Do Muscle Synergies Improve Optimization Prediction of Muscle Activations During Gait?

Determination of muscle forces during motion can help to understand motor control, assess pathological movement, diagnose neuromuscular disorders, or estimate joint loads. Difficulty of in vivo measurement made computational analysis become a common alternative in which, as several muscles serve each degree of freedom, the muscle redundancy problem must be solved. Unlike static optimization (SO), synergy optimization (SynO) couples muscle activations across all time frames, thereby altering estimated muscle co-contraction. This study explores whether the use of a muscle synergy structure within a static optimization framework improves prediction of muscle activations during walking. A motion/force/EMG gait analysis was performed on five healthy subjects. A musculoskeletal model of the right leg actuated by 43 Hill-type muscles was scaled to each subject and used to calculate joint moments, muscle-tendon kinematics and moment arms. Muscle activations were then estimated using SynO with two to six synergies and traditional SO, and these estimates were compared with EMG measurements. SynO neither improved SO prediction of experimental activation patterns nor provided SO exact matching of joint moments. Finally, synergy analysis was performed on SO estimated activations, being found that the reconstructed activations produced poor matching of experimental activations and joint moments. As conclusion, it can be said that, although SynO did not improve prediction of muscle activations during gait, its reduced dimensional control space could be beneficial for applications such as functional electrical stimulation (FES) or motion control and prediction.


Introduction 31
Knowledge of muscle forces during human movement could elucidate basic principles of human 32 motor control (M.R. Pierrynowski and Morrison 1985), facilitate assessment of pathological 33 movement and diagnosis of neuromuscular disorders, and improve estimation of the loads 34 experienced by diseased or injured joints (Hardt 1978  Mehrabi, Schwartz, and Steele 2019). However, the models used in these studies were limited to 51 sagittal plane motion and used a reduced number of muscles because the synergy information was 52 extracted from EMG measurements available from only superficial muscles. 53 Recently, a computational approach called Synergy Optimization (SynO) has been proposed that uses 54 muscle synergies to reduce indeterminacy when estimating leg muscle forces during walking (S. 55 Shourijeh and Fregly 2019). The authors evaluated how the specified number of synergies affected 56 estimated lower body joint stiffness and inverse dynamic joint moment matching. While results 57 obtained from SynO were compared with those obtained from SO, experimental evaluation of the 58 muscle activations predicted by SynO was not performed. Furthermore, because imposition of a 59 synergy structure on predicted muscle activations ties all time frames together, SynO is more complex 60 functional axes in the leg model likely affected inverse dynamics joint moment calculations (Reinbolt 111 et al. 2007), which in turn likely affected muscle activation calculations. Moreover, not having a 112 process for calibrating Hill-type muscle-tendon model properties likely affected the estimated muscle 113 activations (Serrancolí et al. 2016). However, all the methods proposed in this work were used with 114 the same limitations. 115

Synergy Optimization 116
The synergy optimization (SynO) approach used in (S. Shourijeh and Fregly 2019) estimates muscle 117 forces during human walking using synergy-constructed muscle activations, similar to the more 118 complex approach in (Gopalakrishnan, Modenese, and Phillips 2014). In SynO, synergies couple 119 muscle activations across time frames, requiring the optimization to be performed over all the time 120 frames simultaneously. 121 Muscle synergy quantities were used as the design variables for synergy optimization. Each muscle 122 activation synergy was composed of a single time-varying synergy activation defined by p = (f-1)/5+1 123 (nearest integer, f = number of frames) B-spline nodal points along with its corresponding time-124 invariant synergy vector defined by m = 43 weights specifying inter-muscle activation coupling. Thus, 125 for n S synergies (nS = 2 through 6), the number of design variables was nS*(p+m). Each optimization 126 problem was theoretically over-determined. However, in practice, the problems remained under-127 determined since neighboring time frames are not completely independent from one another. 128 Using these design variables, the SynO cost function was formulated as follows: 129 where xm x ( ) SynO finds muscle forces that match the inverse-dynamics joint moments as closely as possible 145 through the moment tracking error term in the cost function. The total variance account (VAF) was 146 used to quantify errors in inverse-dynamics joint moment matching. In contrast, the best pattern's 147 correlations between muscle activation and EMG were found by cross correlation with time delay 148 using the correlation coefficient r (Matlab's function corrcoef) and a maximum delay of 100 ms (S.

Static Optimization 151
In contrast to SynO, SO's muscle activations are independent between time frames, allowing the 152 optimization to be performed one time frame at a time. SO was run for the same conditions as SynO 153 ( Figure 2), using the same solver fmincon and carrying out five global optimizations to obtain the 154 initial guess for the initial time (then, for the remaining time-points, the initial guess is taken as the 155 optimization result of the previous time point with the same criterion to minimize the muscle activity 156 (sum of squares of muscle activations). Unlike SynO, SO finds muscle forces that perfectly reproduce 157 the inverse-dynamics joint moments (in the absence of reserve actuators) through equality constraints. 158

Identification of muscle synergies from Static Optimization 159
To extract a synergy structure from the SO results, we used non-negative matrix factorization (NMF) 160 to decompose the 43 muscles activations estimated by SO: 161 where * a is the vector of the reconstructed muscular activations, i H is the single time-varying 163 synergy activation, and i W the corresponding time-invariant synergy vector for each of n synergies 164 (n = 2 through 6). MATLAB nnmf was modified to constrain the norm-1 of each synergy vector to 165 one to have the same constraint as SynO. Finally, using the rigid tendon Hill-type muscle model, the 166 reconstructed muscle forces and corresponding intersegmental joint moments were derived from * a 167 . This approach was called SO-NMF in this work. 168

Results 169
The joint moments obtained from SynO using 2 through 6 synergies matched the inverse dynamics 170 joint moments well (Table 1 and Figure 3). The worst match was produced when using only 2 171 synergies, though the model was still able to match the inverse dynamics joint moments closely (mean 172 VAF of 90%). With 3 synergies, the mean VAF obtained was higher than 96% for all the subjects. 173 Between 4 and 6 synergies, VAF values were 98% or higher. 174 While SO exactly reproduced the inverse dynamics joint moments through its equality constraints, 175 SO-NMF's muscular activations with 2 through 6 synergies matched the experimental inverse 176 dynamic joint moments poorly (Table 1 and Figure 4). With 2 and 3 synergies, matches for some joint 177 moments were worse than 50% VAF, and the mean match was lower than 70%. Between 4 and 6 178 synergies, mean VAF values were between 76% (with 4 synergies) and 90% (with 6 synergies), and 179 some joint moments remained under 80%. 180 Comparison of muscle activations estimated using SynO with experimental EMG measurements 181 showed significant differences when the number of synergies was increased (example in Figure 5 for 182 one of the subjects). Activations estimated by SynO became more similar to those estimated by SO 183 as the number of synergies was increased. However, the mean correlations r between estimated 184 muscle activations and measured EMG patterns for the five subjects did not present such differences 185 (Table 2). Mean values of the different approaches were close, between 0.56 (4 synergies) and 0.62 186 (6 synergies) for SynO and 0.60 for SO. 187 Reconstructed muscle activations obtained using SO-NMF poorly matched the activations estimated 188 using SO (Table 3 and Figure 6). Using only 2 synergies, a mean r 2 correlation of 0.44 was obtained 189 for the 43 muscles, and a maximum correlation of 0.87 was obtained with 6 synergies. However,  The highest muscle activations were observed for two synergies (blue line), which generated higher 220 co-contraction when seeking to match the intersegmental moments, which would likely produce  Table 2 are reasonable in general, with mean r values for the five subjects 227 varying between 56% and 68%. Surprisingly, no significant differences were observed for different 228 numbers of synergies. The poorest results were obtained for the rectus femoris and the gluteus medius. 229

Correlations observed in
Crosstalk (Jungtäubl et al. 2018) may explain the low correlation for these muscles, especially rectus 230 femoris. Comparing the rectus femoris EMG signal with the vastus intermedius (muscle located under 231 the rectus femoris) estimated activation resulted in a higher correlation (from 0,25 to 0,61). 232 Strangely, the reconstructed activations from SO-NMF matched EMG better than did the original 233 activations from SO. However, the reconstructed inverse dynamics joint moments showed a poor 234 correlation VAF (between 56% and 85%), thus producing an inconsistent actuation. This might have 235 been caused by the use of a reduced number of components when obtaining the synergy information 236 through NMF for a large number of muscles. 237 For SynO as well as for SO-NMF, the best correlations with experimental EMG patterns were 238 obtained using three synergies. As mean intersegmental moment matching with three synergies was 239 good using SynO (96.1% in Table 1 Comparing EMG correlations obtained with SynO and n synergies (n = 2 through 6) and those 254 obtained with SO, there are no clear advantages between the two approaches. Despite its higher 255 dimensional control space, SO presented a mean r correlation of 0.60 with the experimental data. 256 SynO produced better correlations only for the 3 synergy case. 257 In conclusion, this study evaluated the ability of the SynO approach to predict muscle activations 258 obtained from experimental EMG measurements during gait and found that three synergies are 259 theoretically enough to control leg muscles during gait. However, no significant differences in ability 260 to predict experimental EMG patterns were found between SynO with n synergies (n = 2 through 6) 261 and SO, so thus neither approach can be considered preferable for this purpose. While SO is 262 computationally faster and requires muscle forces to match inverse dynamic joint moments through 263 constraints, extraction of synergies by NMF from SO's results generated new intersegmental joint 264 moments that were inconsistent with the experimental joint moments. The SynO approach offers 265 reasonable prediction of muscle activations using an imposed synergy structure and reduced 266 dimensional control space and could be useful for applications such as functional electrical 267 stimulation and motion control and prediction. 268

Conflict of interest 269
No conflicts of interest lie with any of the authors.   considered not good enough, in red). 437 Table 2: Mean across subjects correlation coefficient r values between EMG measurements and: i) 438 muscular activations from SynO, ii) muscular activations from NMF with SO, for n synergies (n = 2 439 through 6) of the 5 subjects (r<0.4, in red, are considered poor and r≥0.6, in green, are considered 440 good). 441 Table 3: Mean correlation coefficient R 2 values between muscular activations calculated by SO and 442 SO-NMF for n synergies (n = 2 through 6). 443