MATRIX FACTORIZATION UNDER THE CONSTRAINT OF CONNECTIVITY BETWEEN OBSERVED AND SOURCE DATA MUSCLE SYNERGY ANALYSIS BASED ON CONNECTIVITY BETWEEN MUSCLE AND BRAIN ACTIVITIES

Abstract

Matrix factorization is a popular method to investigate the hidden elements in observed data for tasks such as speech separation and muscle synergy analysis. The hidden elements may be closely related to the source phenomenon that cause the observed phenomenon. 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. 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. The proposed method factorizes matrices by extracting and combining representations and weights from the regression model in the regression process. 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. In this context, muscle activities are the observed phenomenon and brain activities are the source. 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. 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). 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. 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. 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. 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. Our work thus demonstrates the importance of using not only observed data but also source data to reveal unknown hidden elements.

1. INTRODUCTION

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. For example, the sounds we hear may contain the one voice we are listening to, the voices of other speakers, and various environmental sounds. Separating the elements in the observed phenomena helps us to understand the events individually. Matrix factorization is used to investigate the elements in observed phe-nomena. 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) . In some cases, the observed phenomenon is caused by a functional connection with the source phenomenon. 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. 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. 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) . In this case, muscle activities are the observed phenomenon and brain activities are the source. The corticomuscular system enables us to perform complex body movements by neurologically combining sets of simpler movements. These sets are observed as the basic pattern of muscle activity, namely, muscle synergies. Muscle activity can be observed by using electromyography (EMG), a technique that reveals bioelectric potential signals. 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. 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. Our basic idea is to use the representations in a deep neural network (DNN) model to predict the observed data from the source data. In this paper, we assume both source and observed data can be measured. Specifically, the factorized matrices can function as an activation scalar value and a vector of weights for the observed data at each sample. We can then obtain the factorized matrix under the connectivity constraints between the observed and source data by extracting these values. In this paper, we report the results of applying the proposed method to muscle synergy analysis. 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. To obtain brain activities closely related to muscle activity, we simultaneously measured a monkey's brain signals using both electrocorticography (ECoG) and electromyography (EMG). 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. Our main contributions are as follows. • 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. • We propose a method that utilizes a model representation to predict the observed data from the source data as a factorization matrix. • 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.

2. RELATED WORKS

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. Principal component analysis (PCA) is typically used for this, where the dimensionality-reduced data representation is a factorized matrix (Lee & Seung, 1999) . Another approach is to separate independent components by assuming that the data contains multiple data originating from independent sources. Independent component analysis (ICA) separates signals by independence on the basis of higher-order statistics or temporal correlations (Comon, 1994) . The fast fixed-point algorithm for independent component analysis (FastICA) was proposed as a method to improve the convergence of ICA (Hyvärinen & Oja, 2000) . Non-negative matrix factorization (NMF) factorizes the matrix and restricts it to be positive

