Threshold control of motor actions prevents destabilizing effects of proprioceptive delays

RESEARCH ARTICLE

Jean-Francois Pilon Æ Anatol G. Feldman

Threshold control of motor actions prevents destabilizing effectsof proprioceptive delays

Received: 22 November 2005 / Accepted: 10 March 2006 / Published online: 5 May 2006� Springer-Verlag 2006

Abstract It is usually assumed that proprioceptive feed-back comes to motoneurons too late to contribute to theinitial activity of agonist muscles during fast armmovements, leading to the suggestion that this feedbackis only efficient in slow movements and postural control.The argument does not take into account that thechanges in the motoneuronal membrane potentials andthe associated changes in the state of spinal neuronspreceding the initial activity of muscles deeply affect, in aforward way, the state of reflex systems by shifting theirthresholds, as suggested in the k model for motor con-trol. As a result, the initial muscle activity emerges withfull contribution of these systems so that the effects ofreflex delays become negligible. We tested the hypothesisthat threshold control of muscle activation may beinstrumental in preventing destabilizing effects of pro-prioceptive delays in spinal and trans-cortical pathwaysto motoneurons. The analysis was made by recordingfast elbow movements (peak velocity �300–500�/s) andsimulating them in a dynamic model that incorporatesthe notion of threshold control of intrinsic and reflexmuscle properties. The model was robust in reproducingexperimental movement patterns (R2>0.95). It gener-ated stable output despite substantial proprioceptive (upto 100 ms) and electromechanical (40 ms) delays. Sta-bility was thus ensured for delays not only in segmental(about 25–50 ms) but also in trans-cortical loops(50–70 ms). Our study illustrates that a natural physio-logical process—threshold control—may manifest feed-forward properties hitherto attributed to hypotheticalinternal neural models.

Keywords Sensorimotor integration Æ Thresholdcontrol Æ Stability Æ Equilibrium-point hypothesis ÆFeed-forward processes

Introduction

Static arm positions are well stabilized so that externalforces deflecting the limb from the desired position aremet with resistance resulting from proprioceptive andintrinsic muscle position- and velocity-dependent prop-erties. The stability requirements increase for fast armmovements when the speed of angular rotations of limbsegments may reach 400–600�/s so that the kinetic en-ergy of the arm becomes substantial. Comparativelystrong position- and velocity-dependent feedback isnecessary to dissipate this energy and thus prevent long-lasting terminal oscillations. The capacity of this feed-back to stabilize posture and movement is diminisheddue to delay in proprioceptive feedback to motoneurons.Although rarely mentioned, additional aspects of sta-bility should be emphasized. Each time, only one armposition is stabilized, implying the existence of a vari-able(s) that reflects a decision-making process thatidentifies which arm position is stabilized. The electro-myographic (EMG) activity does not belong to thevariable(s) that reflect(s) the choice of a position forstabilization since the EMG magnitude may be the sameat different positions of the arm when external forces areneutralized (Weeks et al. 1996; Ostry and Feldman2003). Feldman and Latash (2005) found that Fig. 2 inHinder and Milner (2003) shows that the tonic EMGactivity of wrist muscles is the same at different wristpositions. Similarly, variables that depend on EMGactivity such as muscle torques, forces, or stiffness donot determine the position for stabilization (Feldmanand Levin 1995).

The question of which variables reflect the choice ofthe position for stabilization is directly related to theclassic, posture–movement problem in motor controlformulated by Von Holst and Mittelstaedt (1950/1973).

J.-F. Pilon Æ A. G. Feldman (&)Institute of Biomedical Engineering, University of Montreal andRehabilitation Institute of Montreal (CRIR), 6300 DarlingtonAvenue, Montreal, QC, Canada, H3S 2J4E-mail: [email protected].: +1-514-3402085Fax: +1-514-3402154

A. G. FeldmanDepartment of Physiology, Neurological Science Research Center,University of Montreal, Montreal, QC, Canada

Exp Brain Res (2006) 174: 229–239DOI 10.1007/s00221-006-0445-3

They argued that when the body or its segments areintentionally moved from one position to another posi-tion, the posture-stabilizing mechanisms should be redi-rected to the latter position (see also Ostry and Feldman2003). The redirection (resetting) of the stabilizingmechanisms is necessary to prevent the possibility thatmuscle resistance is produced in response to the activemotion away from the initial, formerly stable position.

Physiological variables and mechanisms underlying asolution to the posture–movement problem were unclearuntil animal and human studies (Matthews 1959;Asatryan and Feldman 1965; Feldman and Orlovsky1972; Nichols and Steeves 1986) have shown that centralcontrol levels are able to change a threshold position ofthe arm, at which the proprioceptive feedback becomeseffective in activating appropriate muscles and that theresetting of the posture-stabilizing mechanisms isachieved by shifting the threshold position. Thresholdcontrol offers a solution to the posture–movementproblem. Specifically, by shifting muscle activationthresholds, the system resets the posture-stabilizingmechanisms to a new arm position. The previous posi-tion appears as a deviation from the newly specified one,and the same posture-stabilizing mechanisms generateEMG activity and forces tending to move the joint to thenew position. Thus, the system not only eliminatesresistance to movement from the previous posture buttakes advantage of the posture-stabilizing mechanismsto move to the new posture.

The resetting of posture-stabilizing mechanisms resultsfrom changes in membrane potentials of motoneuronsproduced by central control influences transmittedby descending pathways directly to a-motoneurons orvia c-motoneurons and spinal interneurons of reflexloops (Feldman 1986). In the presence of these loops, thechange in the membrane potentials appears as a changein the position (muscle length or joint angle) at whichmotoneurons begin being recruited (Feldman and Levin1995). Changes in the membrane potentials usuallyprecede the generation of motoneuronal spikes that formthe EMG bursts underlying motor actions in humansand animals (e.g., Kots and Zhukov 1973; Pierrot-Deseilligny et al. 1983; Berkinblit et al. 1980; Kozhinaet al. 1985). As a consequence, shifts in the thresholdposition resulting from changes in the membranepotentials are initiated prior to the onset of EMG activity,forces, and movement. The threshold arm position canbe considered as a virtual position that, when an inten-tional movement is produced, moves ahead of the actualarm position so that the EMG activity and muscle forcesemerge depending on the difference between these twopositions and movement velocity (Gribble and Ostry2000; Ghafouri and Feldman 2001; Archambault et al.2005). Threshold control has also an important, antici-patory aspect. In general, the term anticipation impliesthat the organism tries to prevent undesirable effects ofthe environment or its own actions (e.g., large excursionsof the body during fast arm lifting; Belen’kii et al. 1967).Because of the preventive aspect (elimination of resistance

to intentional movement), threshold control is func-tionally anticipatory.

Starting before the motor output and then guiding it,threshold control of intrinsic muscle and reflex proper-ties may represent a forward mechanism that is instru-mental in preventing the destabilizing effects ofproprioceptive delays in spinal and trans-cortical loops.Here we tested this hypothesis by recording elbowmovements and simulating them in a dynamic modelthat incorporated the notion of threshold control.

Methods

Healthy subjects (n=5; age 22–39) participated in theexperiment after signing informed consent forms ap-proved by the institutional Ethics Committee (CRIR).They sat on a height adjustable chair with a solid backsupport, placed the forearm on a light horizontal ma-nipulandum, and grasped its vertical handle. The axis ofrotation of the manipulandum was aligned with that ofthe elbow joint. In response to a sound signal, eachsubject made five discrete elbow flexion movements(about 50�–60�, peak velocity 300–500�/s) from an initialposition of about 150� (full elbow extension is 180�). Theangular elbow displacement and velocity were computedwithMatlab software based on the position of a reflectivemarker placed on the handle of the manipulandum andrecorded with the Vicon system (sampling rate 120/s).

The recorded elbow movements were compared withthose simulated in a dynamic model with feedback de-lays. The quality of simulations was estimated with agoodness-of-fit criterion based on the coefficient ofcorrelation, Rc

2, between experimental and simulatedcurves, with a level of significance P<0.01.

It has previously been shown that simulations of fastelbow movements in threshold-based models are equallyeffective whether or not the actual anatomical asymme-tries of opposing muscle groups are taken into account(St-Onge et al. 1997; Gribble et al. 1998). It seems un-likely that these asymmetries are essential in overcomingdestabilizing effects of feedback delays. Therefore, todiminish the complexity of equations of the model, weassume here that these groups act symmetrically. Theissue of asymmetries in the action of opposing musclegroups and, in general, the diversity of muscles in termsof function, biomechanics, and anatomy has been ad-dressed in the k model (e.g., St-Onge and Feldman 2004).

Muscle activation and forces underlying intentionalmovement emerge in response to two types of coopera-tive shifts in the thresholds of activation of elbow flexorand extensor groups of muscles: one specifies the posi-tion (R) at which both groups may be silent (a commonthreshold position) and the other specifies the spatialrange (2C) within which these muscles may be co-active(Fig. 1). It has been shown that such shifts are combinedduring rapid elbow movements (Feldman 1980). Thecooperative commands define the central components ofthe thresholds of activation of individual muscle groups

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(Fig. 1; k=kf=R�C for flexors; and ke=R+C for ex-tensors; to derive these formulas, we accepted that flexormuscles lengthen whereas extensor muscles shortenwhen the elbow angle (h) increases).

The equations of the model are listed in Table 1 andare integrated into the model in the order presented inFig. 2. Except for the last equation for the net elbowtorque T+T�+L (T is flexor, T�is extensor torque, L isload torque), all formulas in Table 1 refer to the flexorgroup of muscles. Similar formulas can be obtained forthe extensor muscle group by taking into account thatthese muscles shorten when h increases.

Symbol A in the first row of this table refers to themagnitude of muscle activation at time t. It depends onthe difference between the joint angle hd and dynamicthreshold k* at time t�d (where d is feedback delay); [u] +

is Heaviside step function: [u]+=u if u‡0 and 0 other-wise. The dynamic muscle activation threshold, k*, iscomprised of several components. The first component(k) is defined by the R andC commands (see above); xd isthe angular velocity at time t�d. Parameter l depends onthe dynamic sensitivity of muscle spindle afferents(Feldman and Levin 1995). Parameter q is a componentof threshold that characterizes reflex inter-muscularinteraction, including the reciprocal inhibition of moto-neurons mediated by Ia interneurons that receive pro-prioceptive influences from antagonist muscles. In thepresent simulations, q is proportional to activation A- ofthe antagonist muscle group, with a weight coefficientr=0.05. Pd(A) in the first row of the table represents thestatic torque that would eventually be established if thevalue of A defined by kinematic variables at time t�dwere maintained indefinitely. It resembles the experi-mentally defined invariant torque-angle characteristic(Feldman 1966; Gribble and Ostry 2000).

It is known that, in response to muscle activation, thetorque increases not instantly but gradually due to cal-

cium (Ca)-dependent processes (e.g., Desmedt 1983).The first 2 equations in row 2 of the table describe thisphenomenon. It implies that the muscle torque Md

evoked by tonic muscle activation at time t�d graduallyincreases from the initial, zero level to the level corre-sponding torque Pd that would eventually be achieved ifthe activation were maintained indefinitely. The valuesof time constants (s1=10–20, c=s2=40 ms) were cho-sen in order to simulate muscle responses not only tomaintained, tonic activation but also to phasic muscleactivation (‘‘twitch contractions’’). In addition, it isknown that, in fast movements, muscle contractions areinitiated by doublets or triplets of spikes following withvery short intervals (5 ms), so that the muscle torquerises much faster than in single-pulse activation (Burkeet al. 1976; Partridge and Benton 1981; Thomas et al.1999). To take into account this phenomenon in oursimulations of fast movements, we respectively choosecomparatively small values of s1 indicated above.

The equations for muscle torque M in row 2 take intoaccount that, even if the magnitude of muscle activationdetermined by the number and firing frequency of re-cruited motor units may remain the same during thedelay period, the muscle torque due to elastic propertiesof muscle fibers continues to change depending on thecurrent kinematics. During this period, the slope of theresulting torque-angle characteristic (Fig. 3, thin inclinedline) is considered to be proportional to the level ofactivation (A), with some coefficient j. Symbol N refersto the additional torque resulting from the deviation ofthe current joint angle (h) from the value hd that deter-mined the muscle activation. TorquesMd and N are thussummated to produce the total muscle torque, M.

The slope jA of the torque-angle characteristic of themuscle under constant activation cannot be greater thanthat of the invariant torque-angle characteristic (Fig. 3)since the shape of the latter, in contrast to that of theformer, results from a continuous position-dependentrecruitment of motor units. Based on this condition, wederived the inequality for j (first column, row 3)allowing us to choose values of j from the range(6.0–8.6)·10�3 N m deg�2.

The second equation in row 3 takes into account thatthe muscle torque depends not only on elastic, length-related resistance of muscle fibers (see above) but also onvelocity of change in their lengths as described by asigmoid force–velocity relationship for the contractileapparatus. For negative values of velocity (when themuscle contracts), it resembles the known equation ofHill (1938). When the velocity is positive so that themuscle is stretched, the muscle torque saturates at avalue usually exceeding the static torque value by factor1.1–1.5 (Dudley et al. 1990; Westing et al. 1990). Value1.3 was used in the model. The response of muscle fibersto changes in velocity is practically produced withoutdelay. In the present model, it relates muscle torque T toits value M depending on velocity x with coefficients band vm that define the shape and asymptotic character-istics of the sigmoid curve. The values of coefficients

R

2C

Joint angle

λf

λe

fe

Mus

cle

activ

atio

n

onoff

onoff

Fig. 1 Two central commands (R and C) cooperatively controllingextensor (e) and flexor (f) muscle groups by influencing centralcomponents (ke and kf) of thresholds of these groups. Horizontalbars show the angular zones in which the respective groups areactive (‘‘on’’). In the absence of the C command, the R commandspecifies a common threshold position for both muscle groups (boldvertical line). When present, the C command shifts the activationthresholds of these groups in opposite directions, thus surroundingthe R position with a zone (2C) in which these groups are coactive.Note that the localization of the co-activation zone in the jointspace changes with resetting of the R command

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were taken from the ranges of 80–100 and 600–700�/s,respectively. Newtonian equation of motion in row 3 ofthe table completes the system with T, T�, L and I being,respectively, the flexor torque, extensor torque, loadtorque and inertia of the system (including the forearm,hand, and manipulandum). The values of coefficientsthat were constant in all simulations are shown in row 4.

R and C commands were ramp-shaped (Fig. 4) withthe ramp speeds taken from the ranges of 300–500 and100–300�/s, respectively. By adjusting the rates and theduration of changes of R and C, we matched simulatedand experimental movements. It is known that co-acti-vation of opposing muscle groups gradually decreasesafter the end of movement (e.g., Lestienne 1979).Respectively, the C command was gradually decreasedwhereas the R command was maintained after the end ofmovement. In each simulation, l was constant with avalue taken from the range of 40–72 ms.

In addition to the ability to simulate fast movements,the model was tested in two ways. First, we tested whe-ther or not the model produces stable responses to briefforce pulses (50 N m, 10 ms) applied before or after theend of fast movements. Second, we tested whether or notthe model produces stable fast movements in the pres-ence of electromechanical delay (EMD) in addition todelays in the reflex pathways. The EMD results partly

from comparatively slow release and diffusion of Ca ionsto active sites of cross-bridges of muscle sarcomeres totrigger muscle contraction in response to EMG signals.EMD also includes the time needed for the contractilecomponent to stretch the muscle series-elastic compo-nent to start applying a force to the bone and externalobjects (Cavanagh and Komi 1979). The experimentalvalues of EMD in human arm muscles typically varywithin the range of 20–55 ms (Norman and Komi 1979;Cavanagh and Komi 1979). We took an initial EMDvalue of 40 ms that decayed to 10 ms with a time con-stant of 100 ms, implying that, after the onset of strongmuscle activation associated with fast movements, moreCa ions are available for the initiation of muscle con-traction thus diminishing the EMD.

Results

First, we tested how the model works in the absence ofboth proprioceptive (d=0) and electromechanical delays(EMD=0). In this case, there is no discrepancy betweenthe current joint angle (h) and the angle hd that deter-mines the muscle activation. As a consequence, thecomponent of the muscle torque (N in Fig. 3) resultingfrom delay and intrinsic muscle elasticity is also reduced

Table 1 Equations of the modelA=[hd�k*]+ k*=k-lÆxd+q; q=rÆA Pd(A)=a (exp(a A) – 1) (1)

Md +cÆdMd/dt =D{Pd} D=s1s2Æd2/dt2+(s1+s2)Æd/dt+1 M=Md(A)+N; N=jA (h�hd) (2)

j £ aaÆexp(1)Æ(1+r)

T ¼

0 ; vm � xM 1� x=vmð Þð Þ

1� x=bð Þ ; 0 � x\vm

M 1� 1:3x=b0ð Þð Þ1� x=b0ð Þ ; x\0

8>><

>>:

b0 ¼ 0:3 b vm

b� vm

T+T-+L=IÆdx/dt (3)

A=1.2 N m a=0.05 deg�1 I= 0.1 kgÆm2 (4)

G

Fig. 2 The causal chain of events in the generation of muscle activity, torque and movement resulting from threshold control. Theequations describing these events are shown in Table 1

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to zero. In response to a ramp-shaped R and C com-mands (Fig. 4a), such a reduced model produces stablekinematic output resembling natural fast movements atthe elbow joint, as has previously been shown (St-Ongeet al. 1997). When proprioceptive delay is present(d=30 ms in Fig. 4b and 60 ms in c), but the additionaldelay-related torque component N is still absent (whenj=0 in Fig. 4b), the model produces stable kinematicpatterns. With delay of 70 ms and j=0 (in d), themovement output is characterized by long-lastingdecaying oscillations, not typical for natural elbowmovements.

In the presence of proprioceptive delays, the torquecomponent, N, resulting from intrinsic elasticity ofmuscles during the delay period (j>0, Fig. 5), is non-zero and the model continues to produce stable kine-matic patterns of fast movements in the presence ofdelays of 30 (in Fig. 5a) and 60 ms (in b). These patterns(solid curves in a and b) match empirical patterns(dotted curves; Rc

2 = 0.99, P<0.01). By adjusting therate and duration of R and C commands, one canreproduce experimental movements that vary in term ofthe extent and peak velocity (Fig. 5, respectively, 66.2�and 317�/s in a; 62.2� and 476�/s in b). Moreover, whenthe delay-related function of intrinsic muscle elasticity istaken into account, the range of feedback delays that thesystem can accommodate without compromising itsstability is extended to 100 ms (c). A further increase indelay results in a slowly decaying terminal oscillations(d), not typical for actual fast movements.

In the presence of delays up to 100 ms, the modelproduces stable responses not only to central commands(R and C) but also to external perturbations, whether

Md

Mus

cle

torq

ue

Joint angle

θ θd 0

M+

M

Ν

λ2

A

κ A

Threshold control

* λ1 *

Fig. 3 Threshold control of muscle activation, torque and move-ment production at the elbow joint in the k model. Flexor motorunits begin to be recruited when the actual joint angle exceeds thecentrally controlled threshold joint angle (k*1). The number ofactive motor units and muscle torque increase depending on thedifference between the actual and the threshold angle (right solidtorque–angle curve, the invariant characteristic). To produce anintentional movement, neural control levels reset, at a specifiedrate, the activation threshold to a new position (k*2) thus shiftingthe invariant characteristic (left solid curve). At any current jointangle (h), the muscle activation (A) is defined not by this angle butthe joint angle (hd) transmitted by proprioceptive afferents tomotoneurons with some delay (d). If h and hd coincide, the musclegenerate torque Md defined by the invariant characteristic.Otherwise, due to muscle elastic resistance, an additional torque(N) is produced in proportion to the muscle activation and thedifference between h and hd as shown by the thin diagonal line withslope jA. The total torque, M=Md+N, is somewhat smaller thanthe torque M+ that would be generated at position h in the absenceof delay

–100

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ow a

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(°) C

R

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d = 0 ms N = 0

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ow v

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°/s)

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0 0.5 1 1.5 2Time (s)

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0 0.5 1 1.5 2

d = 60 ms κ = 0

0 0.5 1 1.5 2Time (s)

D

0 0.5 1 1.5 2

d = 70 ms κ = 0

Fig. 4 Fast elbow movements resulting from ramp-shaped R and Ccommands in the model. In all examples, the electromechanicaldelay was zero. a Movement was produced in the absence ofproprioceptive delay (d=0) thus reducing to zero a component ofmuscle torque (N in Fig. 1) resulting from this delay. b, c In the

absence of the intrinsic elastic torque (N or j=0) but in thepresence of proprioceptive delay (d=30 ms in b and 60 ms in c), themodel produces stable kinematic patterns. When d=70 ms andN=0 (in d) the model produces atypical movement patterns withlong-lasting but eventually decaying oscillations

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they are applied before the onset (Fig. 6a, b) or after theoffset of fast movement (c, d).

In all simulations shown in Figs. 4, 5 and 6, EMD=0was used. Figure 7 shows the effect of EMD that ini-tially was 40 ms but gradually (with time constant of100 ms) decreased to 10 ms after the onset of muscleactivation. Thus, the electromechanical delay influencesthe latency, rather than stability of posture and move-ment.

Discussion

Threshold control is a multifaceted phenomenon thatseems to play a major role in the control of posture andmovement, expediently solves the problem of the rela-tionship between these two components of motor ac-tions, and is essential in the organization andmodification of spatial frames of reference in which

0 0.5 1 1.5 2Time (s)

D

B

–600

–400

–200

0

200

Elb

ow v

eloc

ity (

°/s)

d = 60 ms κ = 0.007

0 0.5 1 1.5 2

d = 150 ms κ = 0.0086

0 0.5 1 1.5 2Time (s)

C

0 0.5 1 1.5 2

d = 100 ms κ = 0.0086

–100

–50

0

50

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ow a

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(°)

C

R

θ

A

Rc = 0.99, p < 0.012

ErrRMS = 1.7881 °Rc = 0.99, p < 0.01

2

ErrRMS = 1.8413 °

θ.

d = 30 ms κ = 0.006

Fig. 5 When threshold control is accomplished in the presence ofintrinsic muscle elasticity (j>0) the system remains stable fordelays as high as 100 ms. a, b Simulated (solid lines) andexperimental fast movements (dotted lines) practically match eachother, as estimated by correlation coefficient (Rc

2). The movementextent is practically the same but peak velocity is greater in (b) than

in (a) and a small overshoot is present in (b). c With delay of100 ms, the simulated kinematic patterns are still in the range ofthose characteristic of natural elbow movements. d Delays higherthan 100 ms produce atypical movement patterns characterized bylong-lasting terminal oscillations

–100

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ow a

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(°)

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R

θ

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θ.

d = 60 ms κ = 0.007

B

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0

200E

lbow

vel

ocity

(°/

s)

d = 100 ms κ = 0.0086

0Time (s)

C

0 0.5 1 1.5 2 2.5 3

d = 60 ms κ = 0.007

3

Time (s)

D

0 0.5 1 1.5 2 2.5

d = 100 ms κ = 0.0086

Fig. 6 Even strong pulse perturbations (arrows, 50 Nm during 10 ms) do not destabilize the system at the initial (a, b) or final position (c,d) in the presence of delays 60 (a, c) or 100 ms (b, d)

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neurons function and motor actions are produced(Feldman and Levin 1995; Feldman 2006). Associatedwith changes in the membrane potentials of motoneu-rons (see Introduction), threshold control may sub-stantially influence the state of the neuromuscularsystem before any visible changes in the motor outputand pre-determinate this output for some time ahead,i.e., in a forward way. Our findings confirm the sug-gestion that the forward aspect of threshold control isessential in overcoming destabilizing effects of proprio-ceptive delays in proprioceptive feedback to motoneu-rons. This conclusion is supported by the fact that themodels of arm movements not relying on thresholdcontrol ensure stability of posture and movement forproprioceptive delay of 30 ms (Bhushan and Shadmehr1999) or less but not for the entire range of possibleproprioceptive delays (25–50 ms). In contrast, withthreshold control, even in the absence of the intramus-cular component (j=0) that maintains some level ofelasticity and damping of the system during the delayperiod, stability is ensured for delays up to 60 ms thatcover this range (Fig. 4). Moreover, threshold controlcombined with the intrinsic muscle elasticity anddamping during the delay period (j>0) ensures stabilityof posture and movement for proprioceptive delays up

to 100 ms. Our model also shows that the electrome-chanical delay (40 ms, Fig. 7) may not endanger thesystem stability—it mainly increases the movement la-tency.

It is usually assumed that proprioceptive feedback isnot essential for the generation of the initial EMG burst(Ag1) during fast arm movement (e.g., Gottlieb et al.1989). This assumption is justified by the argument that,because of reflex delay, this feedback comes to moto-neurons too late to contribute to the Ag1. The argumentdoes not take into account that the changes in themotoneuronal membrane potentials preceding the Ag1and associated changes in the state of spinal neuronsdeeply affect (‘‘gate’’) reflex systems by shifting theirthresholds (Feldman and Levin 1995; Adamovich et al.1997). As a result, the Ag1 emerges due to full contri-bution of these systems so that the effects of reflex delaysbecome negligible. The robust reproduction of fast armmovement kinematics in the present model (Fig. 5a, b)also invalidates the argument that proprioceptive sys-tems are only efficient in slow movements or posturalcontrol. In general, proprioceptive signals are conveyedamong numerous pathways with the effects that aremore global and collective than suggested by the notionof linear-signaling-pathways with local actions (cf. Loebet al. 1999). In particular, proprioceptive signals areindispensable for the organization of spatial frames ofreference both at the level of single neurons and differentbrain levels (Feldman and Levin 1995; Feldman 2006).Moreover, the existence and the ability to control thethreshold position result from the integration ofdescending neural inputs with proprioceptive inputs onmotoneurons (Feldman and Levin 1995). Proprioceptivesystems are thus fundamental in making thresholdcontrol possible, as illustrated by numerous deficits ofmotor actions resulting from the loss of threshold con-trol in deafferented subjects (Feldman and Latash 2005).

Although the present study basically focuses on theproblem of stability, we do not think that thresholdcontrol emerged (e.g., in the process of evolution) fol-lowing necessity to solve the stability problem or/and theposture–movement problem. Rather, threshold controlwas a natural outcome of the integration of sensory andindependent control inputs on threshold ele-ments—motoneurons (Feldman 1986). This outcomeconverted the neuromuscular system into a coherent unitso that stability of the whole unit came naturally withthreshold control, without any necessity to build amechanism that compensated for potential deficienciesof system’s components: these deficiencies only appearwhen components are considered in isolation. Also notethat in the context of threshold control, muscle forcescannot be decomposed into pure preflex, reflex andindependent central components, as is obvious from ourdynamic equations.

In the present model, active muscle fibers provide alevel of non-delayed position- and velocity-dependentfeedback. With such feedback, the upper limit of pro-prioceptive delays that the system can tolerate without

0.5 0.6 0.7 0.8 0.9

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(°)

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0.5 1 1.5 20

0.02

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EM

D (

s)

0.5 1 1.5 2

–80

–60

–40

–20

0

20

40

Time (s)

d = 100 ms κ = 0.0086

d = 60 ms κ = 0.007

A

B

C

Fig. 7 Electromechanical delay (EMD) influences the latency butpractically does not influence the kinematic patterns of fast armmovements. a The EMD timing used in simulations shown in (b)and (c). b, c Simulated arm flexion movements with (solid line) andwithout (dashed line) the EMD when the proprioceptive delay was60 (in b) or 100 ms (in c); Also c shows the effect of a briefperturbation (arrow) applied after the end of fast movement

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loosing stability is shifted from 60 to 100 ms. Our resultsthus support the suggestion (Loeb et al. 1999; Brownand Loeb 2000) that intrinsic muscle properties termed‘‘preflexes’’ oppose the destabilizing effects of delays inreflex pathways to motoneurons. We would like toemphasize, however, that only in combination withthreshold control (Fig. 3), can intrinsic muscle and reflexproperties be efficient in overcoming the destabilizingeffects of delays. This conclusion follows from theinability of models that include intrinsic muscle prop-erties but do not include threshold control to stabilizethe system for proprioceptive delays exceeding 30 ms(see above).

Wagner and Blickhan (1999) suggested that, toachieve stabilization, preflexes must be tuned to thegeometric properties of the system. Our study furtheradvances this notion by showing that stability is greatlyimproved if (1) preflexes are integrated with reflexes in asingle, muscle-reflex structure, and (2) this structureproduces muscle activation and forces depending on thedifference between the actual position of the arm and itsvirtual (threshold) position set by the brain. In otherwords, stability is provided by making the neuromus-cular system to function not in an absolute but in arelative frame of reference with the origin defined by thethreshold position set by the brain.

Proprioceptive signals to motoneurons can be medi-ated not only by segmental but also by trans-cortical andother long-latency pathways (e.g., Rothwell et al. 1982,1986). According to our model, the system can accom-modate, without loosing stability, feedback delays ashigh as 100 ms that cover the range of delays in trans-cortical proprioceptive pathways to motoneurons. Oneshould note, however, that trans-cortical responses toproprioceptive inputs are instruction-dependent (Roth-well et al. 1982, 1986) and thus trans-cortical delays maynot be considered as delays characteristic of continuousfeedback. Rather, the trans-cortical system gates pro-prioceptive signals and initiates (‘‘trigger’’) responses tothem depending on the instruction or behavioral situa-tion. As long as the basic, peripheral system is stable,such ‘‘open-loop’’ responses (e.g. corrective ones) do notcreate stability problems, even if their latency exceeds100 ms. Visio-motor reactions (latency of about 200 ms)influencing body posture during quiet standing may alsooperate in open-loop manner by eliciting from time totime discrete transitions between different equilibriumpostures (Zatsiorsky and Duarte 2000).

When no intentional movements is produced andonly a steady state arm position is maintained, controllevels may leave the central components of muscle acti-vation thresholds unchanged (Asatryan and Feldman1965). Even in this case, EMG activity is generateddepending on the difference between the actual andthreshold length (see Table 1, first equation) and thusthe threshold remains a major factor in determiningwhen each muscle is active or silent. In addition, theactivation threshold has a velocity-dependent compo-nent as well as components that are defined by the level

of reflex inter-muscular interaction (Table 1). Thus, thenon-linear threshold nature of the EMG regulation re-mains essential even if no intentional movement is pro-duced. Whether in posture or movement, thresholdcontrol ensures that all posture-stabilizing mechanismsrelated to intrinsic and reflex muscle properties, as wellas co-activation of opposing muscle groups (Fig. 1) aremanifested only when and where, in spatial coordinates,they are required. It is known that experimental esti-mations of stiffness and damping (also called viscosity)of the neuromuscular system depend on models em-ployed for computations of these variables (Feldmanand Latash 2005). Therefore, models that disregardthreshold control may produce unrealistic values ofstiffness and damping, especially when estimated duringarm motion.

Compared to previous simulations of movementbased on threshold control (St-Onge et al. 1997; Gribbleet al. 1998), the present simulation was different in sev-eral aspects. First, we took into account that the torque(Pd) determined by the invariant torque-angle charac-teristic is achieved gradually for a given level of muscleactivation, A (Table 1). In the previous threshold-basedmodels, gradual torque development was taken intoaccount but at a later stage of torque production. Sec-ond, we took into account that during fast movements,motor units are activated by doublets of spikes followingwith a short interval (10 ms or even less), which sharplydecreases the raising time in the gradual torque devel-opments (Partridge and Benton 1981). On the otherhand, we simplified muscle properties by disregardingpassive muscle series and parallel elastic components.The stiffness of the latter contributes to stability of thelimb and, by excluding this stiffness from the presentmodel, we actually underestimated the capacity of thereal system to remain stable. The inclusion of the serieselastic component is essential for simulations of iso-metric torque production but seems less essential for thestability problem considered here. In terms of system’sstability, the action of this component is likely equiva-lent to a small delay that is additive to the electrome-chanical delay (Brown and Loeb 2000). In the presentstudy, the EMD value (40 ms) was somewhat exagger-ated to indirectly accommodate this effect of serieselasticity.

It remains to be tested whether or not our results canbe generalized to multi-joint movements in which sta-bility can be affected, in particular, by variable inertiaand interactive torques between moving segments. Ourpreliminary results show that sit-to-stand movementssimulated in the framework of threshold control remainstable for proprioceptive delay of 50 ms (Goussev et al.,manuscript in preparation).

An efficient way of avoiding instability in the presenceof even larger feedback delays—an internal model-dri-ven control—has been suggested by Miall et al. (1993).They illustrated this method by considering the proper-ties of an abstract linear first-order system that areimitated by an internal model (Smith predictor). No

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properties of a real neuromuscular system that might beessential for system’s stability in presence of delays wereconsidered in their simulations. We think that the sta-bility problem cannot be considered as solved until theposture–movement problem is solved in a physiologi-cally realistic way (see Introduction; Ostry and Feldman2003). It seems impossible to achieve this in the frame-work of standard formulations of the internal modeltheory in which the brain initially plans kinematics andthen, based on some inverse models of the neuromus-cular system interacting with the environment, computesaccelerating and decelerating muscle torques that arethen transformed into motor commands, i.e. EMG sig-nals, to produce the desired movement. If the systemsimply specified muscle torques based on an inverse dy-namic model, these torques would move the arm awayfrom the initially stable position and the movementwould be opposed by posture-stabilizing mechanisms.This resistance would increase in proportion to the dis-tance moved and, for large movement distances, wouldbe comparable with the maximal voluntary torques(Ostry and Feldman 2003). This resistance wouldcounteract the programmed forces and drive the armback, towards the initial position as soon as the torque-EMG program finished. To prevent this, a forwardinternal model is needed. Such a model would predict theforthcoming resistance or compute it on-line based onthe difference between the actual torques or EMG sig-nals (‘‘efference copy’’ of motor commands) and theprogrammed torques. The forward model thus may issueadditional torques and respective phasic and tonic EMGactivity to compensate this resistance and hold the armat the final position. The internal model theory thusimplies that, even in the absence of external forces, thesystem should compensate some intrinsic resistance tomovement by generating appropriate tonic EMG activ-ity of agonist muscles, as can also be seen from numer-ical simulations of arm movements in the framework ofthe internal model theory (see Eqs. 6 and 8 in Sch-weighofer et al. 1998; Bhushan and Shadmehr 1999).This prediction conflicts with systematic observationsthat, in the absence of external loads, the tonic EMGactivity of arm muscles can be practically zero at theinitial arm position and eventually returns to zero afterthe end of movement (Gottlieb et al. 1989; Weeks et al.1996; Ostry and Feldman 2003). Thus, neither inversenor forward internal models nor their combinations ex-plain an important aspect for system’s stability—avoid-ing the resistance of stabilizing mechanisms to activemovement. A combination of internal models withthreshold control in solving the stability problem wouldbe pointless since the system without internal models, aswe demonstrated, already solves the problem. Mostimportant, threshold control and internal model-drivencontrol seem incompatible: the former suggests that thatEMG activity and forces are emergent properties of thesystem and thus are not programmable whereas theexisting formulations of the latter suggest the directlyopposite.

We would like to emphasize that our results do notnegate the necessity of a forward or anticipatorymechanism in overcoming destabilizing effects of pro-prioceptive delays; they just show that threshold con-trol has an anticipatory aspect that ensures stabilitywithout any internal model. In other words, we arepromoting a form anticipation that is dubbed stronganticipation because it is embedded in the system’s or-dinary or physical mode of functioning (Dubois 2001).For example, anticipatory enhancement of the gripforce acting on an object between fingers when the armbegins to move may be produced without any internalmodel of the load dynamics—the system simply relieson the everyday experience (memory) that objects tendto slip between fingers when the arm begins to moveand, respectively, it increases the grip force when armmovements are made. Weak anticipation in Dubois’sterm refers to an anticipation of events predicted orforecast from a symbol manipulating model of a system(see also Rosen 1985, p. 341). It is usually assumed thatsuch a model computes muscle torques and respectivemuscle activations (‘‘motor commands’’) required toproduce the desired movement, an assumption, that isnot helpful in solving the posture-movement problem(see above). One can also argue that even if the nervoussystem were able, with or without internal models, tocompute the requisite torques, it would be unable toactualize these torques. The computed values of tor-ques cannot be actualized unless they are transformedinto appropriate signals to be delivered to motoneu-rons. To achieve this, the system needs to transform thecomputed torques into individual muscle forces, themuscle forces into individual forces of motor units,forces into EMG signals, EMG signals into postsyn-aptic potentials of hundreds of individual motoneu-rons, decompose these potentials into millions ofindividual synaptic potentials evoked in motoneuronsand interneurons by terminals of other spinal and su-pra-spinal neurons. In other words, in order to actu-alize the computational program, the system needs tosolve an exponentially increasing number of redun-dancy problems arising at each step of these inversetransformations. Because of the strong non-linearity ofneural elements (thresholds) some of the inversetransformations may be impossible (Ostry and Feld-man 2003). The programmed specification of torquesand motor commands forces with or without internalmodels appears physiologically unrealistic. A moregeneral conclusion is that although movement andisometric torque production are described in terms ofmechanical variables, control processes cannot be ex-pressed in terms of these variables and, therefore,model-based torque or EMG computations would bean unnecessary exercise for the nervous system (Ostryand Feldman 2003).

According to Gribble and Ostry (2000) and Ostry andFeldman (2003), physiologically natural, threshold-based control may not rely on internal models to pro-duce motor learning, position sense, and maintaining a

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pole on a finger. Usually, the anticipatory generation ofgrip forces (Flanagan and Wing 1997; Blakemore et al.1998; Bhushan and Shadmehr 1999; Witney et al. 1999,2004) is considered as an unequivocal evidence ofinternal models. We have recently demonstrated thatthreshold control may be sufficient in providing antici-patory aspects of grip force generation (Pilon et al.2005a, b). This suggests that anticipatory actions,including those relying on the knowledge of physicalproperties of objects cannot automatically be consideredas evidence for the existence of internal models.

Acknowledgments Supported by NSERC, CIHR, FQRNT, andIGB (University of Montreal), Canada. We thank Mindy Levin forimproving the text.

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