Extended nonnegative matrix factorization for dynamic functional connectivity analysis of fMRI data

Abstract

Dynamic functional connectivity (DFC) analysis using functional magnetic resonance imaging (fMRI) technology has attracted increasing attention in revealing brain dynamics in recent years. Although the nonnegative matrix factorization (NMF) method was applied to dynamic subgraph analysis to reveal brain dynamics, its application in DFC analysis was largely limited due to its nonnegative constraint on the input data. This study proposed the extended NMF (eNMF) method that allowed the input matrix and decomposed basis matrix to have negative values without altering the NMF algorithm. The eNMF method was applied to DFC analysis of both simulated and real resting fMRI data. The simulated data demonstrated that eNMF successfully decomposed the mixed-sign matrix into one positive matrix and one mixed-sign matrix. In contrast to K-means, eNMF extracted more accurate brain state patterns in all cases and estimated better DFC temporal properties for uneven brain state distribution. The real resting-fMRI data demonstrated that eNMF can provide more temporal measures of DFC and was more sensitive to detect intergroup differences of DFC than K-means. Results of eNMF revealed that the female group possibly showed worse relaxation and produced stronger spontaneous cognitive processes although they tended to spend more time in relaxation state and less time in states relevant to cognitive processes in contrast to the male group.

Introduction

The human brain works as a dynamic network system by coordinating the connectivity between brain regions to perform various cognitive functions. Many previous studies have demonstrated that intrinsic functional connectivity between brain regions is not static by using functional magnetic resonance imaging (fMRI) technology (Allen et al. 2014; Preti et al. 2017). Dynamic functional connectivity (DFC) analysis is critical for understanding the intrinsic time-varying functional organization of the human brain.

The sliding-window method is the most common method used to obtain a set of time-varying functional connectivity matrices that can be clustered into sets of repetitively occurring connectivity patterns (brain states) (Allen et al. 2014). The spatiotemporal dynamic measures can be calculated based on the stability and variability of DFC states. The K-means clustering method has been widely used to cluster dynamic connectivity matrices into several clusters to identify recurring connectivity patterns that are presented by cluster centroids (Allen et al. 2014; Tian et al. 2018; Wu et al. 2019). However, the K-means clustering method assumes that only one brain state occurs in each window. Moreover, principal component analysis (PCA), independent component analysis (ICA) and dictionary learning (DL) methods were applied to detect intrinsic brain states from time-varying connectivity matrices (Li et al. 2014; Leonardi et al. 2014; Yaesoubi et al. 2015) without the assumption of K-means. For these three methods, the sign of the extracted brain states is arbitrary, and the weight of the temporal contributions might also be positive or negative, which makes the interpretation of the brain states difficult. In contrast, nonnegative matrix factorization (NMF) decomposes the data matrix into two nonnegative matrices. The nonnegative property of NMF gives it the potential for dynamic functional connectivity analysis without sign confusion, such as PCA/ICA/DL, and enhances the interpretability of the results.

Several studies have applied NMF to dynamic subgraph analysis to reveal brain dynamics (Chai et al. 2017; Khambhati et al. 2018a, b). NMF decomposes functional brain networks into additive subgraphs and time-varying weight coefficients. The nonnegative time-varying weights can be interpreted as the positive contribution of each subgraph to the functional network at each time point. Due to the nonnegative constraint of NMF, the functional network matrix of NMF input must be positive. Previous studies constructed a positive functional network matrix by using wavelet coherence (Chai et al. 2017; Khambhati et al. 2018a; Zhou et al. 2020) or the magnitude of the functional network matrix (Khambhati et al. 2018b). However, Pearson correlation is the most commonly used measure of the functional connectivity network for DFC analysis (Preti et al. 2017). Because the Pearson correlation can be both positive and negative, NMF cannot decompose the functional connectivity matrix directly, which greatly limits the application of NMF in DFC analysis.

Ding et al. (2010) proposed Semi-NMF that relaxed non-negativity constrains of NMF and allowed the input matrix and decomposed base matrix to have negative values (Ding et al. 2010). Furthermore, Semi-NMF applies equally to the Convex-NMF model that is also suitable to mixed-sign input matrices. In contrast to NMF, both Semi-NMF and Convex-NMF are more highly constrained and show worse accuracy for most of nonnegative datasets (Ding et al. 2010; Wang and Mu 2021).proposed a regularized convex nonnegative matrix factorization model (RC-NMF) that introduced graph regularization to constrain nodes with negative edges into different communities and nodes with positive edges into the same communities to detect communities in signed network. RC-NMF is designed to be specific to data with graph structure and increases the model complexity due to the graph regularization. Therefore, it is essential to investigate how to extend NMF simply to the mixed-sign input data without increasing NMF complexity.

This study proposed the extended NMF (eNMF) method that can be used to decompose a mixed-sign matrix into one positive matrix and one mixed-sign matrix without altering the NMF algorithm. The basic idea of eNMF is to split the input matrix into one positive matrix containing all positive elements and one negative matrix containing all negative elements, apply NMF to decompose the new matrix that concatenates the positive matrix and the minus negative matrix into two positive matrices, and transform the two decomposed positive matrices into the original matrix space to obtain the final decomposed matrices. The main contributions of this study are summarized as follows.

• We proposed the eNMF method that generalized NMF to mixed-sign input matrix and removed non-negativity constrain of the input matrix. The eNMF method can decompose any mixed-sign input matrix into one mixed-sign matrix and one positive matrix by transforming the input matrix into a new form without altering NMF algorithm.

• The eNMF method has the same performance as NMF and is less highly constrained than Semi-NMF, convex-NMF and RC-NMF because it doesn’t change the optimization of NMF. Therefore, the eNMF method is simple to be implemented and has the potential to largely enlarge the application of eNMF.

• The simulated data demonstrated that eNMF outperformed K-means in identifying the spatial patterns of the intrinsic brain states. The real resting-fMRI data demonstrated that eNMF can provide more temporal measures of DFC and was more sensitive to detect intergroup differences of DFC than K-means.

• The real fMRI results of eNMF revealed that the female group possibly showed worse relaxation and produced stronger spontaneous cognitive processes although they tended to spend more time in relaxation state and less time in states relevant to cognitive processes in contrast to the male group.

The rest structure of this paper is as follows. The theory of the eNMF method is described in “Theory” section. The experimental design and DFC analysis of the simulated and real fMRI experiments are presented in “Materials and methods” section. The results of all the experiments are presented in “Results” section. The discussion of the results is presented in “Discussion” section. The conclusion of the study is presented in “Conclusion” section.

Theory

Nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is a matrix factorization method with nonnegative constraints for the input and output matrices. It decomposes a nonnegative matrix $V\in R_{+}^{d\times p}$ into two nonnegative matrices $W\in R_{+}^{d\times k}$ and $H\in R_{+}^{k\times p}$ such that the product of $W$ and $H$ is an approximation of $V$:

$$V\approx WH,W\ge 0,H\ge 0$$ 1

NMF finds the approximation by randomly initializing $W$ and $H$ and then iteratively updating $W$ and $H$ to minimize the objective function:

$$argmin{∥V-WH∥}_{F}s.t.W\ge 0,H\ge 0$$ 2

where ${∥·∥}_F$ is the Frobenius norm. Generally, NMF seeks a low-rank approximation of the original matrix $V$. The low-rank $k(k\le min(d,p))$ represents the number of potential features in the original data.

Extended nonnegative matrix factorization

The proposed eNMF method can decompose a mixed-sign data matrix into one mixed-sign matrix and one matrix with a positive constraint. The eNMF method first splits the input data matrix V $\in R^{d\times p}$ into one positive matrix $V_{\mathit{positive}}\in R^{d\times p}$ and one negative matrix $V_{\mathit{negative}}\in R^{d\times p}$.

$$V_{\mathit{positive}}=\frac{V+\left|V\right|}{2}{V}_{\mathit{negative}}=\frac{V-\left|V\right|}{2}$$ 3

The positive matrix $V_{\mathit{positive}}$ contains all of the positive elements of V and sets all of the negative elements of V to zero. The negative matrix $V_{\mathit{negative}}$ contains all of the negative elements of V and sets all of the positive elements of V to zero. The matrix V can be expressed by Eq. (4).

$$V=V_{\mathit{positive}}+V_{\mathit{negative}}$$ 4

A new matrix $V^{\prime }\in R^{2d\times p}$ is formed by concatenating the two matrices $V_{\mathit{positive}}$ and $-V_{\mathit{negative}}$ along the rows. It should be noted that the matrix $V^{\prime }$ does not contain negative values. The input matrix V can be expressed by the Eq. (5).

$$V=V_{\mathit{positive}}+V_{\mathit{negative}}=\left[{\text{I}}_{d\times d}{\text{-I}}_{d\times d}\right]\left[\begin{array}{c}{V}_{\mathit{positive}}\\ -V_{\mathit{negative}}\end{array}\right]=U\times V^{\prime }$$ 5

where U = $\left[{\text{I}}_{d\times d}{\text{-I}}_{d\times d}\right]$ and $V^{\prime }=\left[\begin{array}{c}{V}_{\mathit{positive}}\\ -V_{\mathit{negative}}\end{array}\right]$. The matrix ${\text{I}}_{d\times d}$ is the identity matrix.

Because the matrix $V^{\prime }\in R^{2d\times \text{k}}$ was nonnegative, it can be decomposed into two matrices W $\in R^{2d\times \text{k}}$ and $H^{\prime }\in R^{\text{k}\times \text{p}}$ by NMF (see Eq. 6).

$$V^{\prime }=\left[\begin{array}{c}{V}_{\mathit{positive}}\\ -V_{\mathit{negative}}\end{array}\right]\approx \text{W}\times H^{\prime }=\left[\begin{array}{c}{W}_{\mathit{positive}}\\ W_{\mathit{negative}}\end{array}\right]\times H^{\prime }=\left[\begin{array}{c}{W}_{\mathit{positive}}\times H^{\prime }\\ W_{\mathit{negative}}\times H^{\prime }\end{array}\right]$$ 6

The size of matrices $W_{\mathit{positive}}$ and $W_{\mathit{negative}}$ is $d\times k$. Substitute $V^{\prime }$ in Eq. (5) with Eqs. (6). The matrix V can then be reformulated by Eq. (7).

$$\begin{array}{cc}\hfill V& =\text{U}\times V^{\prime }\approx \left[{\text{I}}_{d\times d}{\text{-I}}_{d\times d}\right]\left[\begin{array}{c}{W}_{\mathit{positive}}\times H^{\prime }\\ W_{\mathit{negative}}\times H^{\prime }\end{array}\right]\hfill \\ \hfill & =W_{\mathit{positive}}\times H^{\prime }-W_{\mathit{negative}}\times H^{\prime }\hfill \\ \hfill & =\left(W_{\mathit{positive}}-W_{\mathit{negative}}\right)\times H^{\prime }\hfill \\ \hfill \end{array}$$ 7

Let $W^{\prime }=W_{\mathit{positive}}-W_{\mathit{negative}}$. Equation (7) can be written as Eq. (8).

$$V\approx W^{\prime }\times H^{\prime }$$ 8

Finally, the matrix V is automatically decomposed into two matrices ($W^{\prime }$ and $H^{\prime }$). The matrix $W^{\prime }\in R^{d\times \mathrm{k}}$ contains both positive and negative elements, while the matrix $H^{\prime }$ only contains nonnegative elements. NMF can be implemented by using an alternating least-squares algorithm (Takács and Tikk 2012).

The algorithm of eNMF is listed in the following.

• Input matrix = V, initialize W and $H^{\prime }$

• Set $V_{\mathit{positive}}=\frac{V+\left|V\right|}{2}$ and $V_{\mathit{negative}}=\frac{V-\left|V\right|}{2}$

• Set $V^{\prime }=\left[\begin{array}{c}{V}_{\mathit{positive}}\\ {-V}_{\mathit{negative}}\end{array}\right]$

• for i = 1:maxiter.

  • Solve for $H^{\prime }$ in matrix equation W^TW $H^{\prime }$=W^T $V^{\prime }$
  • Set all negative elements in $H^{\prime }$ to 0.
  • Solve for W in matrix equation $H_{}^{\prime }{H}^{\prime \mathrm{T}}{W}^T=H_{}^{\prime }{V}^T$
  • Set all negative elements in W to 0.

    end

Set $W_{\mathit{positive}}=W(1:d,:)$ and $W_{\mathit{negative}}=W(d+1:2d,:)$

Set $W_{}^{\prime }=W_{\mathit{positive}}-W_{\mathit{negative}}$

Output $W_{}^{\prime }$ and $H^{\prime }$

Remark 1

Semi-NMF separates the positive and negative parts from the mixed-sign matrix during the updating of H by using multiplicative rules of a constrained optimization problem while holding W fixed. In contrast, eNMF separates the positive and negative parts from the input matrix and generates a new nonnegative input matrix for NMF decomposition by concatenating the positive part and the minus negative part. Therefore, eNMF doesn’t change the optimization problem of NMF and can obtain the same decomposition accuracy as NMF.

DFC analysis of eNMF

The flowchart of DFC analysis of eNMF is presented in Fig. 1. Assume there are $n$ subjects and each subject contains $c$ ROIs. First, the sliding window approach was applied to extract the time-varying functional connectivity matrix from each sliding window of each subject (see Fig. 1A). It is assumed that m windows can be extracted from each subject. For each window of each subject, a $c\times c$ functional connectivity (FC) matrix can be obtained by calculating the Pearson correlation coefficients between each pair of ROIs (see Fig. 1B). Due to the symmetry of the FC matrix, a column vector that stores non-redundant FC values can be obtained by only vectorizing the upper right corner of each FC matrix. For window j of subject i, the FC column vector matrix is denoted as $F_{\mathit{ij}}$. The size of $F_{\mathit{ij}}$ is $\frac{c(c+1)}{2}\times 1$ (see Fig. 1C). The FC column vectors of all subjects’ m windows are concatenated to yield a DFC data matrix $V_{\frac{c(c-1)}{2}\times nm}$ using Eq. (9):

$$V=\left[\underset{subject1}{\underbrace{F_{11},F_{12},\dots ,F_{1m}}},\underset{subject2}{\underbrace{F_{21},F_{22},\dots ,F_{2m}}},\dots \underset{subjectn}{\underbrace{F_{n1},F_{n2},\dots ,F_{\mathit{nm}}}}\right]$$ 9

Fig. 1.

The flowchart of DFC analysis based on eNMF

The input matrix $V_{\frac{c(c-1)}{2}\times nm}$ is decomposed into $W_{\frac{c(c-1)}{2}\times k}_{}^{\prime }$ and $H_{k\times nm}_{}^{\prime }$ by using the eNMF algorithm presented in “Extended nonnegative matrix factorization” section. Each column of the matrix $W_{\frac{c(c-1)}{2}\times k}_{}^{\prime }$ represents an intrinsic functional connectivity pattern (brain state) and can be reshaped into an FC matrix with c $\times$ c size. The parameter $k$ represents the number of intrinsic brain states. Each row of the matrix $H_{k\times nm}_{}^{\prime }$ represents the nonnegative time-varying weights that can be interpreted as the positive contribution of each brain state to the functional network at each time window (see Fig. 1D).

Because the Pearson correlation of FC ranges from − 1 to 1 and the values in $W^{\prime }$ may not be bounded in $\left[-,1,1\right]$, $W^{\prime }$ is scaled to $\left[-,1,1\right]$ by using $W_n=W^{\prime }/a$ and $a=max(\left|W^{\prime }\right|)$. To keep V unchanged, $H^{\prime }$ is scaled to H_n by using $H_n=a{\times H}^{\prime }$. The matrix V is further decomposed into $W_n$ and $H_n$ (see Eq. (10)).

$$V\approx W^{\prime }\times H^{\prime }=\left(\frac{W^{\prime }}{a}\right)\times \left(a,{\times H}^{\prime }\right)=W_n\times H_n$$ 10

eNMF-Elbow for estimation of the brain state number

In this study, we proposed the eNMF-Elbow method to estimate the optimal number of brain states. The eNMF-Elbow method includes three steps. The first step is to recover the individual FC matrices of all sliding windows from the eNMF results. The second step is to classify all of the recovered individual FC matrices into different intrinsic brain states. The third step is to use the elbow criterion to estimate the optimal number of brain states.

Step 1: Recover the individual FC matrices of all sliding windows

After eNMF decomposition, the input data matrix V is decomposed into W_n and H_n (see Eq. 10). The matrix W_n represents the intrinsic group-level FC states. For each window of each subject, it is assumed that the individual FC matrix is a linear sum of all of the intrinsic group-level FC states weighted by each state’s contribution. Let f_ij represent the recovered FC vector for the jth window of the ith subject. The individual FC vector f_ij can be recovered by Eq. (11):

$$f_{\mathit{ij}}=W_n\times h_{\left(i,-,1\right)\ast m+j}^{}$$ 11

where $h_{\left(i,-,1\right)\ast m+j}^{}\in R_{k\times 1}^{+}$ is one column of the matrix H_n. It represents the contributions of all k brain states for the jth window of the ith subject and m is the number of sliding windows. The FC vector $f_{\mathit{ij}}$ can be reshaped to the FC matrix $F_{\mathit{ij}}$ with c $\times$ c size.

Step 2: Classify all of the recovered individual FC matrices into different brain states

For each sliding window of each subject, we suppose that each individual FC matrix can be classified into the brain state with the highest contribution to the individual FC among the k brain states. If the qth element of $h_{\left(i,-,1\right)\ast m+j}^{}$ is the highest among all k elements, the individual FC matrix $F_{\mathit{ij}}$ is classified into the brain state q for the jth window of the ith subject. Therefore, all of the m $\times$ n individual FC matrices of all windows of all subjects are classified into k brain states.

Step 3: Use the elbow criterion to estimate the optimal number of brain states

After classification in step 2, the sum of the squared error (SSE) is calculated with Eq. (12):

$$\mathrm{SSE}=\sum _{q=1}^k\sum _{F_q\in S_q}{\Vert F_q-w_q\Vert }^2$$ 12

where $w_q$ is the qth column of the matrix Wn and represents the qth group-level brain states. $F_q$ represents the individual FC patterns belonging to the qth brain state ($S_q$). Different SSE values are calculated by changing the number of brain states (k). The relationship between SSE and the number of brain states can be plotted, and the optimal number of brain states is estimated by the elbow criterion.

Materials and methods

In this study, simulated experiments were performed to investigate the feasibility and robustness of eNMF and to compare the performance of eNMF and K-means for DFC estimation. Moreover, eNMF and K-means were applied to real resting-state fMRI (rsfMRI) data from the Human Connectome Project (HCP) to examine the effect of gender on the DFC properties of resting fMRI data.

All codes were implemented in MATLAB R2018a (Mathworks). The K-means algorithm and NMF algorithm used source codes from MATLAB. The eNMF algorithm was developed based on the NMF MATLAB code that used an alternating least-squares algorithm.

Simulated experiments

Five simulated experiments were performed to explore the feasibility and robustness of eNMF. The first four simulated experiments were designed to investigate the effects of noise, uneven state distribution, parameter size and sample size on the DFC analysis of eNMF. The performance of eNMF on DFC estimation was compared with that of K-means. The fifth experiment was performed to investigate the feasibility of eNMF-Elbow in estimating the number of brain states.

Robustness to noise

The simulated data of 30 subjects were generated in the first simulated experiment. Each subject’s simulated data consisted of 250 time points in a run. It was assumed that there were 4 brain states underlying each subject’s data, and each state represented the functional connectivity pattern of 40 ROIs. The repetition time (TR) of the fMRI data was set to 2 s.

The simulated fMRI data were generated by the SimTB framework (http://mialab.mrn.org/software/simtb/index.html) under a model of dynamic neural connectivity. The FC patterns of the 4 brain states shown in Fig. 2A were predefined in SimTB. The state time series that represent the active state at each time point were randomly generated by using the Gamma function (https://github.com/OHBA-analysis/HMM-MAR) in HMM. The inputs of the gamma function were the time points of the fMRI time courses, the initial probability of each state ($\pi$) and the transition probability matrix ($A$). The initial probability vector $\pi$ was set to $\pi =\left[0.25,025,0.25,0.25\right]$, and the transition probability matrix $A$ was set to $A=\left[\begin{array}{cc}\begin{array}{cc}0.960& 0.014\end{array}& \begin{array}{cc}0.013& 0.013\end{array}\\ \begin{array}{c}\begin{array}{cc}0.013& 0.960\end{array}\\ \begin{array}{c}\begin{array}{cc}0.01& 0.01\end{array}\\ \begin{array}{cc}0.014& 0.013\end{array}\end{array}\end{array}& \begin{array}{cc}0.013& 0.014\\ 0.970& 0.01\\ 0.013& 0.96\end{array}\end{array}\right]$. In this experiment, the transitions between states were supposed to have similar transition probabilities. Thus, each subject had an even distribution of each state. One subject’s state time series is shown in Fig. 2B. After the FC patterns of 4 states and the state time series were obtained, the simulated resting-state fMRI data of 40 ROIs were generated in simTB. Figure 2C shows the simulated fMRI time series of one subject. Moreover, Gaussian noise with a zero mean value and varied standard deviation (SD) was added to the simulated fMRI data. The SD of the Gaussian noise varied from 0.2 to 0.8 with an increase of 0.2. The simulated fMRI data of 30 subjects with the same noise level consisted of one dataset. Twenty datasets were generated for each noise level. Therefore, a total of 80 simulated datasets (4 noise levels × 20 datasets) were generated in this experiment.

Fig. 2.

The FC patterns of the four states (A), the state time series (B) and the simulated resting-state fMRI data of ROIs (C)

The sliding window approach was used to extract time-varying FC patterns from each subject. The window length was set to 20 TR because previous studies noted that 30–60 s could be a reasonable choice of window length. The rectangular window was shifted with a step of 1 TR. For each subject, a total of 231 windows were produced. In each sliding window, the Pearson correlation coefficient of time courses between each pair of the 40 ROIs was calculated, and a $40\times 40$ FC matrix was obtained. For each dataset, all time-varying FC matrices from all sliding windows of all subjects were used to construct the brain state space. Both eNMF and K-means were applied to all of the time-varying FC matrices of each dataset to estimate the intrinsic brain states from the brain state space. The number of brain states was set to 4 for eNMF and K-means. We ran the eNMF and K-means 100 times and chose the result that had the optimal value of the object function among the 100 iterations.

Pearson correlation was used to match the estimated group-level FC state pattern and the true FC state pattern for eNMF and K-means. The Euclidean distances between each estimated FC pattern and each true FC pattern were calculated for each dataset. The FC Euclidean distance of each dataset was obtained by averaging the state Euclidean distances across 4 states. The mean FC distance of each noise level was obtained by averaging the FC Euclidean distances across 20 datasets.

For eNMF, each column of the matrix H_n in Eq. (10) represents the temporal contribution of each state in each time window, and the state with the highest contribution among the four states in each window can be regarded as the dominant state. In contrast, K-means assumes that only one brain state is active in each window. Therefore, the temporal property of each window’s dominant/active state was investigated for eNMF/K-means. The sojourn time of each subject was calculated by counting the number of consecutive time points spent in a specific dominant/active state. The sojourn distribution of each state was obtained from the sojourn time of 30 subjects. The state Kullback–Leibler (KL) divergence that can quantify the distance between two probability distributions was used to measure the distance between the estimated sojourn distribution and the true sojourn distribution for each state. The KL divergence of each dataset was obtained by averaging the state KL divergences across 4 states. The mean KL divergence of each noise level was obtained by averaging the KL divergences across 20 datasets.

Robustness to an uneven state distribution

The simulated datasets in this experiment were generated in the same way as the first simulated experiment in “Robustness to noise” section, except that the transition probability matrix was changed and the SD of the Gaussian noise level was fixed to 0.6 in this experiment.

To explore the effects of an uneven state distribution on the performance of eNMF, the transition probability matrix $A$ was changed to $A=\left[\begin{array}{cc}\begin{array}{cc}0.960& 0.019\end{array}& \begin{array}{cc}0.019& 0.002\end{array}\\ \begin{array}{c}\begin{array}{cc}0.018& 0.960\end{array}\\ \begin{array}{c}\begin{array}{cc}0.015& 0.014\end{array}\\ \begin{array}{cc}0.013& 0.013\end{array}\end{array}\end{array}& \begin{array}{cc}0.018& 0.004\\ 0.970& 0.001\\ 0.014& 0.960\end{array}\end{array}\right]$ in the second simulated experiment. Because the transition probabilities from the other states to state 4 were set to be the smallest, the occurrence of state 4 was the smallest among the four states, which resulted in an uneven distribution of the 4 states. Each dataset included the simulated data of 30 subjects. A total of 20 simulated datasets were generated in this experiment. The intrinsic brain state extraction by eNMF/K-means and the calculation of the FC Euclidean distance and the KL divergence of each dataset were performed in the same way as the first experiment. The mean FC distance and mean KL divergence were obtained by averaging the distances and divergences across 20 datasets.

Robustness to parameter size

The simulated datasets in this simulated experiment were generated in the same way as the first simulated experiment in “Robustness to noise” section, except that the ROI number was changed and the SD of the Gaussian noise was fixed to 0.6.

To explore the effect of parameter size on the performance of eNMF, the number of ROIs was set to 20, 40 and 60 in the third simulated experiment. Each dataset included the simulated data of 30 subjects. For each ROI number, 20 simulated datasets were generated. Thus, a total of 60 simulated datasets (3 ROI numbers × 20 datasets) were generated in this experiment. The intrinsic brain state extraction by eNMF/K-means and calculation of the FC Euclidean distance and the KL divergence of each dataset were the same as in the first experiment. The mean FC distance and mean KL divergence were obtained by averaging the distances and divergences across 20 datasets.

Robustness to sample size

The simulated datasets in this simulated experiment were generated in the same way as the first simulated experiment in “Robustness to noise” section, except that the sample size was changed and the SD of Gaussian noise was fixed to 0.6.

To explore the effects of sample size on the performance of eNMF, we set the length of the time points to 250, 500 and 1000 in the fourth simulated experiment. The SD of the Gaussian noise level was fixed to 0.6. Each dataset included the simulated data of 30 subjects. For each sample size, 20 simulated datasets were generated. Thus, a total of 60 simulated datasets (3 sample sizes × 20 datasets) were generated in this experiment. The intrinsic brain state extraction by eNMF/K-means and the calculation of the FC Euclidean distance and the KL divergence of each dataset were the same as in the first experiment. The mean FC distance and mean KL divergence were obtained by averaging the distances and divergences across 20 datasets.

For the FC Euclidean distance and KL divergence in the first four simulated experiments, one-way analyses of variance (ANOVA) tests were conducted to test the differences between the two methods.

State number estimation of eNMF-elbow

In this experiment, one simulated dataset that was generated in “Robustness to noise” section was used to verify the feasibility of the eNMF-elbow in estimating the state number. The SD of Gaussian noise was 0.6 for the simulated dataset. The eNMF-Elbow method was applied to estimate the optimal brain state number using the three steps described in “eNMF-Elbow for estimation of the brain state number” section. The performance of the eNMF-Elbow method was compared with several other estimation criteria of the brain state, namely AIC (Akaike 1998), consensus matrix (Brunet et al. 2004), NMFk (Alexandrov et al. 2013), and stability-driven NMF (sdNMF) (Zhou et al. 2020).

HCP fMRI data

Real rsfMRI data were used to examine the effect of gender on the DFC properties and to verify the effectiveness of eNMF.

Data and preprocessing

The Human Brain Connectome Project (HCP) publicly released the resting-state fRMI data of 1200 healthy subjects (https://www.humanconnectome.org/). The rsfRMI data of 100 subjects (age: 22–36 years old, 42 men and 58 women) out of the 1200 subjects were used in this experiment. The HCP ensures the ethics and consent required for the public use of this dataset. Thus, there is no need to obtain further approval from the institutional review board (IRB). Each subject included four runs of fMRI time series, and each run included 1200 volumes with a temporal resolution of 0.72 s and a spatial resolution of 2-mm isotropic. All rsfMRI data were scanned by a 3 T Siemens connectome Skyra. The rsfMRI data provided by the HCP were preprocessed. More details of the preprocessing can be seen in the literature of Smith et al. (2013) and Glasser et al. (2013).

ROI time course extraction

Ninety ROIs were selected based on the anatomical automatic labeling (AAL) atlas. The time courses of the 90 ROIs were extracted from the preprocessed rsfMRI data by using the MATLAB toolbox DPARSF (http://rfmri.org/DPARSF). According to previous studies, the 90 ROIs were divided into nine groups: sensorimotor network (SMN), cingulo-opercular network (CON), auditory network (AUN), default mode network (DMN), visual network (VN), frontoparietal network (FPN), salience network (SN), subcortical network (SCN) and none. The ROI names of each group are listed in Table 1 in the supplementary material.

Table 1.

List of nine brain networks and their corresponding ROIs defined in AAL

Network	Regions	AAL index
SMN	L precentral gyrus	1
	R precentral gyrus	2
	L supplementary motor area	19
	R supplementary motor area	20
	L postcentral gyrus	57
	R postcentral gyrus	58
	L paracentral lobule	69
	R paracentral lobule	70
CON	L inferior frontal gyrus, opercular part	11
	R inferior frontal gyrus, opercular part	12
	L temporal pole: superior temporal gyrus	83
	R temporal pole: superior temporal gyrus	84
AUN	L rolandic operculum	17
	R rolandic operculum	18
	L supramarginal gyrus	63
	R supramarginal gyrus	64
	L heschl gyrus	79
	R heschl gyrus	80
	L superior temporal gyrus	81
	R superior temporal gyrus	82
DMN	L superior frontal gyrus, medial	23
	R superior frontal gyrus, medial	24
	L superior frontal gyrus, medial orbital	25
	R superior frontal gyrus, medial orbital	26
	L anterior cingulate and paracingulate gyri	31
	R anterior cingulate and paracingulate gyri	32
	L posterior cingulate gyrus	35
	R posterior cingulate gyrus	36
	L hippocampus	37
	R hippocampus	38
	L parahippocampal gyrus	39
	R parahippocampal gyrus	40
	L angular gyrus	65
	R angular gyrus	66
	L precuneus	67
	R precuneus	68
	L middle temporal gyrus	85
	R middle temporal gyrus	86
	L temporal pole: middle temporal gyrus	87
	R temporal pole: middle temporal gyrus	88
	L calcarine fissure and surrounding cortex	43
	R calcarine fissure and surrounding cortex	44
VN	L cuneus	45
	R cuneus	46
	L lingual gyrus	47
	R lingual gyrus	48
	L superior occipital gyrus	49
	R superior occipital gyrus	50
	L middle occipital gyrus	51
	R middle occipital gyrus	52
	L inferior occipital gyrus	53
	R inferior occipital gyrus	54
	L fusiform gyrus	55
	R fusiform gyrus	56
FPN	L superior frontal gyrus, dorsolateral	3
	R superior frontal gyrus, dorsolateral	4
	L superior frontal gyrus, orbital part	5
	R superior frontal gyrus, orbital part	6
	L middle frontal gyrus	7
	R middle frontal gyrus	8
	L middle frontal gyrus, orbital part	9
	R middle frontal gyrus, orbital part	10
	L inferior parietal, but supramarginal and angular gyri	61
	R inferior parietal, but supramarginal and angular gyri	62
SN	L inferior frontal gyrus, triangular part	13
	R inferior frontal gyrus, triangular part	14
	L insula	29
	R insula	30
	L median cingulate and paracingulate gyri	33
	R median cingulate and paracingulate gyri	34
	L superior parietal gyrus	59
	R superior parietal gyrus	60
SCN	L caudate nucleus	71
	R caudate nucleus	72
	L lenticular nucleus, putamen	73
	R lenticular nucleus, putamen	74
	L lenticular nucleus, pallidum	75
	R lenticular nucleus, pallidum	76
	L thalamus	77
	R thalamus	78
	L inferior frontal gyrus, orbital part	15
	R inferior frontal gyrus, orbital part	16
None	L olfactory cortex	21
	R olfactory cortex	22
	L gyrus rectus	27
	R gyrus rectus	28
	L amygdala	41
	R amygdala	42
	L inferior temporal gyrus	89
	R inferior temporal gyrus	90

L left, R right, SMN sensorimotor network, CON cingulo-opercular network, AUN auditory network, DMN default mode network, VN visual network, FPN frontoparietal network, SN salience network, SCN subcortical network

Intrinsic brain state extraction

The sliding window approach was used to extract the time-varying FC patterns. The rectangular window with a length of 20 TR was shifted with a step of 1 TR. For each subject, a total of 1181 windows were produced. In each sliding window, the Pearson correlation coefficient of time courses between each pair of the 90 ROIs was calculated, and a $90\times 90$ FC matrix was obtained. All FC matrices of all 100 subjects were used to construct their brain state spaces. The eNMF and K-means were applied to all of the FC matrices of 100 subjects to extract the intrinsic brain states from the brain state spaces. The elbow criterion of K-means and the eNMF-elbow were used to estimate the optimal number of brain states. We ran eNMF and K-means 100 times and chose the result that produced the optimal values of the object function.

DFC analysis unique to eNMF

K-means assumes that only one brain state is active in each window. Different from K-means, eNMF does not have this assumption. For eNMF, the contribution value of each brain state in each window can be obtained from the decomposed matrix H_n in Eq. (10). Thus, some DFC measures unique to eNMF including the signal energy and signal entropy of each brain state can be extracted from the matrix H_n. Suppose $h_t(k)$ represents the contribution value of a subject’s brain state $k$ in window $t$ and ${\mathrm{h}}_t_{}^{\prime }\left(k\right)$ represents the normalized contribution. The mean value $\mathrm{M}$ of $h_t(k)$ can be calculated by averaging the contribution values of state k across the windows, where $\mathrm{M}=\frac{{\sum }_{i=1}^T{h}_i\left(k\right)}{T}$ and $T$ is the number of windows. The normalized contribution ${\mathrm{h}}_t_{}^{\prime }\left(k\right)$ can be calculated by Eq. (13):

$${\mathrm{h}}_t_{}^{\prime }\left(k\right)=\frac{h_t(k)}{M}$$ 13

The normalized energy that measures the contribution degree of one subject’s brain state k can be calculated by Eq. (14):

$$\mathrm{Energy}(k)=\sum _{t=1}^T\left({\mathrm{h}}_t,_{}^{\prime },\left(k\right)\right)_{}^2$$ 14

Normalized entropy that measures the complexity of the signal can be calculated by Eq. (15):

$$\mathrm{Entropy}(k)=\sum _{t=1}^T-P\left(x_t\right)logP\left(x_t\right)$$ 15

where $P\left(x_t\right)$ is a probability mass function of the normalized contribution value of ${\mathrm{h}}_t^{\prime }\left(k\right)$, which can be computed using the histogram method.

One-way analyses of variance (ANOVA) that used group as the between-subject factor were conducted to test the intergroup differences for the energy and entropy of each state. All statistical analyses were carried out in SPSS 20.0 (https://www.ibm.com/analytics/data-science/predictive-analytics/SPSS-statistical-software).

Temporal and spatial DFC analysis for eNMF and K-means

For eNMF, the state with the highest contribution among the four states in each window was regarded as the dominant state. For K-means, each FC pattern in each window was classified as an active state. To investigate the dynamic characteristics of the active/dominant brain states extracted by K-means/eNMF, the mean dwell time and the fraction of time were calculated. The mean dwell time of state k is the average number of consecutive windows in which state k was active/dominant for K-means/eNMF. The fraction of time of state k is the proportion of windows in which state k was active/dominant for K-means/eNMF.

For eNMF, the individual FC matrices of all sliding windows were recovered from the eNMF results during the estimation of optimal brain states by eNMF-elbow. The mean connectivity strength and system segregation were calculated to measure the spatial dynamic characteristics of each brain state. For state k of each subject, a mean FC pattern F_k was obtained by averaging all of the individual FC matrices across the windows in which state k was active/dominant for K-means/eNMF. For each subject, the mean connectivity strength of state k was obtained by averaging the correlations across all connectivities of the mean FC pattern F_k.

The system segregation qualitatively described the within-network correlations in relation to the between-network correlations using the following formula:

$$\text{System}\text{segregation}=\frac{mean\left(Z_w\right)-mean\left(Z_b\right)}{mean\left(Z_w\right)}$$ 16

where $\mathrm{mean}\left(Z_w\right)$ represents the average of the Fisher-transformed correlations within the networks and $\mathrm{mean}(Z_b)$ represents the average of the Fisher-transformed correlations between the networks. The segregation value of each individual FC matrix was calculated for each sliding window of each subject. The system segregation of a state k was computed by averaging the segregation values across the sliding windows in which the state k was active/dominant for K-means/eNMF.

One-way analyses of variance (ANOVA) that used group as the between-subject factor were conducted to test the intergroup differences for the mean dwell time, the fraction of time, the mean connectivity strength and the system segregation of each state. For the mean connectivity strength and the system segregation, the subjects that did not have a state were removed from the ANOVA test of the state.

Results

Simulated experiment

Robustness to noise

Figure 3A shows the FC Euclidean distances of K-means and eNMF at different noise levels. Both K-means and eNMF had increased FC distances with increasing noise levels. In contrast to eNMF, K-means had a faster increase in FC distance than eNMF. The FC distances of eNMF were significantly smaller than those of K-means at all noise levels (F (1,38) = 287.970, p < 0.05 for SD = 0.2, F(1,38) = 1941.722, p < 0.05 for SD = 0.4, F(1,38) = 3364.421, p < 0.05 for SD = 0.6).

Fig. 3.

The Euclidean distance of FC comparisons between K-means and eNMF for the first four simulated experiments. A Different noise levels. B Uneven state distribution. C Different ROI numbers. D Different sample sizes. **p < 0.05

Figure 4A shows the KL divergences of the sojourn time at different noise levels for K-means and eNMF. The impact of noise on the KL divergence of the two methods was smaller than the impact on the FC distances. The two methods did not show significant differences in KL divergence.

Fig. 4.

The KL divergence of the sojourn time comparison between K-means and eNMF for the first four simulated experiments. A Different noise levels. B Uneven state distribution. C Different ROI numbers. D Different sample sizes. *p < 0.1

Figure 5 shows the FC patterns and sojourn time distributions of the four intrinsic brain states that were estimated by K-means and eNMF for one simulated dataset. In contrast to K-means, eNMF produced brain state patterns that were closer to the ground truth. The FC distance of eNMF (distance = 9.19) was smaller than that of K-means (distance = 19.15). Moreover, the sojourn time distribution of eNMF was closer to the ground truth than that of K-means (divergence = 11.19 for K-means, divergence = 9.84 for eNMF).

Fig. 5.

Simulated results of one dataset with noise SD equal to 0.6. A FC patterns of 4 intrinsic brain states for the ground truth (Row 1), K-means (Row 2) and eNMF (Row 3). B Sojourn time distribution of 4 intrinsic brain states for the ground truth (Row 1), K-means (Row 2) and eNMF (Row 3)

Robustness to uneven state size

Figure 3B shows the FC Euclidean distances of the K-means and eNMF for the uneven state distribution. Figure 4B shows the KL divergences of the K-means and eNMF for the uneven state distribution. When the occurrence probabilities of different brain states differed largely, eNMF showed significantly smaller FC distances (F (1,38) = 491.601, p < 0.05) and marginally significant smaller KL divergence (F (1,38) = 3.861, p = 0.057 < 0.1) than K-means.

Robustness to parameter size

The FC Euclidean distances of K-means and eNMF for different ROI numbers are shown in Fig. 3C. The FC distances for the two methods increased with the increasing of ROI numbers. K-means increased faster than eNMF. The FC distances of eNMF were significantly smaller than those of K-means for all ROI numbers (F(1,38) = 4978.840, p < 0.05 for ROI number = 20; F(1,38) = 3364.421, p < 0.05 for ROI number = 40; F(1,38) = 4765.413, p < 0.05 for ROI number = 60). The KL divergences of K-means and eNMF for different ROI numbers are shown in Fig. 4C. The two methods did not show significant differences in KL divergences for different ROI numbers.

Robustness to sample size

The FC Euclidean distances of K-means and eNMF for different sample sizes are shown in Fig. 3D. Increasing the time point length did not affect the FC distances of the two methods. In contrast to K-means, eNMF produced significantly lower FC distances for all sample sizes (F (1,38) = 3364.421, p < 0.05 for 250 time points, F(1,38) = 2921.812, p < 0.05 for 500 time points, F (1,38) = 9327.113, p < 0.05 for 1000 time points). The KL divergences of K-means and eNMF for different sample sizes are shown in Fig. 4D. There were no significant differences between the KL divergences of the two methods.

State number estimation of eNMF-Elbow

Figure 6 shows the results of the state number estimation using eNMF-Elbow, AIC, consensus matrix, NMFk and staNMF for NMF and elbow for K-means. The optimal state number was 4 using the elbow criterion for K-means and eNMF-Elbow for NMF (see Fig. 6A, F), which is consistent with the true state number of the simulated data. The optimal state number was 10 for AIC, 5 for consensus matrix, 2 for NMFk and 2 for staNMF.

Fig. 6.

Variations in different criteria with increasing state number K. A eNMF-Elbow. B AIC. C Consensus matrix. D NMFk. E staNMF. F Kmeans-Elbow

HCP fMRI data

The optimal number of brain states estimated by eNMF-Elbow and K-means-Elbow was 4. Figure 7 shows the FC patterns of the four intrinsic brain states for K-means and eNMF. The four brain states of K-means and eNMF showed similar patterns. Among the four states, state 2 showed overall weak connections, while state 3 showed overall strong connections. All the DFC statistic results of one-way ANOVA are presented in Table 2 in the supplementary material for eNMF and Table 3 in the supplementary material for K-means.

Fig. 7.

FC patterns of the four intrinsic brain states estimated by K-means and eNMF for the HCP fMRI data

Table 2.

DFC statistic results of NMF

	Normalized energy	Normalized entropy	Mean dwell time	Fraction of time	Mean connectivity strength	Mean segregation
State 1	F = 0.24 p = 0.628	F = 0.014 p = 0.9	F = 0.72 p = 0.397	F = 0.5490 p = 0.46	F = 6.373 p = 0.0132**	F = 4.52 p = 0.036**
State 2	F = 3.52 p = 0.064*	F = 1.79 p = 0.18	F = 9.522 p = 0.003**	F = 12.478 p = 6.3e-4	F = 6.852 p = 0.0103**	F = 0.0135 p = 0.9079
State 3	F = 18.09 p = 4.8e − 5**	F = 8.5 p = 0.004**	F = 0.09 p = 0.764	F = 7.473 p = 0.0074**	F = 2.689 p = 0.1043	F = 0.4945 p = 0.4836
State 4	F = 15.92 p = 0.0001**	F = 5.65 p = 0.019**	F = 2.315 p = 0.1313	F = 10.149 p = 0.0019**	F = 6.3983 p = 0.013**	F = 5.746 p = 0.0184**

**p < 0.05 and *p < 0.1

Table 3.

DFC statistic results of K-means

	Mean dwell time	Fraction of time	Mean connectivity strength	Mean segregation
State 1	F = 4.709 p = 0.0324**	F = 0.2103 p = 0.6476	F = 3.139 p = 0.0796*	F = 0.302 p = 0.584
State 2	F = 16.949 p = 8.018e − 05**	F = 16.205 p = 1.12e-4**	F = 9.328 p = 0.0029**	F = 0.004 p = 0.95
State 3	F = 1.426 p = 0.2353	F = 6.973 p = 0.0096**	F = 1.907 p = 0.1704	F = 0.15 p = 0.7
State 4	F = 0.135 p = 0.7141	F = 12.107 p = 7.51e − 4**	F = 1.6827 p = 0.1976	F = 2.356 p = 0.128

**p < 0.05 and *p < 0.1

DFC analysis unique to eNMF

Figure 8 shows the results of normalized energy and entropy that was unique to eNMF. In contrast to the female group, the male group showed marginally significantly higher energy in state 2 (F(1,98) = 3.520, p = 0.064 < 0.1)) and significantly lower energy in state 3 (F(1,98) = 18.092, p < 0.05)) and state 4 (F (1,98) = 15.919, p < 0.05)). Moreover, the normalized entropy of the female group was significantly higher than that of the male group in state 3 (F(1,98) = 8.497, p < 0.05) and state 4 (F(1,98) = 5.648, p < 0.05).

Fig. 8.

Normalized energy (A) and entropy (B) of the intrinsic brain states estimated by eNMF. **p < 0.05 and *p < 0.1

Temporal and spatial DFC analysis for eNMF and K-means

The temporal DFC results of the two methods, including the mean dwell time and fraction of time of each brain state, are presented in Fig. 9A–D. Both eNMF and K-means detected similar significant intergroup differences. For K-means, the female group showed a significantly higher dwell time in state 1 (F (1,98) = 4.709, p < 0.05) and state 2 (F(1,98) = 16.949, p < 0.05) than the male group (see Fig. 9A). Moreover, the fraction of time of the female group was significantly higher in state 2 (F(1,98) = 16.205, p < 0.05) and significantly lower in state 3 (F(1,98) = 6.973, p < 0.05) and state 4 (F(1,98) = 10.149, p < 0.05) than in the male group (see Fig. 9C). For eNMF, the female group produced a significantly higher dwell time than the male group in state 2 (F(1,98) = 9.522, p < 0.05) (see Fig. 9B). The fraction of time of the female group was significantly higher in state 2 (F(1,98) = 12.478, p < 0.05) and significantly lower in state 3 (F(1,98) = 7.473, p < 0.05) and state 4 (F(1,98) = 10.149, p < 0.05) than in the male group (see Fig. 9D).

Fig. 9.

Temporal (A–D) and spatial (E–H) DFC results of the female and male groups for K-means and eNMF. A, B Mean dwell time of the two groups for K-means (A) and eNMF (B). C, D Fraction of time of the two groups for K-means (C) and eNMF (D). E, F Mean connectivity strength of the two groups for K-means (E) and eNMF (F). G, H Mean segregation of the two groups for K-means (G) and eNMF (H). **p < 0.05 and *p < 0.1

The spatial DFC results of the two methods, including connectivity strength and segregation of each brain state, are presented in Fig. 9E–H. For K-means, the male group produced marginally significantly higher connectivity strength in state 1 (F(1,98) = 3.139, p < 0.1) and significantly higher connectivity strength in state 2 (F(1,98) = 9.328, p < 0.05) and state 3 (F(1,98) = 7.203, p < 0.05) (see Fig. 9E). No significant intergroup differences in segregation were found in any of the states (see Fig. 9G). For eNMF, the female group produced significantly lower connectivity strength in state 1 (F(1,98) = 6.373, p < 0.05) and state 2 (F(1,98) = 6.852, p < 0.05) and marginally significantly lower strength in state 4 than the male group (F(1,98) = 6.3983, p < 0.05) (see Fig. 9F). Moreover, the female group showed significantly higher segregation than the male group in state 1 (F(1,98) = 4.520, p < 0.05) and state 4 (F(1,98) = 5.746, p < 0.05) (see Fig. 9H).

Discussion

In this study, we proposed eNMF with a loose positive constraint that decomposed a matrix into one positive matrix and one matrix without a positive constraint and investigated the performance of eNMF in the DFC analysis of the fMRI data. The results of the simulated experiments demonstrated that eNMF showed prominent advantages over K-means in detecting the intrinsic brain states from time-varying FC matrices and significantly better temporal performance of DFC than K-means in cases with an uneven state distribution. The real fMRI data demonstrated that eNMF showed spatial advantages over K-means in detecting more intergroup differences in system segregation and temporal advantages over K-means in extracting more DFC temporal properties, such as energy and entropy, which are unique to eNMF.

Although NMF was previously applied to dynamic subgraph analysis of fMRI data (Khambhati et al. 2018b), the input data were constrained to be positive, which limits the application of NMF in DFC analysis because the functional connectivity between regions generally has both positive and negative correlations. Multivariate data-driven methods, such as ICA, PCA and DL, do not have positive constraints. However, the estimated weight contains both positive and negative values, which makes the interpretation of the brain state confusing (Preti et al. 2017). The core idea of eNMF is to remove the positive constraint on the input data of NMF and only add the positive constraint on the decomposed weight matrix to remove the limitation of NMF. The eNMF divided the input data matrix into a positive matrix and a negative matrix and transformed the input data matrix into the multiplication of one matrix U and one positive matrix $V_{}^{\prime }$ concatenating the positive matrix and the minus negative matrix by row (see Eq. 5). Then, the nonnegative matrix $V_{}^{\prime }$ can be decomposed directly by NMF. After the new data matrix $V_{}^{\prime }$ is decomposed into two positive matrices by NMF, eNMF can transform the two decomposed positive matrices into the original input data space and obtain two new decomposed matrices in the original data space. The two new decomposed matrices contain one positive weight matrix and one mixed-sign matrix. Because the transformation of the input matrix nicely avoids the nonnegative constraint of the input matrix, the eNMF method can use the NMF algorithm simply. Although Semi-NMF also allows the mixed-sign input matrix, the Semi-NMF method needs to solve a constrained optimization problem when the matrix H is updated. Therefore, the Semi-NMF method is more highly constrained and more complicated than the proposed eNMF method. The eNMF method can enlarge the application of eNMF and has the potential in disease diagnoses (ÖzçeliK and Altan 2023).

In the simulated experiment, eNMF successfully extracted the intrinsic four brain states from time-varying FC matrices even though the input FC matrices and the intrinsic brain states contained both positive and negative values (see Fig. 5A). The results suggested the feasibility of eNMF in decomposing an FC matrix with negative values into one positive weight matrix and one mixed-sign matrix. In contrast to ICA, PCA and DL with sign confusion, the positive weight obtained by eNMF has better interpretability. A higher weight of a brain state at a time point indicates a stronger contribution of the brain state to the observed FC matrix.

In contrast to K-means, eNMF showed significantly lower FC distances of brain states for different noise levels, uneven state distributions, different ROI numbers and different sample sizes (see Fig. 3). These results suggested that eNMF showed better robustness and was able to detect significantly more accurate spatial patterns of brain state than K-means in all cases. Moreover, K-means clustering was previously shown to be highly sensitive to noise (Im et al. 2020).Therefore, these results showed much larger impact of noise on K-means than on eNMF (see Fig. 3A). Figure 3C shows that the impact of ROI numbers on K-means was also larger than that on eNMF. More ROIs could introduce more noise into the estimation of the time-varying FC matrices. Thus, the worse robustness of K-means to the ROI number is possibly attributable to more noise in the FC matrices. The better performance of NMF is consistent with the general conclusion that NMF always performs better than K-means in clustering accuracy (Ding et al. 2010) K-means belongs to hard clustering and its clustering objective function attempts to capture the rigid spherical clusters. In contrast, NMF has the flexibility of matrix factorization and belongs to soft clustering that can deal with uncertainties, which possibly results in the advantages of NMF over the K-means in clustering (Ding et al. 2010).

The advantages of eNMF over K-means in the temporal estimation of DFC were not as prominent as in the spatial estimation of brain states. Although eNMF and K-means did not show significant differences in the temporal estimation of DFC for the experiments of robustness to noise, parameter size and sample size, the KL divergence of eNMF was still higher than that of K-means in most cases (see Fig. 4A, C, D). Moreover, it should be noted that the KL divergence of eNMF was marginally significantly lower than that of K-means in cases with an uneven state distribution. In the simulated experiment with an uneven state distribution, the transition probabilities from the other states to state 4 were set to be the smallest, which resulted in the occurrence frequency of state 4 being much smaller than the other states. These results suggested that eNMF had better temporal estimation of DFC than K-means when the occurrence frequencies of different brain states showed large differences. One assumption of K-means is that the clusters of samples have the same size (Bishop 2013). The simulated data with an uneven state distribution violated this assumption of K-means. Therefore, K-means showed worse temporal and spatial estimation of DFC than eNMF.

For the real fMRI data, eNMF detected more distinctive FC state patterns than K-means, although the four states of the two methods were matched (see Fig. 7). More distinctive FC state patterns can provide more information to understand each brain state. Thus, these results may indirectly support the higher spatial detection power of brain states for eNMF versus K-means in the simulated experiment. Brain state is recurring activity patterns distributed across the brain that emerge from physiological or cognitive processes. These patterns are neurobiological phenomena with functional (e.g., behavioral) relevance (Greene et al. 2023). A brain state has the following properties: (1) a brain state is the product of a specified cognitive or physiological state, (2) a brain state is characterized by a widely distributed pattern of activity or coupling, and (3) a brain state affects the future physiology and/or behavior of the organism (Greene et al. 2023). Therefore, we interpreted the cognitive function of each brain state estimated by eNMF based on its more distinctive FC patterns.

For state 1, the connections among the SMN, CON, AUN, VN and SN showed strong positive correlations, while the connections among the DMN, FPN and SCN showed weak correlations that were close to zero. The SMN works as the brain's transducer and processes sensory inputs simultaneously to form a common experience from our senses (Johansson and Flanagan 2009). The CON and SN play roles in the detection and integration of sensory stimuli (Gratton et al. 2018), while the AUN and VN accept the input of auditory and visual information. The strong interactions among SMN, CON, AUN, VN and SN in state 1 may suggest that SN detected and integrated auditory/visual stimuli and SMN oversaw subjects’ responses to the auditory/visual stimuli and formed an experience. Thus, state 1 is possibly related to the integration of auditory and visual stimuli and the interpretation of experiences in and around our body.

State 2 showed overall weaker connections than the other states, and most connections were negative and close to zero. Brain states with overall weak connections were frequently observed in many previous studies (Allen et al. 2014; Denkova et al. 2019; Xu et al. 2021) and possibly are relevant to relaxation. Brain state 3 showed stronger overall positive connections than the other states. The strong positive connections in state 3 may suggest that subjects could be engaged in spontaneous cognitive processing in this state. For state 4, most connections showed a strong positive correlation, except that the connections between the SCN and other subnetworks showed weak correlations close to zero. It has been reported that the SCN plays a pivotal role in affective and social functions in humans (Koshiyama et al. 2018). The weak connections of the SCN with other subnetworks in state 4 may suggest that the subjects were engaged in a cognitive process that was not relevant to emotional or social cognition.

For the DFC analysis of the real fMRI data, eNMF and K-means detected similar intergroup differences in the temporal and spatial measures of DFC. Compared to K-means, eNMF detected more intergroup differences in spatial measures (system segregation). Moreover, eNMF was able to detect the intergroup differences in the energy and entropy of DFC, while K-means cannot calculate these two temporal measures. These results demonstrated the advantages of eNMF over K-means in detecting intergroup differences in DFC. Therefore, we mainly interpreted the eNMF results of the real fMRI data as follows.

In contrast to the male group, the female group showed a significantly higher dwell time/mean dwell time in state 2 and a significantly lower dwell time/mean dwell time in states 3 and 4 (see Fig. 9A–D). Among the four states, state 2 showed weak negative connections, while states 3 and 4 showed strong positive connections. These results may suggest that the female group tended to spend more time in state 2 with weak negative connections, which is consistent with a previous study that found that women spent significantly more time than men in a specific brain state with many negative correlations between network systems (de Lacy et al. 2019). In contrast, the male group tended to spend more time in a state with strong positive connections. Moreover, the male group showed significantly higher connectivity strength in states 1, 2 and 4 and lower system segregation in states 1 and 4 than the female group (see Fig. 8F). The stronger connectivity strength of the male group was consistent with a previous study that observed stronger static internetwork connections in men (de Lacy et al. 2019). Stronger connectivity strength and lower system segregation of the male group may indicate that the men produced more information communications between subnetworks during rest than the female group, which could be attributed to the more frequent switching of the male group than the female group to states with high positive correlations.

Moreover, the female group showed significantly lower energy in state 2 and higher energy in states 3 and 4 than the male group. The mean energy measures the mean contribution of each state during the resting session. In contrast to the male group, the contribution of state 2 to the overall weak connections is smaller, although state 2 more frequently became the dominant state for the female group. Because state 2 could represent relaxation, the lower contribution of state 2 possibly suggests that the relaxation degree of the female group was lower than that of the male group, although the female group preferred to relax when resting. Moreover, the female group showed a higher contribution of states 3 and 4, although the female group was less frequently dominated by states 3 and 4 than the male group. Because states 3 and 4 had strong positive connections, these result may suggest that the female group tended to produce stronger spontaneous cognitive processes than the male group, although the female group did not like to switch to a state relevant to cognitive processes. It should be noted that the female group also showed significantly higher entropy in state 3 and state 4 than the male group. The normalized entropy can describe the randomness of each state’s contribution. In general, women are more emotional and have more intense and frequent emotional experiences than men (Niedenthal et al. 2006). It could be inferred that the higher randomness of the female group in states 3 and 4 is possibly attributable to more variations in emotional experiences during spontaneous cognitive processes for the female group versus the male group.

Conclusion

This study proposed the eNMF method, which can decompose any data matrix into one matrix without a positive constraint and one positive matrix. The eNMF method was successfully applied to the DFC analysis of fMRI data. The simulated experiment demonstrated that eNMF was superior to K-means in detecting spatial FC patterns of intrinsic brain states in all cases and the temporal estimation of DFC for the uneven state distribution. The real resting fMRI data revealed that eNMF can provide more temporal measures of DFC and detect more intergroup differences in both spatial and temporal measures of DFC than K-means.

Author contributions

ZL and LY contributed to the study conception and design. Data analysis was performed by YX. Material preparation was performed by YX, WZ and YD. The first draft of the manuscript was written by ZL and YX. The manuscript was reviewed and edited by ZL and WZ.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethics approval

The HCP ensures the ethics and consent required for the public use of this dataset. The HCP ensures the consent required for the public use of this dataset.