MOVEMENT DISORDERS Emmanuel Guigon Institut des Systèmes Intelligents et de Robotique Université Pierre et Marie Curie CNRS / UMR 7222 Paris, France

[email protected] e.guigon.free.fr/teaching.html

2

2. Methods & advanced data processing multidimensional data analysis, time series analysis

DATA ANALYSIS • Summary statistics — comparison of means — analysis of variance — regression

• But — structure of variability — nonstationarity — multidimensionality

multidimensional data analysis / time series analysis

MULTIDIMENSIONAL DATA ANALYSIS Principal component analysis from muscle activations to isometric joint torque flexion/extension task EMG data — 8 muscles, 6.2 s, 1 kHz

dim(ek)=8 dim(ck)=6200

98% of variance explained by 2 PCs

EXT FLE

muscle 2

8 muscles

6200 samples

X

covariance matrix V = X’X ek / Vek=λkek (k=1…8) eigenvectors of V principal components ck = Xek (k=1…8)

muscle 1 a 1c 1 + b 1

a 2c 2 + b 2

— Kutch and Buchanan, 2001, Neurosci Lett 311:97

Lee and Seung (1999)

MULTIDIMENSIONAL DATA ANALYSIS

Lee (1999) and Seung Lee and Seung

Lee and Seung (1999) Lee and Seung (1999) V⇡ WH

(1999)

V ⇡ WH

V ⇡ WH

1

N = number of synergies

i=1

m(t) =

D = number of muscles

ti ) ci wi (t

N X

Synergies (time-varying) d’Avella et al. (2003) 1

N = number of synergies

pixels cnumber tof i wi (t i) i=1 number of images number of basis images D = number of muscles n m r

m(t) =

N X

r = number of basis images = 5 n number of pixels m number of images V r image database n ⇥ m) number of basis(dim images W basis of images (dim n ⇥ r) H encoded images (dim r ⇥ m)

m = number of images

Synergies (time-varying) d’Avella et al. (2003)

n numbe

N X

c

image databa basis of imag encoded imag

H

V W H

n m r

m(t) =

i=1

D = number

N = number

1

Synergies (time-varying) d’Av

encoded images

(dim r ⇥ m)

image (dim n ⇥ m) V ⇡ WHdatabase V ⇡ WH VSeung image database (dim n ⇥ m) Lee and (1999) Nonnegative matrix factorization algorithm V image database (dim n ⇥ m) V image database (dim n ⇥ m) m = number of images dimensionality reduction W, H Lee and Seung (1999) V ofimage (dim n ⇥ m) VInitialise ⇡ WH m = number images database Lee and Seung (1999) m = number of images Iterate Seung (1999) m = number of images X Viµ r = ofnumber of=basis images = 5 r = number basis images 5 V ⇡ WH V ⇡mWH Wia Wia(dim n ⇥ m) Haµ = number of images V image database r = number of basis imagesµ=(W 5 H)iµ V ⇡ WH r = number of basis images = 5 V image database (dim n ⇥ m) Wia P V image database (dim n ⇥ m) W ia W V basis of image images (dim n(dim ⇥ r) n ⇥(dim image database m) n ⇥=m) database m = number of images r = number of basis images 5 j Wja V imageHdatabase (dim n ⇥(dim m)r ⇥ m) encoded images X n ⇥ m) Viµ V image database (dim W basis of images (dim n ⇥ r) Haµ Haµ Wia (W H)iµ V (dimimage database (dim n⇥ = numberbasis m= numberimages of images i r of = images number of m) basis images 5r) W of images (dim n= ⇥ H encoded r ⇥m m) m = numberVof images W, H W basis of images (dim nUntil ⇥ r)stable(dim image database (dim n ⇥ m) H encoded images r ⇥ m) n number of pixels original image = number H = 5 rencoded (dim r = ⇥ 5m) mr and number of images ofimages images (dim n images ⇥image r)of basis images encoded Lee Seung (1999) =W number ofbasis basis number of basis r = numberr of basis =images 5 images V image database (dim n ⇥ m) H images encoded (dim r ⇥ m) V ⇡ WH W basis of images (dim n ⇥ r) Lee andnSeung (1999) number of pixels V n ⇥ m)image database (dim net ⇥ al. m) (2003) H encoded images (dim r ⇥ m) V image database (dim d’Avella m d’Avella(dim number of images n number of pixels Synergiesdatabase (time-varying) et al. (2003) V image n ⇥ m) V ⇡ WH W (dim n ⇥ r) image database (dim nbasis ⇥ m)of images W basisVof images (dim n ⇥ r) N n number of pixels X N r number images m number of images X W basis of images (dim n ⇥ of r) basis H r ⇥ m) encoded images (dim r ⇥ m(t) m) = ci wi (t ti ) m(t) = ci w ti ) H encoded images (dim i (t n number of pixels m number of images i=1 r number of basis images i=1 H encoded images (dim ⇥ m) of V image (dim n ⇥ m) mdatabase =rnumber images m of muscles number of images r number of—basis images D = number Lee and Seung, 1999, Nature 401:788 D = number of muscles n number of pixels r number of basis images N = number of synergies m number of images rm = number of of basis images = 5 = number images V

image database basis of images n encoded images m r V W H

V

c w (t

m(t) =

n m r

m r

N X

ci wi (t ti )

number of im number of basis im

m(t) =

i=1

D = number of muscles

N = number of synergies

1

Synergies (time-varying) d’Avella et al. (20

Lee and Seung (1999)

Lee and Seung (1999)

MULTIDIMENSIONAL DATA ANALYSIS

N X

number of pixels number of images Synergies (time-varying) d’Avella e number of basis images

(dim n ⇥ m (dim n ⇥ r num (dim r ⇥ m numb number of

W basis of images r = number imagesimages =5 H of basis encoded

m = number ofimage images V database

r = number of basis i image database (dim n ⇥ m)

m = number of i V ⇡ WH

V image database

V ⇡ WH

(

Lee and Seung (1 V image database (d W basis of images (dim n ⇥ r) m= W basis of images V image database m)⇡( H encoded images (dim r ⇥ m)(dim n ⇥ V V image datab H of images encoded images (d W basis (dim n ⇥ r) Lee and Seung (1999) r of = ima num Lee andWSeung (1999) basis V image databa H encoded images (dim r ⇥Vm)im H encoded ima V V⇡ n number of pixels nonnegative Vm = numbe imag matrix number of image da m numbern of imagesV V W basi image databa factorization m images numberofofb r= nnumber of basis number of number pixels r renco = H nnumber r ofmbasis m number of images = nu m = numbe m images r number of basis parts-based V r image datab numb V im r = number representation m = 2, 429 r = 7 ⇥ 7 n = 19 W ⇥ 19 r = basis number of b of ima W b vector m = 2, 429 r = 7 ⇥ 7 n = n H encoded ima quantization e m = 2, 429 r = 7 ⇥ 7 n = 19H ⇥ m 19 V image da Synergies (time-varying) d’Avella et al. (2003) V image r datab W d’Avellabasis of( Synergies (time-varying) d’A Synergies (time-varying) et ima al. W basis of N X encoded Synergies (time-varying) d’Avella etHal. (2003) m(t) = c w (t t ) H encoded ima i i i N n X N prototype X i=1 n= ti ) representation N m(t)m m(t) = c w (t i i X Lee and Seung (1999) i=1 = ci wi (tr ti=1 mnumb i) D = numberm(t) of muscles principal dim Vi=1 = n ⇥ mSynergies = 876869 (time-vary component r D of = muscle numbe D = number analysis n 1D = number of muscles n dim V n⇥m m( m876869 1 m N17689 = numbe dim W 1n ⇥ r r n eigenimage r numb Synergies dim H r ⇥ m (time-varying) 119021 d’Av representation D= Synergies (time-v N X

THE COORDINATION PROBLEM Concepts — Is the CNS concerned with the control of each individual (muscle) or does the CNS simplify control by combining (muscles) into groups and controlling each group, rather than the individual (muscle)? — synergies, structural units, coordinative structures, … The controller organizes relations among elements at a hierarchically lower level, and these relations assure stability of motor performance with respect to a particular motor task

dimensionality reduction 4 dof

3 dof

— Latash et al., 2002, Exerc Sports Sci Rev 30:26 — Macpherson, 1991, in Motor Control: Concepts and Issues, Wiley

SYNERGIES • Definition — “acting together” — common source or coincidence — muscles, joints, multi-segmental / whole-body movements — anatomical vs functional e.g muscle synergy — a set of muscles which act together to produce a desired effect

• Narrow to broad meaning — agonist / which acts with the prime mover / synonymous of motor coordination / patterns of activity that occur at the same time e.g. locomotion (Bernstein) — an extremely widespread synergy incorporating the whole musculature and the entire moving skeleton and bringing into play a large number of areas and conductions pathways of the central nervous system

POSTURAL SYNERGIES Theory — Nashner’s «fixed» postural synergies in subjects standing on a moving perturbation platform

Postural synergies are activated in a mutually exclusive fashion — each muscle was activated by only one postural synergy

Flexibility in patterns of muscle activations (amplitude, prior experience) e.g. ankle/hip strategies

— Nashner, 1977, Exp Brain Res 30:13 — Horak & Macpherson, 1996, in Handbook of Physiology, OUP

SYNERGY AS COORDINATION Rejection and new concept — rejection of the notion of synergy: too constraining and inflexible for the production of natural movements — new concept: more than one muscle synergy can be activated during a postural response and each muscle can also be activated by more than one synergy. By varying the magnitude of the neural command signals to just a few muscle synergies, many different muscle activation patterns can be generated descriptor that indicates some form of motor coordination — no insight into the underlying mechanism of coordination

— Macpherson, 1991, in Motor Control: Concepts and Issues, Wiley — Ting & Macpherson, 2005, J Neurophysiol 93:609-613

MOTOR PRIMITIVES • Compositional elements for movement construction • Kinematic primitives segmentation/decomposition

• Muscle synergies — based on EMG signals — coordinated recruitment of a group of muscles with specific activation balances or specific activation waveforms

MUSCLE SYNERGIES • Principle dimensionality reduction — mapping simplified by a lowdimensional representation of the motor output: if all useful muscle patterns can be constructed by the combination of a small number of elements, selecting the appropriate muscle pattern for a goal requires only determining how these elements are combined

• Organisation — spatial, time-independent synergies — time-dependent, time-invariant synergies — time-varying synergies

• Identification multidimensional data analysis

MUSCLE SYNERGY IDENTIFICATION • In frogs, cats, humans — evidence that the CNS flexibly combines fixed muscle synergies for generating the muscle patterns necessary to perform many motor tasks and behaviors — a variety of muscle patterns used in different behaviors are generated by the combination of a small number of time-invariant and time-varying synergies

• Methods — principal-component analysis (PCA), nonnegative matrix factorization (NMF) — criterion: variance accounted for (VAF) of the experimental EMG after synergy decomposition

⇡

=

=

N X

cij g(wi )

i=1

3 mmobs j

obs

xt

(t = 1...N )

w t i X xi t Xt = (x cij i=1

obs dim m j j= 1=...MP

xt

v u N u1 X F (n) = t (Xt N t=1 log n

slope ↵ log n log F (n)

i=1

Ytn

v u N u1 X F (n) = t (Xt N t=1

hxi)

Ytn

hxi)

t

Xt =

i=1

(xi

(xi

Ytn )2

t

Xt =

t X

(t = 1...N )

j = 1 ... P

(t = 1...N )

Ytn )2

slope ↵

↵ < 1/2 anticorrelated

dim wi = M Synergies N = number of synergies

observed EMG response t

PCA

muscle synergy dim wi = M DFA (t = 1...N ) t hxi) weights t X xt (t = 1...N ) t Ytn v Xt —=close(xto hxi) method matrix factorization i nonnegative DFA u N t u1 X X t ni=1 2 DFA F (n) = (Xt Yt ) N

N = number of synergies

observed EMG response Mdim mobs =M = number of muscles j

Nsynergy muscle i P = number of observedw patterns =5 X pred mobs ⇡ m = cij g(wi ) j j j = 1 ... P i=1

mobs j

cij g(wi )

xt

DFA

⇡

mpred j

i=1

obs 3

DFA

j = 1 ... P P = number of observed patterns = 5log F (n) Synergies

mobs j

mobs 3

N X

cij g(wi )

P = number of observed patterns = 5

M = number of muscles

mobs j

⇡

N = number of synergies

mobs 3

mpred j

=

i=1

N X

evoked from cutaneous stimulation pred obs m ⇡ m = M = number of muscles j j of the spinalized frog hindlimb

P = number of observed patterns = 5

cij g(wi )

Synergies P = number of observedpatterns patterns = 5 Muscle activation

N X i=1

i=1

obs 3

Synergies

FA

cij g(wi )

mpred j

m TIME-INVARIANT SYNERGIES m

ynergies

DFA

=

mobs j

t X

mobs ⇡ j

mpred j

N X

1 In n N D = Diag{p1 , ..., pn } X D=

pred obs m ⇡ tj 1 m 2j g = (¯ x , x¯ , ..., x¯p= )

g = Xt D1

Y=X

cij g(wi )

i=1

t obs 1gm 3

— Tresch et al., 1999, Nat Neurosci 2:162 1

P = number of observed patterns = 5 M = number of muscles

TIME-INVARIANT SYNERGIES N = number of synergies

N = number of synergies discovered = 4

j = 1 ... P

mobs j

observed EMG response wi

muscle synergy

dim mobs =M j

dim wi = M

mobs j

observed EMG response

wi cij

muscle synergy weights

dim mobs =M j dim wi = M

DFA xt

(t = 1...N )

t

1

— Tresch et al., 1999, Nat Neurosci 2:162

contribution of each muscle synergy to responses evoked from different regions of the skin surface for three different animals (a, b, c)

D = number of muscles

TIME-VARYING SYNERGIES

N = number of synergies Synergies (time-varying) d’Avella et al. (2003)

N d’Avella (time-varying) Synergies d’Avella Synergies (time-varying) et al. (2003) (time-varying) d’Avella et al. (2003) et(2003) al. (2003) Synergies (time-varying) d’Avella et al. Synergies (time-varying) d’Avella et al. (2003) N N N i i i X X X N X N m(t) = ci wi (t tim(t) ) m(t) = m(t) ci= wiw (t (tcitw X i )it(t) ti ) = Synergies i=1 i (2003) i (time-varying) i Synergies (time-varying) d’Avella etcal. d’Avella m(t) = c wet(tal. (2003) t)

Xobserved patterns = 5 P = number of m(t) = c w (t t )

Muscle activation dim m(t) = dim wi (t) = Dpatterns D = number of muscles D = number D = number of muscles of muscles i=1

i=1 i=1

N X

i=1

i

Ni=1 X

i

i

number m(t) c w (t oft muscles ) of muscles m(t) = c w (t t ) D D===number D = number of muscles during kicking, jumping and N = number of synergies N = number N = number of synergies of synergies Dw =N of muscles D =synergy number of muscles of synergies (t)= number time varying walking in unrestrained frogs inumber N = number of synergies i

i

i

i=1

i

i

i

i=1

Nnumber == number of et synergies P = number (time-varying) of observed PN =patterns P of5d’Avella observed of observed patterns patterns 5 (2003) = 5 of synergies ynergies al. number of observed synergies N== number ti =number synergy onset P ==number of patterns =delay 5

P =of number of observed patterns = 5 temporal patterns muscle activation

dim m(t) = dim = Dof m(t) dim =N m(t) dimpatterns w=i (t) dim =wDinumber (t) = D of observed patterns = 5 i (t) dim P =wnumber observed dim m(t) = dim wiP (t)== = 5D dim m(t) = dim wi (t) = D

X P = number of observed patterns =5 m(t) = c w (t t ) i i i wi (t) time varying synergy wm(t) w time synergy synergy i (t) = i (t)varying dim dim wtime (t) =varying D dim m(t) = dim w (t) = D wi (t) time varying synergy N number wi (t) of timesynergies varying synergy i=1 ti synergy onset delay ti synergy ti synergy onset delay onset delay ti synergy onset delay ti time synergy onsetsynergy delay (t) time varying synergy w (t) varying D wdim number of muscles m(t) = dim w (t) = D i Dt =synergy number of muscles onset delay t synergy onset delay N number N of synergies N number number of synergies of J number ofsynergies time samples Synergies (fixed) Tresch et al. (1999) N N number of synergies D number D ofDmuscles numbernumber of muscles of muscles number of synergies wiJof(t) varying synergy number of muscles DtimeJsamples number of time muscles N number of synergies NDof of synergies J number numbernumber of time time samples samplesnumber N number of synergies Jvarying number of time samples Ji (t)=wsynergy number ofmuscles time samples N D w number oftime D number of muscles wi (t) time varying (t) time varying synergy synergy i X tJ i w (t) synergy onset delay pred timeofvarying synergy obsti number oftime timevarying samples J wi (t)delay number time samples synergy ti synergy tm delay synergy synergy onset onset delay i i onset ⇡ m = c g(w ) ij i synergy onset delay w (t) ti j timejvarying synergy wti(t) varying synergy synergy onset delay time i=1 t number ofsynergy onset t delay P = observed patternssynergy = 5 onset delay 1 J t ⌧j j 0 T 1 maxJ t1 ⌧Jj jt ⌧0j Tjmax0 Tmax 1 tJobs ⌧j t 1⌧jJ t ⌧j j 0 Tmax 8 8 1 J8m 1 0J t < ⌧ 0j < 0⌧j0< T3 0 ⌧j < 10 J t ⌧ j 0 T ⌧j < < 0 8 i i ⌧j < 0 Wji 0  w ⌧ji (⌧