Defining feasible bounds on muscle activation in a redundant biomechanical task; practical implications of redundancy

Abstract

Measured muscle activation patterns often vary significantly from musculoskeletal model predictions that use optimization to resolve redundancy. Although experimental muscle activity exhibits both inter- and intra-subject variability we lack adequate tools to quantify the biomechanical latitude that the nervous system has when selecting muscle activation patterns. Here, we identified feasible ranges of individual muscle activity during force production in a musculoskeletal model to quantify the degree to which biomechanical redundancy allows for variability in muscle activation patterns. In a detailed cat hindlimb model matched to the posture of three cats, we identified the lower and upper bounds on muscle activity in each of 31 muscles during static endpoint force production across different force directions and magnitudes. Feasible ranges of muscle activation were relatively unconstrained across force magnitudes such that only a few (0∼13%) muscles were found to be truly “necessary” (e.g. exhibited non-zero lower bounds) at physiological force ranges. Most muscles were “optional” having zero lower bounds, and frequently had “maximal” upper bounds as well. Moreover, “optional” muscles were never selected by optimization methods that either minimized muscle stress, or that scaled the pattern required for maximum force generation. Therefore, biomechanical constraints were generally insufficient to restrict or specify muscle activation levels for producing a force in a given direction, and many muscle patterns exist that could deviate substantially from one another but still achieve the task. Our approach could be extended to identify the feasible limits of variability in muscle activation patterns in dynamic tasks such as walking.

Introduction

Musculoskeletal redundancy (Bernstein, 1967) in biomechanical models is often addressed through optimizations that identify a unique muscle activation pattern among many possible. One popular criterion is minimizing muscle stress (Crowninshield and Brand, 1981) which has been widely applied to predict muscle coordination in simulations (Anderson and Pandy, 2001; Thelen et al., 2003; Erdemir et al., 2007). However, measured muscle activity often varies significantly from these predictions (Buchanan and Shreeve, 1996; Herzog and Leonard, 1991; Thelen and Anderson, 2006; van der Krogt et al., 2012). We currently lack methods for analyzing high-dimensional musculoskeletal models that would allow us to quantify the degree to which muscle activity may feasibly vary for a given motor task.

A first step to understanding the variability in muscle activity with respect to musculoskeletal redundancy is to identify absolute biomechanical constraints on muscle activity for a given task. In contrast to optimization, this approach seeks to find the full range of possible solution sets available to the nervous system (Kutch and Valero-Cuevas, 2011). In particular, identifying the explicit bounds on muscle activation can reveal whether predicted or measured muscle activity is due to biomechanical requirements necessary to perform the task, or because of allowable variability in how the task can be achieved. Identifying feasible bounds of muscle activity can also describe the degree to which muscle activity may deviate from optimal solutions.

This study was motivated by experimentally-observed inter- and intra-subject variability during reactive balance control (Horak and Nashner, 1986; Torres-Oviedo et al., 2006; Torres-Oviedo and Ting, 2007). For example in cats, when producing an extensor force vector (Fig. 1A, F_EXT), knee extensor vastus medialis (VM) was recruited consistently across animals, but hip and knee flexor medial sartorius (SARTm) was recruited at different levels across animals (Fig. 1B, F_EXT). Conversely, when producing a flexor force vector (Fig. 1A, F_FLEX), VM recruitment varied across animals but SARTm was recruited consistently in all animals (Fig. 1B, F_FLEX).

Figure 1.

(A) Experimentally-measured hindlimb endpoint force vectors in cat Bi from Torres-Oviedo et al. (2006). Extensor force vector (F_EXT, red) and flexor force vector (F_FLEX, yellow) vectors were essentially identical across cats. (B) Range of experimental muscle activity across producing F_EXT and F_FLEX across three cats. When producing F_EXT, VM was consistently activated in all animals, whereas the activation level of SARTm varied across animals. For F_FLEX, SARTm was activated consistently in all animals and VM was activated at varying levels across animals. (C) Musculoskeletal model of the cat hindlimb (Burkholder and Nichols, 2004) with seven rotational degrees of freedom (3 at the hip, 2 each at the knee and ankle) and 31 muscles. In this static model, the pelvis was fixed to the ground and the endpoint, defined at the MTP joint, was connected to the ground via gimbal joint where moments were constrained to be zero.

Here, we identified feasible ranges of muscle activation during static force production in a detailed model of the cat hindlimb (Fig. 1C; Burkholder and Nichols, 2004; McKay and Ting, 2008). We identified the upper and lower bounds on muscle activity in each of 31 muscles during endpoint force production in different directions and magnitudes. Muscles with non-zero lower bounds were classified as “necessary”, whereas muscles with zero lower bounds were classified as “optional”. Muscles were further classified to have “sub-maximal upper bound” or “maximal upper bound”. To examine the degree to which feasible muscle activation patterns could deviate from an optimal solution, we compared these bounds to muscle activation patterns predicted by minimizing muscle stress (Crowninshield and Brand, 1981), or scaling the pattern required for maximum force generation (Valero-Cuevas, 2000).

Methods

Musculoskeletal model

The static three-dimensional musculoskeletal model of the cat hindlimb (Burkholder and Nichols, 2004) included seven rotational degrees of freedom (Fig. 1C). 31 muscles (Table 1) produced net joint torque τ⃑ (7 × 1), and a resulting endpoint wrench (force and moment vector) F⃑_End (6 × 1) at the metatarsophalangeal (MTP) joint. The MTP was connected to the ground via a gimbal joint (Fig. 1C), representing the experimental condition of a freely standing cat where the foot never lost contact or slipped with respect to the ground (Jacobs and Macpherson, 1996). Endpoint moments were constrained to be zero, a conservative approximation of the small moments that can be supported by the contact area of cat's foot (McKay et al., 2007). The model defined the mapping from muscle activation vector e⃑ (31 × 1) to endpoint wrench F⃑_End:

Table 1. Muscles included in the hindlimb model and abbreviations.

Name	Abbreviation	Name	Abbreviation
Adductor femoris	ADF	Plantaris	PLAN
Adductor longus	ADL	Iliopsoas	PSOAS
Biceps femoris anterior	BFA	Peroneus tertius	PT
Biceps femoris posterior	BFP	Pyriformis	PYR
Extensor digitorum longus	EDL	Quadratus femoris	QF
Flexor digitorum longus	FDL	Rectus femoris	RF
Flexor hallicus longus	FHL	Sartorius	SART
Gluteus maximus	GMAX	Semimembranossus	SM
Gluteus medius	GMED	Soleus	SOL
Gluteus minimus	GMIN	Semitendinosus	ST
Gracilis	GRAC	Tibialis anterior	TA
Lateral gastrocnemius	LG	Tibialis posterior	TP
Medial gastrocnemius	MG	Vastus intermedius	VI
Peroneus brevis	PB	Vastus lateralis	VL
Pectineus	PEC	Vastus medialis	VM
Peroneus longus	PL

$${\mathbf{RF}}_{\mathit{\text{AFL}}}\stackrel{\rightharpoonup }{e}=\stackrel{\rightharpoonup }{\tau }={\mathbf{J}}^{\text{T}}{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{End}}},$$ (1)

where J is a geometric Jacobian (6 × 7), R is a moment arm matrix (7×31) that maps muscle forces to joint torques, and F_AFL is a diagonal matrix (31×31) of scaling factors based on the active force-length property of muscle (Zajac, 1989). To approximate the operating region on the force-length relationship curve commonly observed in habitual postures, all muscles were set to 95% optimal fiber length (Burkholder and Lieber, 2001; Roy et al., 1997; Sacks and Roy, 1982). We found matrices J and R for each of 3 cats Bi, Ni, and Ru based on their average kinematic configuration measured during quiet standing (McKay et al., 2007) using Neuromechanic software (Bunderson et al., 2012).

Target endpoint forces

Five experimentally-derived force vectors in each cat measured during postural responses to translational support perturbation (Torres-Oviedo et al., 2006) were used as target endpoint force vector directions (Fig. 1A). These force vectors represented the active response of the cats following perturbation, measured as the change in the ground reaction force from the background level, averaged over the postural response period 150-200 ms following the perturbation (Jacobs and Macpherson, 1996), where only small angular deviations in joint angles (≤2°) are observed (Ting and Macpherson, 2004). To examine biomechanical constraints across force magnitudes, we scaled each force vector from 0 to the maximum feasible level that could be produced by the model, identified using linear programming. We found the muscle activation pattern e⃑^MAX that maximized force magnitude:

$${\stackrel{\rightharpoonup }{e}}^{\text{MAX}}:\text{Find}\stackrel{\rightharpoonup }{e}\text{s}.\text{t}.\left({\mathbf{RF}}_{\mathit{\text{AFL}}}\stackrel{\rightharpoonup }{e}\right)•\left({\mathbf{J}}^{\text{T}}{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}\right)\text{is maximized},\text{while}\left({\mathbf{RF}}_{\mathit{\text{AFL}}}\stackrel{\rightharpoonup }{e}\right)\times \left({\mathbf{J}}^{\text{T}}{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}\right)=0,$$ (2)

where the cross product constraint in Eq. (2) ensured the preservation of force direction. Activation of each muscle was constrained between 0 and 1, and endpoint moments were constrained to be zero. The maximum feasible force in direction of the experimental force vector is given by:

$${\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}^{\text{MAX}}=\frac{\parallel {\mathbf{RF}}_{\mathit{\text{AFL}}}{\stackrel{\rightharpoonup }{e}}^{\text{MAX}}\parallel }{\parallel {\mathbf{J}}^{\text{T}}{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}\parallel }{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}.$$ (3)

Lower and upper bounds on muscle activation

We used linear programming to identify the lower bound $(e_m^{\mathbf{LB}})$ and the upper bound $(e_m^{\mathbf{UB}})$ on the feasible activation level of each muscle as the magnitude (α) of each of the target endpoint force vectors was scaled from 0 to 1 (Eq. (4) and (5)). Grid spacing Δα=0.1 was used from α=0.0 to 0.9, and grid spacing Δα=0.02 from α=0.9 to 1.0 because initial tests revealed rapid changes for higher values of α. For each muscle and each value of α, the lower and upper bound was identified as follows:

$$e_m^{\mathbf{LB}}:\text{Find}\stackrel{\rightharpoonup }{e}\text{s}.\text{t}.|e_m|\text{is minimized},\text{while}{\mathbf{RF}}_{\mathit{\text{AFL}}}\stackrel{\rightharpoonup }{e}=\mathrm{\alpha }\cdot {\mathbf{J}}^{\text{T}}{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}^{\text{MAX}},$$ (4)

$$e_m^{\mathbf{UB}}:\text{Find}\stackrel{\rightharpoonup }{e}\text{s}.\text{t}.|e_m|\text{is maximized},\text{while}{\mathbf{RF}}_{\mathit{\text{AFL}}}\stackrel{\rightharpoonup }{e}=\mathrm{\alpha }\cdot {\mathbf{J}}^{\text{T}}{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}^{\text{MAX}}.$$ (5)

Each muscle was classified as necessary or optional based on whether, and at what force magnitude the muscle became biomechanically required to generate endpoint force, corresponding to a nonzero lower bound. Similarly, we classified muscles as having sub-maximal upper bound or maximal upper bound based on whether the upper bound was less than or equal to full activation. Considering all combinations of animals, muscles, bounds, endpoint force vectors, and levels of α resulted in 13,206 separate linear programming calls.

Lower and upper bounds identified at α=0 were considered a special case because they do not depend on direction of the endpoint force vector and reveal the feasible muscle activation patterns associated with zero net torque production, which we call the physiological null space.

Comparison to predicted solutions from suggested neural strategies

We compared the feasible range of individual muscle activity to solutions for muscle activation patterns identified by 1) minimizing muscular stress (Crowninshield and Brand, 1981) in the form of sum-squared muscle activation (Thelen et al., 2003) and 2) scaling the muscle activation pattern for the maximal task (Valero-Cuevas, 2000). For the minimum stress strategy, muscle patterns e⃑^min were identified for each level of α via quadratic programming as follows:

$${\stackrel{\rightharpoonup }{e}}^{min}:\text{Find}\stackrel{\rightharpoonup }{e}\text{s}.\text{t}.\text{∑}_{m=1}^{31}{\text{e}}_m^2\text{is minimized},\text{while}{\mathbf{RF}}_{\mathit{\text{AFL}}}\stackrel{\rightharpoonup }{e}=\mathrm{\alpha }\cdot {\mathbf{J}}^{\text{T}}{\stackrel{\rightharpoonup }{F}}_{\mathit{\text{Exp}}}^{\text{MAX}}.$$ (6)

For the scaling strategy, e⃑^MAX identified in Eq. (2) was scaled proportional to α.

Results

Bounds on muscle activation during endpoint force production

The feasible range of muscle activity for each muscle changed non-uniformly as force magnitude α increased from zero to maximal in a given target endpoint force direction (e.g. Fig. 2B, shaded region). This range was defined by the difference between the lower bound (Fig. 2B, bottom trace) and upper bound (Fig. 2B, top trace) at a given α. In each animal, similar patterns of the feasible range of muscle activity was identified across muscles and force directions. Therefore, two force directions are used for detailed illustration of the results: an extensor force F_EXT (Fig. 2A, red) and a flexor force F_FLEX (Fig. 2A, yellow).

Figure 2.

(A) Two representative target endpoint force directions for cat Bi: F_EXT (red, top row) and F_FLEX (yellow, bottom row). (B) Identified feasible range of activation as a function of normalized force magnitude (α) for five muscles in cat Bi: GRA, PT, SART, SOL and VL. Feasible range (shaded) is defined by the difference between the lower bound ( $e_m^{\mathbf{LB}}$, bottom trace) and the upper bound ( $e_m^{\mathbf{UB}}$, top trace). Muscles with zero lower bounds were categorized as optional (OPT), e.g. GRAC, PT, SOL for F_EXT, and GRAC Sol and VL for F_FLEX. Muscle were categorized as necessary (NEC) if they were nonzero at any force level, and further subdivided into categories of becoming necessary gradually, e.g. SART for F_EXT and F_FLEX; only near maximal force, e.g. PT for F_FLEX; or always, e.g. VL for F_EXT. Muscles were also classified as having maximal (MUB) upper bounds e.g. PT, SART, SOL for F_EXT and PT, SART, for F_FLEX; or having sub-maximal (SUB) upper bounds, either monotonically, e.g. GRAC in F_EXT and VL in F_FLEX; nonmonotonically, e.g. GRAC in F_FLEX; or conditionally, e.g. VL in F_EXT and SART in F_FLEX. The vertical line indicates the experimental force levels at which most muscles had zero lower bound and were practically “optional” of the muscles shown, only VL for F_EXT was truly “necessary”. Feasible ranges were wide in general, where activity of a muscle could deviate substantially from the solutions predicted by either minimizing muscle stress (dots), or scaling the pattern required for maximal force (dashed lines). Note that neither strategy predicted the recruitment of optional muscles.

Most muscles had zero lower bound for all force magnitudes ( $e_m^{\mathbf{LB}}=0$ for all α) and were classified as optional (OPT). Muscles for which lower bound became nonzero for some α were classified as necessary; they were either always necessary (NEC_always: $e_m^{\mathbf{LB}}>0$ for all α), or became gradually necessary as α increased (NEC_grad: $e_m^{\mathbf{LB}}>0$ at 0<c<α<1), or became necessary only at the maximum force level (NEC_abrupt: $e_m^{\mathbf{LB}}>0$ only at α≈1). Across cats, 71±7% of muscles were optional for the generation of F_EXT, and 58%–but not the same muscles–were optional for generation of F_FLEX (Table 2). For example, in cat Bi (Fig. 2B), GRAC, PT, SOL were OPT'for F_EXT, and GRAC, SOL, VL were OPT for F_FLEX. VL was NEC_always for F_EXT, SART was NEC_grad in both F_EXT and F_FLEX, and PT was NEC_abrupt for F_FLEX.

Table 2. Muscle classification in terms of lower bound ( $e_m^{\mathbf{LB}}$) behavior.

	Bi		Ni		Ru
	FEXT	FFLEX	FEXT	FFLEX	FEXT	FFLEX
ADF	OPT	OPT	OPT	OPT	OPT	OPT
ADL	OPT	OPT	OPT	OPT	OPT	OPT
BFA	OPT	OPT	OPT	OPT	OPT	OPT
BFP	NECalways	NECgrad	NECalways	NECgrad	NECgrad	NECgrad
EDL	OPT	NECgrad	OPT	NECgrad	OPT	NECgrad
FDL	OPT	OPT	OPT	OPT	OPT	OPT
FHL	OPT	OPT	OPT	OPT	OPT	OPT
GMAX	OPT	OPT	OPT	OPT	OPT	OPT
GMED	NECabrupt	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad
GMIN	OPT	OPT	OPT	OPT	NECabrupt	NECabrupt
GRAC	OPT	OPT	OPT	OPT	OPT	OPT
LG	OPT	NECgrad	OPT	NECgrad	OPT	NECgrad
MG	OPT	NECabrupt	OPT	NECabrupt	OPT	OPT
PB	OPT	NECabrupt	OPT	NECgrad	OPT	OPT
PEC	OPT	OPT	OPT	OPT	OPT	OPT
PL	OPT	NECabrupt	OPT	NECgrad	OPT	NECabrupt
PLAN	OPT	OPT	OPT	OPT	OPT	OPT
PSOAS	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad
PT	OPT	NECabrupt	OPT	NECabrupt	OPT	NECabrupt
PYR	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad
QF	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad
RF	OPT	OPT	OPT	OPT	NECgrad	OPT
SART	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad	NECgrad
SM	OPT	OPT	OPT	OPT	OPT	OPT
SOL	OPT	OPT	OPT	OPT	OPT	OPT
ST	OPT	OPT	OPT	OPT	OPT	NECabrupt
TA	NECabrupt	NECgrad	OPT	NECgrad	NECabrupt	NECgrad
TP	OPT	OPT	OPT	OPT	OPT	OPT
VI	NECabrupt	OPT	OPT	OPT	NECgrad	OPT
VL	NECalways	OPT	NECalways	OPT	NECgrad	OPT
VM	OPT	OPT	OPT	OPT	OPT	OPT

OPT: $(e_m^{\mathbf{LB}})$ is always zero; NEC_always: $e_m^{\mathbf{LB}}$ is always non-zero; NEC_gmd: $e_m^{\mathbf{LB}}$ becomes non-zero gradually; NEC_abrupt: $e_m^{\mathbf{LB}}$ becomes non-zero abruptly at maximal force magnitude. Muscle that changed classifications of necessary versus optional across the two force directions are shown in bold black. Muscles that differ in sub-classifications across force directions are shown in bold gray.

Less than 1/3 of muscles had an upper bound of one for all force magnitudes and were classified as having maximal upper bound (MUB: $e_m^{\mathbf{UB}}=1$ for all α). Muscles with upper bound less than one for some range of α were classified as having sub-maximal upper bound conditionally (SUB_cond: $e_m^{\mathbf{UB}}<1$ for 0<α<c or c<α<1). Muscles for which the upper bound was always less than one were classified as having sub-maximal upper bound ( $e_m^{\mathbf{UB}}<1$ for all α), but were further categorized based on whether the upper bound changed monotonically (SUB_mono) or non-monotonically (SUB_non). Across cats, 32±3% of the muscles had maximal upper bound for generating F_EXT, 22±2% had sub-maximal upper bound for generating F_FLEX (Table 3). For example, PT, SART, SOL were MUB for F_EXT, and PT and SART were MUB for F_FLEX in cat Bi (Fig. 2B). VL was SUB_cond and GRAC was SUB_mono for F_EXT, whereas SOL was SUB_cond, VL was SUB_mono, and GRAC was SUB_non for F_FLEX.

Table 3. Muscle classification in terms of upper bound ( $e_m^{\mathbf{UB}}$) behavior.

	Bi		Ni		Ru
	FEXT	FFLEX	FEXT	FFLEX	FEXT	FFLEX
ADF	SUBmono	SUBmono	SUBmono	SUBmono	SUBmono	SUBnon
ADL	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond
BFA	SUBmono	SUBmono	SUBmono	SUBnon	SUBmono	SUBnon
BFP	SUBmono	SUBnon	SUBmono	SUBnon	SUBmono	SUBnon
EDL	SUBcond	MUB	SUBCond	SUBnon	SUBCond	MUB
FDL	MUB	SUBcond	MUB	SUBcond	MUB	SUBcond
FHL	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond
GMAX	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond
GMED	SUBmono	SUBmono	SUBmono	SUBnon	SUBnon	SUBmono
GMIN	SUBcond	SUBcond	SUBcond	SUBcond	MUB	MUB
GRAC	SUBmono	SUBnon	SUBmono	SUBnon	SUBmono	SUBnon
LG	SUBmono	SUBnon	SUBmono	SUBnon	SUBmono	SUBmono
MG	SUBmono	SUBnon	SUBmono	SUBnon	SUBmono	SUBmono
PB	MUB	SUBcond	MUB	SUBcond	MUB	SUBcond
PEC	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond	SUBcond
PL	MUB	MUB	MUB	MUB	MUB	SUBcond
PLAN	SUBmono	SUBnon	SUBmono	SUBnon	SUBmono	SUBmono
PSOAS	MUB	MUB	MUB	MUB	MUB	MUB
PT	MUB	MUB	MUB	MUB	MUB	MUB
PYR	MUB	MUB	MUB	MUB	MUB	MUB
QF	SUBcond	SUBcond	MUB	SUBcond	SUBcond	SUBcond
RF	SUBmono	SUBnon	SUBmono	SUBnon	SUBcond	SUBcond
SART	MUB	MUB	MUB	MUB	MUB	MUB
SM	SUBmono	SUBnon	SUBmono	SUBnon	SUBmono	SUBnon
SOL	MUB	SUBcond	MUB	SUBcond	MUB	SUBcond
ST	SUBmono	SUBnon	SUBmono	SUBnon	SUBmono	SUBnon
TA	MUB	MUB	MUB	MUB	MUB	MUB
TP	SUBmono	SUBmono	SUBmono	SUBmono	SUBmono	SUBmono
VI	SUBcond	SUBcond	SUBcond	SUBcond	MUB	SUBcond
VL	SUBcond	SUBmono	SUBcond	SUBmono	SUBcond	SUBmono
VM	SUBmono	SUBnon	SUBmono	SUBnon	SUBcond	SUBcond

MUB: $(e_m^{\mathbf{UB}})$ is always one (maximal); SUB_cond: $e_m^{\mathbf{UB}}$ is sub-maximal only at certain range of α SUB_mono: $e_m^{\mathbf{UB}}$ is always sub-maximal and changes monotonnically; SUB_non: $e_m^{\mathbf{UB}}$ is always sub-maximal and changes non-monotonically. Muscle that changed classification of sub-maximal upper bound versus maximal upper bound across the two force directions are shown in bold black. Muscles that differ in sub-classifications across force directions are shown in bold gray.

Muscle classification in terms of the lower and upper bounds depended on the target endpoint force direction (Tables 2 and 3). In total, 20 muscles in Bi, 20 in Ni, 19 in Ru, showed different behavior for F_FLEX as compared to F_EXT. The classification of muscles was relatively consistent across cats for a given force direction: for the upper bound, only 4 muscles were categorized differently across 3 cats for both F_EXT and F_FLEX, and for the lower bound, only 3 muscles were categorized differently across all cats for both F_EXT and F_FLEX. Because the direction of endpoint force vectors was very consistent (cos θ>0.998) across cats for both F_EXT and F_FLEX, these differences in the categorization were due to differences in posture.

Regardless of classifications, the feasible range of muscle activity at physiological levels of force did not identify a clear pattern of muscle activity necessary to achieve the task. At experimentally-observed force magnitudes (Fig. 2B, vertical lines), lower bounds were often zero, suggesting that most muscles were optional at those force levels. Across animals, α_exp was 0.32, 0.77, 0.19 in Bi, Ni and Ru for F_EXT and 0.12, 0.11, 0.11 for F_FLEX.

Comparison of identified bounds and predictions of neural strategies

Because of the large feasible range, muscle activity could deviate substantially from two commonly suggested muscle coordination strategies. Both solutions fell within the feasible ranges of muscle activity, and typically near–but not necessarily at–the lower bound. Both the scaling strategy (Fig. 2B, dashed line) and the minimum stress strategy (Fig. 2B, dotted line) recruited necessary muscles at the earliest nonzero α even though the lower bounds on feasible muscle activity were typically zero at low force magnitudes (e.g. Figure 2B, VL for F_EXT and SART for F_FLEX). Optional muscles were never selected in either strategy (e.g. Figure 2B, GRAC for F_EXT and VL for F_FLEX). Although the upper and lower bounds typically converged on a unique solution for maximum force production (Fig. 2B, bottom row), this was not always the case in ankle muscles for F_EXT (Fig. 2B, top row, PT and SOL), indicating that redundancy remained even at maximum force magnitudes. This resulted from low ankle torque (∼0.004N-m) compared to knee torque (∼0.5N-m) required to produce F_EXT.

Physiological null space at α=0

Approximately 1/3 of muscles had upper bounds of less than one for zero net torque production, defining the physiological null space (Fig. 3). Because the torque generated by each muscle must be counterbalanced by activation of other muscles, those producing large torques (e.g. large moment arms and maximum isometric force) typically had low upper bounds (≤0.53) because of the lower torque-generating capabilities of their antagonists (Ait-Haddou et al., 2004; Jinha et al., 2006).

Figure 3.

Physiological null space, defined as the feasible range at α=0 in cat Ru. While many muscles could be maximally activated, several muscles were limited in the maximum activation that would allow zero net torque production. Upper bounds of muscles that produce large torques (e.g. BFP) were typically limited, because of the lower torque-generating capabilities of their antagonists.

Discussion

Here, we identified the feasible ranges of individual muscle activation during endpoint force generation as a way of understanding the degree to which biomechanical redundancy allows for variability in muscle activation patterns. Feasible ranges of muscle activation were relatively unconstrained across force magnitudes in a cat hindlimb model (7 non-orthogonal DoF's, 31 muscles). Although we identified muscles that became biomechanically “necessary” at higher levels of force (e.g. nonzero lower bound), few muscles were found to be truly “necessary” at physiological force ranges. Thus, biomechanical constraints were generally insufficient to specify muscle activation levels, demonstrating that many possible muscle patterns exist that could deviate substantially from one another. In contrast, the biomechanical bounds on muscle activity in finger force generation (4 orthogonal DoF's, 7 muscles) was shown to be highly constrained, even at sub-maximal force magnitudes (Kutch and Valero-Cuevas, 2011), demonstrating differences in the biomechanical redundancy of the cat hindlimb limb versus the index finger.

The ubiquity of “optional” muscles in both agonist and antagonists across most force levels highlights the necessity to understand neural strategies governing selection of muscle activation patterns. The large space of functionally equivalent solutions is consistent with variations in neural and muscular activity observed across individuals in a variety of neuromotor behaviors (Klein et al., 2010; Prinz et al., 2004; Raphael et al., 2010). Moreover, “optional” muscle were never selected by typical methods that resolve biomechanical redundancy, e.g. minimizing stress (Thelen et al., 2003), or scaling patterns that produce maximal force (Valero-Cuevas, 2000), suggesting that other optimization criteria may need to be considered, such as impedance (Burdet et al., 2001), stability (Bunderson et al., 2008), fiber velocity (Walmsley et al., 1978; Prilutsky et al., 1997), metabolic energy (Alexander, 1989, 2005; Hoyt and Taylor, 1981), or more likely a combination of multiple goals in interplay (Franklin et al., 2008; Ganesh et al., 2010; Todorov, 2004). Alternatively, variations in muscle activation patterns may be due to neural constraints of activating muscles in groups (d'Avella, 2006; Hart and Giszter, 2004; Ting and Macpherson, 2005), or habitual movement patterns (de Rugy et al., 2012). One implication is that altering the biomechanical properties of a muscle, e.g. via weakening or surgery (Valero-Cuevas and Hentz, 2002; Arnold et al., 2005; Hicks et al., 2008; Correa et al., 2012), in a highly redundant system may not affect muscle activation patterns even if the limb's force generating capabilities are altered (Scianni et al., 2009; Damiano et al., 2010).

Despite some limitations in our modeling assumptions, our estimates of feasible muscle range are likely robust and somewhat conservative. Although we specified zero endpoint moment, specifying a different moment values is not likely to alter our results. However, allowing a range of small endpoint moments would increase the set of redundant solutions (McKay et al., 2007), increasing the feasible range of muscle activity. Further, individual variations in morphology of each animal compared to our generic musculoskeletal model (Burkholder and Nichols, 2004) are not expected to change the basic categorizations found. Torque-generating capabilities of muscles based on 95% optimal fiber length were only altered by -9 to +3% when physiological ranges of 80-110% optimal fiber length (Burkholder and Lieber, 2001) were used, and would not change significantly if tendon elasticity were included (Biewener et al., 1998). Finally, the activation-dependent changes in the force-length relationship (Rack and Westbury, 1969), would alter the mapping from muscle force to activation, but would only minimally affect the bounds on feasible muscle activation.

Comparing the predicted feasible muscle activation ranges to experimental data is still difficult due to differences between the model and experimental conditions. Direct comparisons of EMG to feasible limits were not possible because a reference level of muscle force was unknown (and could be estimated by maximum voluntary contraction). Further, because our technique only examines the feasible limits of a single muscle, we cannot use the predictions to identify specific multi-muscle patterns to perform a given task. The measured EMGs may not correspond exactly to the muscles or muscle groupings represented in the model. Finally, for direct comparison to our specific experimental data, we need to take into account background force for standing, which required that the pre-existing force level be considered. For standing, this would decrease extensor force redundancy, increase flexor force redundancy, and likely have a small effect in other directions.

Nonetheless, our approach provides important insight as to the relative variability allowed for a muscle activity that is applicable to both static and dynamic tasks. In contrast to the dimension of the solution space (Bunderson et al., 2008), our approach identifies explicit constraints on muscle activation patterns. The identified bounds could be used to assess confidence of predicted muscle activity as well as possible variations when alternate cost functions or strategies are considered. The feasible muscle range also quantifies the degree to which measured muscle activity is expected to be variable or deviate from predictions. An advantage of our method is that the number of muscles that can be solved is not limited and can be applied to any high-dimensional musculoskeletal models. Our method could also be extended to analysis of dynamic tasks (Ackland et al., 2012; Thelen and Anderson, 2006; van der Krogt et al., 2012) that use methods where each time step of a movement is solved independently, e.g. inverse dynamics or static optimization (Anderson and Pandy, 2001).