Purpose: A muscle synergies model was suggested to represent a simplifying motor control mechanism by the brainstem and spinal cord. The aim of the study was to investigate the feasibility of such control mechanisms in the rehabilitation of post-stroke individuals during the execution of hand-reaching movements in multiple directions, compared to non-stroke individuals.Methods: Twelve non-stroke and 13 post-stroke individuals participated in the study. Muscle synergies were extracted from EMG data that was recorded during hand reaching tasks, using the NMF algorithm. The optimal number of synergies was evaluated in both groups using the Variance Accounted For (VAF) and the Mean Squared Error (MSE). A cross validation procedure was carried out to define a representative set of synergies. The similarity index and the K-means algorithm were applied to validate the existence of such a set of synergies, but also to compare the modulation properties of synergies for different movement directions between groups. The similarity index and hierarchical cluster analysis were also applied to compare between group synergies.Results: Four synergies were chosen to optimally capture the variances in the EMG data, with mean VAF of 0.917 ± 0.034 and 0.883 ± 0.046 of the data variances, with respective MSE of 0.007 and 0.016, in the control and study groups, respectively. The representative set of synergies was set to be extracted from movement to the center of the reaching space. Two synergies had different muscle activation balance between groups. Seven and 17 clusters partitioned the muscle synergies of the control and study groups. The control group exhibited a gradual change in the activation in the amplitude in the time domain (modulation) of synergies, as reflected by the similarity index, whereas the study group exhibited consistently significant differences between all movement directions and the representative set of synergies. The study findings support the existence of a representative set of synergies, which are modulated to execute movements in different directions.Conclusions: Post-stroke individuals differently modulate the activation of synergies to different movement directions than do non-stroke individuals. The conclusion was supported by different muscle activation balances, similarity values and different classifications of synergies among groups.

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Competitive Vertex Recoloring we are required to output a new coloring c i : V → [k] such that under c i , no edge in e 1 , . . . , e i is monochromatic. The cost of a given sequence of colorings c 0 , c 1 , . . . is the total weight of recolorings, i.e., the sum over all vertices of vertex weight times the number of times that vertex was recolored. Due to this formalization, in this paper we refer to the problem interchangeably as Vertex Recoloring or Disengagement.We note that in many cases, disengagement restrictions are temporary. We refer refer to this setting as "fully-dynamic disengagement," as opposed to "semi-dynamic disengagement", in which edges are only added, and never removed. In this work, we mainly focus on the semi-dynamic variant.

Motivated by placement of jobs in physical machines, we introduce and analyze the problem of online recoloring, or online disengagement. In this problem, we are given a set of $n$ weighted vertices and a $k$-coloring of the vertices (vertices represent jobs, and colors represent physical machines). Edges, representing conflicts between jobs, are inserted in an online fashion. After every edge insertion, the algorithm must output a proper $k$-coloring of the vertices. The cost of a recoloring is the sum of weights of vertices whose color changed. Our aim is to minimize the competitive ratio of the algorithm, i.e., the ratio between the cost paid by the online algorithm and the cost paid by an optimal, offline algorithm. We consider a couple of polynomially-solvable coloring variants. Specifically, for 2-coloring bipartite graphs we present an $O(\log n)$-competitive deterministic algorithm and an $\Omega(\log n)$ lower bound on the competitive ratio of randomized algorithms. For $(\Delta+1)$-coloring, where $\Delta$ is the maximal node degree, we present tight bounds of $\Theta(\Delta)$ and $\Theta(\log\Delta)$ on the competitive ratios of deterministic and randomized algorithms, respectively (where $\Delta$ denotes the maximum degree). We also consider the fully dynamic case which allows edge deletions as well as insertions. All our algorithms are applicable to the case where vertices are weighted and the cost of recoloring a vertex is its weight. All our lower bounds hold even in the unweighted case.