# An Algorithm for Choosing the Optimal Number of Muscle Synergies during Walking

^{1}

^{2}

^{BIO}Med Lab, Politecnico di Torino, 10129 Turin, Italy

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## Abstract

**:**

## 1. Introduction

_{i}= [n

_{min}, n

_{max}]). The only constraint is that the number of synergies must not exceed the number of muscles (m) considered in the sEMG acquisition (n

_{max}≤ m); otherwise, the meaning of “synergy” itself would be lost. Afterward, in post-processing, one has to choose the “correct” number of synergies n

_{c}(with n

_{min}≤ n

_{c}≤ n

_{max}), i.e., the input that feeds the factorization algorithm providing “good” results, with a “small-enough” reconstruction error. In recent years, the correct number of muscle synergies (n

_{c}) has been proposed as a meaningful feature for the analysis of motor control strategies in pathological populations [13,14,15,16,17]. A decreased neuromuscular complexity during gait has been assessed in post-stroke patients with respect to a healthy population [13]. Similar results were also found in another work [14], in which a reduced number of muscle synergies (two to four muscle synergies) were observed in the affected side of post-acute stroke patients with respect to a healthy population (four muscle synergies) while executing cycling training. These studies suggest that the number of muscle synergies and their composition could be correlated with motor control capacity and its reduction in pathological conditions [13,14,15,16,17].

_{c}based on the reconstruction accuracy of the factorization, through the variance accounted for (VAF) [1,2,18,19,20,21,22,23,24,25]. To a lesser extent, the coefficient of determination R

^{2}[16,26,27] is also used, which is not conceptually different from VAF. However, this approach requires the selection of an arbitrary threshold for the VAF. The number n

_{c}is defined as the smallest number of synergies that ensures a VAF value above the threshold. In literature, the VAF threshold is commonly set at 90% [1,2,4,7,18,20,21,22,23,24] and less frequently at 95% [19,25]. This method is very simple to implement, but it has several drawbacks: the threshold is arbitrary, it is set without an objective motivation, and there is not a single threshold value shared by all researchers. A few works have explored alternatives to VAF-based criteria. In particular, a statistical approach uses unstructured sEMG signals generated by randomly shuffling the original data across time and muscle [16], while other works consider the variability of muscle synergies between task cycles [28], or a task decoding-based metric [29,30].

## 2. Materials and Methods

#### 2.1. Real Dataset

_{R}and LD

_{L}) muscles.

#### 2.2. Simulated Dataset

- From the dataset of 20 subjects, 15 subjects were extracted, showing n = 4 (5 subjects), n = 5 (5 subjects), and n = 6 (5 subjects) clearly recognizable muscle synergies, as assessed by expert operators (V.A. and M.G.). Hence, for each subject, activation coefficients ($C$) and weight vectors ($W$) were obtained. Figure 1A shows an example of muscle synergies (n = 5) representative of a specific subject.
- For each group of 5 subjects, data augmentation was performed to obtain 25 “simulated subjects”, considering all the possible combinations of $W$ and $C$. In other words, the matrix of weight vectors of the first subject (${W}_{\mathrm{subj}1}$) was combined with the coefficient matrix of every subject in the group (${W}_{\mathrm{subj}1}$${C}_{\mathrm{subj}1}$, ${W}_{\mathrm{subj}1}$${C}_{\mathrm{subj}2}$, … ${W}_{\mathrm{subj}1}$${C}_{\mathrm{subj}5}$), and the same was performed for the other weight matrixes (${W}_{\mathrm{subj}2}$, … ${W}_{\mathrm{subj}5}$), obtaining 25 sets of muscle synergies. Overall, 25 sets were obtained with n = 4, 25 sets with n = 5, and 25 sets with n = 6, for a total of 75 sets.
- For each set of $W$ and $C$, each muscle’s envelope was reconstructed as the product $W$
_{muscle}*$C$, where $W$_{muscle}is the weight vector of a specific muscle. Figure 1B provides an example for the LGS muscle. - For each muscle’s envelope, a simulated sEMG signal (S) was generated by multiplying the envelope by a zero-mean Gaussian process (G
_{S}) with standard deviation σ = 1 a.u. (Figure 1C). At this step, no additive noise was superimposed on the signals. This does not mean that there was “no noise”, but rather that additional noise to the noise originally present in the envelope was not introduced. - Then, different levels of background noise were added to obtain different SNR values (15 dB, 20 dB, 25 dB, and 30 dB), through a zero-mean Gaussian process (G
_{N}) with a standard deviation $\sigma =1/{10}^{SNR/20}$ a.u. [21,34]. Figure 1D shows an example in which SNR was equal to 20 dB. The formula below (1) summarizes how each simulated sEMG signal was generated:

#### 2.3. Muscle Synergy Extraction and Sorting

^{−6}[22], and output variation <10

^{−6}[22]. To explore different solutions, the NMF algorithm was run several times on the same original sEMG data by changing the number of muscle synergies n in the range [1,8].

^{5}, replicas: 15) [22]. Activation coefficients were then sorted accordingly.

#### 2.4. Choosing the Optimal Number of Synergies (ChoOSyn)

- Low similarity across synergies, to avoid selecting muscle synergies containing redundant information.

#### 2.4.1. Intra-Cluster Variability

#### 2.4.2. Weight Similarity

#### 2.4.3. Coefficient Similarity

#### 2.4.4. ChoOSyn

_{c}), there is always a “step” and/or a local minimum. In the following section, this observation will be used to empirically introduce selection rules for obtaining the correct number of synergies. The term “step” refers to a “sharp” increase in the value of the parameter, preceded and followed by “stable” values. The term “local minimum” refers to a situation in which there is an abrupt decrease followed by an abrupt increase in the parameter values [28]. Figure 3 shows examples where both steps and local minima are highlighted (red lines).

#### 2.4.5. ChoOSyn Rules

_{c}), but after exceeding n

_{c}(n > n

_{c}) they become highly variable and with redundant information. Instead, the local minimum represents a condition in which there are low values of the ChoOSyn parameters at the level n, but if n increases or decreases by 1 (n−1 and n+1 levels), the muscle synergies “get worse”.

- There is at least a common choice in the selection(s) provided by the two parameters (Figure 3A). In this case, the common number of synergies is selected.
- The two parameters provide a different selection for the number of synergies (Figure 3B). The number is chosen as the one providing the lowest sum of $ChoOSy{n}_{W}\left(n\right)$ and $ChoOSy{n}_{C}\left(n\right)$ (i.e., with the lowest similarity and highest consistency).

#### 2.5. VAF-Based Methods

- T-VAF (Threshold VAF) (Figure 4A): this method is the most widely used in the literature [1,2,18,19,20,21,22,23,24,25]. It involves the setting of an arbitrary threshold and the subsequent choice of the first number of synergies with VAF above the threshold. The threshold is commonly set at 90% and less frequently at 95%: therefore, we chose to test both 90% and 95% thresholds.
- P-VAF (Plateau VAF) [40] (Figure 4C): this method requires finding the point beyond which the VAF curve reaches a plateau. It uses an arbitrary threshold: the mean-square error obtained by fitting the VAF-curve through a straight line must be smaller than 10
^{−2}. Cheung et al. [40] used a threshold equal to 10^{−5}, but in our simulated dataset 10^{−2}provided the best performance. The first point satisfying this condition is chosen.

#### 2.6. Performance Evaluation

_{c}is the number of correct synergies, and $NS$ is the total number of subjects in the dataset.

_{c}) in the real dataset, we developed a “ground truth” using the judgment of two expert operators. Their judgment was performed blind to the details of the ChoOSyn algorithm as well as to the results of the various methods tested. For each real subject, they analyzed the muscle synergy plots considering different numbers of muscle synergies n and they chose—separately—the number they considered as correct, based on their knowledge of motor control strategies, muscle synergy analysis, and gait biomechanics. It should be noted that expert judgment is subjective, at least to some extent. Cohen’s kappa statistic [41] was used to compute the degree of agreement between the raters. In case of disagreement, the two expert operators discussed the discordant cases to achieve a common ground truth. For the simulated dataset, its own nature guarantees its objectivity, knowing a priori the correct number of muscle synergies.

## 3. Results

#### 3.1. Simulated Data

#### 3.2. Real Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Example of the generation of a simulated sEMG signal for the lateral gastrocnemius (LGS) muscle: the first step is (

**A**) the extraction of muscle synergies ($W$ and $C$ ) from the real data of a representative subject with 5 muscle synergies, the second is (

**B**) the reconstruction of the LGS envelope (obtained as $W$

_{LGS}* $C$ ). Then, (

**C**) a simulated sEMG signal without additive noise is generated. Finally, noise is added to the previous signals. An example of a simulated sEMG signal with SNR = 20 dB is shown in (

**D**).

**Figure 3.**Examples of $ChoOSy{n}_{W}$ and $ChoOSy{n}_{C}$ values calculated on muscle synergies extracted from the data of two representative real subjects. “Steps” and local minima are highlighted by red segments. These examples show how the optimal number of synergies is chosen when the outputs of the two parameters are (

**A**) the same or (

**B**) different.

**Figure 5.**Bar plots representing the mean ± standard error of the two parameters $ChoOSy{n}_{W}$ and $ChoOSy{n}_{C}$. (

**A**–

**C**) Upper plots: simulated dataset. Each bar represents a different noise condition. The dataset is divided into three subsets with (

**A**) 4, (

**B**) 5, and (

**C**) 6 muscle synergies, respectively. (

**D**–

**F**) Bottom plots: real dataset. It is divided into subjects that express (

**D**) 4, (

**E**) 5, and (

**F**) 6 muscle synergies, respectively. We used the ground truth to divide the real dataset into three subsets.

**Table 1.**Simulated dataset—performance of the different methods in terms of the fraction of correctly classified, mean error (ME), and root-mean-squared error (RMSE).

Fraction of CorrectlyClassified | T-VAF(90%) | T-VAF(95%) | E-VAF | P-VAF | ChoOSyn |

No noise | 2/75 | 36/75 | 75/75 | 75/75 | 73/75 |

SNR = 30 dB | 0/75 | 24/75 | 75/75 | 75/75 | 73/75 |

SNR = 25 dB | 0/75 | 18/75 | 75/75 | 75/75 | 74/75 |

SNR = 20 dB | 0/75 | 9/75 | 74/75 | 72/75 | 72/75 |

SNR = 15 dB | 0/75 | 0/75 | 65/75 | 73/75 | 63/75 |

ME ^{1} | T-VAF(90%) | T-VAF(95%) | E-VAF | P-VAF | ChoOSyn |

No noise | −1.29 | −0.52 | 0.00 | 0.00 | −0.03 |

SNR = 30 dB | −1.48 | −0.68 | 0.00 | 0.00 | −0.03 |

SNR = 25 dB | −1.57 | −0.79 | 0.00 | 0.00 | −0.01 |

SNR = 20 dB | −2.11 | −1.04 | 0.01 | 0.04 | −0.05 |

SNR = 15 dB | −3.07 | −1.93 | −0.15 | 0.03 | 0.04 |

RMSE ^{1} | T-VAF(90%) | T-VAF(95%) | E-VAF | P-VAF | ChoOSyn |

No noise | 1.39 | 0.72 | 0.00 | 0.00 | 0.16 |

SNR = 30 dB | 1.56 | 0.82 | 0.00 | 0.00 | 0.16 |

SNR = 25 dB | 1.65 | 0.92 | 0.00 | 0.00 | 0.12 |

SNR = 20 dB | 2.19 | 1.17 | 0.12 | 0.20 | 0.28 |

SNR = 15 dB | 3.15 | 2.00 | 0.53 | 0.16 | 0.53 |

^{1}Unit of measure of ME and RMSE: number of synergies.

**Table 2.**Real dataset—Performance of the different methods as the fraction of correctly classified, mean error (ME), and root-mean-squared error (RMSE).

T-VAF (90%) | T-VAF (95%) | E-VAF | P-VAF | ChoOSyn | |
---|---|---|---|---|---|

Fraction of correctly classified | 8/20 | 7/20 | 12/20 | 6/20 | 17/20 |

ME ^{1} | −0.90 | 0.55 | 0.70 | 0.90 | 0.20 |

RMSE ^{1} | 1.30 | 0.98 | 1.18 | 1.18 | 0.55 |

^{1}Unit of measure of ME and RMSE: number of synergies.

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Ballarini, R.; Ghislieri, M.; Knaflitz, M.; Agostini, V.
An Algorithm for Choosing the Optimal Number of Muscle Synergies during Walking. *Sensors* **2021**, *21*, 3311.
https://doi.org/10.3390/s21103311

**AMA Style**

Ballarini R, Ghislieri M, Knaflitz M, Agostini V.
An Algorithm for Choosing the Optimal Number of Muscle Synergies during Walking. *Sensors*. 2021; 21(10):3311.
https://doi.org/10.3390/s21103311

**Chicago/Turabian Style**

Ballarini, Riccardo, Marco Ghislieri, Marco Knaflitz, and Valentina Agostini.
2021. "An Algorithm for Choosing the Optimal Number of Muscle Synergies during Walking" *Sensors* 21, no. 10: 3311.
https://doi.org/10.3390/s21103311