Original: M ATRIX FACTORIZATION UNDER THE CONSTRAINT OF CONNECTIVITY BETWEEN OBSERVED AND SOURCE DATA
Revised: M ATRIX FACTORIZATION UNDER THE CONSTRAINT OF CONNECTIVITY BETWEEN OBSERVED AND SOURCE DATA

Original: M USCLE SYNERGY ANALYSIS BASED ON CONNECTIVITY BETWEEN MUSCLE
Revised: M USCLE SYNERGY ANALYSIS BASED ON CONNECTIVITY BETWEEN MUSCLE

Original: AND BRAIN ACTIVITIES
Revised: AND BRAIN ACTIVITIES

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Matrix factorization is a popular method to investigate the hidden elements in observed data for tasks such as speech separation and muscle synergy analysis.
Revised: Matrix factorization is a popular method to investigate the hidden elements in observed data for tasks such as speech separation and muscle synergy analysis.

Original: The hidden elements may be closely related to the source phenomenon that cause the observed phenomenon.
Revised: The hidden elements may be closely related to the source phenomenon that cause the observed phenomenon.

Original: However, conventional methods do not always factorize the observed phenomenon elements with the connectivity between the observed and source phenomena because they only use the observed phenomenon.
Revised: However, conventional methods do not always factorize the observed phenomenon elements with the connectivity between the observed and source phenomena because they only use the observed phenomenon.

Original: This paper proposes a matrix decomposition method that constrains the connectivity between observed and source data by using the representations from a decoding model from source data to observed data.
Revised: This paper proposes a matrix decomposition method that constrains the connectivity between observed and source data by using the representations from a decoding model from source data to observed data.

Original: The proposed method factorize matrices by extracting and combining representations and weights from the regression model in the regression process.
Revised: The proposed method factorizes matrices by extracting and combining representations and weights from the regression model in the regression process.

Original: We applied our method to the corticomuscular system, which is made up of corticospinal pathways between the primary motor cortex and muscles in the body and creates muscle synergies that enable efﬁcient connections between the brain and muscles.
Revised: We applied our method to the corticomuscular system, which is made up of corticospinal pathways between the primary motor cortex and muscles in the body and creates muscle synergies that enable efﬁcient connections between the brain and muscles.

Original: In this context, muscle activities are the observed phenomenon and brain activities are the source.
Revised: In this context, muscle activities are the observed phenomenon and brain activities are the source.

Original: Many previous studies have analyzed muscle synergies using only observed muscle activity, but there may be unrevealed muscle synergies under the constraint of the connectivity between brain and muscle activity.
Revised: Many previous studies have analyzed muscle synergies using only observed muscle activity, but there may be unrevealed muscle synergies under the constraint of the connectivity between brain and muscle activity.

Original: We therefore simultaneously recorded the brain activity from multiple regions of an extensive cortical area and the activity of multiple muscles of a monkey’s forelimb while it performed a reach and grasp task throughout the course of recovery from a partial spinal cord injury (SCI).
Revised: We therefore simultaneously recorded the brain activity from multiple regions of an extensive cortical area and the activity of multiple muscles of a monkey’s forelimb while it performed a reach and grasp task throughout the course of recovery from a partial spinal cord injury (SCI).

Original: Analysis from a dataset of the monkey before SCI showed that some of the muscle synergies calculated from the proposed method using brain and muscle activities, did not exhibit a high degree of similarity to synergies obtained from the conventional method.
Revised: Analysis from a dataset of the monkey before SCI showed that some of the muscle synergies calculated from the proposed method using brain and muscle activities, did not exhibit a high degree of similarity to synergies obtained from the conventional method.

Original: The proposed method results obtained from the monkey after SCI showed an adaptive change in the number of muscle synergies associated with the degree of functional recovery.
Revised: The proposed method results obtained from the monkey after SCI showed an adaptive change in the number of muscle synergies associated with the degree of functional recovery.

Original: Speciﬁcally, the numbers of muscle synergies obtained by the proposed method initially increased immediately after SCI and then gradually decreased, while those obtained by a conventional method maintained the same number before and after SCI.
Revised: Speciﬁcally, the numbers of muscle synergies obtained by the proposed method initially increased immediately after SCI and then gradually decreased, while those obtained by a conventional method maintained the same number before and after SCI.

Original: These results suggest that our method is able to capture the unrevealed connectivity in the corticomuscular system that contributes to functional recovery: in other words, that it can factorize the observed data under the constraint of the connectivity between the observed and source data.
Revised: These results suggest that our method is able to capture the unrevealed connectivity in the corticomuscular system that contributes to functional recovery: in other words, that it can factorize the observed data under the constraint of the connectivity between the observed and source data.

Original: Our work thus demonstrates the importance of using not only observed data but also source data to reveal unknown hidden elements.
Revised: Our work thus demonstrates the importance of using not only observed data but also source data to reveal unknown hidden elements.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: As human beings, we observe a complex mixture of real-world events as phenomena, and each observed phenomenon contains elements of the events in the real-world.
Revised: As human beings, we observe a complex mixture of real-world events as phenomena, and each observed phenomenon contains elements of the events in the real-world.

Original: For example, the sounds we hear may contain the one voice we are listening to, the voices of other speakers, and various environmental sounds.
Revised: For example, the sounds we hear may contain the one voice we are listening to, the voices of other speakers, and various environmental sounds.

Original: Separating the elements in the observed phenomena helps us to understand the events individually.
Revised: Separating the elements in the observed phenomena helps us to understand the events individually.

Original: Matrix factorization is used to investigate the elements in observed phe- nomena.
Revised: Matrix factorization is used to investigate the elements in observed phe- nomena.

Original: For example, a speciﬁc speech sound can be extracted from a sound containing a mixture of multiple sounds (Smaragdis, 2007), the appropriate recommendation can be calculated from a customer’s purchase history (Li et al., 2006), and muscle synergies can be extracted from multiple muscle activities (Shourijeh et al., 2016).
Revised: For example, a speciﬁc speech sound can be extracted from a sound containing a mixture of multiple sounds (Smaragdis, 2007), the appropriate recommendation can be calculated from a customer’s purchase history (Li et al., 2006), and muscle synergies can be extracted from multiple muscle activities (Shourijeh et al., 2016).

Original: In some cases, the observed phenomenon is caused by a functional connection with the source phenomenon.
Revised: In some cases, the observed phenomenon is caused by a functional connection with the source phenomenon.

Original: This connectivity is a functional connection that causes an observed phenomenon from a source phenomenon, and the factors of connectivity include such as human connections and anatomical connections.
Revised: This connectivity is a functional connection that causes an observed phenomenon from a source phenomenon, and the factors of connectivity include such as human connections and anatomical connections.

Original: In the case of muscle synergy analysis, the corticospinal pathways between the primary motor cortex and muscles function as a corticomuscular system (Liu et al., 2019), and cortical events normally propagate to the peripheral muscles.
Revised: In the case of muscle synergy analysis, the corticospinal pathways between the primary motor cortex and muscles function as a corticomuscular system (Liu et al., 2019), and cortical events normally propagate to the peripheral muscles.

Original: Studies on humans and monkeys have reported that brain and muscle activity are closely associated with cortico-motoneuronal connections(Lemon, 2008; Baldissera & Cavallari, 1993; Lemon & Grifﬁths, 2005).
Revised: Studies on humans and monkeys have reported that brain and muscle activity are closely associated with cortico-motoneuronal connections(Lemon, 2008; Baldissera & Cavallari, 1993; Lemon & Grifﬁths, 2005).

Original: In this case, muscle activities are the observed phenomenon and brain activities are the source.
Revised: In this case, muscle activities are the observed phenomenon and brain activities are the source.

Original: The corticomuscular system enables us to perform complex body movements by neurologically combining sets of simpler movements.
Revised: The corticomuscular system enables us to perform complex body movements by neurologically combining sets of simpler movements.

Original: These sets are observed as the basic pattern of muscle activity, namely, muscle synergies.
Revised: These sets are observed as the basic pattern of muscle activity, namely, muscle synergies.

Original: Muscle activity can be observed by using electromyography (EMG), a technique that reveals bioelectric potential signals.
Revised: Muscle activity can be observed by using electromyography (EMG), a technique that reveals bioelectric potential signals.

Original: Since the framework of conventional matrix factorization uses only observed data to factorize observed data, it might not factorize the observed phenomenon elements with the connectivity between the observed and source phenomena.
Revised: Since the framework of conventional matrix factorization uses only observed data to factorize observed data, it might not factorize the observed phenomenon elements with the connectivity between the observed and source phenomena.

Original: We have therefore developed a method to capture the elements in the observed data by considering the connectivity between both the observed data and the source data.
Revised: We have therefore developed a method to capture the elements in the observed data by considering the connectivity between both the observed data and the source data.

Original: Our basic idea is to use the representations in a deep neural network (DNN) model to predict the observed data from the source data.
Revised: Our basic idea is to use the representations in a deep neural network (DNN) model to predict the observed data from the source data.

Original: In this paper, we assume both source and observed data can be measured.
Revised: In this paper, we assume both source and observed data can be measured.

Original: Speciﬁcally, the factorized matrices can function as an activation scalar value and a vector of weights for the observed data at each sample.
Revised: Speciﬁcally, the factorized matrices can function as an activation scalar value and a vector of weights for the observed data at each sample.

Original: We can then obtain the factorized matrix under the connectivity constraints between the observed and source data by extracting these values.
Revised: We can then obtain the factorized matrix under the connectivity constraints between the observed and source data by extracting these values.

Original: In this paper, we report the results of applying the proposed method to muscle synergy analysis.
Revised: In this paper, we report the results of applying the proposed method to muscle synergy analysis.

Original: As stated earlier, the corticomuscular system is composed of corticospinal pathways between the primary motor cortex and muscles (Liu et al., 2019), and we can consider the muscle activities as observed data and the brain activities as source data.
Revised: As stated earlier, the corticomuscular system is composed of corticospinal pathways between the primary motor cortex and muscles (Liu et al., 2019), and we can consider the muscle activities as observed data and the brain activities as source data.

Original: To obtain brain activities closely related to muscle activity, we simultaneously measured a monkey’s brain signals using both electrocorticography (ECoG) and electromyography (EMG).
Revised: To obtain brain activities closely related to muscle activity, we simultaneously measured a monkey’s brain signals using both electrocorticography (ECoG) and electromyography (EMG).

Original: We measured the muscle and brain activities of a monkey before and after partial spinal cord injury to investigate the potential of the proposed method under the conditions of a stable and a dynamically changing nervous system.
Revised: We measured the muscle and brain activities of a monkey before and after partial spinal cord injury to investigate the potential of the proposed method under the conditions of a stable and a dynamically changing nervous system.

Original: Our main contributions are as follows.
Revised: Our main contributions are as follows.

Original: • We show a novel matric factorization framework under the constraint of connectivity between both observed data and source data, in contrast to the conventional approach that uses only observed data.
Revised: • We show a novel matric factorization framework under the constraint of connectivity between both observed data and source data, in contrast to the conventional approach that uses only observed data.

Original: • We propose a method that utilizes a model representation to predict the observed data from the source data as a factorization matrix.
Revised: • We propose a method that utilizes a model representation to predict the observed data from the source data as a factorization matrix.

Original: • We demonstrate the potential of the proposed method to capture unrevealed matrix factors by applying it to muscle synergy analysis with a comparison to the non-negative matrix approach.
Revised: • We demonstrate the potential of the proposed method to capture unrevealed matrix factors by applying it to muscle synergy analysis with a comparison to the non-negative matrix approach.

Original:  R ELATED WORKS
Revised:  R ELATED WORKS

Original: Matrix Factorization in Statistics: One approach taken in statistics is to factorize the data by extracting the components that best represent the variation in the data.
Revised: Matrix Factorization in Statistics: One approach taken in statistics is to factorize the data by extracting the components that best represent the variation in the data.

Original: Principal component analysis (PCA) is typically used for this, where the dimensionality-reduced data representation is a factorized matrix (Lee & Seung, 1999).
Revised: Principal component analysis (PCA) is typically used for this, where the dimensionality-reduced data representation is a factorized matrix (Lee & Seung, 1999).

Original: Another approach is to separate independent components by assuming that the data contains multiple data originating from independent sources.
Revised: Another approach is to separate independent components by assuming that the data contains multiple data originating from independent sources.

Original: Independent component analysis (ICA) separates signals by independence on the basis of higher-order statistics or temporal correlations (Comon, 1994).
Revised: Independent component analysis (ICA) separates signals by independence on the basis of higher-order statistics or temporal correlations (Comon, 1994).

Original: The fast ﬁxed-point algorithm for independent component analysis (FastICA) was proposed as a method to improve the convergence of ICA (Hyv¨arinen & Oja, 2000).
Revised: The fast ﬁxed-point algorithm for independent component analysis (FastICA) was proposed as a method to improve the convergence of ICA (Hyv¨arinen & Oja, 2000).

Original: Non-negative matrix factorization (NMF) factorizes the matrix and restricts it to be positive
Revised: Non-negative matrix factorization (NMF) factorizes the matrix and restricts it to be positive

Original: (Lee & Seung, 2000; Wang & Zhang, 2013).
Revised: (Lee & Seung, 2000; Wang & Zhang, 2013).

Original: Since the real-world events we observe are often composed of linear sums of non-negative values, they have a high afﬁnity with NMF analysis, so it has accordingly been used in a variety of applications (Rajapakse & Wyse, 2003; Smaragdis, 2007;
Revised: Since the real-world events we observe are often composed of linear sums of non-negative values, they have a high afﬁnity with NMF analysis, so it has accordingly been used in a variety of applications (Rajapakse & Wyse, 2003; Smaragdis, 2007;

Original: Shourijeh et al., 2016).
Revised: Shourijeh et al., 2016).

Original: One study on muscle synergy analysis showed that NMF is the most popular method for analysis (Rabbi et al., 2020).
Revised: One study on muscle synergy analysis showed that NMF is the most popular method for analysis (Rabbi et al., 2020).

Original: However, these methods factorize the observed data using only the observed data.
Revised: However, these methods factorize the observed data using only the observed data.

Original: Deep Matrix Factorization:
Revised: Deep Matrix Factorization:

Original: The deep matrix factorization approach has shown promise in studies that focus on interpretability.
Revised: The deep matrix factorization approach has shown promise in studies that focus on interpretability.

Original: Several methods to improve the interpretability by using representations in an autoencoder process as multi-factorized matrices have been proposed (Ye et al., 2018;
Revised: Several methods to improve the interpretability by using representations in an autoencoder process as multi-factorized matrices have been proposed (Ye et al., 2018;

Original: Song et al., 2015; Wang & Zhang, 2021).
Revised: Song et al., 2015; Wang & Zhang, 2021).

Original: To ensure meaning in the multilayered factor matrix,
Revised: To ensure meaning in the multilayered factor matrix,

Original: Trigeorgis et al.
Revised: Trigeorgis et al.

Original: (2014) proposed a method that pre-trains each layered factor matrix to represent elements such as facial pose, expressions, and identity.
Revised: (2014) proposed a method that pre-trains each layered factor matrix to represent elements such as facial pose, expressions, and identity.

Original: Sparseness is also essential to capture the essence of the data.
Revised: Sparseness is also essential to capture the essence of the data.

Original: Chang et al.
Revised: Chang et al.

Original: (2021) proposed making the factorized matrix sparse in not only decoding but also encoding.
Revised: (2021) proposed making the factorized matrix sparse in not only decoding but also encoding.

Original: One concept similar to our own is the use of implicit feedback (e.g., purchase history and unobserved ratings) in recommendation systems (Xue et al., 2017).
Revised: One concept similar to our own is the use of implicit feedback (e.g., purchase history and unobserved ratings) in recommendation systems (Xue et al., 2017).

Original: While we feel that such feedback can contribute to improving accuracy, it is not the same as a source phenomenon.
Revised: While we feel that such feedback can contribute to improving accuracy, it is not the same as a source phenomenon.

Original: And technically, there are few approaches for matrix factorization to use both parameters in the regression process and the weights inside the model.
Revised: And technically, there are few approaches for matrix factorization to use both parameters in the regression process and the weights inside the model.

Original: Applications: There are a variety of general applications of the matrix factorization approach.
Revised: Applications: There are a variety of general applications of the matrix factorization approach.

Original: In the biological ﬁeld, metagenes and molecular pattern discovery have been studied by applying NMF (Brunet et al., 2004; Devarajan, 2008).
Revised: In the biological ﬁeld, metagenes and molecular pattern discovery have been studied by applying NMF (Brunet et al., 2004; Devarajan, 2008).

Original: In the engineering ﬁeld, image processing tasks such as face recognition (Rajapakse & Wyse, 2003), acoustic processing (e.g., speech-music separation) (Smita et al., 2007), and speech separation (Chien & Chen, 2006) have been studied.
Revised: In the engineering ﬁeld, image processing tasks such as face recognition (Rajapakse & Wyse, 2003), acoustic processing (e.g., speech-music separation) (Smita et al., 2007), and speech separation (Chien & Chen, 2006) have been studied.

Original: As for medical applications, fetal heart rate determination has been explored (Szalai & Mozes, 2014).
Revised: As for medical applications, fetal heart rate determination has been explored (Szalai & Mozes, 2014).

Original: In the kinematics and kinetics ﬁeld, muscle synergy analysis is a signiﬁcant application of matrix factorization
Revised: In the kinematics and kinetics ﬁeld, muscle synergy analysis is a signiﬁcant application of matrix factorization

Original: Shourijeh et al.
Revised: Shourijeh et al.

Original: Matrix factorization has also been applied to community structure detection (e.g., social networks, collaboration networks, and citation networks) to study interaction (Ye et al., 2018).
Revised: Matrix factorization has also been applied to community structure detection (e.g., social networks, collaboration networks, and citation networks) to study interaction (Ye et al., 2018).

Original:  M ATRIX FACTORIZATION UNDER CONSTRAINTS BETWEEN OBSERVED AND SOURCE DATA
Revised:  M ATRIX FACTORIZATION UNDER CONSTRAINTS BETWEEN OBSERVED AND SOURCE DATA

Original: Suppose that the observed data are arranged as a matrix Y =
Revised: Suppose that the observed data are arranged as a matrix Y =

Original: [ y 1 , . . .
Revised: [ y 1 , . . .

Original: , y T ] T ∈ R n × T , which is n channels sampled T times in time series.
Revised: , y T ] T ∈ R n × T , which is n channels sampled T times in time series.

Original: The observed vector at time t represents y t =
Revised: The observed vector at time t represents y t =

Original: [ y 1 t , y 2 t , . .
Revised: [ y 1 t , y 2 t , . .

Original: y nt ] .
Revised: y nt ] .

Original: The goal of the proposed method is to approximately factorize the observed data as
Revised: The goal of the proposed method is to approximately factorize the observed data as

Original: [Equation]
Revised: [Equation]

Original: where V =
Revised: where V =

Original: [ v 1 , . .
Revised: [ v 1 , . .

Original: . , v r ] T ∈ R n × r and U =
Revised: . , v r ] T ∈ R n × r and U =

Original: [ u 1 , . . .
Revised: [ u 1 , . . .

Original: , u r ] ∈ R r × T are the factor matrices to reconstruct the matrix Y , and r is the number of factorized components.
Revised: , u r ] ∈ R r × T are the factor matrices to reconstruct the matrix Y , and r is the number of factorized components.

Original: Suppose the source data is the m channels sampled T times in time series, and the source data at time t is represented as the matrix X t
Revised: Suppose the source data is the m channels sampled T times in time series, and the source data at time t is represented as the matrix X t

Original: =
Revised: =

Original: [ x (1 ,t ) , . . .
Revised: [ x (1 ,t ) , . . .

Original: , x ( m,t ) ]
Revised: , x ( m,t ) ]

Original: ∈ R m × T , which is m channels sampled T times in time series.
Revised: ∈ R m × T , which is m channels sampled T times in time series.

Original: Considering the time difference ( τ ) between observed and source data, the vector of source data at time t consists of data from time t to t − τ .
Revised: Considering the time difference ( τ ) between observed and source data, the vector of source data at time t consists of data from time t to t − τ .

Original: The i -th source data vector at time t is x ( i,t ) =
Revised: The i -th source data vector at time t is x ( i,t ) =

Original: [ x ( i,t ) , x ( i,t − 1) , x ( i,t − 2) , . . .
Revised: [ x ( i,t ) , x ( i,t − 1) , x ( i,t − 2) , . . .

Original: , x ( i,t − τ ) ] .
Revised: , x ( i,t − τ ) ] .

Original: The basic idea underlying the factorization of matrices under constraints between observed and source data is to factor matrices from a model that includes connectivity constraints between observed and source data.
Revised: The basic idea underlying the factorization of matrices under constraints between observed and source data is to factor matrices from a model that includes connectivity constraints between observed and source data.

Original: We obtain this model by constructing a DNN prediction model from the source data to the observed data, where the factorized vectors v and u can be assigned as the weight for each channel of observed data and the activation for the weights, respectively.
Revised: We obtain this model by constructing a DNN prediction model from the source data to the observed data, where the factorized vectors v and u can be assigned as the weight for each channel of observed data and the activation for the weights, respectively.

Original: 3. F RAMEWORK AND A RCHITECTURE
Revised: 3. F RAMEWORK AND A RCHITECTURE

Original: We designed a regression model M that computes the observed data y t from the source data X t .
Revised: We designed a regression model M that computes the observed data y t from the source data X t .

Original: As shown in the Fig.
Revised: As shown in the Fig.

Original: 1, the representation in the model is designed to contain activation (scalar) and weights (vector).
Revised: 1, the representation in the model is designed to contain activation (scalar) and weights (vector).

Original: It is not only the source data that contributes to the factorization at time t ; data over a certain time range up to t is also considered to be involved in some cases.
Revised: It is not only the source data that contributes to the factorization at time t ; data over a certain time range up to t is also considered to be involved in some cases.

Original: Temporal convolutional networks (TCN) have outperformed recurrent networks such as LSTM and GRU (Bai et al., 2018) for handling multi-channel time-series data and have been applied to gesture recognition muscle activity (Rahimian et al., 2021; 2022).
Revised: Temporal convolutional networks (TCN) have outperformed recurrent networks such as LSTM and GRU (Bai et al., 2018) for handling multi-channel time-series data and have been applied to gesture recognition muscle activity (Rahimian et al., 2021; 2022).

Original: Inspired by such research, we utilize TCN to incorporate the time- features in method.
Revised: Inspired by such research, we utilize TCN to incorporate the time- features in method.

Original: TCNet :
Revised: TCNet :

Original: We implement TCNet as the TCN in the regression model to capture the features from sequence data.
Revised: We implement TCNet as the TCN in the regression model to capture the features from sequence data.

Original: As in (Bai et al., 2018; Rahimian et al., 2021; 2022), the TCNet layer consists of a sequence of TBlock layers, as shown in Fig.
Revised: As in (Bai et al., 2018; Rahimian et al., 2021; 2022), the TCNet layer consists of a sequence of TBlock layers, as shown in Fig.

Original: The TBlock is the dilated causal convolution, with the dilation factor exponentially increased (1, 2, 4, ...) and the number of ﬁlters c and the kernel size k .
Revised: The TBlock is the dilated causal convolution, with the dilation factor exponentially increased (1, 2, 4, ...) and the number of ﬁlters c and the kernel size k .

Original: Each TBlock consists of two repeated sequences consisting of extended causal convolutions (“DCconv”), weight norm, ReLU and Dropout, and ﬁnally a residual connection.
Revised: Each TBlock consists of two repeated sequences consisting of extended causal convolutions (“DCconv”), weight norm, ReLU and Dropout, and ﬁnally a residual connection.

Original: The parameters for TCNet are the number of ﬁlters c , the kernel size k , and the number of dilations d .
Revised: The parameters for TCNet are the number of ﬁlters c , the kernel size k , and the number of dilations d .

Original: When d is set to three, the dilation of
Revised: When d is set to three, the dilation of

Original: TBlock is 1, 2, and 4 sequentially.
Revised: TBlock is 1, 2, and 4 sequentially.

Original: This layer outputs a matrix ∈ R ( c × r ) to Mean for each input data X t .
Revised: This layer outputs a matrix ∈ R ( c × r ) to Mean for each input data X t .

Original: Mean, Scalarization, and Sigmoid layers : We designed these layers so that the captured features from TCNet would have a positive scalar value (activation u t ).
Revised: Mean, Scalarization, and Sigmoid layers : We designed these layers so that the captured features from TCNet would have a positive scalar value (activation u t ).

Original: The Mean layer calculates a vector ( R c ) by averaging each ﬁlter’s result.
Revised: The Mean layer calculates a vector ( R c ) by averaging each ﬁlter’s result.

Original: The Scalarization layer is a linear layer to calculate a scalar value from a vector.
Revised: The Scalarization layer is a linear layer to calculate a scalar value from a vector.

Original: The Sigmoid layer is set to keep the computed scalar value u t within a positive range.
Revised: The Sigmoid layer is set to keep the computed scalar value u t within a positive range.

Original: This scalar value u t is calculated at each time by sliding the window of the source data.
Revised: This scalar value u t is calculated at each time by sliding the window of the source data.

Original: Linear layer : The Linear layer inputs a positive scalar value and calculates the reconstructed observed vector ˆ y t =
Revised: Linear layer : The Linear layer inputs a positive scalar value and calculates the reconstructed observed vector ˆ y t =

Original: [ˆ y 1 t , ˆ y 2 t , . .
Revised: [ˆ y 1 t , ˆ y 2 t , . .

Original: ˆ y nt ] as
Revised: ˆ y nt ] as

Original: [Equation]
Revised: [Equation]

Original: where w = [ w 1 , w 2 , . . . , w n ] is the weight for each element of an observed vector.
Revised: where w = [ w 1 , w 2 , . . . , w n ] is the weight for each element of an observed vector.

Original: When acquiring a vector of non-negative weights, the weights are clipped so that they are between 0 and 1 during training.
Revised: When acquiring a vector of non-negative weights, the weights are clipped so that they are between 0 and 1 during training.

Original: 3. M ATRIX FACTORIZATION THROUGH REGRESSION CALCULATIONS
Revised: 3. M ATRIX FACTORIZATION THROUGH REGRESSION CALCULATIONS

Original: First, the regression model M is trained using X t and y t t ∈ (1 , . . .
Revised: First, the regression model M is trained using X t and y t t ∈ (1 , . . .

Original: , T ) .
Revised: , T ) .

Original: Similar to the previous approach, (Shourijeh et al., 2016;
Revised: Similar to the previous approach, (Shourijeh et al., 2016;

Original: Rajapakse & Wyse, 2003), the proposed matrix factorization is veriﬁed by the restoration degree of the observed data, so there is no need to separate source and observed data into training and test data.
Revised: Rajapakse & Wyse, 2003), the proposed matrix factorization is veriﬁed by the restoration degree of the observed data, so there is no need to separate source and observed data into training and test data.

Original: The regression model M predicts the observed data y t from the source data X t as M ( X t ) .
Revised: The regression model M predicts the observed data y t from the source data X t as M ( X t ) .

Original: An activation u t is obtained from the representation (scalar) from the layer of Sigmoid , and a weights vector v is obtained by extracting the weights at the Linear layer in a model M .
Revised: An activation u t is obtained from the representation (scalar) from the layer of Sigmoid , and a weights vector v is obtained by extracting the weights at the Linear layer in a model M .

Original: u t is the activation of the component at time t .
Revised: u t is the activation of the component at time t .

Original: X t is a sample extracted from the time series source data with a τ -width time window.
Revised: X t is a sample extracted from the time series source data with a τ -width time window.

Original: When the window locates at t = 1 , Xis extracted from the time series source data.
Revised: When the window locates at t = 1 , Xis extracted from the time series source data.

Original: Regression model M inputs X 1 and outputs y 1 , and u 1 is obtained from this calculation process.
Revised: Regression model M inputs X 1 and outputs y 1 , and u 1 is obtained from this calculation process.

Original: By sliding the window, regression model M inputs
Revised: By sliding the window, regression model M inputs

Original: X t , t ∈ (1 , . .
Revised: X t , t ∈ (1 , . .

Original: . , T ) and outputs y t , t ∈ (1 , . .
Revised: . , T ) and outputs y t , t ∈ (1 , . .

Original: . , T ) .
Revised: . , T ) .

Original: u t , t ∈ (1 , . .
Revised: u t , t ∈ (1 , . .

Original: . , T ) is obtained from each time.
Revised: . , T ) is obtained from each time.

Original: By concatenating u t from each time, we get the activation vector u =
Revised: By concatenating u t from each time, we get the activation vector u =

Original: [ u 1 , u 2 , . .
Revised: [ u 1 , u 2 , . .

Original: , u T ] .
Revised: , u T ] .

Original: In this way, we can obtain an activation and a weight vector of a component.
Revised: In this way, we can obtain an activation and a weight vector of a component.

Original: Input: X t and y t , t ∈ { 1 , . . .
Revised: Input: X t and y t , t ∈ { 1 , . . .

Original: , T } , and r .
Revised: , T } , and r .

Original: repeat
Revised: repeat

Original: Update parameters in M i , i ∈ { 1 , . .
Revised: Update parameters in M i , i ∈ { 1 , . .

Original: , r }
Revised: , r }

Original: Clip w in Linear : 0 < w <until objective function is converged u i ⇐ [ u t ] , t ∈ { 1 , . . .
Revised: Clip w in Linear : 0 < w <until objective function is converged u i ⇐ [ u t ] , t ∈ { 1 , . . .

Original: , T } from Sigmoid in M i v i ⇐ w from Linear in M
Revised: , T } from Sigmoid in M i v i ⇐ w from Linear in M

Original: i
Revised: i

Original: V ⇐ [ v i ] , i ∈ { 1 , . . .
Revised: V ⇐ [ v i ] , i ∈ { 1 , . . .

Original: , r }
Revised: , r }

Original: U ⇐ [ u i ] , i ∈ { 1 , . . .
Revised: U ⇐ [ u i ] , i ∈ { 1 , . . .

Original: , r }
Revised: , r }

Original: Output: ˆ Y = V U
Revised: Output: ˆ Y = V U

Original: Algo. 1 explains the factorization procedure.
Revised: Algo. 1 explains the factorization procedure.

Original: The reconstructed observed data ˆ y t can be more accurately reconstructed by ensembling M as
Revised: The reconstructed observed data ˆ y t can be more accurately reconstructed by ensembling M as

Original: [Equation]
Revised: [Equation]

Original: where r is the number of factorized components and M
Revised: where r is the number of factorized components and M

Original: i is the i -th model.
Revised: i is the i -th model.

Original: In computing matrix factorization, the number of components r needs to be given.
Revised: In computing matrix factorization, the number of components r needs to be given.

Original: The appropriate number of factorized components r is determined according to evaluation indices such as the degree of restoration.
Revised: The appropriate number of factorized components r is determined according to evaluation indices such as the degree of restoration.

Original: M i , i ∈ (1 , . . .
Revised: M i , i ∈ (1 , . . .

Original: , r ) are trained in parallel to minimize the objective function using the same dataset across models.
Revised: , r ) are trained in parallel to minimize the objective function using the same dataset across models.

Original: From each M i , we obtain the activation vector u i and the weight vector v i .
Revised: From each M i , we obtain the activation vector u i and the weight vector v i .

Original: The factorized matrix V is obtained by combining the weight vectors as [ v 1 , . .
Revised: The factorized matrix V is obtained by combining the weight vectors as [ v 1 , . .

Original: , v r ] .
Revised: , v r ] .

Original: The factorized matrix U is also obtained by combining the activation vectors as [ u 1 , . . .
Revised: The factorized matrix U is also obtained by combining the activation vectors as [ u 1 , . . .

Original: , u r ]
Revised: , u r ]

Original: The objective function is
Revised: The objective function is

Original: [Equation]
Revised: [Equation]

Original: where L ( y t , ∑ ri =1 M i ( X t )) is the reconstruction loss and R ( w ) is the regularization term for the weight.
Revised: where L ( y t , ∑ ri =1 M i ( X t )) is the reconstruction loss and R ( w ) is the regularization term for the weight.

Original: The parameters of the models are updated to minimize the objective function are denoted as M i .
Revised: The parameters of the models are updated to minimize the objective function are denoted as M i .

Original: At each iteration of learning model, each element of the weights is clipped from 0 to 1 to obtain the positive factors of the matrix.
Revised: At each iteration of learning model, each element of the weights is clipped from 0 to 1 to obtain the positive factors of the matrix.

Original: The weight vector w in M i is used as v i .
Revised: The weight vector w in M i is used as v i .

Original: V and U are obtained by concatenating of vectors v i and u i , i ∈ { 1 , . . .
Revised: V and U are obtained by concatenating of vectors v i and u i , i ∈ { 1 , . . .

Original: , r } , respectively.
Revised: , r } , respectively.

Original: Reconstructed observed data ˆ Y =
Revised: Reconstructed observed data ˆ Y =

Original: [ y t ] , t ∈ { 1 , . . .
Revised: [ y t ] , t ∈ { 1 , . . .

Original: , T } is reconstructed by V and U .
Revised: , T } is reconstructed by V and U .

Original:  E XPERIMENT
Revised:  E XPERIMENT

Original: We applied the proposed method to muscle synergy analysis.
Revised: We applied the proposed method to muscle synergy analysis.

Original: As stated in the introduction, our bodies contain corticospinal pathways between the primary motor cortex and muscles that function as a corticomuscular system (Liu et al., 2019).
Revised: As stated in the introduction, our bodies contain corticospinal pathways between the primary motor cortex and muscles that function as a corticomuscular system (Liu et al., 2019).

Original: Normally, cortical events propagate to the peripheral muscles, and the corticomuscular system enables us to perform complex body movements by neurologically combining sets of simpler movements.
Revised: Normally, cortical events propagate to the peripheral muscles, and the corticomuscular system enables us to perform complex body movements by neurologically combining sets of simpler movements.

Original: These simpler movements are observed as the basic pattern of muscle activity, namely, muscle synergies.
Revised: These simpler movements are observed as the basic pattern of muscle activity, namely, muscle synergies.

Original: Muscle synergy analysis is designed to capture this efﬁcient neural connection between the brain and muscle.
Revised: Muscle synergy analysis is designed to capture this efﬁcient neural connection between the brain and muscle.

Original: However, the conventional approach uses only muscle activities (observed phenomena) to capture the muscle synergies, and there may still be unexplored muscle synergies that remain hidden.
Revised: However, the conventional approach uses only muscle activities (observed phenomena) to capture the muscle synergies, and there may still be unexplored muscle synergies that remain hidden.

Original: In applying the proposed method to muscle synergy analysis in this experiment, we consider the muscle activities as the observed phenomenon and the brain activities as the source phenomenon.
Revised: In applying the proposed method to muscle synergy analysis in this experiment, we consider the muscle activities as the observed phenomenon and the brain activities as the source phenomenon.

Original: To test the proposed method, we measured the muscle and brain activities simultaneously in a monkey.
Revised: To test the proposed method, we measured the muscle and brain activities simultaneously in a monkey.

Original: 4. D ATA MEASUREMENT
Revised: 4. D ATA MEASUREMENT

Original: We measured the biopotential signals derived from the muscles and brain activities.
Revised: We measured the biopotential signals derived from the muscles and brain activities.

Original: The invasively measured electrocorticographic (ECoG) signal was selected as the brain activity data because it contains more detailed brain activity than a non-invasively measured signal.
Revised: The invasively measured electrocorticographic (ECoG) signal was selected as the brain activity data because it contains more detailed brain activity than a non-invasively measured signal.

Original: Thus, to obtain both the brain activity and the muscle activity, we recorded ECoG and electromyographic (EMG) signals simultaneously from the contralateral hemisphere and the ipsilateral forelimb muscles of a monkey, respectively.
Revised: Thus, to obtain both the brain activity and the muscle activity, we recorded ECoG and electromyographic (EMG) signals simultaneously from the contralateral hemisphere and the ipsilateral forelimb muscles of a monkey, respectively.

Original: The monkey used in this experiments was an adult Japanese macaque monkey (Macaca fuscata;
Revised: The monkey used in this experiments was an adult Japanese macaque monkey (Macaca fuscata;

Original: Monkey D: female, 6.0 kg, 7 years old).
Revised: Monkey D: female, 6.0 kg, 7 years old).

Original: The Ethical description is in A.1.
Revised: The Ethical description is in A.1.

Original: Processing : ECoG signals were down-sampled to 500 samples per second, similar to previous works (Shin et al., 2012;
Revised: Processing : ECoG signals were down-sampled to 500 samples per second, similar to previous works (Shin et al., 2012;

Original: Hidenori et al., 2012).
Revised: Hidenori et al., 2012).

Original: Fourth-order Butterworth bandpass ﬁlters were applied to each ECoG signal, dividing them into multiple speciﬁc bands ( δ (1 . 5 − 4 Hz ) , θ (4 − 8 Hz ) , α (8 − 14 Hz ) , β 1(14 − 20 Hz ) , β 2(20 − 30 Hz ) , γ 1(30 − 50 Hz ) , γ 2(50 − 90 Hz ) , γ 3(90 − 120 Hz ) and γ 4(120 − 150 Hz ) ).
Revised: Fourth-order Butterworth bandpass ﬁlters were applied to each ECoG signal, dividing them into multiple speciﬁc bands ( δ (1 . 5 − 4 Hz ) , θ (4 − 8 Hz ) , α (8 − 14 Hz ) , β 1(14 − 20 Hz ) , β 2(20 − 30 Hz ) , γ 1(30 − 50 Hz ) , γ 2(50 − 90 Hz ) , γ 3(90 − 120 Hz ) and γ 4(120 − 150 Hz ) ).

Original: The nine bandpass ﬁlters split each of the 15-channel ECoG signals into nine signals to produce 135 channels of bandpass-ﬁltered signals.
Revised: The nine bandpass ﬁlters split each of the 15-channel ECoG signals into nine signals to produce 135 channels of bandpass-ﬁltered signals.

Original: We performed normalization and smoothing the same as in (Shin et al., 2012; Hidenori et al., 2012).
Revised: We performed normalization and smoothing the same as in (Shin et al., 2012; Hidenori et al., 2012).

Original: A bandpass ﬁlter that passes a 20–500-Hz frequency was applied to the EMG signals for removing motion artifacts.
Revised: A bandpass ﬁlter that passes a 20–500-Hz frequency was applied to the EMG signals for removing motion artifacts.

Original: Bandpassed signals were then rectiﬁed and passed through a fourth-order low-pass ﬁlter with a cut-off frequency of 4 Hz.
Revised: Bandpassed signals were then rectiﬁed and passed through a fourth-order low-pass ﬁlter with a cut-off frequency of 4 Hz.

Original: Finally, the signals were down-sampled to 500 Hz.
Revised: Finally, the signals were down-sampled to 500 Hz.

Original: To account for the nerve signal transmission delay between brain and muscles, delayed signal values is also used.
Revised: To account for the nerve signal transmission delay between brain and muscles, delayed signal values is also used.

Original: The value of discrete-time step-size ∆ t is set to 20 ms as in the previous study (Shin et al., 2012).
Revised: The value of discrete-time step-size ∆ t is set to 20 ms as in the previous study (Shin et al., 2012).

Original: The muscle activity at time t is predicted using 10 time points starting 200 ms before the target time t.
Revised: The muscle activity at time t is predicted using 10 time points starting 200 ms before the target time t.

Original: The i -th ECoG data vector at time t is x ( i,t ) =
Revised: The i -th ECoG data vector at time t is x ( i,t ) =

Original: [ x ( i,t ) , x ( i,t − 20) , x ( i,t − 40) , . . .
Revised: [ x ( i,t ) , x ( i,t − 20) , x ( i,t − 40) , . . .

Original: , x ( i,t − 200) ] , and the ECoG data matrix at time t is X t =
Revised: , x ( i,t − 200) ] , and the ECoG data matrix at time t is X t =

Original: [ x (1 ,t ) , . . .
Revised: [ x (1 ,t ) , . . .

Original: , x (135 ,t ) ] .
Revised: , x (135 ,t ) ] .

Original: Baseline : We utilized non-negative matrix factorization (NMF) as a baseline and used the Python library scikit-learn.
Revised: Baseline : We utilized non-negative matrix factorization (NMF) as a baseline and used the Python library scikit-learn.

Original: All parameters were set to the same as those for the pre-set.
Revised: All parameters were set to the same as those for the pre-set.

Original: Metric : Variance accounted for (VAF) is the main metric of the degree of reconstruction and used to determine the number of synergies (Steele et al., 2013; Delis et al., 2013), where the dimension of factors is determined when VAF exceeds 0.9.
Revised: Metric : Variance accounted for (VAF) is the main metric of the degree of reconstruction and used to determine the number of synergies (Steele et al., 2013; Delis et al., 2013), where the dimension of factors is determined when VAF exceeds 0.9.

Original: The dimension of factors is used as the synergy number in some cases (Hug et al., 2010; Fr`ere & Hug, 2012; Chvatal & Ting, 2013).
Revised: The dimension of factors is used as the synergy number in some cases (Hug et al., 2010; Fr`ere & Hug, 2012; Chvatal & Ting, 2013).

Original: The speciﬁc calculation is decribed in A.4.
Revised: The speciﬁc calculation is decribed in A.4.

Original: In our analysis, the similarity of the muscle synergy components resulting from the two methods (proposed and NMF) is calculated by the cosine similarity of the weights of the synergies.
Revised: In our analysis, the similarity of the muscle synergy components resulting from the two methods (proposed and NMF) is calculated by the cosine similarity of the weights of the synergies.

Original: Model selection : We performed three times optimization and adopted the model with the largest
Revised: Model selection : We performed three times optimization and adopted the model with the largest

Original: VAF because the VAF of the proposed method varies slightly each time.
Revised: VAF because the VAF of the proposed method varies slightly each time.

Original: 4. M USCLE S YNERGY A CQUISITION P OTENTIAL USING ONLY MUSCLE ACTIVITY ( CONVENTIONAL FRAMEWORK )
Revised: 4. M USCLE S YNERGY A CQUISITION P OTENTIAL USING ONLY MUSCLE ACTIVITY ( CONVENTIONAL FRAMEWORK )

Original: for each number of synergies.
Revised: for each number of synergies.

Original: In the proposed method, we experimentally explored the number of ﬁlters from [1, 1] to [128, 128].
Revised: In the proposed method, we experimentally explored the number of ﬁlters from [1, 1] to [128, 128].

Original: First, we examined the proposed method’s availability of muscle synergies by using only muscle activities (conventional framework).
Revised: First, we examined the proposed method’s availability of muscle synergies by using only muscle activities (conventional framework).

Original: The muscle synergies were determined based on the degree of reconstruction (VAF) and the number of synergies.
Revised: The muscle synergies were determined based on the degree of reconstruction (VAF) and the number of synergies.

Original: Theoretically, as the number of synergies increases, the VAF becomes higher because the reconstruction accuracy improves.
Revised: Theoretically, as the number of synergies increases, the VAF becomes higher because the reconstruction accuracy improves.

Original: Researchers have shown that the VAF of NMF increases with the increase of the number of muscle synergies in the case of using only muscle activities (Rabbi et al., 2020;
Revised: Researchers have shown that the VAF of NMF increases with the increase of the number of muscle synergies in the case of using only muscle activities (Rabbi et al., 2020;

Original: Gui & Zhang, 2016).
Revised: Gui & Zhang, 2016).

Original: If the proposed method can calculate muscle synergies by factorizing the input matrix, the VAF should increase with the number of muscle synergies.
Revised: If the proposed method can calculate muscle synergies by factorizing the input matrix, the VAF should increase with the number of muscle synergies.

Original: We tested the
Revised: We tested the

Original: VAF of the proposed method to see if it would increase and exceed 0.9, as the number of synergies increases only when using muscle activity.
Revised: VAF of the proposed method to see if it would increase and exceed 0.9, as the number of synergies increases only when using muscle activity.

Original: The proposed method inputs an EMG vec- tor y t =
Revised: The proposed method inputs an EMG vec- tor y t =

Original: [ y 1 t , . .
Revised: [ y 1 t , . .

Original: . , y 14 t ] , and the output is also supposed to be y t without changing the architecture as following
Revised: . , y 14 t ] , and the output is also supposed to be y t without changing the architecture as following

Original: [Equation]
Revised: [Equation]

Original: We used data from a one-day experiment that included 90 trials in the intact state (no neurological disorders).
Revised: We used data from a one-day experiment that included 90 trials in the intact state (no neurological disorders).

Original: In the proposed method, the number of ﬁlters c of the temporal convolutional networks in the activation layer is a key parameter to determine the potential of synergy acquisition.
Revised: In the proposed method, the number of ﬁlters c of the temporal convolutional networks in the activation layer is a key parameter to determine the potential of synergy acquisition.

Original: Thus, we experimentally tested eight types of ﬁlters here: [1, 1], [2, 2], [4, 4], [8, 8], [16, 16], [32, 32], [64, 64], and [128, 128].
Revised: Thus, we experimentally tested eight types of ﬁlters here: [1, 1], [2, 2], [4, 4], [8, 8], [16, 16], [32, 32], [64, 64], and [128, 128].

Original: The vector length of the number of ﬁlters is the number of dilations in the temporal convolutional network, which is two in all eight types.
Revised: The vector length of the number of ﬁlters is the number of dilations in the temporal convolutional network, which is two in all eight types.

Original: For example, in the case of [8, 8], the ﬁrst “ 8 ” is the number of ﬁlters at the ﬁrst dilation, and the second “ 8 ” is the number of ﬁlters for the second dilation.
Revised: For example, in the case of [8, 8], the ﬁrst “ 8 ” is the number of ﬁlters at the ﬁrst dilation, and the second “ 8 ” is the number of ﬁlters for the second dilation.

Original: Figure 3 shows the VAF results of the proposed method (eight different numbers of ﬁlters) and
Revised: Figure 3 shows the VAF results of the proposed method (eight different numbers of ﬁlters) and

Original: NMF by increasing the synergy number for the same data.
Revised: NMF by increasing the synergy number for the same data.

Original: Most of the VAF results calculated by the proposed method increased with the synergy number, as did those of NMF.
Revised: Most of the VAF results calculated by the proposed method increased with the synergy number, as did those of NMF.

Original: The dashed horizontal line indicates a VAF of 0.9, which determines the muscle synergy number.
Revised: The dashed horizontal line indicates a VAF of 0.9, which determines the muscle synergy number.

Original: For the same number of muscle synergies, VAF tended to be higher for a larger number of ﬁlters.
Revised: For the same number of muscle synergies, VAF tended to be higher for a larger number of ﬁlters.

Original: By conﬁrming that
Revised: By conﬁrming that

Original: VAF increases with increasing the number of synergies, we veriﬁed the possibility that the proposed method can analyze muscle synergies.
Revised: VAF increases with increasing the number of synergies, we veriﬁed the possibility that the proposed method can analyze muscle synergies.

Original: 4. N UMBER OF FILTERS
Revised: 4. N UMBER OF FILTERS

Original: We also tested whether the proposed method can acquire muscle synergy under connectivity constraints between brain and muscle activity.
Revised: We also tested whether the proposed method can acquire muscle synergy under connectivity constraints between brain and muscle activity.

Original: Fig.
Revised: Fig.

Original: 4 shows the VAF per muscle synergy number for the proposed method and NMF.
Revised: 4 shows the VAF per muscle synergy number for the proposed method and NMF.

Original: Most of the proposed method’s VAF values increased with the number of muscle synergies not depending on the number of ﬁlters.
Revised: Most of the proposed method’s VAF values increased with the number of muscle synergies not depending on the number of ﬁlters.

Original: The higher number of ﬁlters tended to make VAF higher.
Revised: The higher number of ﬁlters tended to make VAF higher.

Original: We can see that the determined number of muscle synergies, where VAF exceeds 0.9 at ﬁrst, depends on the number of ﬁlters.
Revised: We can see that the determined number of muscle synergies, where VAF exceeds 0.9 at ﬁrst, depends on the number of ﬁlters.

Original: At the numbers of ﬁlters [32, 32] and [64, 64], the determined number of muscle synergies is eight.
Revised: At the numbers of ﬁlters [32, 32] and [64, 64], the determined number of muscle synergies is eight.

Original: At the number of ﬁlters [128, 128], the determined number of muscle synergies is six.
Revised: At the number of ﬁlters [128, 128], the determined number of muscle synergies is six.

Original: Deﬁning the number of ﬁlters depends on how the muscle synergies are used.
Revised: Deﬁning the number of ﬁlters depends on how the muscle synergies are used.

Original: To compare the components of muscle synergies from the proposed method and NMF, we set the number of ﬁlters to [32, 32], which is the minimum number of ﬁlters when the number of muscle synergies is eight.
Revised: To compare the components of muscle synergies from the proposed method and NMF, we set the number of ﬁlters to [32, 32], which is the minimum number of ﬁlters when the number of muscle synergies is eight.

Original: 4. C OMPARISON OF MUSCLE SYNERGY COMPONENTS
Revised: 4. C OMPARISON OF MUSCLE SYNERGY COMPONENTS

Original: We examined whether the proposed method can capture the unrevealed muscle synergy components that are difﬁcult for the conventional approach to extract.
Revised: We examined whether the proposed method can capture the unrevealed muscle synergy components that are difﬁcult for the conventional approach to extract.

Original: As a muscle synergy component, a vector of weights and activations is obtained from a muscle synergy.
Revised: As a muscle synergy component, a vector of weights and activations is obtained from a muscle synergy.

Original: A, B, and C in Fig.
Revised: A, B, and C in Fig.

Original: 5 respectively indicate the eight components from the proposed method using only EMG, NMF, and the proposed method using ECoG and EMG.
Revised: 5 respectively indicate the eight components from the proposed method using only EMG, NMF, and the proposed method using ECoG and EMG.

Original: The left panels in the ﬁgure show the weight of the muscles of each synergy, while the right panels show the activation of each synergy, which is calculated by averaging all trials.
Revised: The left panels in the ﬁgure show the weight of the muscles of each synergy, while the right panels show the activation of each synergy, which is calculated by averaging all trials.

Original: To compare the similarity of each component, we calculate the similarity matrix.
Revised: To compare the similarity of each component, we calculate the similarity matrix.

Original: D in Fig.
Revised: D in Fig.

Original: 5 shows the similarity matrix using the weights from these three approaches.
Revised: 5 shows the similarity matrix using the weights from these three approaches.

Original: The upper panel in D shows the similarity matrix of the weights between the proposed method (only using
Revised: The upper panel in D shows the similarity matrix of the weights between the proposed method (only using

Original: EMG) and the NMF method, while the lower panel in D shows that between the proposed method using ECoG and the EMG and NMF methods.
Revised: EMG) and the NMF method, while the lower panel in D shows that between the proposed method using ECoG and the EMG and NMF methods.

Original: Using only EMG data, the similarity of almost all components of muscle synergy was close to 1, meaning that they were all identical for the proposed and NMF methods.
Revised: Using only EMG data, the similarity of almost all components of muscle synergy was close to 1, meaning that they were all identical for the proposed and NMF methods.

Original: The averaged computation times of the proposed method (basically using GPU:
Revised: The averaged computation times of the proposed method (basically using GPU:

Original: NVIDIA RTX3090) and NMF (CPU: AMD Ryzen Threadripper 3970X) are 6209.3 sec and 12.
Revised: NVIDIA RTX3090) and NMF (CPU: AMD Ryzen Threadripper 3970X) are 6209.3 sec and 12.

Original: sec, respectively.
Revised: sec, respectively.

Original: The computational complexity of the proposed method is higher than that of the
Revised: The computational complexity of the proposed method is higher than that of the

Original: NMF.
Revised: NMF.

Original: The low similarity of synergies of 6 in both the proposed method (using only EMG) and NMF may have been due to the high degree of freedom of activation in the proposed method.
Revised: The low similarity of synergies of 6 in both the proposed method (using only EMG) and NMF may have been due to the high degree of freedom of activation in the proposed method.

Original: As a result, synergies 4 and 7 of the proposed method represent the corresponding synergies in NMF, while also representing synergy 6 in NMF by the cooperative activation of the two synergies.
Revised: As a result, synergies 4 and 7 of the proposed method represent the corresponding synergies in NMF, while also representing synergy 6 in NMF by the cooperative activation of the two synergies.

Original: It may be possible to solve this problem by setting a constraint term such that the activation acts independently between the synergies.
Revised: It may be possible to solve this problem by setting a constraint term such that the activation acts independently between the synergies.

Original: Some of the similarities between the proposed components (using ECoG and EMG) and NMF did not exhibit a high degree of similarity.
Revised: Some of the similarities between the proposed components (using ECoG and EMG) and NMF did not exhibit a high degree of similarity.

Original: For example, the similarities of the proposed method’s synergy 1 and 2 to synergy 1 in NMF were 0.76 and 0.82, respectively.
Revised: For example, the similarities of the proposed method’s synergy 1 and 2 to synergy 1 in NMF were 0.76 and 0.82, respectively.

Original: The results indicate that the proposed method separates synergy 1 of NMF into two synergies.
Revised: The results indicate that the proposed method separates synergy 1 of NMF into two synergies.

Original: These separated components may be the result of capturing the connectivity between brain and muscle activities.
Revised: These separated components may be the result of capturing the connectivity between brain and muscle activities.

Original: The same can be said for synergy 2 and 3 of the proposed method.
Revised: The same can be said for synergy 2 and 3 of the proposed method.

Original: 4. A DAPTIVE CHANGE ANALYSIS
Revised: 4. A DAPTIVE CHANGE ANALYSIS

Original: We next investigated whether the proposed method can capture the adaptive change of muscle synergy components during nervous recovery.
Revised: We next investigated whether the proposed method can capture the adaptive change of muscle synergy components during nervous recovery.

Original: First, brain and muscle activity was measured from an intact (no neurological disorders) monkey.
Revised: First, brain and muscle activity was measured from an intact (no neurological disorders) monkey.

Original: Then, we performed partial SCI surgery on the monkey and measured its activity for 30 days (not every day) after the surgery.
Revised: Then, we performed partial SCI surgery on the monkey and measured its activity for 30 days (not every day) after the surgery.

Original: We explored the number of components of each of the experiment days in the proposed and NMF methods.
Revised: We explored the number of components of each of the experiment days in the proposed and NMF methods.

Original: Figure 6 shows the number of components (upper
Revised: Figure 6 shows the number of components (upper

Original: and task precision (
Revised: and task precision (

Original: ) in each day.
Revised: ) in each day.

Original: We can see the precisions during the intact period were almost 100 %, then moved to nearly% for a month after the surgery, and then improved to around 80 % after three months.
Revised: We can see the precisions during the intact period were almost 100 %, then moved to nearly% for a month after the surgery, and then improved to around 80 % after three months.

Original: These results suggest that the monkey performed the task before the surgery based on the pathways between the brain and muscles acquired so far.
Revised: These results suggest that the monkey performed the task before the surgery based on the pathways between the brain and muscles acquired so far.

Original: Immediately after the partial SCI surgery, it was difﬁcult for the monkey to use the pathways because many of the previously acquired pathways were disconnected and it was more difﬁcult to move the upper limb.
Revised: Immediately after the partial SCI surgery, it was difﬁcult for the monkey to use the pathways because many of the previously acquired pathways were disconnected and it was more difﬁcult to move the upper limb.

Original: The monkey regained the function to perform the task by adaptively changing the pathways using the residual nervous connections, and improved the precision of the task.
Revised: The monkey regained the function to perform the task by adaptively changing the pathways using the residual nervous connections, and improved the precision of the task.

Original: The numbers of components of the proposed and NMF methods are plotted in red and blue lines, respectively.
Revised: The numbers of components of the proposed and NMF methods are plotted in red and blue lines, respectively.

Original: The days dashed in red (proposed) are days where muscle synergy was not calculated below 14 (the number of measured muscles).
Revised: The days dashed in red (proposed) are days where muscle synergy was not calculated below 14 (the number of measured muscles).

Original: The numbers of components of NMF’s muscle synergy were stable both before and after SCI (Fig.
Revised: The numbers of components of NMF’s muscle synergy were stable both before and after SCI (Fig.

Original: While those of the proposed method were similar during the intact period, they exceededabout a month after the surgery and then gradually decreased and converged around three months after the surgery.
Revised: While those of the proposed method were similar during the intact period, they exceededabout a month after the surgery and then gradually decreased and converged around three months after the surgery.

Original: We presume that the number of components increased immediately after the surgery because the monkey could not use the pathways obtained so far, and had to use the residual pathways (not sophisticated).
Revised: We presume that the number of components increased immediately after the surgery because the monkey could not use the pathways obtained so far, and had to use the residual pathways (not sophisticated).

Original: The number of components decreases and converges in the adaptive change of the nervous system, so the transition of the proposed method’s results might be explainable from the neurological point of view.
Revised: The number of components decreases and converges in the adaptive change of the nervous system, so the transition of the proposed method’s results might be explainable from the neurological point of view.

Original: This result indicates that the proposed method can potentially be used to analyze muscle synergy with adaptive nervous change, which is difﬁcult to extract in the conventional approach.
Revised: This result indicates that the proposed method can potentially be used to analyze muscle synergy with adaptive nervous change, which is difﬁcult to extract in the conventional approach.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we proposed a matrix factorization method under the constraint of connectivity between observed and source data.
Revised: In this paper, we proposed a matrix factorization method under the constraint of connectivity between observed and source data.

Original: The core idea of our method is to use the representation in a model that predicts the observed data from the source data as a factored matrix.
Revised: The core idea of our method is to use the representation in a model that predicts the observed data from the source data as a factored matrix.

Original: Applying the proposed method to muscle synergy analysis using a monkey’s brain and muscle activities demonstrated its potential to reveal unknown muscle synergy components, which are difﬁcult for the conventional approach to capture.
Revised: Applying the proposed method to muscle synergy analysis using a monkey’s brain and muscle activities demonstrated its potential to reveal unknown muscle synergy components, which are difﬁcult for the conventional approach to capture.

Original: Speciﬁcally, the proposed method’s transitions of the number of components after partial SCI surgery indicated the adaptive changes in the corticomuscular system.
Revised: Speciﬁcally, the proposed method’s transitions of the number of components after partial SCI surgery indicated the adaptive changes in the corticomuscular system.

Original: Through two analysis experiments, we demonstrated the importance of using not only observed data but also source data to extract the elements hidden in observed data, and showed that the proposed method’s potential to capture these hidden elements.
Revised: Through two analysis experiments, we demonstrated the importance of using not only observed data but also source data to extract the elements hidden in observed data, and showed that the proposed method’s potential to capture these hidden elements.

Original: A A PPENDIX A. E THICS
Revised: A A PPENDIX A. E THICS

Original: All animal experimental procedures were performed in accordance with the guidelines of the National Institute of Health and the Ministry of Education, Culture, Sports, Science, and Technology of Japan and were approved by the Institutional Animal Care and Use Committee of the Tokyo
Revised: All animal experimental procedures were performed in accordance with the guidelines of the National Institute of Health and the Ministry of Education, Culture, Sports, Science, and Technology of Japan and were approved by the Institutional Animal Care and Use Committee of the Tokyo

Original: Metropolitan Institute of Medical Science (Approval No. 18034).
Revised: Metropolitan Institute of Medical Science (Approval No. 18034).

Original: The monkey was monitored closely and animal welfare was assessed daily or, if necessary, several times a day.
Revised: The monkey was monitored closely and animal welfare was assessed daily or, if necessary, several times a day.

Original: A. EC O G AND EMG SIGNALS
Revised: A. EC O G AND EMG SIGNALS

Original: We implanted a 15-channel grid electrode array in which the diameter of each electrode was 1 mm and the inter-electrode distance was 5 mm (Unique Medical, Japan) beneath the dura mater.
Revised: We implanted a 15-channel grid electrode array in which the diameter of each electrode was 1 mm and the inter-electrode distance was 5 mm (Unique Medical, Japan) beneath the dura mater.

Original: To implant the grid electrode array on the cortical surface over the sensorimotor cortex, the cortices around the arcuate sulcus and the central sulcus on the right side were exposed by a craniotomy.
Revised: To implant the grid electrode array on the cortical surface over the sensorimotor cortex, the cortices around the arcuate sulcus and the central sulcus on the right side were exposed by a craniotomy.

Original: As to the muscle activity, electrodes were implanted in 14 muscles of the left forelimb: three elbow muscles [triceps brachii (TRI), biceps brachii (BB), brachialis (BR)], ﬁve wrist muscles [extensor carpi radialis (ECR), extensor carpi ulnaris (ECU), ﬂexor carpi radialis (FCR), ﬂexor carpi ulnaris (FCU), palmaris longus (PL)], ﬁve digit muscles [extensor digitorum communis (EDC), extensor digitorum 4,5 (ED45), ﬂexor digitorum superﬁcialis (FDS), ﬂexor digitorum profundus (FDP), abductor pollicis longus (APL)], and one intrinsic hand muscle [adductor pollicis (ADP)].
Revised: As to the muscle activity, electrodes were implanted in 14 muscles of the left forelimb: three elbow muscles [triceps brachii (TRI), biceps brachii (BB), brachialis (BR)], ﬁve wrist muscles [extensor carpi radialis (ECR), extensor carpi ulnaris (ECU), ﬂexor carpi radialis (FCR), ﬂexor carpi ulnaris (FCU), palmaris longus (PL)], ﬁve digit muscles [extensor digitorum communis (EDC), extensor digitorum 4,5 (ED45), ﬂexor digitorum superﬁcialis (FDS), ﬂexor digitorum profundus (FDP), abductor pollicis longus (APL)], and one intrinsic hand muscle [adductor pollicis (ADP)].

Original: ECoG and EMG signals were recorded simultaneously using a CerebusTM data acquisition system (BLACKROCK
Revised: ECoG and EMG signals were recorded simultaneously using a CerebusTM data acquisition system (BLACKROCK

Original: MICROSYSTEMS, Utah, USA) at a sampling rate of 2,000 Hz.
Revised: MICROSYSTEMS, Utah, USA) at a sampling rate of 2,000 Hz.

Original: A. B EHAVIOR TASK
Revised: A. B EHAVIOR TASK

Original: The monkey performed the task of reaching and grasping for a piece of food multiple times in one experimental day.
Revised: The monkey performed the task of reaching and grasping for a piece of food multiple times in one experimental day.

Original: On each day, the decision to perform the measurement experiment and the number of trials were determined on the basis of the monkey’s condition.
Revised: On each day, the decision to perform the measurement experiment and the number of trials were determined on the basis of the monkey’s condition.

Original: In total, we conducted the measurement experiment for 41 days.
Revised: In total, we conducted the measurement experiment for 41 days.

Original: Each recording session consisted of ˜100 trials.
Revised: Each recording session consisted of ˜100 trials.

Original: A. VAF CALCULATION
Revised: A. VAF CALCULATION

Original: Variance accounted for (VAF) is a metric of the degree of reconstruction.
Revised: Variance accounted for (VAF) is a metric of the degree of reconstruction.

Original: This calculation uses true value vectors y m =
Revised: This calculation uses true value vectors y m =

Original: [ y m 1 , . . .
Revised: [ y m 1 , . . .

Original: , y mT ] , m ∈ (1 , . . .
Revised: , y mT ] , m ∈ (1 , . . .

Original: , 14) and reconstructed vectors ˆ y =
Revised: , 14) and reconstructed vectors ˆ y =

Original: [ˆ y m 1 , . . .
Revised: [ˆ y m 1 , . . .

Original: , ˆ y mT ] , m ∈ (1 , . . . , 14) , where m is the number of measure muscles.
Revised: , ˆ y mT ] , m ∈ (1 , . . . , 14) , where m is the number of measure muscles.

Original: The VAF is calculated as following
Revised: The VAF is calculated as following

Original: [Equation]
Revised: [Equation]

Original: where V ar ( y m − ˆ y m ) is the variance of y m − ˆ y m , the square of the standard deviation. 
Revised: where V ar ( y m − ˆ y m ) is the variance of y m − ˆ y m , the square of the standard deviation. V ar ( y m ) is the variance of y m .