# Modification of Hand Muscular Synergies in Stroke Patients After Robot-Aided Rehabilitation
## 2 Materials and Methods
### EMG Acquisition and Muscle Synergies Extraction  
SEMG signals were acquired from six muscles of the hand: flexor digitorum superficialis (FDS), extensor digitorum superficialis (EDS), flexor pollicis brevis (FPB), abductor pollicis brevis (APB), abductor digiti minimi (ADM), and exstensor digiti minimi (EDM). Before placing the electrodes, the hairs were removed from the skin surface and cleaned thoroughly with alcohol. The sEMG data were collected at 1 kHz and stored in a text file for offline analysis. A visual inspection was performed to avoid the presence of artifacts that contaminated the signal. They were filtered with a sixth-order Butterworth filter with cutoff frequencies of 30,450 Hz and a second-order Butterworth notch filter (50 Hz) and then rectified to extract muscle synergies. Muscle synergies were extracted from EMG signals by using the non-negative matrix factorization (NNMF) algorithm [33] as  
\[E(t)=\sum_{i=1}^{n}(W_{i}*H_{i}(t))+e \tag{1}\]  
where \(E(t)\) is an N \(\times\) M EMG signals matrix (N muscles and M number of the samples), W is the N \(\times\) S synergy matrix (S number of synergies), and H is the T \(\times\) M coefficient matrix. Each column of W contains the weights of each muscle for the corresponding synergy, and each row of H represents how much the corresponding synergy was activated or used to generate force. With this assumption, the contribution of a single muscle to the performed task is given by the linear combination of the product between the muscle gain \(W_{i}\) and the corresponding activation coefficient \(H_{i}(t)\).  
Figure 4: The Gloreha Sinfonia, the robotic exoskeleton for the neuro-rehabilitation of the hand.  
For muscle synergies extraction, the NMF algorithm was initialized with random matrices with elements belonging to a uniform distribution with values between 0 and 1, then updated until convergence, so that the coefficient of determination \(R^{2}<0.01\%\). The synergy extractions were iterated 50 times for each limb of each subject and the synergies with the highest \(R^{2}\) were selected for further comparisons, because the obtained solutions for the synergies and their coefficients found by the algorithm may represent a local extremum on the \(R^{2}\) surface ([15]). For this reason, the largest \(R^{2}\) was selected for subsequent analysis, in order to ensure positioning as close as possible to the absolute maximum on the surface of the \(R^{2}\). The value of 50 was chosen as it is considered a good trade-off between the reconstruction of the EMG signal, evaluated with the \(R^{2}\), and the computational burden for the iterated calculation. The number of extracted synergies was chosen on the basis of \(R^{2}\). In other words, the EMG factorization is performed in order to minimize the difference,  
\[e(t)=E(t)-W_{i}*H_{i}(t), \tag{2}\]  
between the acquired EMG signal and the reconstructed one. The number of synergies N extracted from the EMG signal is the minimum value that returns \(R^{2}\) at least equal to 80% ([15]). More in detail, in order to select the minimum number of synergies N, suitable for a complete representation of the recorded EMG signal \(E(t)\), the sets of muscle synergies were extracted for each subject with N ranging from 1 to 6 in order to identify the minimum number of synergies for which the condition \(R^{2}>80\%\) is satisfied. At the end of this analysis, the mean value of synergies N on all the patients was considered, in order to obtain comparable synergy sets.