of muscle redundancy, a central tenet in our field. This is
made possible by computational geometry and stochastic techniques we have been developing to under-
stand the interactions among (i) model topology (the number and type of and connectivity among the
elements of the model); (ii) parameters values (the individual and specific numerical values assigned to
each model parameter); and (iii) the requirements of real-world tasks for tendon-driven biomechanical
systemswithnumerouskinematicdegreesoffreedomandmuscles.
2
The notion of computational models as instantiations of specific hypotheses, the stochastic explo-
ration of model capabilities to test these hypotheses, and the relationship between generic vs. subject-
specific models has been addressed elsewhere [151, 150, 98, 65, 155]. However, increasing the physi-
ological realism and utility of these techniques requires extending them to ever higher dimensions (i.e.,
larger numbers of muscles and kinematic degrees-of-freedom, DoFs), and to real-world tasks involving
the production of static forces and fast motions—while limiting computational cost. But working with
ever-greater numbers of muscles and DoFs inevitably challenges our ability to visualize the complex and
high-dimensionalstructureofthesetoffeasiblemuscleactivationpatterns. Italsosignificantlychallenges
our ability to find unique solutions (if they even exist) to these computational problems, or defend their
optimality/uniqueness.
We have found these stochastic modeling techniques particularly useful to test the classical notion of
muscle redundancy, which has often been called the central problem of motor control [16]. The classical
notionofmuscleredundancyisthoughttoarisebyvirtueofhaving(many)moremusclesthanDoFs. With
many muscles acting upon the same number or fewer joints, some argue that the central nervous system
(CNS) must solve an optimization problem to select and implement specific muscle activation patterns
from a theoretically infinite set of possibilities [93, 105]; while others argue for near- or sub-optimal
solutionsbeinggoodenough[80,105]. Iffewermusclesactuatedalimb,theargumentsgo,feasibleforces
andmotionscouldbeproducedwithoutsignificantneedforsuchoptimizations.
Several of us have argued that this classical interpretation of the number of muscles in vertebrate
limbs is paradoxical with respect to evolutionary biology, and the clinical reality of motor dysfunction:
extant vertebrates tend to have many more muscles than DoFs, even though it is energetically expensive
to develop and maintain muscle mass—and injury to even a few muscles can cause dysfunction. Using
the same argument of energetic efficiency invoked for optimization in motor control—but at the scale of
evolutionary time—we, and others, have argued that we likely have barely enough muscles for versatile
real-worldbehavior[64,79,147,72,74]. Thisviewiscloselyalignedwiththecomputationalneuroethol-
ogy approach [11, 7, 28] that argues that perhaps we need all our muscles because of the sheer variety
3
of tasks—each distinguished by the type and number of constraints they must meet—over the course of a
day/week/lifespan. Put differently, if we have too many muscles in our limbs, which ones would you like
to donate or paralyze? Therefore, it is important that our research into muscle redundancy work toward
reconcilingthesedifferentviews.
Still, for most tasks in healthy individuals, some redundancy is bound to remain; regions of feasible
activation solutions that are not a single point will consist of a neighborhood or subspace that naturally
contains an infinite number of solutions (i.e., points). The nervous system is still confronted with the
need to choose a specific solution to implement at any point in time; however, that collection of feasible
solutions remains highly structured due to both the mechanics of the limb and the constraints of the task
[113, 72, 74, 156, 133, 17]. The purpose of this work, therefore, is to begin to address the need posed
by us [72, 74, 156], and others [113, 79, 133], to improve computational methods for understanding and
visualizing the dimensionality and structure of feasible solutions sets for limbs with large numbers of
muscles performing tasks with realistic constraints. Here we do so for 3-D musculoskeletal models of
a cat hindlimb and a human arm with 31 and 17 muscles, respectively, using MATLAB (v2013b, The
Mathworks,NatickMA).
1.3 Cathindlimbmodel: Methods
The purpose of this cat hindlimb model is to present a novel way to visualize the structure of the set
of all feasible muscle activations to produce maximal and submaximal static paw forces in every 3-D
direction. In addition, we compare solutions among three subject-specific models to explore the effect
of between-subject anatomical variability on muscle activation. The models consist of three feline (Felis
catus) hindlimbs, each with 31 muscles actuating 7 kinematic DoFs from the hip to the ankle. We used
thebonelengthsandmomentarmsforthecathindlimbsoriginallypresentedbyMcKayandTingin2008
[87], and modified by Sohn et al. 2013 [113], that were graciously shared with us by the authors. The
speciesaveragemodelforthecathindlimbisshowninFigure1.3.
4
As described in detail elsewhere (e.g., [156, 86, 148]), a feasible force set (FFS) describes the set of all
static forces that can be produced at the endpoint of a limb. Briefly, the feasible mechanical output of
the endpoint of a limb is 6-dimensional: 3 forces (the FFS) and 3 torques (the feasible output torque
set)—whicharisesfromthefactthatarigidbody(i.e.,theendpointofalimb)hassixdegreesoffreedom,
three displacements and three rotations. Together they form a 6-dimensional feasible output wrench [89].
Intheroboticsliterature[88],feasibleforceandtorqueoutputsareplottedseparatelyastheyhavedifferent
units. ThustheFFScanbeatmost3-D,andisasubsetofthefeasiblewrenchset. Forthetaskofproducing
pure output force as in this model, we enforce the constraint that the endpoint produce no output torques
[156]. Thus the FFS is the complete representation of the maximal mechanical output of the limb. For
limb models constrained to move on a plane, the FFS is a convex 2-D polygon (Figure 1.4.2, Left). For
modelsthatcanmovein3-Dspace,theFFSisaconvex3-Dpolyhedron(Figure2,Center)withitsorigin
attheendpointofthelimb[156,148].
Importantly, as described elsewhere [156, 86, 148, 71], the FFS is produced by the feasible activation
set (FAS)—the set of all muscle activations that meet the constraints of the task. For linear constraints as
inthiscase,theFASisaconvexpolytopeinn-dimensionalspace,wherenisthenumberofindependently
controlled muscles acting on the limb. The FAS is at the center of studies of muscle redundancy because
itcontainsaninfinitenumberofpoints. Sometimesthissubspaceiscalledthenullspaceofthetaskasany
pointinitcan,byconstruction,meetitsconstraints[24]. Butitisneverthelessahighlystructuredsubsetof
n-dimensionalspace. Acriticalresultofourworkisthatwepresentameanstovisualizeandcharacterize
theFFSbyexaminingonemuscleatatime.
1.3.1 Vectormappingofthefeasibleforceset
It is challenging to understand and visualize a 3-D FFS, as it is an irregularly shaped convex polyhedron
(Figure2,Center). Likewise,thosedifficultiesareexacerbatedfortheFASasitisalsoanirregularlyshaped
polytope,butinhighdimensions. AsmentionedintheIntroduction,itiscriticaltounderstandthestructure
of the FFS and FAS as they lie at the heart of many debates about muscle redundancy, muscle synergies,
5
disability, rehabilitation, motor learning, etc. One approach to connect the structure of the FFS and the
FAS is by computing their bounding boxes (i.e., the extreme points in every dimension [113, 72, 59]).
However, this overestimates both their size and volume, and ignores the complexity of their structure.
Another possibility is to find the largest sphere the polytope can encase [59], but this underestimates their
size and volume, and assumes a uniform structure. We now propose an alternate method that helps us
visualize the structure of the FFS, in a ‘vectormap’. After identifying the maximum feasible force in a
givendirection(Figure2,Left),weassignthatvalueofforcetoa3-Dpoint,wherecolordenotestheforce
intensity. A spherical heatmap is formed with all of the computed directions and respective maximum
forces;Figure2(Right)showsthevectormaprepresentationoftheFFS.
Traditionally,polyhedraliketheFFScannotbecombinedorcomparedquantitativelybecausethever-
tices do not align across different individual musculoskeletal models. As vectormaps are composed of
consistent unit vectors for force output (or muscle activation, see next section), they can be averaged and
compared. Forexample,theycanbecomparedacrossindividualsofaspeciestoidentifyregionsthathave
highervariabilitywithinapopulation. Thecoloronthesurfaceofthespherecanthenbeusedtorepresent
themeanorstandarddeviationofmaximaloutputforceormuscleactivation(Figure3).
1.3.2 Vectormappingofthefeasibleactivationset
We present a way to visualize the structure of the FAS, a convex polytope in n-dimensional space, on a
muscle-by-muscle basis. For each muscle we can generate activation vectormaps where color represents
itsuniqueactivationlevelforeverypointonthesurfaceoftheFFS(Figure4). Thisispossiblebecauseany
pointonthesurfaceoftheFFS(i.e.,themaximalforceineverydirection)isgeneratedbyauniquemuscle
activation pattern [156]. This unique activation pattern assigns the color to that point on the vectormap of
eachmuscle. Inthecaseofthecathindlimbthereare31muscles,andtherefore,31vectormapsofunique
muscleactivations.
Importantly, submaximal forces in each 3-D direction (i.e., points within the FFS) can be produced
by an infinite number of solutions [21, 30]. The structure of those solutions can be approximated by the
6
bounding box approach in [113, 72]. We extend that prior work by creating vectormaps of the lower and
upperboundsofactivationforeachmuscle,foralldirectionsin3-D(Figure4).
1.4 Cathindlimbmodel: Results
1.4.1 Intra-speciesdifferencesinthefeasibleforceset
With one FFS per cat, we find that force capability distributions for the three cats can differ in specific
3-D directions. Figure 3 shows between-cat comparisons: a species average (top), and standard deviation
(bottom) plots across the three FFSs. We see that species average and individual FFSs are of the same
general shape (c.f. Figure 2, Right and Figure 3, Top) with maximal force magnitudes remaining in the
same general direction (towards the posterior direction) and of similar magnitude (c. 60 N) as for the
individual cat in Figure 2. However, the standard deviation among the FFSs of the three cats can show
importantdifferencesintherangeof20Ninthosesamedirectionsasthemaximalmagnitude. Butinmost
other3-Ddirectionsthedifferencesremainbelow5N.
1.4.2 Structureoffeasibleactivationsets
Figure4showswhattoourknowledgeisthefirstportrayalofthestructureoftheFASforforceproduction
in every 3-D direction. For the sake of brevity, we only show the results for three muscles. The plots for
all31musclesareavailableathttps://valerolab.org. Thevectormapsonthefarrightshowtheuniquelevel
of activation for maximal feasible forces in all directions. While in several 3-D directions that activity
of a muscle can remain unchanged, we also see discontinuities where muscle activity is not smoothly
distributed across 3-D directions as, for example, the ‘fingers’ of higher activations penetrating into areas
ofloweractivationsfor vastus lateralis.
To extend prior work [113, 72], we also found the lower and upper bound vectormaps for all muscles
forsubmaximalforcesinall3-Ddirections. Thisistheboundingboxapproachin[113,72],butextended
to every direction in 3-D. These plots provide a detailed view of the structure of the 31-dimensional FAS
7
fordifferentforcemagnitudes,viewingonemuscleatatime. ThesmallervectormapstotheleftinFigure
4showthelowerandupperboundsasoneincreasesforcemagnitudeinalldirectionsin10%steps,starting
at 50% of maximal force. The lower and upper bounds naturally converge for maximal output, but they
converge at different rates across muscles and directions of force output—sometimes towards the upper
bound, and sometimes towards the lower bound. These vectormaps of the FAS enable us to understand
the rate at which redundancy is ‘lost,’ or not for every direction of force production. They also enable
future studies where, say, the loss of the soleus muscles, or its hypertonia, are simulated by driving its
activation to the lower or upper bound, respectively, to visualize the feasible range of compensations by
othermuscles.
1.5 Humanarmmodel: Methods
The purpose of this human arm model is to understand the constraints imposed on time varying muscle
activationduringthekinematicsofahigh-speedathleticmovement. Specifically,ourmodelpredictsmus-
cle fiber lengths and velocities during a specific athletic activity—in this case throwing a flying disc with
a backhand motion, like throwing a Frisbee R . A five-DoF, 17-muscle arm model of the right arm was
modeled after [1], and consisted of three joints (shoulder, elbow, and wrist) articulating three limb seg-
ments (upper arm, lower arm, and hand) with lengths of 0.35m, 0.27m, and 0.11m, respectively (Figure
5). The three DoFs at the shoulder included internal/external rotation, abduction/adduction, and horizon-
tal abduction/adduction, and the DoF at both the elbow and wrist is flexion/extension. We note that our
simplified model does not consider all DoFs at the elbow and wrist. This limitation affects the calculation
ofjointanglesandfibervelocities,butlikelydoesnotchallengeourresultsasinsomecasesfibervelocity
would be somewhat lower, but also somewhat higher. We added 17 muscles/muscle groups with resting
fiberlengthandmomentarmdatafromvarioussources[48,51,56,90]. Themomentarmdataareshown
inFigure5.
8
1.5.1 Kinematicsofthrowingaflyingdiscandresultingmusclefibervelocities
The time-history of joint angles of the throwing motion were also obtained from [58]. We considered
the initiation of forward motion, release, and follow-through portions of the throw to last, conservatively,
450ms; and approximated it as 45 unique postures at 10ms time steps, as illustrated in Figure 6. We
combined measured limb kinematics with moment arm values to predict the instantaneous normalized
musclefibervelocitythroughoutthethrow(Fig. 5).
Consideratendon-drivenlimbwithnmuscles(17inthiscase)andmjoints(orDoFs,fiveinthiscase),
andalimbposturedefinedbyjointangles✓ =[q 1
...q m
]
T
. ThemomentarmmatrixR(✓ )
mxn
canbedefined
forthistendon-drivensystem,havingentriesconsistingofthemomentarmsr(✓ )
i,j
:i=1,...,m,j=1,...,n
,atthe i
th
jointand j
th
muscle[5]formingtheposture-dependentmomentarmmatrix1.1
R(✓ )=
0 B B B B B B B B B B @ r(✓ )
1,1
r(✓ )
1,2
r(✓ )
1,3
... r(✓ )
1,n
r(✓ )
2,1
r(✓ )
2,2
r(✓ )
2,3
... r(✓ )
2,n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
r(✓ )
m,1
r(✓ )
m,2
r(✓ )
m,3
··· r(✓ )
m,n
1 C C C C C C C C C C A (1.1)
As per the right-hand-rule, r(q )
i,j
is positive when pulling j
th
tendon induces a counterclockwise ro-
tation at the i
th
joint, and negative otherwise. A postural change is a rotation of joints from a reference
limb posture ✓ 0
=[q 1
...q m
]
T
to a new limb posture ✓ 0
=[q 1
...q m
]
T
and is denoted bmD q =✓ ✓ 0
=
[D q 1
...D q m
]
T
. fullydeterminestheexcursionsD s
ofallnmuscles[36],wherenegativeandpositiveexcur-
sionvaluescorrespondtoeccentricandconcentriccontractions,respectively:
In this case we obtain the over-determined system where the changes of angles of a few variables (the
jointangles)specifytheexcursionsofallthemanymuscles.
9
D s=R
T
(✓ )D ✓ =
0 B B B B B B B B B B B B B B @ r(✓ )
1,1
r(✓ )
2,1
··· r(✓ )
5,1
r(✓ )
1,2
r(✓ )
2,2
··· r(✓ )
5,2
r(✓ )
1,3
r(✓ )
2,3
··· r(✓ )
5,3
.
.
.
.
.
.
.
.
.
.
.
.
r(✓ )
1,17
r(✓ )
2,17
··· r(✓ )
5,17
1 C C C C C C C C C C C C C C A ✓ (1.2)
Tobeclear,thisistheveryoppositeofredundancy.
When the interval between postures is allotted a given amount of time, the instantaneous velocity of
themusclefibersis
⌫ =
D s
D t
(1.3)
Please note that the velocity of the muscle fibers is not necessarily the velocity of the musculotendon.
Muscle fiber pennation angle and tendon elasticity can both contribute to this [160]. For the sake of
simplicity,andwithoutlossofgenerality,weassumemusclefibersspanthelengthofthewholemuscleand
haveasmallpennationanglesothatwecanconsiderthemtobeequivalent. Arecentmodelingstudy[42]
also suggests that ‘paradoxical’ contractions—where the extreme case of muscle fibers shortening while
themusculotendonasawholeislengtheningduetotendonstretch—arebriefeventslimitedmostlytolarge
eccentric contractions to reverse movement direction. Due to these reasons, we assumed the velocities of
themusclefibersandtendonsweremostlyequivalentduringthemidsectionoftheuni-directionalthrowing
motion we consider in our analysis. As is customary, we calculated the normalized muscle fiber length
velocitiesbydividingfibervelocitiesbytherestingmusclefiberlength(l
O
)ofeachmuscle[160].
¯ n =
n l
o
(1.4)
10
1.6 Humanarmmodel: Results
1.6.1 Musclefibervelocitiesforflyingdiscthrow
Figure7showsthenormalizedmusclefibervelocitiesforallmusclesduringa450msflyingdisc-throwing
motion. Noticethatmultiplemuscleshavenormalizedmusclefibervelocitiesexceeding±5fiberlengths/s
(deep blue and deep red in Figure 7 Top, respectively). Because these high velocities are considered
to be unrealistically fast [160, 55], we used Monte Carlo simulations to explore the robustness of our
findings (Figure 7, Bottom). As is often done in musculoskeletal modeling [151], we explored the effect
of modeling uncertainty by iteratively running our model while sampling parameter values from uniform
distributions spanning ±25% of the nominal moment arm values. Given that the joint kinematics and
segment lengths come from direct measurements, our stochastic approach focused in the uncertainty of
moment arm values obtained from the literature as they may or may not be appropriate for the arm of the
subject who performed the flying disc throw. Note we fixed the duration of the motion to 450ms because,
although slow in comparison to competitive athletes, it provides a conservative estimate of muscle fiber
velocities and thus a more reasonable and defensible set of results. We guaranteed convergence of the
Monte Carlo simulation by testing the variability of the running mean of normalized fiber velocity of the
infraspinatus[151].
This muscle experienced the largest lengthening velocities, and as such, was at the greatest risk for
injury. Only twelve iterations sufficed for the running mean of the maximal infraspinatus normalized
fibervelocitytovarylessthan2%. RunningtheMonteCarlosimulationformoreiterationsunnecessarily
increases processing time without refining the results of maximal fiber velocities for this task. The results
of our Monte Carlo simulation (Figure 7, Bottom) provide confidence in the assertion that the task of
throwing a flying disc using a stroke that lasts 450ms will induce multiple muscles to exhibit normalized
fibervelocitiesexceeding ±5fiberlengthspersecond.
11
1.7 Discussion
Inthisinvitedmethods-drivenpaper,wepresenttwoexamplesofcomputationalmethodstotestneurome-
chanical hypotheses in the context of subject-specific and generic models, and apply these methods to
explore different aspects of the classical notion of muscle redundancy. In the first example, three indi-
vidual models of a cat hindlimb with 31 muscles allowed us to investigate the intra-species variation in
maximal force production. This was made possible by novel computational and visualization techniques
to complement a computational geometry approach to the control of tendon-driven limbs. The results
presented in this manuscript, and supplemental results online at https://valerolab.org, allows us to, for the
first time, describe detailed features of intra-species differences in maximal force production, and of the
structure of the 31-dimensional feasible set of muscle activation patterns for submaximal and maximal
forces in all 3-D directions. In the second example, we used stochastic Monte Carlo methods to demon-
strate that the kinematics of the everyday recreational and sports task of throwing a flying disc inevitably
leadstounexpectedlyfasteccentricandconcentricmusclefibervelocities. Thesetwoexampleschallenge
differentaspectsoftheclassicalnotionofmuscleredundancy,andleadtospecificnewtestablehypotheses
to move our field forward. It is useful to first mention that the analytical support for the perspective that
musculature is not as redundant as we have come to believe comes from examining the set of feasible
muscle activations that gives rise to the set of feasible limb outputs [156, 71]. This is the counterpart to
usinganoptimizationapproachtofindasingleuniqueandoptimalsolutiontothattask[147,156]. Rather,
it seeks to find the set of all feasible muscle activation strategies that, naturally and by construction, are a
well-defined region in the high-dimensional space formed by the intersection of all operating mechanical
constraintsofthetask,giventheanatomyofthelimb. Therefore,thenumberofconstraintsthatdefinethe
task is as important as the number of muscles in the limb—where more muscles allow meeting a greater
varietyandnumberoffunctionalconstraints[64,79,158].
Thus an argument against the classical notion of muscle redundancy is that the number of muscles in
vertebratelimbshasevolvedunderfunctionalconstraintsofversatilereal-worldbehavior[64,79,147,72,
12
74, 156]. We can perform ‘complex’ tasks (complexity defined as satisfying many constraints simultane-
ously or sequentially [79]) because we have many muscles—and muscle redundancy is most prominently
seeninlaboratorytasksthataretoosimple,andnotequivalenttotasksinthenaturalenvironment[28]. This
view is compatible with the above reasoning that a task is defined by the type and number of constraints
that must be met. The geometric approach to define feasible outputs and their associated feasible neural
inputs (FFS and FAS, respectively) provides a rigorous computational approach to the concept of muscle
redundancy. Thusmuscleredundancyisreallymoreafeatureofthetaskthanofthelimb[64,79,158].
1.7.1 Structureofthefeasibleactivationandfeasibleforcesetsofthecathindlimb
We present the vectormap as an innovative way to visualize and analyze the structure of the irregular FFS
polyhedra and FAS polytopes that result from the interaction of the biomechanics of the limb and the
constraints of the task. This allows us not only to interpret individual feasible sets, but also provide a
coordinate system (i.e., the surface of the sphere) to combine or compare feasible sets. This differs from
prior approaches that have compared their relative volume, shape, or bounding box, as described above.
Figure3identifiesthespecific3-Ddirectionsandregionsoffeasibleforcegenerationthatexhibitthehigh-
est variability across three individuals of a species. This has applications to, for example, understanding
howphenotypical(i.e.,anatomical)changesleadtobehavioralchangesinfeasibleforceandactivationon
whichevolutionaryselectionmayact.
Itisofcriticalinteresttothefieldofneuralcontroltounderstandwhyextantvertebrateshave‘somany’
muscles—yetwepreviouslylackedmeanstovisualizethestructureofthesetoffeasiblemuscleactivations
foragiventask. Themaindifficultyisthatselectingagivenmuscleactivationpatternnecessitatesselecting
a point from within the set of all feasible activations determined by the mechanics of the limb and the
constraints of the task [74, 156, 133]. As described above, prior work approximated the structure of
feasible activations for force production in a given direction by their bounding box [113, 72, 74]. In
Figure 4 we present how it is now possible to visualize the lower and upper bounds of feasible levels of
activation for eachandevery muscle when producing submaximal force in every 3-D direction. It can be
13
quite striking that even for very near maximal activation (i.e., at 90%), the range in between these upper
and lower bounds can be exceptionally wide, as in Figure 4. This had been reported in a single direction
of force production by [113], but here we can show the rate of convergence to the unique solution for
maximalforceforeverydirectionofforceproduction.
The wide (or narrow) latitudes in allowable coordination patterns for submaximal force seem to very
clearly demonstrate that trying to find and justify a ‘unique’ solution to these types of problems is highly
dependentonthetaskandthecostfunctionchosen. Notethatinothermusclesand/ordirectionsthisrateof
loss of redundancy can proceed at different rates, directly affecting the latitude the nervous system has to
selectagivencoordinationstrategy—andthenecessarycorrelationsinactivationsamongmuscles[72,74].
Thestructureofthesolutionspace,thelatitudeitaffords,andthenecessarycorrelationsinmuscleac-
tivations are all at the root of the study of muscle redundancy, muscle synergies, learning and adaptation,
uncontrolled manifolds, etc. Importantly, these vectormaps of feasible activation ranges for submaximal
forces motivate EMG studies to understand whether and how vertebrates actually make use of them (e.g.
duringlearningandadaptation). Thistiesintothespatiotemporalexploration-exploitationofthenull-space
of a task. As discussed in detail elsewhere ([41] and references therein), traversing the solution manifold
is likely an active spatio-temporal process where the neural controller can choose to inhabit a particular
region or subset of the solution space to meet the requirements of the task. Thus, the nature of motor
control may be more related to exploring and learning the feasible set of activations, and using memory
and improvements via fast and slow gradients, than the current thinking emphasizing optimization to find
uniquesolutions. Asubtlepointisthatmusclesynergieswillnaturallybedetectedfromsuchexplorations-
exploitationsofawell-structuredfeasibleactivationspace. Ourhopeisthatthesetechniquesmayhelpthe
evolutionofthis[74,133,17,132]andotherdebatesinmotorcontrol.
Whilequestionsremainaboutwhichmusclesarenecessaryoroptionaltoproducesubmaximalforceoutput
foragivensetofconstraintsandwhy[2],theycanonlybeansweredaswebegintoaddallspatio-temporal
constraints [99, 38] for natural behavior in the real world [64, 79]—as opposed to tasks in the laboratory
setting. But for now, we at least demonstrate that we have the tools to visualize and compare changes in
14
the structure of the FAS. In fact, for the case of maximal force output for which the activation levels are
unique,wecanalreadygleanimportantlessonsthatmotivatetestablehypotheses(Figure4,farRight).
Anexamplethatcomestomindlookingatthethreemusclesshown(andmoreavailableonline),isthatthe
interactionbetweenlimbmechanicsandtaskconstraintsleadstoirregularandcomplexlevelsofactivation
across 3-D directions of force production. This counters the widespread view that muscles are engaged
in a manner consistent with spatially smooth cosine tuning functions [44]. Therefore, these tools begin to
addresstheneedforcomputationaltoolspointedoutin[113,74,17]tocharacterizeandexploretheextent
towhichmechanicalconsiderationsdeterminetheneuralcontrolofnumerousmuscles.
1.8 Muscleactivationforfasteverydayrecreationalandsportstasks
The velocities of individual muscle fibers, and how they are determined by the kinematics of a task, are a
particular example of task constraints that have often been overlooked. We find the common recreational
task of throwing a flying disc (and reasonably other similar tasks such as throwing a ball, etc.) invariably
leads to muscle fiber velocities greater than c. 5 muscle fiber lengths per second (Figure 7). Such high
concentricandeccentricmusclefibervelocitiesarethoughttoincapacitateactiveforceproductionorlead
to tearing injuries, respectively [160, 55]. We employed a process of elimination to systematically inves-
tigate our model and its parameters to give us confidence in our interpretation of the results. Intuitively,
wecanassumethatthebone(segmentlengths)andjointkinematicswereobtainedexperimentallyandare
physiologically reasonable. The muscle fiber lengths and moment arms we considered in our study were
obtained from published data [48, 51, 56, 90]. Due to the between subject variability, we applied a Monte
Carloanalysistoconsiderarangeofmomentarmvaluesforeachmuscleandstillfindhighfibervelocities
(Figure 7, bottom). While we do not show them, we find similar results in a Monte Carlo analysis of the
muscle fiber lengths. Moreover, even though our model is limited in that it did not include the acceler-
ation and deceleration phases of the movement, adding them could only increase muscle fiber velocities
15
we report. Likewise, assuming a less conservative total time for the movement would only exacerbate the
high velocities we find. Despite this systematic Monte Carlo analysis, we still find muscle fiber velocities
greater than 5 fiber lengths per second. Thus we are compelled to challenge the traditional understanding
of the force-velocity properties of muscles and motivate future research in muscle mechanics: somehow,
such high fiber velocities are likely present in everyday tasks do not lead a complete loss of force pro-
duction capabilities in the concentric phase, or injury in the eccentric phase. This is not the first time we
questionthefunctionalroleoftheforce-velocitypropertiesofmusclesforeverydaytasks[64].
Another fundamental result from these simulations is that they emphasize the need to study the temporal
structure of muscle activation in the context of muscle redundancy [99, 38]. Consider Eq. 2 defining the
over-determined physical relationship between changes in joint angles and tendon excursions that drive
changesinmusclefiberlengths. Thisrelationshipdefinestheobligatorycorrelationsamongtendonexcur-
sions where a sequence of (a few) joint angles uniquely and completely determines the excursions of all
(numerous)musculotendons. Thisistheoppositeofmuscleredundancyasthereisasingleanduniqueset
of tendon excursions that can satisfy the kinematics of a given movement. This begs the question of how
the nervous system coordinates eccentric and concentric contractions to produce such fast movements. If,
foranyreason,anymusclefailstolengthen(i.e.,contracteccentrically)tosatisfytherotationsofthejoints
itcrosses,thedesiredmotionwill,atbest,bedisrupted,oratworst,thelimbwillfreeze.
Whatinhibitsstretchreflexestoallowsuchcoordinatedeccentriccontractions? Alpha-gammaco-activation,
reciprocal-inhibition,andgatingofspindleafferentinformationaresomeofneuralinteractionsthoughtto
be necessary to modulate/inhibit stretch reflexes [45]. Thus the nervous system must issue neural com-
mands, coordinated throughout the entire duration of the movement, to (i) alpha-motoneurons to produce
thenecessaryjointtorquesasperthestandardforce-sharingmotorcontrolproblem(e.g.,[93,24]);(ii)co-
ordinatereciprocal-inhibitionofalpha-motoneuronpoolsacrossshorteningandlengtheningmuscles(e.g.,
[57]);(iii)inhibitthestretchreflexinmusclesneedingtoundergoeccentriccontractions(e.g.,[161]);while
(iv)satisfyingthetimeconstantsofmuscleexcitation-contractiondynamics[160]toensurethecontinuity
of these neural commands as the motion progresses. This compounding of multiple spatial and temporal
16
constraints naturally leads to a shrinking of the set of feasible motor commands for natural movements
(seeabovediscussionand[64]).
In fact, clinicians have long been aware of how disorders of reflexes or the neural circuits of ‘afferented
muscles’ lead to disruptions or failures of movements (for an overview see [101, 102]. We now propose
that these so-called dystonias may in fact be a natural consequence of the nervous system failing to meet
the stringent temporal demands on alpha-gamma neural drive for the eccentric and concentric contrac-
tionsessentialtosmoothlimbmovement. Thisagainsupportstheviewthatextantvertebrateshavebarely
enough neural degrees of freedom for versatile real-world behavior [64, 79, 147, 72, 74] as the muscle
activations to produce smooth movements is likely not redundant, or at the very least not as redundant as
currentlythought.
One last comment is on the over-determined nature of producing the necessary muscle excursions for a
limb movement. As mentioned above, over-determined systems either have one unique solution (if it ex-
ists), or no solution at all. When no solution exists, a practical alternative is found by violating some or
all constraints as in the method of least squares for a set of equations in which there are more equations
than unknowns. This may actuallybegin to explain why musclesand tendons have non-trivialamounts of
passive elasticity—to provide tolerance to errors in the neural control of their excursions when eccentric
and concentric contractions are not controlled accurately enough by the CNS. From the engineering per-
spective, such elasticity complicates control as it adds delays and internal actuator dynamics, and reduces
actuator bandwidth. But in the case of biological tendon-driven limbs, this built-in tolerance to excursion
errorsmaybeacriticalcomplimentto,andenablerof,theneuralcontrolofsmoothmovements.
1.9 Acknowledgments
Funding Sources: Research reported in this publication was supported by NIAMS of the National In-
stitutes of Health under award numbers R01AR050520 and R01AR052345 grants to FVC. The content
17
is solely the responsibility of the authors and does not necessarily represent the official views of the Na-
tional Institutes of Health. The Swiss Federal Institute of Technology-Zurich (Eidgen¨ ossische Technische
Hochschule-Z¨ urich) for the support of HFI. We thank Lena Ting for generously sharing her cat hindlimb
datawithus.
18
Figure 1.1: Bone lengths, joint axes of rotation, and moment arm matrix for the species average cat
hindlimb model, in cm. Positive values are shown in red and negative values in blue, as per the right-
hand-rule.
19
Figure 1.2: Left: The polygon of the 2-D feasible force set in the sagittal plane. The color-coded vec-
tormapping of radial lines indicate the magnitude of the maximal feasible force along that direction, then
vectormappedontotheperimeterofthecirclesurroundingtheFFS.Theverythinlinesemanatingfromthe
originarethelinesofactionofeachofthe31muscles. Center: thepolyhedronofthe3-DFFS,againwith
the vectormapping of force magnitude values onto a circle in the sagittal plane. Right: The color-coded
vectormappingontothesurfaceofasphereindicatingthemaximalfeasibleforceineverydirectionin3-D.
NotetheFFSisratherflatonthesagittalplane, butelongatedtowardstheposteriordirection. Alldataare
forthecatcalledBirdyin[113].
20
Figure 1.3: Top: Vectormap of the average of maximal feasible force across all sampled output vectors
in three feline hindlimbs. Bottom: A vectormap displaying regions of the feasible force space that have
higherstandarddeviationacrossthreecathindlimbs. ForceinNewtonsrepresentedbycolorscale.
21
Figure1.4: Structureofthefeasibleactivationsetforthreemuscles. ThelargevectormapsonthefarRight
show their unique activation level for maximal force output in every 3-D direction. Because multiple
activation levels can produce submaximal forces, the small vectormaps to the Left show the lower and
upperboundsofthosefeasibleactivationlevelsforforcemagnitudes(a)graduallyincreasingfrom50%of
maximalinevery3-Ddirection.
22
Figure1.5: Momentarmvaluesforhumanarmmodel. Themomentarmsfromthe17musclesconsidered
inthismodelandtheirassociationswiththefiveDoFsareillustrated,incm. Themomentarmsaregrouped
by DoF and are shown below the associated joint. Positive values are shown in red and negative values in
blue.
23
Figure1.6: Topviewofthe3-Dhumanarmmodel. Thisfigureillustratestheinitiationofforwardmotion
through follow-through of the flying disc throw. The reference posture is shown in black and the release
pointinthethrowisshowninred. Theinterpolatedjointanglesforthe45posturesdescribingthismotion,
obtainedfrom[58],areshowninthebottompanel.
24
Figure 1.7: Normalized instantaneous fiber velocities during the throw for the nominal model. Top: The
musclesarelistedonthey-axisandthe45posturesmakingupthethrowareshownonthex-axis. Excessive
muscle velocities are shown in red (shortening) and blue (lengthening). Bottom: The same data are
illustrated with individual traces for each muscle that show the fiber velocity. Muscles controlling the
shoulder, elbow, and wrist are illustrated in blue, red, and green, respectively. Instantaneous fiber velocity
is given on the y-axis and the postures during the throw are on the x-axis. Regions of the traces outside
ofthehorizontaldashedlinesindicateexcessivemusclevelocities. Inbothfigures,thereleasepointofthe
throwisindicatedwithaverticaldashedline.
25
Chapter2
Structureofthesetoffeasibleneuralcommandsforcomplexmotor
tasks
Valero-CuevasFJ
1
,CohnBA
3
,Szedl´ akM
4
,FukudaK
4
andG¨ artnerB
4
1
DepartmentofBiomedicalEngineering,UniversityofSouthernCalifornia,LosAngeles,CA
2
Divisionof
Biokinesiology and Physical Therapy, University of Southern California, Los Angeles, CA
3
Department
of Computer Science, University of Southern California, Los Angeles, CA
4
Swiss Federal Institute of
Technology-Zurich,Zurich,Switzerland
2.1 Abstract
The brain must select its control strategies among an infinite set of possibilities; researchers believe that
it must be solving an optimization problem. While this set of feasible solutions is infinite and lies in
highdimensions,itisboundedbykinematic,neuromuscular,andanatomicalconstraints,withinwhichthe
brain must select optimal solutions. That is, the set of feasible activations is well structured. However, to
date there is no method to describe and quantify the structure of these high-dimensional solution spaces.
Boundingboxesordimensionalityreductionalgorithmsdonotcapturetheirdetailedstructure. Wepresent
a novel approach based on the well-known Hit-and-Run algorithm in computational geometry to extract
the structure of the feasible activations capable of producing 50% of maximal fingertip force in a specific
26
direction. Weusearealisticmodelofahumanindexfingerwith7muscles,and4DOFs. Foragivenstatic
forcevectorattheendpoint,thefeasibleactivationspaceisa3Dconvexpolytope,embeddedinthe7Dunit
cube. It is known that explicitly computing the volume of this polytope can become too computationally
complex in many instances. However, our algorithm was able to sample 1,000,000 uniform at random
points from the feasible activation space. The computed distribution of activation across muscles sheds
lightontothestructureofthesesolutionspaces—ratherthansimplyexploringtheirmaximalandminimal
values. Although this paper presents a 7 dimensional case of the index finger, our methods extend to sys-
temswithatleast40muscles. Thiswillallowourmotorcontrolcommunitytounderstandthedistributions
offeasiblemuscleactivations,providingimportantcontextualinformationintolearning,optimizationand
adaptationofmotorpatternsinfutureresearch.
2.1.0.1 AuthorContribution
In a collaboration with ETH, this work represented the seminal work that led to the introduction of Feasi-
bilityTheory,andIcontributedallcode,analyses,whileMSandBG,andFVcontributedtothetheoretical
implementationsandtheneuromuscularimplications,respectively.
2.2 Introduction
Muscle redundancy is the term used to describe the underdetermined nature of neural control of muscula-
ture. The classical notion of muscle redundancy proposes that, faced with an infinite number of possible
muscle activation patterns for a given task, the nervous system uses optimization to select a specific so-
lution. Here, each of the N muscles represents a dimension of control, and a muscle activation pattern
is a point in [0,1]
N
[144]. Thus researchers often seek to infer the optimization approach and the cost
functions the nervous system likely uses to find points in activation space to produce natural behavior
[23,92,106,129,32,54].
27
Implicit in these optimization procedures is the notion that there exists a well structured set of feasi-
ble solutions. Thus several of us have focused on describing and understanding those high-dimensional
subspacesembeddedin [0,1]
N
[73,75,110,144,62].
Forthecaseofmuscleredundancyforsubmaximalandstaticforceproductionwithalimb,theproblem
is phrased as one of computational geometry: find the convex polytope of all feasible muscle activations
given the mechanics of the limb and the constrains of the task [9, 144, 139, 62]. This convex polytope is
calledthefeasibleactivationset. Todate,thestructureofthishigh-dimensionalpolytopeisinferredbyits
bounding box [73, 110, 62]. But the bounding box of a convex polytope will always exclude the details
of its shape. Empirical dimensionality-reduction methods have also been used to calculate basis vectors
for such subspaces [27, 33, 69]. But those basis vectors only provide a description of the dimension,
orientation,andaspectratioofthepolytope,butnotofitsboundariesorinternalstructure.
Here we present a novel application of the well-known Hit-and-Run algorithm [109] to describe the
internalstructureofthesehigh-dimensionalfeasibleactivationsets. Weapplyourtechniquetoaschematic
example with three muscles to describe the method, and then use a realistic model of an index finger with
sevenmusclesandfourjoints[144].
2.3 Methods
2.3.1 Hit-and-Runalgorithm
The boundaries of the convex polytope defining the feasible activation set are defined by the mechanics
of the limb and the constraints of the task, as is described in Subsection 3.3.1. The goal of the Hit-and-
Run algorithm is to uniformly sample a convex body [109]. In the case of a schematic tendon-driven
limb with three muscles, the feasible activation space is the unit cube (as muscles can only be activated
positively from 0 to a maximal normalized value of 1). As explained in [139], when task constraints are
introducedtothesystem,thefeasibleactivationsetisfurtherreduced;inthiscontext,ataskisastaticforce
vector produced at the endpoint of the limb, which is represented as a set of inequality constraints. Thus
28
if this simple limb meets all constraints, the feasible activation set of the polygon P contains all feasible
activationsa2 R
n
thatsatisfy
f=Aa,a2 [0,1]
n
,
where f2 R
m
is a fixed force vector, and A = J
T
RF
o
2 R
m⇥ n
—where J, R, and F
o
are the matrices of
the Jacobian of the limb, the moment arms of the tendons, and the strengths of the muscles, respectively
[144,139]. Pisboundedbytheunitn-cubesinceallvariablesa
i
,i2 [n]areboundedby0and1frombelow,
aboverespectively. Considerthefollowing1⇥ 3fabricatedexample,wherethetaskisa1Nunidimensional
force.
1=
10
3
a
1
53
15
a
2
+2a
3
a
1
,a
2
,a
3
2 [0,1],
thesetoffeasibleactivationsisgivenbytheshadedsetinFigure1TODO.
TheHit-and-Runwalkon Pisdefinedasfollows(itworksanalogouslyforanyconvexbody).
1. InnerPoint: Findagivenstartingpointpof P(Figure2.2a).
2. Direction: Generate a random direction from p (uniformly at random over all directions) (Figure
2.2a).
3. Endpoints: Find the intersection points of the random direction with the edges of the polytope
(Figure2.2b).
4. NewPoint: Pickapointuniformatrandomalongthelinesegmentdefinedbytheendpoints(Figure
2.2c).
5. Repeatfrom (a)theabovestepswiththenewpointasthestartingpoint.
Tofindastartingpointin
f=Aa,a2 [0,1]
n
,
29
Figure 2.1: The feasible activation set for a three-muscle system meeting one functional constraint is a
polygoninR
3
. Notethatmuscleactivationsareassumedtobeboundedbetween0and1.
30
Figure2.2: GraphicaldescriptionoftheHit-and-Runalgorithm.
31
weonlyneedtofindafeasibleactivationvector. FortheHit-and-Runalgorithmtomixfaster,wewantthe
starting point to be centrally located within the polytope. We use the following standard trick with slack
variablese i
.
maximize  n
i=1
e i
subjectto f = Aa
a
i
2 [e i
,1e i
], 8 i2 {1,...,n}
e i
0, 8 i2 {1,...,n}.
(2.1)
The recursive nature of the algorithm means that consecutive points are autocorrelated; it’s important
that each point sampled from the polytope is uniform at random, so we subset points separated by a num-
ber of iterations. For convex polygons in higher dimensions (over 40 dimensional), experimental results
suggestthatO(n)stepsoftheHit-and-Runalgorithmaresufficient. InparticularEmirisandFisikopoulos
papersuggestthat(10+
10
n
)nstepsareenoughtoconvergeupontheuniformdistribution[43]. Intheindex
fingermodelweexecutedtheHit-and-Runalgorithm1,000,000times,selectingonlyevery100thpoint.
2.3.2 Realisticindexfingermodel
Weusedourpublishedmodelin[144]tofindmatrixA2 R
4⇥ 7
,wherea2 R
7
. Thesevenmusclesareflexor
digitorumprofundus(FDP),flexordigitorumsuperficialis(FDS),extensorindicisproprius(EIP),extensor
digitorum communis (EDC), lumbrical (LUM), dorsal interosseous (DI), and palmar interosseous (PI).
The four degrees of freedom were ad-abduction, flexion-extension at the metacarpophalangeal joint, and
flexion-extension at the proximal and distal interphalangeal joints. The force direction we simulated is in
thepalmardirectioninthepostureshowninFigure2.3.
32
Palmar
Figure 2.3: The index finger model simulated 50% of maximal force production in the palmar direction.
Adaptedfrom[144].
2.4 Results
Figure 2.4 shows the distributions of activations resulting from the solutions computed with Hit-and-Run
sampling. This is the first time (to our knowledge) that the internal structure of the feasible activation set
hasbeenvisualizedforasub-maximalforce.
Noticethatthelowerandupperboundsoftheactivations(i.e.,thedashedlinesindicatingtheirbound-
ingbox),areunhelpfulindeterminingtheactualdensitydistributionoffeasibleactivations. Theactivation
needed for the maximal force output (thick gray line) is very often not the mode of the activations at
50% of output. It’s important to note that these histograms are unidimensional- they do not illustrate the
between-muscleassociations.
2.5 Discussion
Ourresultsandmethodologyraisethefollowingideas:
• The Hit-and-Run algorithm can explore the feasible activation space for a realistic 7-muscle finger
inawaythatiscomputationallytractable.
33
Figure2.4: Weshowonehistogramforeachmuscleoftheindexfingertoillustratehowthemuscleisused
across all feasible solutions. For this set of distributions, the task was 50% of maximal force output in the
palmar direction. Muscles are FDP, FDS,EIP,EDC, LUM, DI, and PI are shown in that order from top
tobottom. Theorangedottedlinesarethelowerandupperboundsofactivation.
34
• For some muscles, we find that the bounding box exceptionally misconstrues the internal structure
ofthefeasibleactivationset.
• TheHit-and-Runalgorithmiscost-agnosticinthesensethatnocostfunctionisneededtopredictthe
distribution of muscle activation patterns. Therefore, we can provide spatial context to where ‘opti-
mal’ solutions lie within the solution space; this approach can be used to explore the consequences
ofdifferentcostfunctions.
• The distribution of muscle activations may be intricately related to strong modes which critically
affectthelearningofmotortasks.
With the spatial context of the feasible activation space, we can explore the statistical tendencies of a
musculoskeletalsystem,andbetterdefinethelandscapeuponwhichoptimizationoccurs. Thisapplication
ofHit-and-Runprovidesatooltogeneratetestablehypothesesofhowcoordinationhabitsmaycomeabout,
howtheyarelearned,andhowdifficultoreasyitistobreakoutofthem.
35
Chapter3
FeasibilityTheoryreconcilesandinformsalternativeapproachesto
neuromuscularcontrol
BrianA.Cohn
1
,MaySzedl´ ak
2
,BerndG¨ artner
2
andFranciscoJ.Valero-Cuevas
3,4
3.0.0.1 Affiliations
1
University of Southern California, Department of Computer Science, Los Angeles, CA.
2
ETH Zurich,
DepartmentofTheoreticalComputerScience,Zurich,Switzerland.
3
UniversityofSouthernCalifornia,DepartmentofBiomedicalEngineering,LosAngeles,CA.
4
UniversityofSouthernCalifornia,DivisionofBiokinesiologyandPhysicalTherapy,LosAngeles,CA.
3.1 Abstract
We present Feasibility Theory, a conceptual and computational framework to unify today’s theories of
neuromuscularcontrol. Webeginbydescribinghowthemusculoskeletalanatomyofthelimb,theneedto
controlindividualtendons,andthephysicsofamotortaskuniquelyspecifythefamilyofallvalidmuscle
activations that accomplish it (its ‘feasible activation space’). For our example of producing static force
with a finger driven by seven muscles, computational geometry characterizes—in a complete way—the
structureoffeasibleactivationspacesas3-dimensionalpolytopesembeddedin7-D.Thefeasibleactivation
36
space for a given task is the landscape where all neuromuscular learning, control, and performance must
occur. This approach unifies current theories of neuromuscular control because the structure of feasible
activation spaces can be separately approximated as either low-dimensional basis functions (synergies),
high-dimensional joint probability distributions (Bayesian priors), or fitness landscapes (to optimize cost
functions).
3.1.0.1 AuthorContribution
In a collaboration with ETH, this work expands upon the techniques from Chapter 2. I contributed all
code, analyses, and simulations, MS and BG contributed to the theoretical equation implementations and
theneuromuscularimplications,andFVsupportedthedeepanalysisofneuromuscularimplicationsofthe
results.
3.2 Introduction
How the nervous system selects specific levels of muscle activations (i.e. a muscle activation pattern)
for a given motor task continues to be hotly debated. Some suggest the nervous system either combines
low-dimensionalsynergies[75,118,18,39,96,119,3],learnsprobabilisticrepresentationsofvalidmuscle
activation patterns[66, 67, 15, 100], or optimizes physiologically-tenable cost functions[23, 92, 106, 129,
32, 54]. At the core of this problem lies the nature of ‘feasible activation spaces’, and the computational
challenge of describing and understanding their high-dimensional structure (for an overview, see[149]).
A feasible activation space is the family of valid solutions (i.e. muscle activation patterns) that meet the
mechanicalconstraints
1
ofagivenmotortask. Fig.3.1illustratestheseneuromechanicalinteractionsthat
definethefeasibleactivationspaceforaparticulartask.
Themostthenervoussystemcando,therefore,isselectandapplyaspecificmuscleactivationpattern
from within the feasible activation space. This is because muscle activation patterns outside of this space
are, by definition, inappropriate for the task. In fact, the feasible activation space defines the landscape
1
Mechanicalconstraintsisaformalwaytocallthephysicaldemands,requirements,orcharacteristicsofagivenphysicaltask.
37
Cost associated with each
valid activation pattern
(c)
Activation patterns defining
the structure of the polytope
muscle 1 muscle 2 muscle n
Mechanical
definition of the task
Biomechanical and
muscular properties
n-muscle system
n=7 for index finger
Feasible Activation Set
Convex Polytope embedded
in n-dimensional space
Full-dimensional
manifold of all valid
muscle activation patterns
Cortical neuron activity
> 1,000,000 dimensions
Spinal neuron activity
> 1,000 dimensions
Theories of sensorimotor control
Presumed fitness landscapes
for all valid
muscle activation patterns
(a)
(b)
(d)
(e)
(f) (g) (h)
Low-dimensional
approximation to
feasible activation set
Low-dimensional control
e.g., Muscle synergies
Uncontrolled manifold
Probabilistic control
e.g., Bayesian inference
Exploration-Exploitation
Optimization
e.g., Optimal Control
Model-Predictive Control
(Bio)Mechanics Neural commands
One valid muscle
activation pattern
dimensionality
reduction
uniform
sampling
(i)
cost
function 1
Figure 3.1: Emergence and interpretation of feasible activation spaces for a particular motor task.
The descending motor command for a given task is issued by the motor cortex (a), which projects onto
inter-neuronsandalpha-motorneuronpoolsinthespinalcord(b). Thecombineddrivetoallalpha-motor
neurons of a muscle can be considered its total muscle activation level (a value between 0 and 1). If we
consider that muscles can, to a large extent, be controlled independently and in different ways, then the
overall motor command can be conceptualized as a multi-dimensional muscle activation pattern (i.e. a
point)inahigh-dimensionalmuscleactivationspace[23,116,70,144,129](c). Forthatmuscleactivation
pattern to be valid, it has to elicit muscle forces (d) capable of satisfying the mechanical constraints of
the task—in this case defining a well-directed sub-maximal fingertip force (e). Given the large number
of muscles in vertebrates, there can be muscle redundancy: where a given task can be accomplished with
a large number of valid muscle activation patterns. We propose that our novel ability to characterize the
high-dimensionalstructureoffeasibleactivationspaces(i)allowstoustocompare,contrast,andreconcile
today’sthreedominantapproachestomuscleredundancyinsensorimotorcontrol(f,g,h).
38
uponwhichallneuromuscularlearningandperformancemustoccurforthattask. Studyingneuromuscular
control is, therefore, equivalent to studying how the nervous system finds, explores, inhabits, and exploits
thecontentsandstructureoffeasibleactivationspaces[75,118,18,47,39,96,119].
But the ‘curse of dimensionality’[12, 13, 9] makes it computationally challenging to calculate, de-
scribe, and understand the nature and structure of high-dimensional feasible activation spaces[142, 23,
116, 70, 125, 104, 39]—even for an isolated human finger or cat leg generating everyday static forces[75,
62, 149, 110]. This is due to the computational complexity of algorithms to map the geometric details of
objectsembeddedinhighdimensions[46,109,83].
Current theories of neuromuscular control
2
are alternative responses to overcome the curse of dimen-
sionality in this context. These alternative approaches, however, are seldom combined and often the in-
sights from one realm are not readily applicable to the others. Here we emphasize how the mechanics of
thebodyandthephysicsofthetaskconstitutethecommongroundforalltheories.
Wenowpropose‘FeasibilityTheory’,whichisaconceptualframeworktocharacterizefeasibleactiva-
tion spaces in detail. While prior work has described how to find such feasible activation spaces for static
force production[144, 75, 157, 145, 85], we now explain why the structure of a feasible activation space
can be approximated with low-dimensional synergies and probability distribution functions, and can be
associatedwithmultiplefitnesslandscapesoverwhichtooptimize.
3.3 Methods
In the case of the seven muscles of the human index finger producing static fingertip force, we show that
the family of feasible commands, the feasible activation space, is a 3-dimensional polytope embedded in
7-dimensional muscle activation space [144]. A ‘polytope’ is the formal name for bounded polyhedra in
dimensions higher than three. With 4 task constraints applied to 7 muscles, the result is a 3-dimensional
polytope embedded in the 7-dimensional muscle activation space. By construction of anatomy, producing
2
Neuromuscularcontrolisvariouslyreferredtoasneural,motor,sensorimotor,etc. control.
39
static force with a fixed posture naturally leads to a relationship between muscle forces and endpoint
torques. The linear constraint equations that define this relationship (and in parallel the polytope that
arises from the constraints) accurately represent the set of feasible motor commands [144, 149, 110]. Our
computational approach hinges on the efficient sampling and complete representation of the geometric
structureofhigh-dimensionalpolytopes,whichfullycharacterizesthefamilyofallvalidmuscleactivation
patterns—eachofwhichsolvesthesametask. Bydefinition,thispolytopeisthenullspaceofthetask.
The methods to obtain feasible activation spaces for ‘tendon-driven’ limbs are described in detail in
the textbook Fundamentals of Neuromechanics and references therein[149]. This tendon-driven approach
explicitly and distinctly avoids the conceptual approach to calculate net torques at each joint. Rather,
it emphasizes studying the individual actions of all muscles at all levels of analysis, from their neural
activationtotheircontributionstofingertipforce. Wedescribethembrieflyhere.
Consider a tendon-driven limb, such as a finger, with n independently controllable muscles, where we
define the neural command to each muscle as a positive value of activation between 0 (no activation) and
1 (maximal activation), where a value of 1 would produce the maximum possible tendon force for that
muscle. Wedonotdifferentiatebetweenconcentricoreccentriccontraction—wedefinemuscleactivation
as the net static tendon tension, normalized by the maximum tendon tension possible by that muscle. We
can then visualize the set of all feasible neural commands (i.e. all possible muscle activation patterns) as
the points contained in a positive n-dimensional cube with sides of length equal to 1. A specific muscle
activation pattern is a point (i.e. an n-dimensional vectora) in this n-dimensional cube[23, 116, 70, 144].
Nowconsideraspecifictask,suchasproducingavectorofstaticforcewiththefingertip,aswhenholding
anobject. Clearly,notallmuscleactivationpatternsinsidethen-dimensionalcubecanproducethatdesired
static fingertip force vector: bone lengths, kinematic degrees of freedon, anatomical routing, posture, and
musclestrengthinequitiesdefinethesubsetofpointsinthen-cubewhichproduceafingertipforcevectorof
a specific magnitude and direction. As described in[23, 116, 70, 149] the musculoskeletal anatomy of the
limb, the need to control individual tendons, and the physics of a motor task uniquely specify a polytope
embedded in R
n
(i.e. the feasible activation space). This polytope contains the family of (potentially
40
infinite)validmuscleactivationpatternsthatcanproducethisstaticforceproductiontask. However,these
valid muscle coordination patterns are not arbitrarily different because, by construction, the geometric
structureofthepolytopethatcontainsthemdefinesstrictspatialcorrelationsamongthem[75].
Systemoflinearequationstosimulatestaticforceproductionbyatendon-drivensystem
Consider producing a vector of static force with the endpoint of the limb in a given posture. The con-
straintsthatdefinethattask(i.e. thedirectionandmagnitudeoftheforcevectorattheendpoint)arelinear
equations[149]thatcomefromthemappingbetweenneuralactivationofindividualmusclestostaticend-
pointforcesandtorquesthelimbcanproduce. Thismappingislinearlymodeledbytheequation
0 B B B B B B B B B B B B B B B B B B @ f
x
f
y
f
z
t x
t y
t z
1 C C C C C C C C C C C C C C C C C C A =w=Ha=H
0 B B B B B B B B B B B B B B @ a
1
a
2
a
3
...
a
n
1 C C C C C C C C C C C C C C A ,a2 [0,1]
n
(3.1)
where H is the matrix of linear constraints defined by the musculoskeletal anatomy of the limb[62], a is
the input vector of n muscle activations, and f2 R
m
is the m-dimensional limb output ‘wrench’ (i.e. the
forcesandtorquesthefingercanproduceattheendpoint).
Theoutputwrench,w,isatmost6-dimensional(i.e. 3forcesand3torques)dependingonthenumber
of kinematic degrees of freedom of the limb, and usually m< n because limbs have more muscles than
kinematic degrees of freedom[149]. Muscles can only pull, so elements of a cannot be negative, and are
cappedat1(i.e. 100%ofmaximalmuscleactivation).
What are the muscle coordination patterns that produce a given task? As explained in[149], the task
of producing a static fingertip force vector is defined by specifying the desired values for the elements
of the endpoint forces and torques of w. Each value yields a constraint equation, which in turn defines a
41
hyperplaneofdimensionn1,andtheircombinationdefinesthetaskcompletely. Thefeasibleactivation
space of the task, if it is well posed[25], is defined by the points a that lie within the n-cube and at the
intersectionofallconstrainthyperplanes.
Geometrically speaking, the feasible activation space is a (nm)-dimensional convex polytope P
embedded inR
n
that contains all n-dimensional muscle coordination patterns (i.e. points a) that satisfy
all constraints, and therefore can produce the task. Increasing task specificity by adding more constraints
naturally decreases the dimensionality and changes the size and shape of the feasible activation space[70,
110,61].
TheHit-and-Runalgorithmuniformlysamplesfromfeasibleactivationspaces
Calculatingthegeometricpropertiesofconvexpolytopesinhighdimensionsiscomputationallychalleng-
ing. Taking the generalized concept of an n-dimensional volume as an example of a geometric property
of interest, the exact volume computations for n-dimensional polytopes is known to be tractable only in
a polynomial amount of time (i.e. #P-hard)[41]. Currently available volume algorithms can only handle
polytopes embedded in small dimensions like 10 or slightly more[21]. Studying vertebrate limbs in gen-
eral,however,canrequireincludingseveraldozenmuscles,suchasourstudiesofa17-musclehumanarm
anda31-musclecathindlimbmodel[62];andothermodelshaveover40musclesofthehumanlowerlimb
[8,75,52,36].
Similardifficultiesarisewhencomputingothergeometricpropertiessuchastheshapeandaspectratio
of P in high dimensions. We and others have described polytopes P by their bounding box (i.e. the
range of values in every dimension)[110, 73], but that singularly overestimates the shape and volume of
the feasible activation space as discussed in[62]. Consider a 3-muscle system with only one constraint,
producing a 2-dimensional polygon as the feasible solution space. The bounding box of the polygon has
a volume—even though a plane has zero volume—, and can be almost as large as the positive unit cube
itself. Similarproblemsariseintheinterpretationoftheinscribedandcircumscribedball[60].
42
We applied the Hit-and-Run method to sample points from the feasible activation space. We have
presentedadetailedexplanationoftheTheory(InChapter9of[149]),andhavejustifiedtheutilityofthis
method on tendon-driven models of the index finger [145]. This complete probabilistic method describes
the structure of feasible activation spaces P with a set of uniformly-at-random muscle activation patterns
that produce the same wrench. This enables us to derive descriptive statistics, histograms, and point
densities of the set of valid muscle activation patterns a uniformly sampled from the polytope. To do so,
weusetheHit-and-Runmethod.
This approach can scale up to ⇠ 40 dimensions (i.e. limbs with ⇠ 40 independent muscles). This
suffices to study extant vertebrate limbs, and thus compare, contrast, combine—and reconcile—today’s
threedominantapproachestoneuromuscularcontrol.
3.3.1 Exampleofatendon-drivensystem
Realistic 3-D model of a 7-muscle human index finger We applied this methodology to our pub-
lished model of an index finger for static fingertip force production. The model is described in detail
elsewhere[146]. Briefly,theinputtothemodelisa7-Dmuscleactivationpatterna,andtheoutputisa4-D
wrenchw(i.e. staticforcesandtorques)atthefingertip:
w =Ha (3.2)
H =J
T
RF
o
,H2 R
4⇥ 7
(3.3)
where
43
a=
0 B B B B B B B B B B B B B B B B B B B B B B @ a
FDP
a
FDS
a
EIP
a
EDC
a
LUM
a
DI
a
PI
1 C C C C C C C C C C C C C C C C C C C C C C A (3.4)
In Cartesian coordinates, the 4-D output wrench corresponds to the anatomical directions shown in
Fig.3.1e.
w=
0 B B B B B B B B B B @ f
x
f
y
f
z
t x
1 C C C C C C C C C C A =
0 B B B B B B B B B B @ f
radial
f
distal
f
palmar
t radial
1 C C C C C C C C C C A (3.5)
The biomechanical model H includes three serial links articulated by four kinematic degrees of free-
dom(ad-abduction, flexion-extensionat themetacarpophalangealjoint, andflexion-extensionat theprox-
imal and distal interphalangeal joints). The action of each of the seven muscles (FDP: flexor digitorum
profundus, FDS: flexor digitorum superficialis, EIP: extensor indicis proprius, EDC: extensor digitorum
communis,LUM:lumbrical,DI:dorsalinterosseous,andPI:palmarinterosseous)oneachjointtoproduce
torqueisgivenbythemomentarmmatrixR2 R
4⇥ 7
. Lastly,J2 R
4⇥ 4
andF
0
2 R
7⇥ 7
aretheJacobianofthe
fingertip with 4 kinematic degrees of freedom, and the diagonal matrix containing the maximal strengths
of the seven muscles, respectively[149, 138]. The finger posture was defined to be 0
ad-abduction and
45
flexion at the metacarpophalangeal joint, and 45
and 10
flexion, respectively, at the proximal and
distalinterphalangealjoints.
44
Feasible activation space for a static fingertip force task Our goal is to find the family of all feasible
muscle activation patterns that can produce a given task. In particular, the task we explored is produc-
ing various magnitudes of a submaximal static force in the distal direction f
distal
— in the absence of
any t radial
, shown in Fig. 3.1e. Therefore the feasible activation space is a polytope P in 7-dimensional
activationspacethatmeetsthefollowing four linearconstraintsina[144,149,138]
f
radial
= 0 (3.6)
f
distal
= desired magnitude as % of maximal (3.7)
f
palmar
= 0 (3.8)
t palmar
= 0 (3.9)
Thesefourconstraintsonthestaticoutputofthefingeryielda3-dimensional(i.e. 74=3)polytope
P embedded in 7-dimensional activation space. For details on how to create such models, apply task
constraintsandfindsuchpolytopesviavertexenumerationmethods,see[149].
For the index finger model used in this paper, the published maximal feasible force in the distal direc-
tion is 28.81 Newtons. We defined the normalized desired distal task intensity as a value ranging between
0 and 1, i.e. each submaximal force can be produced by any of the points contained in its corresponding
feasible activation space. For the production of a maximal force, the feasible activation space shrinks to a
singlepoint[116,23,25,138].
3.3.2 Analysisoffeasibleactivationspaces
Parallelcoordinatevisualization Forustounderstandthestructureofthefeasibleactivationspace,we
aimtovisualizethedata. Ifwehadasimplemodelwithonlythreemuscles(andonetaskforcedimension),
wecouldplotthefeasibleactivationspaceasaplanewithina3Dcube,asillustratedinFig.3.2a. However,
in our model, we have seven muscles. In our 3D reality, we cannot create a 7D scatter plot to highlight
45
how muscle activation patterns are spatially located across the muscle dimensions, so we must project the
datainadifferentway.
Parallelcoordinatesareacommongraphicalapproachtovisualizeinteractionsamonghigh-dimensional
data [10, 68]. To build familiarity with this visualization method, consider the results of a simple 3-
dimensional(3-muscle)toyexampleshowninFig.3.2a. Thisisthedimensionalityofafingerwithonly3
muscles, aiming to create a unidimensional pressing force. We begin by drawing n parallel vertical lines
for each of the dimensions n (i.e. 3 muscles). With the axis limits of each line set between 0 and 1 (at
thebottomandtopoftheplot,respectively),eachmuscleactivationpattern(Fig.3.2a)isthenrepresented
by a zig-zag line that connects to the coordinates between 0 and 1 on each axis, as shown in Fig. 3.2b.
The blue zig-zag line that is connected at the top of m
1
in Fig. 3.2b represents the muscle activation point
equal to (m
1
=0.8,m
2
=0.9,m
3
=0.4). You can see its corresponding location in the 3D cube, mapped
to the parallel coordinate zig-zag line (the gray dotted line connects the two representations of the muscle
activationpattern).
Neural and metabolic cost functions As mentioned in the Introduction, the field of neuromuscular
controlhasalonghistoricaltraditionofusingoptimizationtofindmuscleactivationpatternsthatminimize
effort, which requires the (often contentious) definition of cost functions[116, 23, 92, 32]. Therefore, we
used four representative cost functions to calculate the relative fitness of each of the muscle activation
patterns sampled—in effect also calculating the fitness landscape across all possible solutions. The cost
functions are defined at the level of neural effort (L
1
, and L
2
norms, representing the normalized sum of
descending neural a -drive to the motor neuron pools); and at the level of metabolic cost, thought to be
approximatedbyneuraldriveweightedbythestrengthofeachmuscle(L
w
1
and L
w
2
norms)[92,32].
To visualize the costs associated with each valid muscle coordination pattern, we simply added three
vertical lines at the far right of the parallel coordinates plot, one for each of the three cost functions,
Fig. 3.2c. The variables a
i
and F
0i
represent the activation of the i
th
muscle in a given muscle activation
pattern, and the maximal strength of each muscle[92, 32]. Maximal muscle strengths are approximated
46
bythemultiplyingeachmuscle’sphysiologicalcross-sectionalarea,incm
2
,bythemaximalactivemuscle
stress of mammalian muscle, 35 N/cm
2
[159]. These four cost functions are but four examples from the
literature; an investigator is free to use this visualization of the feasible activation space with any cost
functiondeemedrelevanttotheirstudy.
Histograms of the activation level of each muscle across all valid solutions Muscle-by-muscle his-
tograms are another straightforward way to visualize the many points sampled from the convex polytope.
Histograms are particularly helpful because they illustrate the structure of the space of all feasible activa-
tions, allowing us to see which muscle activation patterns are on the edge of the space, which solutions
exist in the middle of the space, and how the bounds of the space and the distribution change across dif-
ferent tasks (in this case, as the task force increases). They visualize the relative number of solutions (i.e.
density of solutions) that required a particular level of activation from a particular muscle within its range
of[0,1]. Inaddition,theupperandlowerboundsofthehistogramsshow,infact,thesizeofthesideofthe
boundingboxofthepolytopeineverydimension(i.e. foreachindependentlycontrolledmuscle).
Dimensionality reduction Investigators have repeatedly reported that electromyographical signals (i.e,
experimental estimates of muscle activation patterns) tend to exhibit strong correlations with one another.
In these experimental descriptions of dimensionality reduction of neuromuscular control, only few inde-
pendent functions—sometimes called synergies—suffice to explain the majority of the variability in the
observedmuscleactivationpatterns[75,118,18,39,119,3,69]. Principalcomponentsanalysis(PCA)isa
widely used technique to extract these few independent basis functions (correlation vectors called princi-
palcomponents,PCs)fromhigh-dimensionaldata[27]. Inthiscase,PCsareoftencalledtheexperimental
representationsofsynergiesofneuralorigin[75].
Therefore, we applied PCA to points (i.e. muscle coordination patterns) sampled from the feasible
activation space at each force level. This provides the PCs that describe the correlations among valid
muscle activation patterns for a given task. For example, the feasible activation space P in a 3-muscle
systemwithoneconstraintisa2-dimensionalpolygonembeddedin3-dimensionalactivationspace. Thus,
47
applyingPCAtopointssampledfromthepolygonwillextract2synergies(i.e. 3-dimensionalcorrelation
vectors PC1 and PC2) that wholly explain the feasible activation space. By extension, in the case of
fingertip force production in Fig. 3.1, the feasible activation space is a 3-dimensional polytope embedded
in the 7-dimensional activation space. And PCA should extract, by construction, as many synergies as
there are dimensions in the feasible activation space. For static force production with the index fingertip
(i.e. 7musclesand4constraints),weknowthat3principalcomponentswilldescribe100%ofthevariance
inpointssampledfromthefeasibleactivationspace(i.e. 7-dimensionalcorrelationvectorsPC1,PC2,and
PC3).
ApplyingPCAtoourdataallowsustotestwhetherandhowitsresultschangewhenappliedtofeasible
activationspacesfordifferentmagnitudesoffingertipforce. WeappliedPCAtofeasibleactivationspaces
for fingertip task intensities ranging from 0 to 90% of maximal. Specifically, we applied the prcomp
function in R, and specified that the calculation operates on the covariance matrix of the raw data. We
compare both the variance explained by each PC and their loadings (e.g. correlations among muscles) as
theforcelevelincreases[152]. Lastly,wetestedwhetherthedispersion(i.e.,thetwocentralquartiles)and
median of our PCA estimates are sensitive to the number of points sampled from each feasible activation
space. This is important in practice because experimental studies tend to record and analyze a practical
number (e.g., 10) of repetitions of the same motor task from a given subject, and aggregate data from
differentsubjects[153]. Althoughwehavereportedthatsubjectstendtoexhibitsimilarmuscleactivations
for a given task [138], performing dimensionality reduction on such few trials and across multiple non-
identical subjects (i.e., samples in Figure 3.5) may lead to imprecise (i.e., uncertain) estimates of the
synergieswhensamplingfromhigh-dimensionalspaces.
48
3.4 Results
Weusedourrealisticindexfingermodeltocalculatethefeasibleactivationspaceforthetaskofproducing
static fingertip force in the distal direction (see Fig. 3.1). By showing how this same space can be inter-
preted from three dominant perspectives, we propose a conceptual paradigm to unify today’s theories of
neuromuscular control. The model contains the contribution of each of the seven muscles of the finger
to the resultant static fingertip force vector [149]. As described briefly in the Methods, all valid muscle
activation patterns to produce a given fingertip force vector (i.e. all ways in which one can combine the
actionsofthesevenmusclestoproduceagivenfingertipforcevector)arecontainedinalow-dimensional
polytope embedded in 7-dimensional space. Hit-and-Run is a method for uniform polytope sampling that
collects thousands of muscle activation patterns, which become a valid geometric approximation to the
structure of the feasible activation space[145]. We examined how these feasible activation spaces (and
their alternative representations) change with increasing task intensity (i.e. fingertip force magnitude,
Fig. 3.1e). In particular, we studied task intensities between 0% (i.e. pure co-contraction without output
force)and100%ofmaximalstaticforce(i.e. auniquesolution[144]).
3.4.1 Parallelcoordinatevisualizationnaturallyreveals
thestructureofthefeasibleactivationspace
Parallel coordinate visualization effectively reveals correlations that exist among the 1,000 valid muscle
activation patterns for each intensity of desired fingertip force, and activation pattern cost, Fig. 3.2 and
Fig.3.3.
Parallel coordinate visualization provides deep insight into the interactions among muscles that can
produce a given task. Because it allows interactive exploration of the feasible activation space, one can
restrict the activation level of any one or multiple muscles to see the associated activation levels of the
remaining muscles (i.e. see a subsample of the feasible activation set). Figure 3.4 shows how, for 80% of
task intensity, only 46% (i.e.
461
1,000
) of all possible solutions survive when we only keep solutions where
49
EIP and EDC are below 80% of maximal excitation. We chose to limit the extensors, as they are both
innervated by the radial nerve and are susceptible to limitation from, for example, neuropathy or stroke.
This robustness-related system behavior is visible in other muscle pairs via the interactive parallel coor-
dinates plot. We find that even a minor neural or muscle dysfunction can disproportionally compromise
the solution space—even for sub-maximal forces. These results further challenge the definition of muscle
redundancy as discussed in detail in[73, 149, 85], in that our description of redundancy may need to in-
corporate the structure of the feasible activation space to best describe how motor control can occur with
perturbationtooneormoremuscles.
Whileweknowfromexperiencethat alimitationononemuscleyieldscompensationfromtheothers,
Fig. 3.4 explains why, and how much to expect. All data used for Fig. 3.4 are for a task intensity of 80%.
Given this task, limiting PI to 20% of maximal activation eliminates 30.1% of the valid solutions, while
limiting DI to 20% eliminates 42.8% of them. The level of resilience to muscle limitation is intricately
linked to muscle redundancy. When we select only the lowest 5% of L2 weighted costs (3.4, middle
figure) there exist many ‘near-optimal’ solutions that are dramatically different (note the broad ranges
and criss-cross patterns in the second panel of in Fig. 3.4). This wide space exists in spite of this strong
criterion. ConsidertherangeofactivationforDIandPIinFig. 3.4whichliesbetween0and0.52and0.39,
respectively. Limiting DI to 20% pulls PI’s maximum down by nearly 0.20, and the converse has nearly
thesameeffect. However,inbothcases,themedianactivationamongsurvivingsolutionschangesnomore
than 0.06. This emphasizes that understanding feasible activation spaces requires an understanding of its
internaldensityandnotjustitsbounds.
Evaluatingtheslopeofthelinesconnectingmusclesenablesanintuitiveunderstandingofinter-muscle
correlations. ThePearsonproduct-momentcorrelationcoefficientswere0.99,-0.50,and-0.06intheadja-
centmusclepairsFDP—FDS,LUM—DI,andEIP—EDC,respectively. Theinteractiveparallelcoordinate
visualizationalsoallowsforanypairwisecomparisonbysimplydraggingandreorderingtheverticalaxes.
Thisisaneffectivead-hocmethodtoviewingtheinter-musclecorrelationsforexploratorydataanalysis.
50
3.4.2 Low-dimensionalapproximationstothefeasibleactivationspace
We applied Principal Component Analysis (PCA) to sampled muscle activation patterns for 10 levels of
task intensity. However, to replicate the fact that experimental studies can only collect a finite amount of
data from each subject, we did this in an iterative fashion as follows. We collected 10,000 points sampled
uniformlyatrandomfromeachfeasibleactivationspaceviaHit-and-Run[145]. Fromthese10,000points,
we sampled 10, 100, and 1,000 points at random (to simulate ‘experimental’ sample sizes), and applied
PCA to each set of sampled points. For each of the sample sizes, we replicated the sampling 100 times,
producing a distribution of principal component results, and thus, a distribution of variance-explained
metrics for PC1 (and the same for the other components). This bootstrap analysis serves to inform how
many samples one must collect from a subject to get an effective set of principal components. The H
matrixwasfixedacrossallreplicatesandsamples.
Figure 3.5 shows the box plots describing the variances explained by the three principal components
(PC1, PC2, and PC3) across task intensities. The third PC, PC3, explains the remainder of the variance
(13—15%)fortheresulting3-dimensionalpolytope. Recallthatthe4taskconstraints(f
radial
,f
distal
,f
palmar
,t palmar
)
applied to 7 muscles yield a 3-dimensional polytope embedded in the 7-dimensional muscle activation
space [144]); as such, the sum of all three PCs is exactly 100%. The supplemental website (linked in the
Data Availability Statement below) contains alternate versions of Figure 3.6 with varying input transfor-
mations.
The box plots in Fig. 3.5 quantify how different amounts of data change the estimates of variance
explainedbyaPCwithtaskintensity(c.f. labelsavs.bvs.c). Weseethisdispersionissmallinthecenter
and right columns. Note that the ratio of variance explained between PC1 and PC2 between 50 to 80% of
taskintensityrevealschangesintheaspectratioofthefeasibleactivationspacewithtaskintensity.
Importantly, we observe how using experimentally realistic samples sizes of 10 same-task repetitions
persubject(theleftmostcolumninFig.3.5)notonlydoesnotcapturethischange,butitsstandarddeviation
is large enough to blur the notable differences that are known to appear with larger (but experimentally
51
unrealistic) sample sizes. The impact of impoverishing the number of independent samples fed to PCA
reminds us that inadequate amounts of data obfuscate the underlying changes in the structure of the data
analyzed(Fig.3.5).
There were also changes in the loadings of the PCs, especially above 60% task intensity. While the
ratioofvarianceexplainedbetweenPC1andPC2givesasenseoftheaspectratioofthefeasibleactivation
space,theloadingsofPC1andPC2speaktoitsorientation[149,152]. Figure3.6showshowtheloadings
of the PC1 and PC2 vectors change across labels a, b, and c, Fig. 3.5. These loadings indicate that the
orientation of the feasible activation space in 7-dimensional space change mildly at forces less than 65%
ofthemaximaltaskforce,andchangemoredramaticallywithhigherforces.
These changeswe see in(i) the lowerand upper bounds ofactivations, and in(ii) the relativevariance
explained and (iii) loadings for all three PCs, demonstrate that the size, shape, and orientation of the
feasible activation space changes with task intensity. The muscle activation distribution ‘between the
bounds’ has profound implications for prior work which chiefly examines the ultimate upper- and lower-
boundsofactivationfortasksindifferentdirections[107,62]. Moreover,detectingchangesinthesehigh-
dimensional structures is done in the best-case scenario, as it exists in the absence of experimental noise,
within-andacross-subjectvariability,andmeasurementerror. AswillbeelaboratedintheDiscussion,this
implies that PCs (i.e. synergies) are laborious to obtain experimentally, and even then do not necessarily
generalizeacrossintensitylevels.
3.4.3 Changes in the probabilistic structure of the feasible activation space with
increasingtaskintensity,orhowmuscleredundancyislost.
The maximal static fingertip force vector in a given direction is produced by a single and unique combi-
nation of muscle activations. In contrast, any sub-maximal magnitude of that same vector is produced by
an infinite number of solutions[116, 23, 149, 138]. Our analysis of feasible activation spaces at different
52
taskintensitiesalsoallowsustocharacterizehowthisredundancychanges,andiseventuallylost. Thehis-
togram heatmaps in Fig. 3.7 illustrate the changes and shrinking of within-muscle histograms (the space
upon which probability density functions must operate) of valid activation levels across task intensities,
converging to a single solution at maximal force output. These surface plots show how the normalized
histograms (of 1,000 valid activation levels for each muscle at each intensity level) change at each of 100
equally-spaced levels of task intensity between 0 and 1. Following a muscle’s column from bottom to
top shows the activation histograms converge, naturally, to a spike at the unique value for maximal force
production.
The low flat areas on the sides of each surface plot (e.g., clearly visible for DI) represent muscle
activation levels that are not valid for that task intensity. That is, there exist no valid muscle activation
patternsthatcontainthatmuscleatthatlevel,andthusnopointsarefoundthere.
These plots show within-muscle probability functions and their and rate of convergence to the unique
solution for maximal force output across muscles. This is in contrast with the parallel coordinate plots in
Fig.3.3thatshowsthecorrelationacrossmuscles. Importantly,thehistogramsofactivationlevelsforeach
muscle need not be symmetric, nor have the same shape (skewness and kurtosis) as the magnitude of the
output force increases. For some muscles, the convergence accelerates after 60% or 80% of task intensity
(as in LUM and EIP), while others converge monotonically along the entire progression (e.g., DI and PI).
Thepeaks(i.e.modesormostcommonvalues)ofeachhistogramateachtaskintensityrepresentstheslice
of the polytope that has the largest relative volume along that muscle’s dimension (i.e. greatest frequency
of that level of muscle activation across all valid solutions). Importantly, for most muscles (FDP, FDS,
EIP, EDC, and LUM), the mode is not necessarily located at the same relative level of activation needed
formaximalforceoutput—evenwhenscalingitlinearlywithtaskintensity. Thatis,thehistogramathigh
levels of force is not simply a shifted version of the histogram at low levels of force. The histograms for
DIaretheexception,whosemodesseemtoscalelinearlywithtaskintensity.
Thesehistogramsandtheparallelcoordinatevisualizationsdemonstratethattheprobabilisticandcor-
relation structure, respectively, of feasible activation spaces, do not necessarily generalize across task
53
intensities. Nor can they be inferred from their bounding boxes alone (i.e. upper and lower activation
bounds for each muscle). An immediate example is how, for most task intensities, both EIP and LUM
have similar lower and upper bounds near 0 and 1, respectively—yet their distributions are thoroughly
distinct.
3.5 Discussion
3.5.1 Summary
Feasibility Theory, as a conceptual and computational approach, is a means to pierce the curse of dimen-
sionalitytoestablishaphysics-basedgroundtruthforneuromuscularcontrol. Thispracticalapproachcan
nowcharacterize—inanarguablycompleteway—thespaceofallvalidwaystoactivatemultiplemuscles
toproduceagiventask. Thisinitialpresentationislimitedtothecaseofstaticforceproduction. Additional
work is needed to extend to sequences of tasks, as has been done for optimization during gait analysis—
wherethedynamicalconstraintsduringmovementareappliedinthecontextofstaticoptimization[6,107].
But we can already say that feasible activation spaces are, in fact, the high-dimensional landscapes upon
which all neuromuscular learning, control, and performance must occur. These landscapes are predicated
uponthestrongexperimentalevidenceforlinearityintension-to-forcetransductionincadaveric[73],live
[63], and modeled [122] studies. Therefore, they provide an integrative and unifying perspective that
demonstrates how today’s dominant theories of neuromuscular control are alternative approximations to
feasibleactivationspacesfromoptimization,synergistic,andprobabilisticperspectives. FeasibilityTheory
unifies these alternative approaches to motor control in the sense that feasible activation spaces represent
anobjectiveconceptualandcomputationalcommongroundforthesetheories.
Note that these changes in the structure of the feasible activation space do not imply a given control
strategy. Theymerelyestablishtheboundswithinwhichaspeciesevolveacontrolpolicyforagivenbody
54
morphology. It is possible that the nervous system operates within a very small subset of this space—
which could be described by different principal components and even probability distribution functions.
FeasibilityTheory,however,allowsustoformallyphraseandtestsuchhypotheses.
3.5.2 Thevalueofacostfunction
Optimizationistheoldestcomputationalapproachtofindingvalidmuscleactivationpatternsthatproduce
limb function (e.g.,[23]). While optimization is, of course, a reasonable hypothesis to explore neuromus-
cularcontrol[129],somecriticizeitasamathematicalabstractionthatanthropomorphizesneuronswiththe
abilitytochoose,evaluateandfollowcostfunctionsinhigh-dimensions[35,80]. Thereis,nevertheless,an
intimate relationship between optimization and feasible activation spaces[25]. Optimization is analogous
to finding the best solution in the dark—guided by repeated small steps based on evaluations of cost- and
constraint-function. Computing the feasible activation space is then a means to ‘turn on the lights’ to see
all possible valid solutions independently of cost[149]. Our complete sampling of high-dimensional fea-
sible activation spaces [109, 83] allows us to compare and contrast families of solutions as per alternative
cost functions instead of individual optimal solutions for a particular cost function. Fig. 3.3 demonstrates
a complete description of families of valid coordination patterns and their relationship to alternative cost
functions. Importantly,similarvalidmuscleactivationpatternscanhavedissimilarcostsandviceversa.
Thus, Feasibility Theory allows us to compare, in detail, alternative ‘cost landscapes’ across the en-
tire set of feasible motor commands. By not having to insist on (or settle for) individual optimal—or
near-optimal—solutions, we now have the same ability the nervous system has to explore, compare and
contrast multiple valid (be they optimal or suboptimal) ways to coordinate muscles. Importantly, the rela-
tionshipsamongvalidmuscleactivationpatternsemergenaturallyfromthephysicalpropertiesofthelimb
and definition of the task. This cost-agnostic approach allows us to re-evaluate our assumptions about
what the nervous system cares—and does not care—about. Lastly, this cost-agnostic approach also pro-
vides a powerful tool for inverse optimization, i.e. uncovering latent cost functions from data[137]. Our
comparisonacrosscostfunctionsusingparallelcoordinatesisalreadyaformofinverseoptimization.
55
3.5.3 Freedomunderconstraints
We have so far only used ‘hard’ task constraints which must be met exactly. However, Feasibility Theory
also holds for soft constraints. For example, if a tendon-driven system is required to produce a 3D force
vector in general distal direction and of a general magnitude (defined, say, as a sphere of 1.0 N radius
centered on the nominal force), then we can apply these tolerances to the constraints defining the task.
In effect, Feasibility Theory allows us to study both soft and hard constraints, where the latitude of the
accuracyofthetasknaturallydefinestheprecisionwithwhichmuscleactivationpatternsmustbeselected.
One can define the task intensity to be, say, anywhere between 50 and 60%, and study the concomitant
increase in options available to produce forces within that range. Thus, one can characterize the changes
in the feasible activation space as the task constraints are relaxed or tightened. Similarly, adding task
constraints, such as the need to produce a particular stiffness at the endpoint [61], naturally reduces the
dimensionalityofthefeasibleactivationspace.
3.5.4 HowtoapplyFeasibilityTheoryinanexperiment
The most important input to this analysis is the relationship between muscles and the endpoint wrench.
With this relationship composed as the H matrix as in 3.1, and a desired wrench w, Hit-and-Run can be
usedtoproduceparallelcoordinateplotsanddensityhistogramsforstaticforceproductionwithvertebrate
limbs. Forexample,usingameasureofmuscleactivation(suchasfine-wireEMG),anexperimentalistcan
compare the muscle activation pattern chosen by a research participant in comparison to the full feasible
activation space that could achieve the same force, and see how those patterns change across fatigue,
disability of a muscle, or manipulation of the feedback. After a tendon-transfer surgery, for example, the
subjectmayinitiallyinhabitonlyaspecificpartofthefeasibleactivationspacetoproduceatask,butmust
usefeedbackfromtheparallelcoordinateplottofindsolutionswhichtakelesseffort. Ineffect,visualizing
theentirefeasibleactivationspacecouldhelpusunderstandhowrehabilitationcanbeguidedtowardsmore
advantageouslocalminima[131].
56
In parallel, a scientist with a cost function to test on a model can quickly identify how different cost
function parameters can affect the space of feasible activations, and see how specific the global optima is,
with respect to other muscle activation patterns. Importantly, anthropometric differences affect the shape
of the feasible activation space, so those subject-specific differences must be either incorporated or may
beaddressedthroughsensitivityanalysis(suchasMonte-Carlomanipulationofmomentarmvalues,asin
[62]).
3.5.5 Extensiontodynamicalforceproductionormovement
Limbsarevaluableformorethanjusttheirabilitytoproduceisometricforces. First,thereistheextension
to ‘non-static isometric’ force production (e.g., rotating a grasped object with respect to gravity), which
mustcontendwithtime-varyingmuscleactivation-contractiondynamicsandtargetgraspwrench(i.e. such
that the object is always securely held against a time-varying gravity vector [95]). With changes in joint
angles, the end-effector Jacobian, moment arm matrix, and vector of maximal feasible contraction levels
per muscle will vary nonlinearly—and with kinematic redundancy as a possibility for a given endpoint
location, we can introduce multiple feasible activation spaces that are capable of producing a given task
force. Even a simple task in the workspace likely exhibits redundancy at different levels of abstraction,
whereredundancyissourcedfromfeasibleactivationspacesandjointnullspacessimultaneously.
As muscles exhibit state dependence, the ability of an animal to produce precise dynamic forces is
affected by the tendon tensions from moment to moment. The inter-muscle dynamics across a human
index finger, for example, would necessarily require a feasible activation trajectory—which may or may
notberepresentablebyaconvexhull. ApplyingFeasibilityTheorytonon-staticisometricforceproduction
may require detailed investigation into the dynamics of musculoskeletal force transduction. In parallel to
the dynamics, non-convexities may emerge from neural constraints or even nonlinearities and hysteresis
ofmusclefunction.
Secondly, Feasibility Theory can be extended to address dynamical behavior by applying it to a se-
quence of slices in time. That is, a dynamical task can be equivalently analyzed as a sequence of ‘slices’
57
[6, 135, 26, 107]—where one can define a feasible activation space at each slice to determine how the
nervoussystemmustchangeactivationpatternssuchthatitisalwaysimplementingavalidsolution[107].
When strung together, these individual spaces give rise to a ‘spatiotemporal tunnel’—the time-varying
extensionofthefeasibleactivationspace(Fig. 3.8).
3.5.6 Structure,correlation,andsynergies
The physical properties of the limb and the definition of the task together give rise to a low-dimensional
structure of the feasible activation space [149]. Therefore, experimental recordings of muscle activations
during limb function will exhibit a dimensionality that is smaller than the number of muscles[75, 3, 133].
Thus, applying PCA to the points sampled from the feasible activation space will inevitably find that few
PCscanexplainthevarianceinthedata[20].
Our application of PCA at increasing task intensities (i.e. as muscle redundancy is lost) allows us to
demonstrate—for the first time to our knowledge—several important features and limitations of dimen-
sionality reduction. For example, we see that the aspect ratio (Fig. 3.5) and orientation (Fig. 3.6) of the
feasible activation spaces change as their size shrinks (Fig. 3.7). Thus, such descriptive synergies [20]
extractedfromlimitedexperimentalobservationslikelydonotgeneralizewellacrosstaskintensities. Pro-
ducing further insights into the feasibility-synergy relationship necessitates more objective metrics of the
feasibleactivationspace’sstructure.
The intensity-dependent structure of feasible activation spaces also has important consequences for
motor control and learning. Producing force vectors at the endpoint of a finger or limb with accurate
magnitude and direction are critical for versatile manipulation and locomotion[31, 144, 40]. If a given
synergycanproducesuchaccurateforcevectorsonlyforagiventaskintensity(andthusinaccuratevectors
atotherintensities),thentheattractivenessoftask-specificsynergiestosimplifytheneuromuscularcontrol
of the limb is reduced. Although we do not present an analysis of task-irrelevant synergies, data from this
paper can be concatenated prior to PCA analysis to explore how principal components vary across the
entiredistaltask.
58
To compensate, the nervous system would need to learn, recall, and implement intensity-specific syn-
ergies. Prior experimental work has shown that the nervous system produces accurate fingertip forces of
different magnitudes by, instead, likely scaling a remembered muscle activation pattern to produce forces
of different magnitudes[138], together with full-dimensional error correction [143]. The observation of
higher forces yielding more variable PC loadings indicates that lower dimensional substructures could
approximate low- and medium-level forces for a given direction, motivating further analyses of PCA ef-
fectivenessacrosstask-intensity(andwithNMF,forexample).
Our results also show how experiments with realistically moderate numbers of participants and test
trialslikelydonotcontainsufficientinformationtoproducerobustestimatesofdescriptivesynergiesacross
taskintensities. Asperthecurseofdimensionality,samplinguniformlyatrandomfromhigh-dimensional
spaces is exponentially difficult. Thus, even for this anatomically complete 7-muscle finger model, PCA
depends strongly on the number of independent observations, such as uncorrelated trials from one subject
or different subjects. Figure 3.5 shows that 100 to 1,000 such ideal data points from a simulated ‘test
subject’ are needed to produce accurate estimates of changes in the PCs with task intensity (c.f. labels a
vs. b vs. c). Future studies should explore how many experimental data points are sufficient from a given
subject when recording from only a subset of the many (20+) muscles of human limbs in the presence
of experimental noise, inherent stochasticity of EMG, and within- and between-subject variability. Some
studieshavebeguntoasksubjectstoexploredifferentwaystoperformagiventask[14,76](i.e. estimate
the structure of the feasible activation space), but in practice, such studies cannot likely collect sufficient
datauniformlyatrandomtoobtainaccurateestimatesofthedescriptivesynergies[75].
PCAisoneofseveralmethodstoextractlower-dimensionalrepresentationsofmotorpatterns[34,127,
27]. Alternativetechniquesdonotimposeorthonormalityconstraintsorover-estimatetherealdimension-
alityofnonlinearunderlyingmanifolds[27]. Similarly,Non-NegativeMatrixFactorization(NMF)would
not be subject to the flips in sign observed in Figure 3.5 [134]. We noted that for a given task intensity, a
muscle’s activation across the sampled solutions can have different variance than the other muscles, and
59
these variances change as task intensity increases (and the feasible activation space shrinks) (See the sup-
plemental website for the task-variance figure). While PCA helps us uncover how these shapes change
in this study, PCA can be leveraged to uncover different intramuscular relationships (e.g., analyzing the
eigenvalue decomposition of the correlation matrix, as opposed to using PCA on the covariance matrix).
Bootstrapping or data shuffling technique for sensitivity analysis are also applicable to dimensionality
reductiontechniques[152].
Feasibility Theory allows us to put dimensionality reduction in perspective. First, as a natural conse-
quence of the definition of a task (i.e. the need to meet specific mechanical constraints). And second,
as an approximation to the structure of the latent feasible activation space embedded in high-dimensions.
Whileourresultssuggestcautionwheninterpretingsynergiesobtainedexperimentally,weunderscorethat
dimensionalityreductionis,nevertheless,ausefulapproachtocapturethegeneralgeometricpropertiesof
feasibleactivationspaces.
3.5.7 Towardprobabilisticneuromuscularcontrol
Our results are particularly empowering for the emerging field of probabilistic neuromuscular control[66,
67, 100]. Suppose that the nervous system uses some form of probabilistic or Bayesian learning and
control strategy. Such approach requires two enabling—and biologically plausible—elements: trial-and-
error iterative exploration to build prior distributions, and memory-based exploitation of the probability
densityfunctionsusedtoapproximatethefeasibleactivationspaces[66]. Theparallelcoordinateplotsand
histograms in Fig. 3.2 and 3.7 provide, to our knowledge, the first complete[109, 83] characterization of
such multi-dimensional conditional motor control spaces for a realistic tendon-driven system performing
a well-defined task (i.e. activation of one muscle is contingent upon the activations of the other muscles).
With a better understanding of the physical task, future studies into optimal motor control can leverage
the feasible activation space to contextualize motor control policies, whether they are experimentally-
observed or theoretically predicted [15]. As mentioned above, the muscle activation patterns that the
nervoussystemsactuallyusewillnecessarilybeasubsetofthesefeasibleactivationspaces.
60
Feasibility Theory critically empowers the study of fundamental aspects of probabilistic control. For
example,anorganismcanonlyexecutesomanytrial-and-erroriterationsduringlearning,likelytoofewto
completely and exhaustively sample the high-dimensional feasible space of interest. This makes it much
morelikelythat,byvirtueofbeingmoreeasilyfound,anorganismwillfindandpreferentiallyexploitthe
strong modes (i.e. narrow and high peaks in Figs. 3.3, 3.4, and 3.7) of the multi-dimensional probability
density functions than any other region of feasible activation spaces. Thus, first, the maximal ranges of
feasibleactivationsdescribedbytheboundingbox[110,62]mayhavelittlepracticalbearingonhowthose
tasksarelearnedandexecuted. Andsecond,thosesamestrongmodeswouldrepresentstrongattractorsto
createandreinforcemotorhabits. Habitualcontrolhasbeenproposedbasedonexperimentalandempirical
data as an alternative to a strict optimization approach to neuromuscular control[35, 44]. Our work now
provides the computational means to link habitual to probabilistic control in isometric force production.
This allows us to generate testable hypotheses of how these motor habits are defined by the structure of
thefeasibleactivationspace,howeasilytheyarelearnedbytheorganism,andhowdifficultoreasyitisto
breakoutofthem[97].
Motor function likely emerges from trial-and-error [1] or imitation [91, 22] to identify, remember and
adopt easily-found, good enough solutions in the feasible activation space—independently of their cost.
It is then possible to use some heuristic approach to improve performance to transition to less likely—but
potentially ‘better’ solutions as per some metric relevant to the individual—subregions of the solutions
space. But this likely requires numerous iterations in practice, which explains why only a few of us are
expertsatagivenmotortask,orwhyrehabilitationissodifficult[1,81,50].
3.5.8 FeasibilityTheoryasatheoryofmotorcontrol
Feasibility Theory goes beyond Bayesian control by underscoring how the physics of the body, and the
properties of the task are the arbiter that guides the biological process of finding, exploring, inhabiting,
and exploiting low-dimensional solution spaces embedded in high-dimensions. Feasibility Theory es-
pousesheuristiclocalsearches—drivenbythememoryoflikelihoodsofdifferentindividualsolutions—to
61
create what ultimately are useful, yet likely sub-optimal, motor habits. These processes hinge on trial-
and-error, memory, pattern recognition, and reinforcement that come naturally to neural systems. Even
though Feasibility Theory is presented in the context of neural control of the human hand, it applies to
tendon-drivenorganismsingeneral.
Importantly, organisms perform strict optimization or synergy control at their peril. A feasible acti-
vation set is low-dimensional because it loses one dimension with each functional constraint that is being
met[61,144]. Thus,movingalongsuchlow-dimensionalspacestofindanewvalidsolutionisequivalent
tomovingalongaline(whichhaszerovolume)in3-dimensionalspace. Takingastepfromanyonevalid
pointtoanothervalidpointonthefeasiblespacerunstheriskof‘fallingoff’andfailingatthetask—arisk
that is exponentially exacerbated in higher-dimensions. Thus, searching for improvements in the neigh-
borhood of a known solution necessarily risk task failure and potential injury. These are all arguments in
supportoftheevolutionaryanddevelopmentallyusefulstrategytousegood-enoughcontrolbasedonhabit
orsensorimotormemoryratherthanoptimizationorsynergycontrol[35,45].
Thislineofthinkinghasconsequencestoneurorehabilitation. Neurologicalconditionsdisruptfeasible
activation spaces, be it by affecting anatomy of the limb, muscle strength, and independence with which
musclesarecontrolled. Functionalrecoveryfollowingthedisruption,ifnotdestruction,ofthelandscapeof
validmuscleactivationpatterns,requiresre-learningexistentorbuildingnewprobabilitydensityfunctions.
Olderadultssufferingfromreducedperceptuo-motorlearningratesarepresentedanevenmoreconstrained
feasibilityspace[29].
Aprobabilisticlandscapeforneuromuscularfunctionbeginstoexplainwhyneurorehabilitationinag-
ingadultsissodifficult(e.g.,[53,81])andwhymotorlearninginchildrentakesthousandsofrepetitions[1].
Butitempowersustoleverageknowledgeofthefamiliesoffeasiblesolutionstocreatenewrehabilitation
strategiesandtestablehypothesesaroundthem.
62
ConflictofInterestStatement
The authors declare that the research was conducted in the absence of any commercial or financial rela-
tionshipsthatcouldbeconstruedasapotentialconflictofinterest.
AuthorContributions
*BCandMScontributedequallytothiswork.
BC–Studydesign,computationalimplementation,experimentalanalysis,andbiologicalinterpretation.
MS–Mathematicalbasisandcompositionofcomputationalgeometrytechniques.
BG–Mathematicalbasisandcompositionofcomputationalgeometrytechniques.
FV–Studydesign,experimentalanalysis,theoreticalandbiologicalinterpretation.
Funding
ResearchwassupportedbytheNationalInstituteofArthritisandMusculoskeletalandSkinDiseasesofthe
NationalInstitutesofHealth(NIH)underAwardsNumberR01AR-050520andR01AR-052345toFVC,
National Science Foundation Graduate Research Fellowship (NSF GRF) to BC, and the Swiss National
Science Foundation (SNF Project 200021 150055 / 1) to BG, KF and FVC. The content is solely the
responsibility of the authors and does not necessarily represent the official views of the NIH, NSF, or the
SNSF.
Acknowledgments
We thank K Fukuda for his integral support in designing this research collaboration. We thank J Pugliesi,
TKim,CLim,ABaugus,PVachhani,andABolingforsupportinreviews,documentation,andcode.
63
Table3.1: ApplicabilityandcompatibilityofFeasibilityTheorywithdominanttheoriesofneuromuscular
control
DimensionalityReduction PCA, NMF, etc. describe the general shape and structure of the feasi-
ble activation space. The resulting basis functions serve as an approx-
imation of the input-output relationship of the system (i.e., descriptive
synergies).
MotorPrimitives/Synergies
/ModularOrganization If the basis functions mentioned above are of neural origin, they would
be the means by which the nervous system inhabits the feasible activa-
tionspaceandexecutesvalidsolutions(i.e.,prescriptivesynergies).
UncontrolledManifold(UCM)
Theory The UCM Theory emphasizes that the temporal evolution of muscle
activation patterns in the interior of the feasible activation space need
not be as tightly controlled as those at its boundaries. This is because
moving between interior points has no impact on the output as they
constitutethenull-spaceofthetask(i.e.,theyare‘goal-equivalent’asin
[104]). In contrast, Feasibility Theory describes details of the structure
ofthefeasibleactivationspace.
Exploration-Exploitation Heuristic and trial-and-error approaches can be used to find points
within the Feasible Activation Space because it is a needle-in-a-
haystackproblem. Bydefinition,thereisasmalllikelihoodoffindinga
point on a low-dimensional manifold embedded in a high-dimensional
space (e.g., the volume of a line is zero). Thus, the families of valid
solutions found are preferentially adopted (e.g., as motor habits[35]).
Such a heavily iterative approach is compatible with reinforcement
learning[142],motorbabbling[130],thehundredsofthousandsofsteps
children take when learning to walk[1], or the mass practice a patient
needsforeffectiverehabilitation[78].
ProbabilisticNeuromuscular
Control If muscle activation patterns within the feasible activation space can be
found(byanymeans),theycanbecombinedtobuildprobabilitydensity
functions (i.e., Bayesian priors). A likely valid action for a particular
situationcanthenbeselectedviaBayes’Theorem(e.g.,[66]).
Optimization/Minimal
InterventionPrinciple
/OptimalControl Everypointinthefeasibleactivationspaceis,bydefinition,valid. How-
ever, if a cost function is used to evaluate each point in it, the feasible
activation space is transformed into a fitness landscape. Optimization
methodscanthennavigatethisfitnesslandscapetofindlocalandglobal
minima(e.g.,[32,6,129]).
3.6 DataAvailabilityStatement
Thedatasetsgeneratedandanalyzedforthisstudycanbefoundfreelyavailableathttps://github.com/bc/space,
and at the supplemental website http://valerolab.org. We designed a web-based parallel coordinate visu-
alization that lets users interactively limit muscles, select solutions, and calculate effects on the feasible
activationspacefromeachpost-hocconstraint(Fig. 3.4). Ourcompanionsiteincludesampledocumenta-
tion, code implementation in Scala (with a comprehensive test suite), and all data visualization code in R,
includinganoverheadviewofFigure3.7.
64
Figure 3.2: Parallel coordinates characterize the high-dimensional structure of a feasible activation
spaces. Considerfourpoints(i.e. muscleactivationpatterns)fromthepolygonthatisafeasibleactivation
space (a). The activation level for each muscle (i.e. the coordinates of each point) are sewn across three
verticalparallelaxes(b). Asiscommonwhenevaluatingmusclecoordinationpatterns,eachpointcanalso
be assigned a cost as per an assumed cost function. The associated cost for each muscle activation pattern
canalsobeshownasanadditionaldimension. Weshowthreerepresentativecostfunctions(c). Activation
levelsareboundbetween0and1,andcostsarenormalizedtotheirrespectiveobservedranges.
65
Figure 3.3: Activation patterns of the seven muscles of the index finger across six intensities (mag-
nitudes) of afingertip force vector inthe distal direction. The connectivity across parallel coordinates
visualizes the correlations among muscle activation patterns at different task intensities. At the extremes
of 0% and 100% we have, respectively, the coordination patterns that produce pure co-contraction and no
fingertipforce,andtheoneuniquesolutionformaximalfingertipforce[144]. Inbetween,weseehowthe
structureofthefeasibleactivationspaceschanges,andthatmuchredundancyislostratherlate(atintensi-
tiesgreaterthan80%,inagreementwith[110]). Inbluearetheactivationvalues,andinredarenormalized
costs for four common cost functions in the literature. For each task intensity, we produced 1,000 points
that are uniformly distributed in the polytope via the Hit-and-Run method. The muscles are FDP: flexor
digitorum profundus, FDS: flexor digitorum superficialis, EIP: extensor indicis proprius, EDC: extensor
digitorum communis, LUM: lumbrical, DI: dorsal interosseous, PI: palmar interosseous. Color is used
solelytodifferentiatemuscleactivations(blue)fromcostvalues(red).
66
EIP < 80%, EDC < 80%
cost functions
FDP FDS EIP EDC DI PI LUM L1 L2 L1W L2W
muscles
Bottom 10% of the range
of each cost
50 points with lowest L2W Cost
points
points
points
Figure 3.4: Exploration of the feasible activation space for task intensity of 80%. Here we show
three informative examples of constraints applied to the points sampled from the feasible activation space
(n=1,000; axes match those of Fig. 3.3). With this interactive visualization, we can easily see how the
size(i.e. numberofsolutions)andcharacteristicsofthefamilyofvalidmuscleactivationpatternschange.
Forexample,intheeventof(Top)weaknessofagroupofmuscles(54%reduction),(Middle)selectionof
the lowest 5% of a given cost function (95% reduction), and (Bottom) enforcing the lowest 10% of cost
range across multiple cost functions (99.6% reduction). In all cases, the family of valid muscle activation
patterns retains a wide range of activation levels for some muscles. While it is challenging to understand
the structure of the feasible activation space with a static plot of the parallel coordinates, interactively
manipulating the muscle ranges on one or multiple axes makes it very easy to view and describe how
muscleactivationschangeinthefaceofdifferentconstraints.
67
Variance
explained
by PC1
Variance
explained
by PC3
Variance
explained
by PC2
0 3.2 19.2 25.6 28.8
Max
0 3.2 19.2 25.6 28.8
Max
Distal force produced at the fingertip (N)
0 3.2 19.2 25.6 28.8
Max
0.5
0.6
0.7
0.8
0.2
0.3
0.4
0.05
0.10
0.15
0.20
n = 10 n = 100 activation points (samples) passed to PCA
PCA generated
with fewer samples
PCA generated
with many samples
n = 1000
Figure3.5: Approximatingthestructureoffeasibleactivationspacesviaprincipalcomponentsanal-
ysis (PCA) is sensitive to both the task intensity and the amount of input data used. Rows show
the variance explained by the first (top) through third (bottom) principal components with increasing data
points for a given replicate (left to right). Hit-and-Run sampling provides the ground truth for the high-
dimensionalstructureofthefeasibleactivationsetateachtaskintensity. Eachboxplot,acrossallsubplots,
isformedfrom100metrics(replicates),whereeachmetricisthePCvarianceexplainedforareplicate‘sub-
ject’ which performed the task n times (where n is one of 10, 100, or 1000 task repetitions). We find that
PCA approximations to this structure do not generalize across tasks intensities (i.e. the polytope changes
shape as redundancy is lost), and numbers of points. That is, > 100 muscle activation patterns should be
collected from a given subject to confidently estimate the real changes in variance explained as a func-
tion of task intensity. Compare points labeled a, b, c, corresponding to 11, 66, and 88% of task intensity,
respectively.
68
FDP
FDS
EIP
EDC
LUM
DI
PI
PC1 Loading
PC2 Loading
PC3 Loading
Task Intensity (N)
Max Null b c
Discontinuities in loading value are
due to flips in sign, not large
changes in the structure of the
PCA yields different loadings as the
structure of the feasible activation
The way loadings change as intensity
Force:
PC2 Loading:
Force:
PC2 Loading:
+
Reflections of loading value
between discontinuities show how
Reflections of loading value between
discontinuities show how sign-flipping is
Figure 3.6: PCA loadings change with task intensity For each of 1,000 task intensities, we collected
1,000 muscle activation patterns from the feasible activation space and performed PCA. The facet rows
show the changes in PC loadings, which determine the direction of all PCs in 7-dimensional space. Note
that the signs of the loadings depend on the numerics of the PCA algorithm, and are subject to arbitrary
flips in sign [27]—thus for clarity we plot them such that FDP’s loadings in PC1 are positive at all task
intensities. DottedverticallinesconnectloadingsofPC2andPC3inspiteofflipsinsign. Adiscontinuity
here is not indicative of a major change to the feasible activation space. It instead, is a result of how PCA
selects loadings. The shape of the activation space has tilted at these points, thereby flipping the sign.
Note that the values are the same before and after the jump, less the sign. These loadings (i.e. synergies)
change systematically, as noted for representative task intensities a, b, c in Fig. 3.5, and more so after b.
Thisreflectschangesinthegeometricstructureofthefeasibleactivationspaceasredundancyislost.
69
FDP
FDS
EIP
EDC
LUM
DI
PI
Muscle Activation
0
0%
40%
60%
80%
20%
100%
100%
0%
1
Task Intensity
Percentage of
the feasible
activation space
Task Intensity
Figure3.7: Thewithin-muscleprobabilisticstructureoffeasiblemuscleactivationacross1,000levels
of fingertip force intensity. The cross-section of each density plot is the 50-bin histogram of activation
for each muscle, at that task intensity. The changes in the breadth and height for each muscle’s histogram
reveal muscle-specific changes in their probability distributions with task intensity. Height represents the
percentage of solutions for that task. The axis going into the page indicates increasing fingertip force
intensity up to 100% of maximal. Color is used to provide perspective. It is interesting to note that, for
example, both extensor and flexor muscles are used to produce this ‘precision pinch’ force. This is to be
expectedastheactivityintheextensorsisnecessarytoproperlydirectthefingertipforcevector[141].
70
m! m"
m#
Figure 3.8: Spatiotemporal Tunneling. A dynamical movement can be decomposed into a sequence of
slicesintime,whereeachslicehasacorrespondingfeasibleactivationspace. Strungtogether,thesequence
of feasible activation spaces form the ‘spatiotemporal tunnel’ through which the neuromuscular system
must operate. In this 3-dimensional schematic example, the black line represents one valid time-varying
sequence of activations for three muscles. Because this sequence exists within each feasible activation
space,itnecessarilymeetstheconstraintsofthedynamicaltaskateachinstant.
71
Chapter4
Spatiotemporaltunnelsconstrainneuromuscularcontrol
BrianA.Cohn
1
andFranciscoJ.Valero-Cuevas
2,3
1
UniversityofSouthernCalifornia,DepartmentofComputerScience,LosAngeles,CA
2
UniversityofSouthernCalifornia,DepartmentofBiomedicalEngineering,LosAngeles,CA
3
UniversityofSouthernCalifornia,DivisionofBiokinesiologyandPhysicalTherapy,LosAngeles,CA
4.1 Abstract
Animals must control their limb endpoint forces for tool use and manipulation. And while decades of
research has elucidated much about how intentions lead to physical forces and movements, and what
correlations exist between muscles, these methods do not address core questions about why these rela-
tionshipsoccur,andwhatneuromuscularandphysicalrequirementsmostconstraintheirpossibilities. Our
prior work has more faithfully enumerated the full dimensional neuromuscular control landscapes upon
whichlearningmustoccur,butwedidnotaddresstheconstraintsofhowmusclescanperformwithinthese
spaces. In this study, we address this gap by stochastically exploring the addition of a simple time param-
eter,andratherthanoptimizeorreducethedimensionalityofournullspace,wefullyenumeratethespace
as a product polytope in 49 dimensions—7 muscles over a 7-step task, with the constraint matrix being
solvedinonestep. Asaresult,wecanselectonlythosetrajectorieswhichmeettheactivation-contraction
72
constraint across the entire trajectory, and sample from the space in a new manner that is traditionally
intractable for analyzing muscular systems of this size. By defining the physical realities that govern how
evolutionmustderiveneuromuscularstructures,wediscoveredhowpowerfulevensimpletimeconstraints
are, and how they warp the ways muscles interact, correlate, and optimize. This theoretical and compu-
tational work offers new tools to generate hypotheses across the interplay of high-dimensional neurology
andfull-dimensionaltaskphysics,highlightinghowthetime-varyingnatureofactivationsdefineandlimit
howneuromuscularsystemscanevolveandlearnmotorpatterns.
4.1.0.1 AuthorContribution
BC designed analyses and code, FV supported the deep analysis of neuromuscular implications of the
results.
4.2 Introduction
For the case of a human tendon-driven fingertip generating endpoint forces, having more muscles than
output constraints raises a long-explored question: how do animals find and select a muscle activation
pattern that work, given a high dimensional space to choose from [4]? For much of the past 40 years of
neuromuscular control, there have been deep analyses of the cortical circuits, evaluations of the muscle
activity, and knock-out studies of different points along the pathways providing motor drive. Prior work
with Feasibility Theory [30] has effectively sampled from the Feasible Activation Space, a representation
of the way muscles can combine their activations to effect an output wrench[149]. Many effective proce-
duresexistforanalyzingandinterpretingthehighdimensionalnatureofunder-constrainedmotorcontrol,
including extraction of a series of lower-dimensional vectors [3, 133, 2], approximation of their distribu-
tion as Bayesian priors [66], or by application of a heuristic cost-function to find optimal subtrajectories
[129, 125]. Feasibility Theory posits that the full dimensionality of tendon-driven systems must be pre-
served to yield a fair and common ground in contextualizing theories of neuromuscular control, chiefly
73
the aforementioned synergistic, Bayesian, and optimization-based methods. However, forces change over
time, and it’s still difficult to this day to fully describe why an animal chooses a particular motor pattern
in redirecting or scaling force. Those internal ‘functions’ are often approached from the musculoskeletal
side,bymeasuringdirectlyfromthemusclesactivity,viameasurementoftensionsovertime,andthrough
spinal and cortical imaging. Much incredible progress has been made to incorporate the time-varying dy-
namics into these activation-to-wrench approximations, in the case where dimensionality is reduced by
optimization[108],orataseriesofpointsinatransectfromthemostoptimaltrajectorytotheleast[111].
Musclescannotactimmediately—theyareasingleelementofadynamicalsystemwithmanymoving
parts. With motoneurons being discharged stochastically, and with limited chemical energy at any given
moment, there are neurological, chemical, physiological, and physical constraints to creating changes
in musculotendon tension, which acts as a series elastic element. Muscle-length and muscle velocity
significantly impact the force applied at the tendon’s point of insertion, and different parts of the muscle
have different response speeds (i.e. slower and faster twitch models of motor control). Furthermore,
integumentalstructures(includingskin)andpassivemusculoskeletalstructures(includingligaments)serve
asadynamicalsystemwhichhavetheirowntime-varyingdynamicsandstate. Forthecontextofthisstudy,
wedonotfullymodelthesedelaysandlimitation–rather,wemakeaupper-boundobservationofthefastest
changeinmuscleactivationovertimeasdescribedintheMethods.
Ifwecouldadequatelyreducethesizeofthefeasibleactivationspace—allwaystoachievethetask—
we would be able to close the boundaries for hypotheses on motor control, reducing the space of viable
hypotheses, and allowing for immediate feedback for scientists looking to build models which explain
for the behavior of vertebrate tendon-driven control. This paper provides a novel computational tool for
stochastically, andfairly sampling the feasible activation space, even in light of the incorporation of time
as a new variable. We provide strong evidence that even a rather low-dimensional to low-dimensional
tendon driven limb (a human index finger) has an incredibly broad distribution of activation trajectories
for the simple task of redirecting a 10N force 30
, and that redundancy remains even after adding novel
constraints of muscle activation-contraction dynamics to the full-dimensional feasible activation space.
74
segmental
feedback
alpha drive
M1 drive
t=50ms
t=100ms
t=150ms
t=250ms
t=200ms
Neurophysiological constraints Biomechanical constraints for t=0ms and t=300ms
Any trajectory from the feasible trajectory
from pure
palmar wrench
(degrees)
0
Time (ms)
250
50
10N
150ms
100ms, 200ms
50ms, 250ms
0ms, 300ms
150
100
200
300
30°
0°
H from
R Index Finger
palmar
(-z)
all tasks are isometric with a
dorsal
proximal
(+x)
distal
A -dimensional convex polytope that describes
wrench outputs
All tasks lie on the xz plane;
torque
y
set to 0
1N
1N
0 1
of muscle
i
uniform at random sampling
# points
sampled
feedback
Figure 4.1: Overview of the primary objective of this work. Our objective is to computationally sur-
vey the Feasible Trajectory Space in the context of activation-contraction constraint, to better inform our
perspectivesofdescendingneuromuscularcontrolparadigms.
75
Werefertothisbehaviorasaspatiotemporaltunnel—thewell-structuredrepresentationoffeasiblemuscle
activations that must be traversed by both evolutionary and learning timeframes. To achieve a series of
isometricforces,thelimbmustmeettheconstraintoftime(Figure4.1).
4.3 Methods
Asin[30,113,145]wedefinethelineartransductionoftendontensionsintooutputendpointwrenchas
H⇤ ¯ x= ¯ w. (4.1)
Where H (a [4,7] matrix in this paper for 4 output dimensions and 7 input tendon activations) represents
thelinearactivation-to-wrenchrelationship,suchthat H⇤ x= ¯ w
output
.
Wrenchesarefourdimensionalastheindexfingercanproduceatorque(i.e. scratching) ¯ w=(f
x
, f
y
, f
z
,t
y
)[156].
AsthedataforH werecollectedinthesameposture,andasthereisstrongevidencesupportingthelinearity
of tendon-driven isometric force transduction in fixed postures, we do not need to model the intermediary
Jacobian or the Moment-arm matrix [123, 144, 140, 113]. We define x2 [0,1]
7
where 1 represents 100%
activation.
Note that the term ‘muscle activation’ can take on different meanings depending on the level of the
analysisbeingused. Inourcase,weuseitasshorthandforthetotalsignalneededtoproduceagivenlevel
ofneuraldrivetoproduceforceateachmuscle. Thereasonwedothisisthatitencompassesthemetabolic
cost,intensity,andfeasibleratesofchangeofboththeneuraldriveandmuscleforce. Assuch,itincludes:
• Presynapticinputtoapopulationofa motoneurones
• Theneuralcommandsentbythea motoneurontothepopulationofmusclefibersinitsmotorunits
[115]
• The biochemical processes required for the release and uptake of acetylcholine at the motor end-
plateofeachmusclefiber[136]
76
• Ca+releaseanduptakebythe sarcoplasmic reticulum[136]
• Thecross-bridgecycleatthesarcomeretoproduce,holdandchangethelevelofmuscleforce
We make the simplification, without loss of generality, to not distinguish between muscle types and con-
siderequaltimeconstantsfortheincreaseanddecreaseofneuraldriveandmuscleforce.
And while many approaches minimize ¯ c
T
¯ x where ¯ c represents a vector of linear weights to combine
with ¯ xtoformametricofcost,e.g. if ¯ c=(1,1,1,1,1,1,1),¯ c
T
¯ xwouldcomputethe’sumcostofactivation’,
or ¯ c=(0,0,0,0,0,0,1)wouldcomputethe‘sumofjustpalmarinterosseus’. Nonlinearobjectivefunctions
have also been used to better understand weighted L
2
and L
3
metabolic cost functions[30]. For this paper,
rather than minimization or optimization on an arbitrarily defined cost function (a model choice in itself)
ourapproachinsteadsamplesfromthenullspaceof ¯ xuniformly-at-random(u.a.r). Weleveragethesame
computationalgeometrytechnique’Hit-And-Run’asin[30,145],whichisoriginallydescribedin[83,82].
Synaptic drive applied to motor units create forces, which ultimately generate muscle forces, and
accumulate to tendon force. The tendon is compliant and together, the musculotendon is a big dynamic
systemwithmanyphysiologicalandphysicalconstraints. It’saserieselasticelement.
4.3.0.1 HitandRunsamplingofthefeasibleactivationspace
Visualization and analytics of these high dimensional structures requires unique approaches to highlight
different aspects of feasible activation spaces, and there has been some success in using 2D and 3D vi-
sualization to decompose neural control of force [145, 30]. As the dimensionality of the space increases,
the ratio of out-of-polytope to in-polytope volumes within the unit cube expands exponentially, thereby
making 2D and 3D approaches computationally intractable with systems with more than 2 muscles. Like
in prior work, we sample the space with the Hit-and-Run algorithm—a Markov chain propagating within
thepolyhedronthatyieldsauniform-at-randomdistributionwithinthevolumeofagivenconvexpolytope.
This method is agnostic to measures of metabolic or neurologic cost, and allows for contextualization of
thesolutionsoptimizationmayselect.
77
4.3.0.2 Definingthetemporalconstraints
Onecorelimitationofourpriorwork[30]isthesingle-momentanalysis,thatdoesnottakeintoaccountthe
amountofchangetheCNSmustperformtomovefromsolutiontosolution,fromtasktotask. Musclesdo
notactwithinfinitelyfastresponsetimes;torespectthis,weincorporateanelementoftemporalconstraint
in our model by limiting a muscle’s change in activation between ±d % over a 50ms interval. Given the
observation that deactivation in vertebrate muscle is often slower than activation[115], we set this limit to
the faster of the two, forming a conservative bound. We refer to this metric as the activation-contraction
constraint, and as we take the absolute value of the deltas, this metric is always set between [0,1].A
constraint value of 0.25 means that in 50ms, activations can change their output by no more than 25% of
theirmaximaltension.
4.3.1 Specimen
Ouractivation-to-wrenchmodelH wassourcedfromanexperimentusingcadaverfingers[154],withorig-
inaldata(n=11)from[116]. Torevealtheeffectsofactivation-contractionconstraintsonatime-dependent
feasible activation space, we leveraged a stochastic Monte Carlo technique to fairly extract activation
trajectories—Hit-and-Run[83]. Inadditiontobeingnormalizedbetweenanactivationof0and1(muscles
can’t go negative as they can only pull), muscles were constrained in their ability to change their output
activation from moment to moment. For each moment in time, the endpoint vector had to meet the re-
quirements of its desired output wrench within a series of seven tasks. Formally, we add new constraints
in the way the activations can change, which are ultimately classifiable as Lipschitz constraints [114, 19].
Formally, we sample u.a.r. from the null space on x, given A and b where x2 [0,1]
n
. Our Lipschitz Con-
straints (referred to hereafter as ’activation-contraction constraints’ as they serve to link different motor
patternsovertimetodifferentoutputwrenches.
|x
i+1
x
i
| d x2 [0,1]
7
(4.2)
78
0 B B B B B B B B B B @ f
x
f
y
f
z
t y
1 C C C C C C C C C C A =w=Ha=H
0 B B B B B B B B B B B B B B @ a
1
a
2
a
3
...
a
7
1 C C C C C C C C C C C C C C A ,a2 [0,1]
7
(4.3)
We set the task to a series of 7 individual wrenches performed over the course of 300ms, which starts
at a pure f
x
force (towards palmar), with a 30 degree rotation towards proximally (rotated about the axis
definedbytheulnardirection),andasymmetricalreturn. Theprogressisshapedasasinglecosineperiod,
with the peak being the 4
th
index. Wrenches (0,6),(1,5),(2,4) are identical—providing a symmetric set of
taskstostayconstantwhiletheactivation-contractionconstraintdemandsmaychange.
We sampled 100,000 activation trajectories per activation-contraction constraint condition, where the
constraintsweresetfrom1to0.05(Figure4.3).
4.3.1.1 Analyzingunseededvsseededactivationtrajectorydistributions
To address this difficulty in analyzing the distributions of muscle activations, we present the following
‘’unseededvsseeded’trajectoryanalysisinFigure4.2,withmethodsdescribedinFigure4.3.
Wecomputethepossibletrajectorieswhenthefirstmomentisfixedtoaseed-point,andcomparethose
‘’seeded’trajectoriestothe‘’unseeded’trajectoriesthatwerenotfixed. It’simportanttonotethatunseeded
trajectories are still sampled under the same activation-contraction constraint as their seeded counterpart,
that all unseeded trajectories meet those activation-contraction constraint, and that all seeds must have a
startingpointthatexistsintheunseededpolytope. WedescribeoursamplingprocedureinFigure4.3cand
d. Fromout100,000unseededtrajectorieswecomputedunderaactivation-contractionconstraintof 0.12.
Fromthose,weselected10atrandomandcalledthoseour‘seeds’. Foreachseed,wetrimmedoffthet=50
tot=300activationvalues, andappendedanewconstrainttotheoriginalconstraintmatrix,soallsampled
79
FDP FDS EIP EDC LUM DI PI
0 50 100 150 200 250 300
0.00 1.00
1.00
0.00
0.25
0.50
0.75
1.00
Muscle activation (0 to 1 is 0 to 100%)
Within bin Volume wrt to mode
Seed ID
Time (ms)
Muscle
X1325
X22583
X23343
Figure4.2: Consequencesofselectingaspecificinitialmuscleactivationpatternforamaxactivation-
contraction speed of 0.25 Here we show the distributions of three trajectory seeds selected across a uni-
form sample of the unseeded H (Eq. 4.3). As in 4.10, lines are drawn by connecting the midpoints of
100 histogram bins. We observe strong hysteresis in the positioning of muscle activation when the seed
trajectory locks the activation high or low on a given muscle, and that selecting a seed point implies that
youcannoteasilyreturntoanotherseedpoint.
80
Figure 4.3: Method for generating unseeded and seeded trajectories Unseeded trajectories can orig-
inate in any valid solution at t = 0 show their evolution across the subsequent polytopes (i.e., solution
spaces) subject to the temporal constraints of activation-contraction dynamics of muscle. A seeded tra-
jectory, on the other hand, is pulled from the same constraint matrix, but with an additional constraint:
all of the points selected from a seed start at a same seed point (i.e., valid solution at t = 0). A seed
point can be extracted from the unseeded trajectories. Seeded points can only evolve in time into sub-
regions of the subsequent solution spaces that are reachable given the starting point and the temporal
constraints of activation-contraction dynamics of muscle. Importantly, unseeded trajectories all meet
activation-contractionconstraintsaswell
81
trajectorieswereforcedtomatchtheseedint=04.3. Wecomputed10,000trajectoriesperseed. Weshow
all 10 in an interactive video described in Figure 4.10, and show three examples in Figure 4.2. Each show
the [7,7] by-muscle, by-moment distributions of the unseeded (in black) and seeded (a different color for
eachseed).
Figure 4.4: Quantifying the evolution over time of the distribution of solutions for unseeded and
seeded trajectories Here we detail our method for analyzing and visualizing the effect of selecting a
solution seeded in t =0. We began by extracting one hundred thousand activation trajectories from H as
in Eq. 4.3. With 10 of those trajectories, we extracted only the first value, then ran a further sampling
paradigmonamodifiedconstraintequationwherethefirstactivationpattern(of7muscleactivations)had
to match the seed’s activations at t=0. As we want to visualize the effect of selecting a seed point, but
cannoteasilyplota4Dstructureembeddedin7D,weappliedprincipalcomponentanalysistoeachofthe
7 moments of time across the unseeded distribution. We then projected both the unseeded, and seeded
activation trajectories across the first two PCs, highlighting where in the lower-dimensional space those
solutionsweremostprobable.
4.4 Results
The time history of feasible activations for a given action is highly restricted under activation-contraction
constraintsimposedbymusclephysiology4.5.
First, Figure 4.5 demonstrate that, as the activation-contraction speed limit is reduced, the trajectories
become more spatially constrained in the regions of the feasible activation sets they can inhabit/exploit.
82
FDP FDS EIP EDC LUM DI PI
0.00
0.25
0.50
0.75
1.00
0 300
Time (ms)
0 300
Time (ms)
1.0
0.5
0.0
0.5
1.0
0.5
0.0
0.5
1.0
FDP
FDS
EIP
EDC
LUM
DI
PI
FDP FDS EIP EDC LUM DI PI
0.00
0.25
0.50
0.75
1.00
1.0
0.5
0.0
0.5
1.0
1.0
1.0
0.5
0.0
0.5
1.0
FDP FDS EIP EDC LUM DI PI
0.00
0.25
0.50
0.75
1.00
a
i
a
i
1.0
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
FDP
FDS
EIP
EDC
LUM
DI
PI
FDP
FDS
EIP
EDC
LUM
DI
PI
a
i
a
i
a
i
a
b
c
Di!erentiate
0 300
Time (ms)
0 300
Time (ms)
0 300
Time (ms)
a
i
a
i
a
i
a
i
0 300
Time (ms)
s.t. constraint
0.50
s.t. constraint
0.25
s.t. constraint
0.10
{
{
{
Figure 4.5: Ten example trajectories with three levels of activation-contraction constraints For each
level,weshowa)tenexampletrajectories,whereeachcolorisadifferenttrajectory. b)Thosetrajectories,
differentiated to show how quickly the activations were changing with the upper and lower activation-
contractionconstraintsshownasdottedlines,andc)adistributionofthetrajectory‘activation-contraction
speeds’,groupedbymuscle. Notethatcolorsonpartc)donotrelatetoa)andb). Outliersarenotshownon
c. Note that unlike Figure 4.6 which shows the max(|˙ a
i
|), this figure shows the raw differentiated muscle
activationsas ˙ aandthusissignedfrom ±1.
83
This allows us to describe the effects of the activation-contraction constraint under which muscle coordi-
nation happens to be able to produce a force and change its direction. Figure (4.5a,b shows 10 example
trajectories (from the 10,000 calculated) as they traverse the 49-dimensional space (i.e., 7 muscles over
7 timepoints or tasks). For completeness, Figure 4.5c also provide boxplots of the per-muscle activation
levels across all 10,000 trajectories sampled. Note that the limit on activation-contraction speed natu-
rally affects muscles with larger feasible activation ranges (shown as the top row of Figure 4.8). But also
note that as the maximal activation-contraction speed is reduced, those same muscles will visit/exploit
increasingly smaller subspaces of their feasible activation sets. This spatiotemporal interaction is best
seen in EIP, which has a naturally large range of feasible activation, which are suitably exploited when
theactivation-contractionconstraintisless-constraining. Butthenshrinksastheconstraintbecomesmore
strict. However, changes also spill over to muscles with naturally smaller ranges of feasible activations
such as FDP. This muscle has few trajectories with an activation-contraction rate greater than 0.25 to
beginwith,butbecomeslimitedinrangeastheactivation-contractionspeedisreduced4.5c.
A closer look further confirms that muscles that have a narrow range of feasible activations will be
least sensitive to changes in activation-contraction constraints. Figure 4.6 shows the distribution of the
maximalchangeinactivationforallmuscles,andweseethatonlythemusclesthathavegreaterrangesof
activation have non-overlapping central quartiles between 0.75 and 0.5 activation-contraction speed. It is
hard to drag race in a driveway. We observe how, for example, FDP, FDS, and PI are more affected by
thereductionofmaximalactivation-contractionspeed.
Producing a fingertip force and changing its direction requires selecting a specific solution and im-
plementing a specific sequence of activation patterns. Our ‘seeded analysis’ reflects the consequences of
choosing an initial activation pattern (a ‘seed’) to subsequent feasible activation patterns (Figure 4.2). We
show this for three choices of initial seeds selected at random (top row), where the subsequent feasible
activations for each muscle are limited in where they can go given the activation-contraction speed of
0.25. As we do observe many changes in the profiles of trajectories when time is considered, and do not
observe changes in some of the by-muscle distributions (as seen in Supplementary Figure 4.8, traditional
84
FDP FDS EIP EDC LUM DI PI
0.00
0.25
0.50
0.75
1.00
Spatiotemporal Constraint
Maximal within-trajectory absolute difference between timepoints
1 0.75 0.5 0.25 0.1 0.05
Figure 4.6: Theeffectofdifferingactivation-contractionconstraintsonthedistributionofmax(|˙ a
i
|),
compared across muscles When we sample trajectories, we get a bunch of n-dimensional trajectories,
where n=7 muscles. From each of those trajectories, we differentiate them (e.g. ˙ a
i
= a
LUM
i+1
a
LUM
i
),
and we show here the distributions of e.g. ˙ a
LUM
. These speeds are grouped by the applied activation-
contraction constraint. The case with no activation-contraction constraints is a 1.0; a 0.1 means a muscle
isspatiotemporallyconstrainedsothatitcannotchangebymorethan10%within50ms.
85
techniques for visualizing these spaces, including density distributions and parallel coordinates as used in
[30]couldbemisleadingontherawactivations,whenincorporatingtheconceptoftime.
PC1
PC2
PCs computed from unseeded
trajectories sampled from t=0
t=250ms t=300ms t=200ms
t=50ms t=0ms t=100ms t=150ms
R Index Finger
10N palmar force
30°
Figure 4.7: Spatiotemporal tunnels for each of 10 seed points The ‘seed’ activation you choose in the
firstmomenthighlyconstrainswhereyourmuscleactivationscangoacrossthefollowingsixtasks. Shown
for a activation-contraction constraint of 0.12 (in that no muscle can change more than 12% in tension
from slice to slice). Each slice of the tunnel is a task, where the points have been projected onto the un-
seeded PCs (PC1 and PC2), which were computed separated for each slice, providing polytope-relevant
changesinthedistributionsofseededdistributionswithrespecttotheunseededtrajectorydistribution.
Finally, the hypothesis illustrated conceptually in Figure 4.1 is highly supported by data in Figure
4.7. We show how, for ten randomly selected seed points, the activation-contraction constraints shown
in Figure 4.2 limit the evolution of muscle activations over time to produce a force and cyclically change
86
its direction. To create an adequate visualization, we had to find a method to fairly represent, project,
and render the 49-dimensional space of trajectories onto a page as a 2D representation. We achieved this
objective through a series of relative PCA mappings: we take many trajectories, split them into individual
moments, compute the first two principal components, then project those points onto PC1 and PC2. The
result is a view of the feasible activation space as a distribution landscape. From there, we make use of
this visualization by projecting seeded trajectories onto those same PC’s (See Figure 4.4), all culminating
inavisiblewindowintotheeffectofaddingactivation-contractionconstraints,Figure4.7. These‘tunnels’
representarepresentationoftime-constrainedactivationsovertime,givenastartingpointint=0. Asingle
seedpointdefineswheretheactivationmustmove,highlylimitingthespaceoffeasibleactivationpatterns
thatcanbeusedtoachievetherestofthetask;aspatiotemporaltunnelexists.
4.5 Discussion
Activation-contractionconstraintshighlyaffecttheneuromuscularcontrollandscapeuponwhichalllearn-
ing, motor control, and evolution must operate. This work has strong implications to many elements of
control systems where a null space exists for a relatively static task. One such example is the positioning
of the tongue in production of varying vowel sounds, where the 10 muscles can yield multiple motor
patternswhich”resultinsimilartongueposition,shape,and/orcontactswiththepalate”[49]. Inextension
tochangingvowelsoverashorttimespan,aperspectivebornefromspatiotemporalfeasibilityimpliesthat
speaking a single word requires finding, selecting, and repeating feasible activation trajectories so as to
be intelligible. Even a few millimeters of variance in tongue and jaw position can yield distinguishably-
different vowels[162]. With production of sound being highly dependent on anatomy at the species and
within-population level [128], neuromuscular strategies for manipulating the characteristics of sound de-
fine the ”universal inventory of phonemic contrasts available for use in language”[120]. This example
servesasjustonenewperspectivegeneratedbyourresearch.
87
Variability across all valid solutions does not necessarily decrease in light of activation-contraction
constraints becoming more strict. The utility of a muscle has been described many times as a description
of the bounds within which that muscle can be used [113, 30, 145, 62]. Our work highlights how those
boundsaretoooptimisticaboundfortime-varyingevaluationsofspatiotemporalfeasibility,eveninavery
simplisticforceredirectiontask.
Many experimental studies suggest that a pair of muscles can be highly correlated with one another,
and these techniques have been leveraged to build lower dimensional representations of motor patterns,
including with our work [30]. Looking at this simple force-redirection action, and the strong spatiotem-
poral effects, it’s clear that any synergies or motor primitives would also be subject to similar constraints
at a higher level. Control of these spaces with lower dimensional approximations, if learned or evolved,
could be able to leverage the consistency of activations across small tasks, suggesting that planning the
activation for the last set of the movement in a feed-forward manner represents a selection from a greatly
reducedfeasibleactivationspace,asperFigure4.7. Asthesemusclesdon’thavetochangeverymuch,this
alludes to there being solutions that could be less tiresome to implement and repeat—solutions that could
be preferentially turned into motor habits. We see the effects upon the control strategies that are possible,
andastheseconstraintstwistandchangeoverthecourseofontogeny,weenlightensomeelementsofhow
motor control is intertwined between the physics and anatomy we inherit, and the muscles and control
strategieswebuildandlearn.
The nature of activation-contraction constraints have implications on our interpretations of sensori-
motor manifolds and the latent representation that animals have for neuromuscular control spaces[77].
Random sampling of the raw 7-dimensional activation space is statistically untenable, and when poised
with a real task of force redirection, sampling a point on the feasible trajectory space is impossible. If an
individual has variability due to hierarchical sensorimotor control, it is possible to apply spatiotemporal
feasibilitymodellingtounderstandhowthosetremorousactivationsaffectthesetofsolutionsforachieving
a given redirection or scaling task. Furthermore, examining motor noise in this context creates a smaller
88
subset of possible actions that the CNS can perform, so we can address some of the mechanisms on the
physicalendpoint. Thisworkishighlycompatiblewithsensoryrepresentationsaswell[121,126].
Ultimately,theseneuromuscularlandscapeshaveexistedthroughouthistory;thisworkisconduciveto
comparing posture-specific spatiotemporal structures, as well as cross-species comparisons of spatiotem-
poral tunneling in the context of evolutionary biology, for example, in comparing the index finger manip-
ulability between humans and bonobos[124]. With the wealth of cadaveric, computational, and in-vivo
studies,thereisawidevarietyoffuturecomparisonstosupportongoingresearchintomuscleredundancy
[122,94].
4.6 SupplementaryInformation
These methods were (computationally) performant within 49 dimensional trajectory space (taking c. 40
minutesper100,000sampledpoints),butwearehesitanttomakeclaimsforthismethodsupportinghigher
dimensionality. Evaluationoftheseclaimsinamodelwithmanymoremusclese.g. n=31musclesinacat
hindlimbmodel[113,140]isaviablenextstep. Primarilythelongestoperationistheremovalofredundant
constraints, and the selection of the first hit-and-run point, and although we have made multiple steps
toward parallelizing and memoising some of these operations across multiple runs, some complexities of
spatiotemporalsamplingmaybeintractableattheircore,requiringinventionofnewmethods.
We describe how activation distributions (undifferentiated) change with respect to their variance in
Figure4.9, highlightinghowthevarianceintheby-muscle, by-momentdistributionsareaffectedbytime,
and how that effect is permuted by activation-contraction constraint. Intensifying activation-contraction
constraintfrom1to0.5forEIPmeantbothareductioninthegeneralvariance,butovertimethelinedips
deeper under activation-contraction constraint up to 0.25, but then becomes more shallow and consistent
as constraints move toward 0.05. The same trend is observed with EDC and LUM. The variance of the
muscle activation space for FDP, FDS, DI and PI are much more consistent across differing activation-
contraction constraint. This suggest that when we observe the structure of the feasible activation space
89
under activation-contraction constraint, those constraints may not change the distribution or span of a
givenby-muscleutilitydistribution.
WewillmakecodeforcomputingspatiotemporalfeasibilitysamplingavailableunderanMITLicense
athttps://github.com/bc/feasibilitytheory.
90
FDP FDS EIP EDC LUM DI PI
1 0.75 0.5 0.25 0.1 0.05
0 300
0.00
0.25
0.50
0.75
1.00
Time (ms)
Muscle Activation
Spatiotemporal
constraint
1
0.75
0.5
0.25
0.1
0.05
Figure 4.8: Activation distributions under differing activation-contraction constraints Taking all of
thepointscollected,wegroupthembymuscle,task,andbythespatiotemporalconstraintunderwhichthey
were collected. You can see each color represents a different spatiotemporal constraint, and the boxplots
representthewayeachmusclewasused,atthattaskindex. Alltrajectoriessampledareunseeded.
91
FDP FDS EIP EDC LUM DI PI
0.00
t=0
Time (ms)
Muscle
300
0.02
0.04
0.06
0.08
Variance of the FAS
Spatiotemporal
Constraint
(no const.)
(strict const.)
1
0.75
0.5
0.25
0.1
0.05
Figure 4.9: Supplemental Figure: Variance of a across trajectories (within a given muscle) does
not necessarily go down as the feasible activation space is under more strict activation-contraction
constraint Given the velocity constraints, we extract a long series of activations for each muscle, at
each task index. Per muscle, we computed the variance of each series, creating a visualization of the
feasible activation space as the task is performed, and across differing activation-contraction constraint
(and the degenerate case). dimensions are barely affected by either the change in the task, nor by the
activation-contraction constraint. variance, and also had a bigger effect in their variance shrinking un-
der more activation-contraction constraint. Temporal constraint led to reduction in variance across those
muscles,indicatingthatthedistributionacrossthemusclemaybecomemoreuniformlydistributed
92
Amount of points
sampled from the polytope
Activation
Muscle
10 Different Seed IDs
0
Time (ms)
100ms
200ms
300ms
Figure 4.10: Interactive seed trajectory explorer In comparing unseeded distributions with seeded dis-
tributions, we designed an interactive data exploration supplement to highlight how different the seeded
trajectories could be, and how they were often highly constrained by their activation in t=0. Bottom: we
provided a slider so the user could change the seeds, and see how the distributions compared with the
unseededdistribution(whichremainedconstantacrossallseeds,forthisgivenredirectiontask). Linesare
drawnbyconnectingthemidpointsof100histogrambins.
Figure4.11: Spreadofdifferentseedpointsundervaryingactivationcontractionconstraints
93
4.7 Conclusionandongoingwork
Forvertebratelimbswithmanymoremusclesthandegreesoffreedom,withbillionsofneuralconnections,
and with many ways to solve a given force, movement, or manipulation task, the question remains: how
does the nervous system solve these highly under-constrained problems, and why do they solve it that
way? It’s remained a challenge to understand the root of the variance—even the best models are not
always right. With this work’s alternative approach to motor control, rather than observing how muscles
work across a task, we define why muscles must work a certain way. The work presented herein serves
as a strong foundation to build new neuromuscular control studies. Work from Cohn et. al. 2018 has a
place in contextualizing studies of motor control, [117, 111, 103, 37, 112] and this work led to a good
application of feasibility theory at the intersection of tendon-driven control and task-based optimization
[84]. The whole of this work is focused on a simple isometric task, as it is a stable system, and building
a strong framework requires strong assumptions as a foundation—this focus on isometric of course raises
questions about posture-dependent control strategies, and the implications for movement. Work from [6]
suggeststhataseriesofstatictaskshaveasimilaranalyticalsolutionwithrespecttodynamicalsolutions–
and although these methods could be applied directly as a series of static tasks (with a changing H matrix
going down the diagonal of the constraint matrix), there lie many opportunities in expanding dynamical
system representation into a linear subspace that can be sampled fairly. By working our way from the
task, through the bones and joints of the hand, and finally into the requirements on the tendon tensions,
we clearly render a new view of the solution set the nervous system (brain and spinal cord) has to pick
from. Indoingso,weprovidedeepcontextintothechoiceofneuromuscularcontrol. Ultimately,thiswork
highlights the way animals control their muscles for simple tasks, across both timescales of evolutionary-
derivationandlifelong-learning.
94
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Abstract (if available)
Abstract
Feasibility Theory is a conceptual and computational approach to understand the dimensionality of how tendon-driven limbs are controlled. How do the brains of animals control their bodies? This remains one of the deepest mysteries in biology, a concept with an enormous consequence upon how we understand, diagnose, and treat diseases or injuries that rob animals of manipulation and locomotion. Engineers and scientists have tackled this problem from rigorous mathematical and scientific perspectives and much progress has been made—primarily descriptive approaches that attempt to predict how the nervous system solves a motor task (e.g. modeling observed behavior based on recordings of muscle activity). However, this rigor has a downside: the efficient ways we know how to solve problems mathematically (via formal optimization) can surreptitiously bias the scientific community into hypothesizing that the brain also solves problems this way. These methods describe how the muscles function, but they do not describe why certain patterns of control are evolved (over millennia), learned (over a lifetime), or chosen (within just one motor task). The work presented herein delves into the problem from a full-dimensional perspective of motor control—requiring a truly Big Data exploration into, first, a view of the feasible options for control, and second, a set of constraints which faithfully describe the ways in which tendon-driven limbs truly must be controlled. I present new mathematically formal ways to show that many muscles are indeed needed even for simple tasks, and indicate how even the most ‘optimal’ solutions are highly prone to disruption, even with minor disability of just one muscle. I propose new computational methods for reconciling alternative approaches to motor control, including techniques in dimensionality reduction, Bayesian representation, and optimization. Ultimately, this work now enables a new perspective towards exploring how motor control affects health and quality of life.
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Asset Metadata
Creator
Cohn, Brian Alexander
(author)
Core Title
Feasibility theory
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Computer Science
Publication Date
03/16/2021
Defense Date
05/04/2020
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English
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Valero-Cuevas, Francisco J. (
committee chair
), Culbertson, Heather (
committee member
), Nikolaidis, Stefanos (
committee member
)
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bc@alumni.usc.edu,brianaco@usc.edu
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Tags
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muscles, manipulation