Propagations of spontaneous brain activity in awake rats

Abstract

Slow propagations of spontaneous brain activity have been reported in multiple species. However, systematical investigation of the organization of such brain activity is still lacking. In this study, we analyzed propagations of spontaneous brain activity using a reference library of characteristic resting-state functional connectivity (RSFC) patterns in awake rodents. We found that transitions through multiple distinct RSFC patterns were reproducible not only in transition sequences but also in transition time delays. In addition, the organization of these transitions and their spatiotemporal dynamic patterns were revealed using a graphical model. We further identified prominent brain regions involved in these transitions. These results provide a comprehensive framework of brainwide propagations of spontaneous activity in awake rats. This study also offers a new tool to study the spatiotemporal dynamics of activity in the resting brain.

1. Introduction

Spontaneous brain activity is essential to brain function in health and disease (Raichle, 2010). This brain activity is typically measured by resting-state functional magnetic resonance imaging (rsfMRI), and is predominantly used to non-invasively quantify inter-areal resting-state functional connectivity (RSFC) (Biswal et al., 1995). Conventional rsfMRI analysis mainly estimates temporal co-variations of spontaneous blood-oxygenation-level dependent (BOLD) signals between brain regions. This analysis has revealed that the spatial patterns of spontaneous brain activity are far from random and are organized in highly structured brain networks (Beckmann et al., 2005).

Besides stable RSFC spatial patterns, the temporal dynamics of RSFC have been investigated with numerous methods such as sliding window (Allen et al., 2014; Calhoun et al., 2014; Hutchison et al., 2013), joint time-frequency analysis (Chang and Glover, 2010) and co-activation patterns (Liu and Duyn, 2013). In particular, it has been discovered that temporal transitions between spatially organized brain networks follow specific orders, indicating that spontaneous brain activity is not only spatially non-random, but also temporally well organized (Anemüller et al., 2006; Eavani et al., 2013; Karahanoğlu and Van De Ville, 2015; Ma and Zhang, 2018; Ponce-Alvarez et al., 2015; Vidaurre et al., 2017; Zalesky et al., 2014). Effort has also been spent in investigating the integrative dynamics of spatial and temporal patterns of spontaneous brain activity (Mitra et al., 2015a; Takeda et al., 2016; Thompson et al., 2014). For instance, Mitra et al. used a method, termed resting-state lag analysis (RS-LA), to analyze overlapped propagation patterns (aka lag threads) among voxels by decomposing the matrix of delays between time series of each pair of voxels in human rsfMRI data (Mitra et al., 2015a; Mitra and Raichle, 2016). In addition, Majeed and colleagues used a template-updating algorithm and found several spatiotemporal propagation patterns, named quasi-periodic patterns (QPPs), in both dexmedetomidine-anesthetized rats and awake humans (Majeed et al., 2011). Notably, coherence between simultaneously recorded calcium and hemoglobin signals has been reported in anesthetized and awake rats (Matsui et al., 2016; Mitra et al., 2018), suggesting that spatiotemporal propagations of rsfMRI activity has neural basis, instead of being a pure vascular phenomenon.

Despite the important advancement achieved by previous studies, systematic investigation of the spatiotemporal dynamics of spontaneous brain activity is still lacking. To tackle this issue, here we used a method based on the graph modeling of temporal transitions between whole-brain spontaneous BOLD activity defined by a library of characteristic RSFC patterns (Ma and Zhang, 2018). We have recently demonstrated that the sequences of transitions between these RSFC patterns were highly organized (Ma and Zhang, 2018). In the present study we further tested the hypothesis that besides the sequences, the time delays in these transitions are non-random. We also examined whether single-step transitions can serve as building blocks to construct multi-step transitions, which altogether provide a comprehensive framework of spatiotemporal propagation patterns in the time scale of tens of seconds. We applied the algorithm to rsfMRI data acquired in awake rats using the method established in our lab (Liang et al., 2013, 2011; Zhang et al., 2010) to avoid the potential effects of anesthesia on spatiotemporal dynamics of the rsfMRI signal (Hamilton et al., 2017; Liang et al., 2015a, 2015b, 2012a, 2012b; Ma et al., 2017; Smith et al., 2017).

2. Methods

2.1. Animals

A total of 71 male adult Long-Evans rats (300g–500g) were used in the present study. Part of data were used in previous publications (Ma and Zhang, 2018; Ma et al., 2018) and reanalyzed for the purpose of the present study. All rats were housed in Plexiglas cages with a 12h light : 12h dark schedule and a controlled temperature between 22 °C and 24 °C. Food and water were provided ad libitum. The experiment was approved by the Institutional Animal Care and Use Committee (IACUC) of the Pennsylvania State University.

2.2. rsfMRI data acquisition

All rats were first acclimated to a ‘mock’ MRI environment for seven days before imaging using the procedure described in (Liang et al., 2012a, 2011; Ma et al., 2018; Zhang et al., 2010). The acclimation procedure minimized motion and stress of the animal during scanning (Dopfel et al., 2019; Dopfel and Zhang, 2018; Gao et al., 2017; Liang et al., 2014). Before imaging, animals were briefly anesthetized with 2–4% isoflurane and restrained by a head holder with a built-in volume coil, shoulder bars and a body tube. Isoflurane was then discontinued, and rsfMRI data acquisition started at least 30 min after that. All animals were fully awake during imaging. Data were acquired at the High Field MRI Facility at the Pennsylvania State University on a 7T Bruker 70/30 BioSpec running ParaVision 6.0.1 (Bruker, Billerica, MA). Anatomical images were acquired using a rapid imaging with refocused echoes (RARE) sequence with the following parameters: repetition time (TR) = 1500 ms; echo time (TE) = 8ms; matrix size = 256 × 256; field of view (FOV) = 3.2 × 3.2 cm²; slice number = 20; slice thickness = 1mm; and RARE factor = 8. rsfMRI data were acquired with a single-shot gradient-echo echo-planar imaging sequence with TR = 1000 ms; TE = 15 ms; matrix size = 64 × 64; FOV = 3.2 × 3.2 cm²; slice number = 20; flip angle = 60°; and slice thickness = 1 mm. Six hundred volumes were acquired for each rsfMRI scan, and two to four scans were acquired for each session.

2.3. rsfMRI data preprocessing

Detailed description of the data preprocessing pipeline can be found in (Ma et al., 2018). Briefly, relative framewise displacement (FD) (Power et al., 2012) of each rsfMRI frame was calculated. Frames with FD > 0.2 mm, as well as the frames immediately before and after them were removed (3.66% of all frames in total). The first ten rsfMRI frames of each scan were also removed to ensure the steady state of magnetization. Subsequently, the first frame of each rsfMRI scan was linearly co-registered to a standard rat brain atlas using Medical Image Visualization and Analysis (MIVA, http://ccni.wpi.edu/). Motion correction was conducted using SPM12 (http://www.fil.ion.ucl.ac.uk/spm/). Spatial smoothing (Gaussian kernel, FWHM = 0.75 mm), nuisance regression (regressors: three translational and three rotational motion parameters as well as signals from the white matter and ventricles) and band-pass filtering (0.01–0.1Hz) were performed using in-house MATLAB scripts (Mathworks, Natick, MA).

2.4. Analysis of spontaneous brain activity propagations

2.4.1. Overview of the method

Spatiotemporal propagations of spontaneous brain activity were modeled as temporal transitions of whole-brain resting-state BOLD patterns, in reference to a library of characteristic RSFC patterns using the method described in our previous publication (Ma and Zhang, 2018). We first constructed templates of all possible single-step and multi-step RSFC pattern transitions. Second, we assessed the validity of these transitions by determining their occurrence ratios and statistical significance using our rsfMRI data. Third, we identified robust transitions in spatiotemporal propagations of spontaneous brain activity. Finally, the reproducibility of both sequences and time delays in all transitions identified was assessed.

2.4.2. Section I: Constructing templates of all possible spatiotemporal propagations

In this section, we described the method to construct templates of all potential spatiotemporal propagation patterns composed of a sequence of characteristic RSFC spatial patterns with measured between-pattern time delays.

2.4.2.1. Characteristic RSFC spatial patterns

We used a library of characteristic RSFC spatial patterns, which included a collection of a specified number (m = 25, 40 and 60 in the present study) ‘seedmaps’ obtained with m non-overlap brain parcels, respectively. The parcellation method was described in (Ma et al., 2018). Briefly, for each brain voxel, its RSFC map was obtained using seed-based correlational analysis with the voxel as the seed. All brain voxels were clustered by K-means clustering (K = m), with the spatial correlation between their RSFC maps defined as the distance. As a result, m brain parcels (i.e. clusters) were obtained and provided a whole-brain functional partition in awake rats, in which voxels within each parcel had similar RSFC patterns, whereas voxels between parcels had somewhat dissimilar RSFC patterns (Ma et al., 2018). The seedmap of each parcel, generated by the correlational analysis with the parcel as the seed, represented a characteristic RSFC spatial pattern. The spatial correlations between these seedmaps at m = 25, 40, and 60 were shown in Fig. S1. Notably, these parcel numbers were used as examples of low-dimensionality parcellation of the rat brain. Other parcel numbers can also be used with similar analysis.

2.4.2.2. Calculation of time delays between two RSFC patterns

The time delay between each pair of RSFC patterns was determined by detecting the lag between the consecutive occurrences of these two RSFC patterns, as illustrated in Fig. 1a. For each characteristic RSFC pattern (e.g. pattern p_i), we first calculated its framewise Pearson spatial correlation, yielding a time series of correlations with the pattern for each rsfMRI scan. The step was repeated for all RSFC patterns. As previously described (Ma and Zhang, 2018), we matched each frame to one of the 40 RSFC patterns, and for that frame only the pattern with the largest spatial correlation was kept and correlation values with the other patterns were set to zero. If the frame was not significantly correlated with any RSFC patterns (p value < 0.05/total number of rsfMRI frames), all its correlation values were also set to zero (< 0.1% of the total frames met this criterion). This step provided m thresholded correlation time series for each rsfMRI scan. The occurrences of pattern p_i in a scan can then be identified by peaks (i.e. nonzero spatial correlations) in its thresholded correlation time series. Epochs for pattern p_i including 50 frames before and 50 frames after each occurrence of p_i were extracted. This process was repeated for all m characteristic RSFC patterns. Time shifted cross-correlations between epochs for pattern p_i and pattern p_j were then calculated with the step size of 1 s (the same as TR), and averaged across all epochs in all subjects. The time delay between pattern p_i and pattern p_j was determined using time lag corresponding to the peak closest to the zero-lag point in the averaged cross-correlation curve. The directional information was maintained using the sign of the time lag (i.e. positive time lags indicate transitions from pattern p_i to pattern p_j, and negative time lags indicate transitions from pattern p_j to pattern p_i). This process yielded an m-by-m time-delay matrix, which contained m²-m time lags (excluding elements in the diagonal).

Fig. 1. Steps to identify single/multi-step transition paths between characteristic RSFC patterns with significant occurrence ratios.

(a) Calculating the time delay in single-step transitions. Spatial correlations between each fMRI frame and characteristic RSFC patterns are determined. Time delays between each pair of RSFC patterns are obtained by time shifted cross-correlations between their epochs (the sketch time-delay matrix was capped at 10 sec). (b) Calculating the occurrence ratios (ORs) of single/multi-step transitions. For a multi-step transition with a given sequence of reference RSFC templates, mean time delays relative to the first pattern are calculated by applying the ‘lag thread’ algorithm (Mitra et al., 2015a) to pairwise time delays of these RSFC templates obtained in (a). The OR of the transition is the number of instances, where rsfMRI frames match all RSFC templates in the transition with intervals between frames equal to the corresponding time delays, divided by the total number of matching trials. (c) A toy model of recursively searching for multi-step transitions with significant ORs, starting with searching for 1-step transitions with significant ORs, and then recursively searching for significant multi-step transitions whose sub-transitions are also significant.

2.4.2.3. Calculation of time delays in multi-step transitions

The pseudocode implementing the algorithm to calculate time delays in multi-step transitions was provided in Appendix (textbox 1, also see Fig. 1b). Specifically, for any n-step transition path going through RSFC patterns p₁ p₂ p₃ … p_n+1, an antisymmetric time delay matrix T0 was constructed, with the element at row i and column j representing the time delay from pattern p_i to pattern p_j calculated in the previous step. The time delays of this n-step transition path were then determined by principal component loadings from the principal component analysis (PCA) conducted on column-wise demeaned T0, based on the method reported in (Mitra et al., 2015a).

2.4.3. Section II: Determining the occurrence ratios of RSFC-pattern transitions

This section describes the method to identify RSFC pattern transitions that had occurrence ratios (ORs) significantly above chance based on our rsfMRI data. The pseudocode implementing the algorithm to calculate the ORs of transitions was provided in Appendix (textbox 2, also see Fig. 1b).

Occurrences of each single/multi-step transition were determined by sliding the corresponding RSFC templates, separated by the time delays calculated in Section I, along rsfMRI time series (step size=1). To avoid any artifactual smoothness, we did not interpolate any frames between RSFC templates. Also, to eliminate the influence of spatial similarity between RSFC templates (Ma and Zhang, 2018), before calculating the correlation between a RSFC template and a rsfMRI frame we regressed out other RSFC templates in the path that were positively correlated with the template from the rsfMRI frame.

A match of a transition path was deemed when the spatial correlations of all its templates with the corresponding rsfMRI frames were within top 20 percentile of all spatial correlations across all transitions of the same length. Using the percentile threshold allowed for comparisons across transition paths of different lengths. We also controlled the significance level of the percentile threshold selected. For instance, the p values of top 20 percentiles for 1, 2, 3-step transitions when using 40 RSFC patterns were 1×10⁻⁸, 9×10⁻⁵, and 2×10⁻³, respectively. The OR of each transition was calculated using the total count of matched occurrences divided by the total number of matching trials.

The statistical significance of the OR for each transition was decided using Monte Carlo simulation. Raw rsfMRI frames were randomly permuted and preprocessed using all steps described in rsfMRI data preprocessing, resulting in a control rsfMRI dataset. The OR of the transition in the control dataset was calculated, and this permutation process was repeated 100 times, providing the null distribution of the transition’s OR. The distribution was subsequently fitted by a Gamma distribution. The p value of the OR for the transition was calculated by the portion of ORs higher than the measured one.

2.4.4. Section III: Searching for robust transitions

We next identified robust spatiotemporal propagations of spontaneous brain activity, defined by RSFC transitions that not only themselves but also any of their subsets of transitions all had significant ORs, shown as follows:

$$S=\left\{p\right|ORofpissignificant\&allsub-segmentsofp\in S\}$$ (1)

, where p indicates a transition with any step number. Equivalently, an n-step transition met the criterion if and only if its two (n-1)-step sub-transitions (Sub-transition 1: pattern₁ to pattern_n-1; Sub-transition 2: pattern₂ to pattern_n) both had significant ORs. To search for such transitions, we developed a recursive algorithm, starting from identifying significant single-step transitions and step-by-step moving to longer transitions. It has to be noted that to construct an n-step parent transitions with two (n-1)-step sub-transitions, both the sequences and the time delays between RSFC patterns in the two (n-1)-step transitions needed to match. Therefore, we first described a time-delay-error testing method to ensure matching of time delays in the parent and the two sub-transitions.

2.4.4.1. Time-delay-error testing method

According to the method proposed by Mitra et al., the first principal lag thread corresponds to the strongest transition paths between RSFC patterns (Mitra et al., 2015a). Therefore, in our analysis we only focused on the first principal lags for all transition paths.

For a n-step transition going through RSFC patterns p₁ p₂ p₃ … p_n+1, the time delays of its first principal lag thread relative to p₁ can be denoted as [0, t₁, t₂, …, t_n]. The time delays between two adjacent RSFC patterns were given by: [t₁, t₂ − t₁, …, t_n – t_n−1], denoted as [τ₁, τ₂, …, τ_n]. Using this definition, the time delays between two adjacent RSFC patterns in the left sub-transition (p₁→p_n) and the right sub-transition (p_2→p_n+1) were denoted as [τ’₁, τ’₂, …, τ’_n−1)] and [τ”₁, τ”₂, …, τ”_n−1], respectively. The metric that measured the time delay errors was defined by the sum of squares of time delay error percentages (SSEP), shown as follows:

$$SSEP=\sum _{i=0}^{n-1}\left({\left(\frac{{\tau }_i-{\tau }_i^{\prime }}{{\tau }_i}\right)}^2+{\left(\frac{{\tau }_{i+1}-{\tau }_i^{″}}{{\tau }_{i+1}}\right)}^2\right)$$ (2)

Next, we estimated the distribution of SSEP with Monte Carlo simulation. The null distribution of $\frac{{\tau }_i-{\tau }_i^{\prime }}{{\tau }_i}$ was decided by sampling the difference between two measures of a single-step time delay divided by the mean of that time delay (referred to as time delay error percentage (TDEP)) using bootstrapping. Specifically, for a dataset of m subjects, we randomly sampled m subjects with replacement and calculated the time delays of significant single-step transitions. We repeated the sampling for 200 times to create 100 pairs of samples. The TDEP in each pair of samples was calculated, generating the TDEP distribution for each single-step transition (in total 100×k samples, k is the number of significant single-step transitions). Subsequently, we sampled with replacement from the TDEP distribution 1000 times to obtain the distribution of SSEP. Lastly, for a multi-step transition, the p value of its SSEP was the portion of the SSEPs in the distribution larger than the SSEP measured. Notably, SSEP tested type 2 error in time delays.

2.4.4.2. Recursive searching algorithm

The pseudocode implementing the recursive searching algorithm was provided in Appendix (Textbox 3). First, all single-step transitions with significant ORs were identified (p < 0.05, false discovery rate (FDR) corrected). Next, the algorithm went through all two-step transitions composed of two significant single-step transitions. All two-step transitions with p_OR < 0.05 (FDR corrected) and p_SSEP > 0.1 were identified and considered as significant. The same process was repeated to identify transitions with more steps, until no more significant multi-step transitions were found.

2.4.5. Section IV: Clustering spatiotemporal propagation patterns

Each single/multi-step transition identified in Section III corresponded to a spatiotemporal propagation pattern (4D data), which was reconstructed by averaging all epochs 5 sec before to 20 sec after the occurrences of the transition in the rsfMRI data. To reduce the redundancy and provide a general view of spatiotemporal propagation patterns, we separately clustered these patterns defined by libraries of various numbers of RSFC patterns (25, 40, and 60) using spectral clustering followed by K-means clustering. Spectral clustering was first used to obtain low-dimensional features in spatiotemporal propagation patterns before k-means clustering was applied.

Specifically, a symmetric affinity matrix W between spatiotemporal propagation patterns was calculated, where the affinity scores were defined by the Pearson correlation coefficients between these patterns plus one to meet the constraint of non-negative scores in spectral clustering. Then, a diagonal degree matrix D was calculated where each entry in the diagonal was the sum of the same row (or column) of W. Next, a generalized eigen-decomposition (D-W)v= λDv was solved, where λ was the eigenvalue and v was the eigenvector. Each eigenvector, whose length was equal to the number of patterns to be clustered, was a feature used for further clustering. Except for the smallest eigenvalue, which was equal to zero, the smaller the eigenvalue was, the larger between-cluster difference the corresponding eigenvector captured. Therefore, we sorted the eigenvalues ascendingly and extracted the 2^nd to n^th eigenvectors as feature vectors. The number n was determined by the elbow of the eigenvalue curve (Fig. 5a).

Fig. 5. Clustering the spatiotemporal patterns of all significant transition paths.

(a) Eigenvalues in the spectral clustering sorted in an ascending order. Red dots represent eigenvalues of selected eigenvectors for further K-means clustering. (b) Normalized SumD of K-means clustering at a given K. Red dots indicate the number of clusters selected. (c) The spatiotemporal Pearson correlations between the cluster spatiotemporal patterns. (d), (e), and (f) show correlations between the cluster spatiotemporal patterns calculated with different number of RSFC patterns (40 vs 25, 40 vs 60, 60 vs 25, respectively).

Subsequently, K-means clustering was applied on these feature vectors (i.e. n-1 dimensional features) to cluster the spatiotemporal propagation patterns into K clusters, with the Euclidean distance between feature vectors as the distance. We experimented K-means clustering with different Ks (2, 3, …, 30) and plotted the curve of sums of point-to-centroid distance (SumD) normalized by the total variance of feature vectors (i.e. SumD when K=1), which was essentially the portion of variance unexplained by the clustering relative to the total data variance. The K was then chosen by selecting the elbow in the normalized-SumD curve (Fig. 5b).

Finally, the spatiotemporal pattern of each cluster was presented in the form of a t map, calculated by one sample t-test on all epochs within the cluster for each voxel at each time point.

2.4.6. Section V: Reproducibility of sequences and time delays in single- and multi-step transitions

To evaluate the reproducibility of the sequences and time delays in all single-step and multi-step transitions identified at the group level, we randomly split all 71 rats into two subgroups with 35 rats in subgroup 1 and 36 rats in subgroup 2.

First, the Dice index between the single-step transitions for the two subgroups was calculated, which was defined as

$$Dice\left(S_1,S_2\right)=\frac{2\left|S_1\cap S_2\right|}{\left|S_1\right|+\left|S_2\right|}$$ (3)

, where S₁ and S₂ were sets of single-step transitions separately obtained from the two subgroups. Pearson correlation of time delays of overlapped single-step transitions between the two subgroups was also calculated.

For multi-step transitions, we first calculated the Dice index of single-step transitions involved between the two subgroups. We also evaluated the reproducibility of the time delays averaged across these single-step transitions between the two subgroups.

3. Results

In the present study we tested the hypothesis that besides the sequences (Ma and Zhang, 2018), time delays between RSFC pattern transitions were nonrandom. Furthermore, we used these single-step transitions as building blocks to construct significant multi-step transitions. We demonstrated the network structure and ORs of most robust transitions and displayed prominent regions involved in these transitions. Lastly, we clustered spatiotemporal propagation patterns reconstructed from these transitions.

3.1. Characteristic RSFC patterns from functional parcellations

We obtained three libraries of 25, 40, and 60 characteristic RSFC patterns through functional parcellations on all 71 rats. Our previous study (Ma et al., 2018) used 40 characteristic RSFC patterns obtained from 41 rats with the same method. We confirmed high reproducibility of both the functional parcellation and RSFC patterns between the previously used 41 rats and newly included 30 rats: The Dice index between the corresponding parcels was 0.509 ± 0.244 (mean ± std), consistent with our previous report (Ma et al., 2018). The absolute spatial correlation between the corresponding RSFC patterns was 0.870 ± 0.079 (mean ± std).

Each row in Fig. S1 shows the Pearson spatial correlations between RSFC patterns within each library (25, 40, and 60 parcels from top to bottom, respectively), where Figs. S1a, c, and e show the correlation coefficient matrices and Figs. S1b, d, and f show the histograms of off-diagonal matrix entries, which were consistent across different parcel numbers.

3.2. Single- and multi-step transitions between RSFC patterns with non-random time delays

Fig. S2 shows two examples of time-shifted cross-correlations between epochs of different RSFC patterns (m = 40) that were used to determine the time delays. Fig. 2a shows the matrix of time delays (n= 71) in significant single-step transitions between 40 characteristic RSFC patterns, grouped by the brain system their seed regions belonged to. Figs. S3a and e show the time delay matrices using 25 and 60 characteristic RSFC patterns, respectively. In total we observed 130, 255, and 311 single-step transitions with significant ORs with 25, 40, and 60 characteristic RSFC patterns, respectively. Figs. 2b–c show the matrices obtained from the two subgroups (m = 40, n= 35 rats in subgroup one and n = 36 subgroup two). The Dice index of significant single-step transitions between the two subgroups was 0.51. In addition, the time delays of overlapped single-step transitions between the two subgroups were reproducible (r = 0.24, p = 1.3×10⁻³), as shown in Fig. 2d. The Dice indices for the two subgroups using 25 and 60 RSFC patterns were 0.55 and 0.48, respectively, and the correlations between time delays were r = 0.31, p = 5 × 10⁻³ and r = 0.22, p = 1.3 × 10⁻³, respectively. These data demonstrate not only the sequences but also the time delays in single-step transitions between RSFC patterns were non-random.

Fig. 2. Time delays of single-step transitions with significant occurrence ratios.

(a) Time delay (TD) matrix calculated with the full dataset (71 rats). The RSFC pattern index (Ma and Zhang, 2018) and the brain system each parcel belongs to are displayed around the matrix. The direction of each transition is from the RSFC pattern in the row to the RSFC pattern in the column. (b) and (c), TD matrices calculated from the two subgroups, respectively. (d) Reproducibility of the TDs of overlapping transitions in (b) and (c).

We also identified 88, 140, and 396 2-step transitions, and 5, 30, and 143 3-step transitions with significant ORs and matched SSEPs with libraries of 25, 40, and 60 RSFC patterns, respectively. Variance explained by the first lag thread was 91.3% ± 2.00% and 88.7% ± 2.35% (mean ± std) for 2-step and 3-step transitions, respectively, for 25 RSFC patterns; 94.2% ± 3.56% and 89.5% ± 2.31% for 40 RSFC patterns; and 94.6% ± 4.30% and 91.7% ± 3.11% for 60 RSFC patterns. Between the two subgroups, the Dice indices of transitions involved in these 2 and 3-step transitions were 0.60 and 0.38 respectively for 25 RSFC patterns, 0.53 and 0.55 for 40 RSFC patterns, and 0.50 and 0.48 for 60 RSFC patterns. With 25 RSFC patterns, the Pearson correlations of the time delays averaged across these transitions between the two subgroups were r = 0.377, p = 1 × 10⁻⁴) and r = 0.768, p = 1.5 × 10⁻² for 2 and 3-step transitions, respectively. With 40 RSFC patterns, the correlations were r = 0.342, p = 3.5 × 10⁻⁵ and r = 0.561, p = 1.3 × 10⁻³); with 60 RSFC patterns, these values were r = 0.180, p = 5.2 × 10⁻⁵ and r = 0.199, p = 5.2 × 10⁻⁴. These data suggest that single-step transitions are building blocks for significant multi-step transitions with non-random time delays.

3.3. Network structure of significant single/multi-step transition paths

Fig. 3 displays the network structure of all significant single- and multi-step transition paths. In this figure, nodes represent RSFC patterns, color coded by the brain system their seeds belonged to, and edges indicate single-step transitions (Fig. 3a) or single-step transitions that were components of significant multi-step transitions (Figs. 3b–3c). Arrows indicate transition directions. The gray level of each edge in Fig. 3a indicates the p value of OR of the single-step transition. The darker it is, the lower the p value is. In Figs. 3b–3c, the gray level of each edge shows the lowest p value of OR among all multi-step transitions involving the edge. The edge width was proportional to the number of multi-step transitions passing the edge.

Fig. 3. Single/Multi-step transitions between RSFC patterns.

Each node represents a RSFC pattern, color coded based on the brain system the seed belongs to. Edges indicate significant single-step transitions (a) or those involved in significant multi-step transitions (b-c). Arrows indicate the transition direction. Nodes without edges are not shown. The gray level of each edge indicates the p value of occurrence ratio of the single-step transition (a), or the lowest p value of occurrence ratios of multi-step transitions passing the edge (b-c). The edge width in (b-c) is proportional to the total number of multi-step transitions passing the edge. PF: prefrontal areas; CG: cingulate areas; RSC: retrosplenial areas; M: motor areas; SS: somatosensory areas; OLF: olfactory areas; AUD: auditory areas; V: visual areas; HC: hippocampal and retrohippocampal regions; ST: striatal regions; AMY: amygdala regions; THA: thalamic regions; HYPO: hypothalamic regions; dMB: dorsal mibrain regions; vMB: ventral midbrain regions.

Fig. S3 displays the network structure calculated from the libraries of 25 (a-c) and 60 (d-f) RSFC patterns, respectively. We observed consistent network topology across different parcel numbers. Single-step transition networks tended to be separated into two dominant community structures, where transitions were more likely to occur within communities than between communities. One community included RSFC patterns with the seed regions in the prefrontal cortex (PFC), striatum, thalamus, hippocampus, and dorsal midbrain, and the other community contained the visual, somatosensory (SS), motor and cingulate cortices. In addition, RSFC patterns of the thalamus, auditory/temporal association, somatosensory and cingulate cortices were nodes connecting the two communities. For two- and three-step transitions, this topological pattern remained similar, although less transitions, especially inter-community transitions, were involved. Notably, separate transition paths could share the same segments (i.e. transitions) and some segments were visited more frequently than others, as indicated by different edge widths in Figs. 3b–c and Figs. S3 c–d, g–h.

3.4. Involvement of individual RSFC patterns in transitions

We examined whether single/multi-step transitions more likely visited some RSFC patterns than others. Notably, transitions with high spatiotemporal similarity tended to involve the same RSFC patterns. To avoid such redundancy, we did not simply count the number of transitions involving a certain RSFC pattern. Instead, for each RSFC pattern, we regressed out the spatiotemporal patterns of all transitions involving this RSFC pattern from rsfMRI data, and calculated the corresponding reduction of variance in rsfMRI data. Higher reduction in variance was quantitatively associated with more involvement of the RSFC pattern in transitions.

Specifically, for each transition, we first correlated its reconstructed spatiotemporal pattern to rsfMRI data using a sliding window with a step size of 1. In this correlation time series, values at time points where no transition occurred were to zero, and the occurrence was determined using the same method in OR calculation. Subsequently, we generated a 4D regressor for this transition by convolving the correlation time series with the spatiotemporal pattern of the transition, which was then regressed out from rsfMRI data with non-negativity constraints on regression coefficients. For each RSFC pattern, we calculated the reduction of variance after regressing out all transitions that included (Fig. 4a), started with (Fig. 4b), or ended with (Fig. 4c) the RSFC pattern. The same procedure was repeated in the permuted control rsfMRI dataset, and the reduction of variance for each RSFC pattern was statistically compared between the real and control datasets using t-tests.

Fig. 4. Variance explained by the transition paths that (a) included, (b) started with, or (c) ended with individual RSFC patterns, compared to that in the permuted dataset.

The corresponding seed regions are displayed. Transparent color indicates the number of transition paths are zero. Distance to Bregma is displayed at the bottom of each slice.

Fig. 4 shows the involvement of individual RSFC patterns in transitions (seed regions were displayed in t values). Overall, heterogeneous distributions of these measures were observed among RSFC patterns. The RSFC pattern of dorsal hippocampus (HC) was the most involved in transitions, followed by the auditory cortex (AUD) and anterior cingulate area (ACA). The RSFC pattern of the ACA was the most frequent starting pattern of transitions, followed by the AUD and dorsal HC. The RSFC pattern of the dorsal HC was the most frequent ending pattern of transitions, followed by the SS and striatum. Prominent involvement of these RSFC patterns in transition paths indicate potential pivotal roles of their corresponding functional networks in mediating the spatiotemporal propagations of spontaneous brain activity.

3.5. Clusters of spatiotemporal propagation patterns reconstructed from significant transitions

We clustered significant transitions and calculated the cluster-wise spatiotemporal patterns as described in Methods. Figs. 5a–b show the process of selecting the number of eigenvectors and clusters, respectively, in the case of 40 RSFC patterns. Since an elbow at the 7^th eigenvalue was observed (Fig. 5a), the 2^nd to the 7^th eigenvectors were selected as the features (i.e. 6 values were used to represent the spatiotemporal pattern of a transition) to cluster transitions. Then we repeatedly conducted K-means clustering with different Ks (2 to 30) and calculated the normalized SumD (sums of point-to-centroid distances) given each K. As shown in Fig. 5b, the elbow of the SumD curve was around 10. As a result, the spatiotemporal propagation patterns of all transitions were clustered into 10 clusters. Cluster-wise t-maps of the spatiotemporal patterns were calculated. Fig. 5c shows the spatiotemporal Pearson correlation between these t-maps (the absolute Pearson correlation was 0.487±0.265 (mean ± std)). The clustering process was repeated for 25 and 60 RSFC patterns, and resulted in 10 clusters in both cases as well. The pairwise correlation between the cluster-wise t-maps calculated with 25, 40, and 60 patterns were shown in Figs. 5d (40 vs 25), 5e (40 vs 60), and 5f (60 vs 25), demonstrating good consistency.

Fig. 6 shows the 10 cluster-wise spatiotemporal patterns calculated with the library of 40 RFSC patterns. Fig. 6a shows the propagation from the default-mode network (DMN), which consisted of ACA, IL, PL, VO, dorsal thalamus, dorsal HC, retrosplenial cortex (RSP), and temporal association cortex (TeA), to the sensorimotor cortex (SM). Fig. 6b shows the propagation from anterior midbrain to the whole cortex excluding the PFC. Fig. 6c shows the propagation from the SS to the dorsal HC. Fig. 6d shows the propagation along the cortex in mid- and posterior brain, as well as the dorsal thalamus and anterior midbrain accompanied by deactivation of the Re and BF. The pattern in Fig. 6e shows DMN activation in the beginning, followed by activation of the PFC, anterior ventral thalamus, posterior dorsal thalamus, and dorsal HC and deactivation of the amygdala (AMY), hypothalamus, motor cortex and visual cortex. The pattern in Fig. 6f started from activation of the dorsal and ventral brain as well as deactivation of the PFC and ventral striatum, followed by activation of the cortex excluding the PFC, AUD, and TeA. Fig. 6g shows the propagation from the midbrain to the BF, ventral thalamus, striatum, and PFC. In addition, the AMY, motor cortex, RSP, and ACA remained deactivated during the propagation. Fig. 6h shows the propagation from the PFC, ventral thalamus, BF, ventral striatum to the dorsal HC and TeA; and the AMY, motor cortex, RSP, and ACA remained deactivated during the propagation. Fig. 6i shows the propagation from the PFC, striatum, BF, and ventral thalamus to the dorsal HC. Fig. 6j shows the propagation from the dorsal HC to the posterior midbrain, ventral striatum, ventral thalamus, and BF.

Fig. 6. Cluster-wise spatiotemporal propagation patterns of all significant transition paths.

Transitions begin at 0 second. Patterns from 2 seconds before to 9 seconds after the beginning of transitions are shown.

3.6. Propagations of spontaneous brain activity was not caused by motion and/or signal-to-noise ration (SNR) variance across the brain

To rule out the potential impact of subject motion on the parcellation and reference RSFC patterns, we divided the dataset (71 rats) into two groups based on the mean FD of each rat, with 35 rats in the low-motion group (i.e. all animals’ motion levels were below median) and 36 rats in the high-motion group (i.e. all animals’ motion levels were above median). Between these two subgroups, we validated the reproducibility of the parcellation with the Dice index between the corresponding parcels (m=40) of 0.543 ± 0.211 (mean ± std), and the spatial Pearson correlation between the corresponding characteristic RSFC patterns of 0.901 ± 0.089 (mean ± std), suggesting that motion had minimal impact on the parcellation and resulting characteristic RSFC patterns. In addition, framewise FD did not co-vary with the corresponding framewise spatial correlations with any characteristic RSFC patterns across all rsfMRI scans (FDR > 0.72). Furthermore, we found that FDs during propagation epochs and those during randomly selected equal-length epochs were comparable (Fig. S4). Taken together, these results demonstrated that motion was unlikely the dominant factor driving the propagations of spontaneous brain activity.

We also confirmed that the observed spatiotemporal propagation patterns were not related to the variation of SNR in fMRI signal. It has been suggested that activation could appear first in regions with stronger signal (higher SNR) and next in those with weaker signal (lower SNR), forming an artificial propagation, while signals in these regions were actually in phase (Logothetis et al., 2009). Our method should be insensitive to SNR variations within the brain since all discovered transitions had significantly higher ORs in real rsfMRI data than those in the permuted data, in which SNR variations were reserved but the temporal sequences of rsfMRI frames were shuffled. However, to rigorously rule out this possibility, we generated a whole-brain SNR map averaged across all frames in all scans (Fig. S5a), where the SNR of each voxel was defined as its raw fMRI signal divided by the standard deviation of voxels outside of the brain (four corners of the FOV on every slice were selected). To investigate whether transitions tended to travel from high-SNR brain region to low-SNR brain region (or vice versa), we fit the whole-brain SNR distribution by two Gamma functions and split the brain into low- and high-SNR regions based on the SNR cutoff obtained (SNR cutoff = 41.6, Fig. S5b). Then, we counted transition steps that traveled across the two regions, in all single- and multi-step transitions (Fig. 3). We found that the numbers of high-to-low and low-to-high transitions were well balanced (Fig. S5c). Furthermore, we ran a paired Student’s t-test on the SNR in source and destination regions of each transition step and found no significant difference between SNR in source and destination regions (t = −0.64, p = 0., Fig. S5d). Finally, to tease out the possibility that SNR variance drove propagation patterns with other mechanisms (Laumann et al., 2016), we generated a simulated dataset with shifted signal phase but the same SNR variance. For each individual voxel, we maintained its spatial location but circularly shifted its fMRI signal in each scan by a random number between −100 s (left shift) and 100 s (right shift). The shift was evenly distributed across all voxels. We applied our method to the simulated dataset using either the characteristic RSFC patterns calculated from the simulated dataset or the ones from the real dataset. We found no one-step transition with significant OR in both cases. Taken together, these results suggest that the observed spatiotemporal propagations overall were not driven by SNR variance across the brain.

4. Discussion

In this study, we investigated the spatiotemporal dynamics of spontaneous brain activity in the awake rat brain. We found that not only the sequences, but also the time delays in transitions between RSFC patterns were nonrandom. In addition, we identified a number of robust transitions among multiple RSFC patterns with nonrandom and reproducible sequential orders and time delays, revealed a network structure of these transition paths, and showed prominent brain regions involved and their temporal evolutions during the propagation of spontaneous brain activity. Furthermore, we clustered the spatiotemporal patterns of the RSFC transitions to provide a more general view. These observations collectively provide important evidence demonstrating well-organized coordination among large-scale brain systems both spatially and temporally.

4.1. Method applied to uncover the spatiotemporal propagation patterns in rsfMRI data

There are two key components in our method to investigate the spatiotemporal dynamics of spontaneous brain activity: dimensionality reduction in rsfMRI spatial patterns and graph model of transitions between RSFC patterns.

Given the high dimensionality of rsfMRI spatial patterns, an effective dimensionality reduction method is a crucial step for systematic investigation of spatiotemporal patterns of rsfMRI data. In the present study, our strategy was to use a library of characteristic spatial patterns and match each rsfMRI frames to one of the spatial patterns. These patterns were generated using a RSFC-based functional parcellation method based on rsfMRI data collected in awake rats (Ma et al., 2018). In this parcellation scheme, voxels with similar RSFC patterns were clustered, obtaining non-overlap m (m = 25, 40, or 60) parcels covering the whole brain. Each parcel provided a characteristic RSFC pattern, yielding a library of m characteristic RSFC patterns. Our previous study showed that the parcellation and RSRC patterns generated were high consistent across animals even for rsfMRI data collected at different magnetic field strength (Ma et al., 2018). Because RSFC patterns were similar within parcels and dissimilar across parcels, reflected by high within-parcel homogeneity (Ma et al., 2018), this library of characteristic RSFC patterns provided a comprehensive survey of spatial patterns of rsfMRI data across animals, and thus offered a reliable basis for reducing the dimensionality of rsfMRI spatial patterns, based on the premise that they were proper references of instantaneous rsfMRI spatial patterns, demonstrated in previous reports that BOLD patterns of single rsfMRI frames also represented instantaneous RSFC patterns (termed coactivation patterns) (Liu and Duyn, 2013). Our previous paper also showed that frames matched to each characteristic RSFC pattern indeed had very high spatial similarity with the pattern (spatial correlations > 0.72 for all characteristic RSFC patterns) (Ma and Zhang, 2018).

The dimensionality reduction in reference to characteristic RSFC patterns enabled studying spatiotemporal propagations using a graphical model, where nodes represented RSFC patterns and binary edges represented significant transitions between these patterns. Searching for the propagation patterns was then equivalent to searching for paths (i.e. multiple connected edges) between nodes. As a result, multi-step transitions could be constructed by connecting significant single-step transitions.

It needs to be noted that a path in the transition graph only indicate this transition can theoretically exist, but does not guarantee the actual occurrence of these transitions. This gap was filled by looking for segments in our rsfMRI data that resembled the concatenated RSFC patterns constructed based on the transition path, which took into account both RSFC patterns involved and time delays between them (Mitra et al., 2015a). Moreover, identifying multi-step transitions in which all sub-transitions were also statistically validated enabled us to reconstruct pivotal spatiotemporal propagations in spontaneous brain activity.

4.2. Spatiotemporal propagations of spontaneous brain activity might represent a general phenomenon in the mammalian brain

Infraslow (0.01–0.1 Hz) propagations of spontaneous brain activity were observed in both humans and animals. With the QPP algorithm, Majeed et al. (Majeed et al., 2011) observed the propagation between the default-mode network and the task-positive network with the propagation duration of around 20 seconds. Using the RS-LA method, Mitra et al. (Mitra et al., 2015a) observed eight faster (within 2 sec) and reproducible spatiotemporal propagation patterns, likely representing transitions both within and across resting-state networks.

In rodents, Majeed et al. (Majeed et al., 2011) observed two propagation patterns in anesthetized rats: lateral-to-medial propagation across the cortex and within-CPu propagation. In addition, Belloy and colleagues (Belloy et al., 2018) reported propagations among the CPu, CG and somatosensory systems in anesthetized mice with a similar duration. Mitra et al. also observed infraslow propagations of spontaneous neural activity from the motor cortex to the visual cortex in awake mice by using calcium/hemoglobin optical imaging (Mitra et al., 2018).

Correspondingly, in awake rats we revealed propagations involving similar brain structures in a similar time scale. For instance, we observed the propagation from the motor cortex to the visual cortex (Fig. 7a), resembling the spatiotemporal propagation pattern previously reported in awake mice (Mitra et al., 2018). We also observed the propagation from the CPu to the CG (Fig. 7b) in the time scale of 5 sec, similar to that demonstrated by Belloy et al. (Belloy et al., 2018). In addition, the lateral-to-medial propagation in the somatosensory cortex and CG we showed (Fig. 7c) was consistent with the finding reported in (Majeed et al., 2011), and the ventral-to-dorsal propagation in CPu we found (Fig. 7d) was consistent with the finding by Thompson et al. (Thompson et al., 2014).

Fig. 7. Examples of spatiotemporal propagation patterns consistent with the literature.

(a) Propagation from the motor cortex to the visual cortex. (b) Propagation from the caudate putamen to the motor and cingulate cortices. (c) Lateral-to-medial propagation in the somatosensory and cingulate cortices. (d) Ventral-to-dorsal propagation in the caudate putamen. (e) Dorsal-to-ventral propagation in the caudate putamen.

In addition to these replicated patterns, our study identified new propagation patterns that were not reported in anesthetized animals, likely due to the difference in animals’ state during rsfMRI data acquisition. For instance, we for the first time observed propagations from the DMN to SM (Fig. 6a), from the SS to HC (Fig. 6d), and from the midbrain to cortex (Fig. 6b) in rodents, consistent with the findings in humans (Mitra et al., 2018). We also observed the unreported dorsal-to-ventral propagation in the CPu (Fig. 7e) as well as multiple subcortical transitions between striatum, HC, and midbrain (Figs. 6g, i, and j). These results collectively provide a comprehensive framework of spatiotemporal propagations of spontaneous brain activity. Consistent finding in brain activity propagation between humans and rodents suggests that it might represent a general phenomenon in the mammalian brain.

4.3. Possible neurological underpinnings of spatiotemporal propagations

There has been accumulating evidence suggesting the neural origin of the spatiotemporal propagations of spontaneous brain activity. Coherence was reported between the calcium signal and the hemoglobin signal measured using optical imaging, indicating that infraslow (0.01–0.1 Hz) spatiotemporal propagation patterns in the cortex of anesthetized mice reflected neural activity (Matsui et al., 2016). A more recent study showed that the propagation trajectories in the infraslow band were state-dependent (wake versus anesthesia) and different from those in delta (1–4 Hz) activity (Mitra et al., 2018). Combined with laminar electrophysiological recording, the study further demonstrated that infraslow activity traveled through specific cortical layers and had a different cross-laminar temporal dynamics from those of higher frequency activity. Our results further validate that the cortical propagation could be a part of brain-wide activity propagations.

It has also been revealed that the propagation patterns could be altered by the arousal level (Mitra et al., 2015b). Interestingly, our results showed that a number of system/regions such as the DMN, PFC, VTA, HYPO, ZI, BF, and midline nucleus of the thalamus, which were believed to relate to sleep-wake/arousal modulation (Drew et al., 2018; Schiff, 2008; Schwartz and Kilduff, 2015; Van Der Werf et al., 2002), were prominently involved during spatiotemporal propagations. Furthermore, we observed the activation of the whole cortex (excluding the PFC and visual cortex), accompanied by deactivation of the midline thalamus nucleus and BF in brain activity propagation. The pattern is similar to that reported in humans and monkeys (Liu et al., 2018), which was believed to be related to variations in arousal. These data collectively suggest that whole-brain spatiotemporal propagations might be modulated by different physiologic states.

4.4. Potential limitations

Even though modeling the spatiotemporal dynamics with transition paths among characteristic RSFC patterns allowed for a systematic investigation of spontaneous brain activity, the method has limitations. First, the method cannot exclusively identify all multi-step transitions. Instead, we focused on the most robust transitions in spontaneous rsfMRI data. Second, although using a library of predefined reference patterns has the advantages of straightforward interpretation of results and easy comparison across studies, this method may bias the dynamic transitions discovered towards some dominant reference patterns, while mask subtle spontaneous dynamics. Third, the spatiotemporal propagations can be affected by animals’ experience inside the scanner, such as auditory input and situational memory. Like all human rsfMRI experiments, the subject’s experience inside the scanner is difficult to control. For instance, auditory stimuli and situational memory are inevitable in fMRI studies, and the number of independent variables impacting the brain activity is difficult to estimate. With these complexities in mind, we used an indirect method to estimate whether the subject’s experience determined the propagation of RSFC patterns by postulating that the subject’s experience was dependent on the time spent inside the scanner. Given that multiple scans were acquired for each animal, we measured the correlation between the ORs of all RSFC pattern transitions for each individual scan and the time spent inside the scanner at the midpoint of the scan. We did not find any significant correlation between these two quantifies, suggesting that the observed spatiotemporal transitions were irrelevant to the time animals spent in the scanner (i.e. the length of exposure to external stimuli).

5. Conclusions

This study aimed to investigate the infraslow spatiotemporal propagations of spontaneous brain activity in awake rats. We used a method that identified nonrandom multi-step transition paths and validated the corresponding spatiotemporal propagation patterns in rsfMRI data. Furthermore, we systemically revealed the network structure of these transition paths and propagation patterns. The study has offered a new tool to investigate the spatiotemporal propagations of spontaneous brain activity and provided new insight into the spatiotemporal organization of activity in the resting brain.

Appendix

TEXTBOX 1.

Algorithm 1: Calculating time delay matrices of multi-step transition paths

Input:

A multistep transition path p, where p is a vector of reference RSFC pattern labels (from 1 to number of patterns n_pat).

A time-delay array T (n_pat x n_pat). T[i][j] represents averaged time delays between reference pattern i and j.

Output:

A time-delay vector td (l x 1), where the elements indicate the time delays to the first pattern

% Get the corresponding set of time delays, a submatrix T0 of T.

for m = 1…l:

for n = 1…l:

if m <= n:

T0[m, n] = T[pi[m], pi[n]]

T0[n, m] = − T[pi[m], pi[n]]

% Apply the lag thread algorithm

T0 ← demean each column of T0

coeff ← factor loadings (component coefficients) of the 1st principal component from principal component analysis of T0

td ← T0 x coeff

td ← td - td[1]

TEXTBOX 2.

Algorithm 2: Calculating occurrence ratios of transition paths

Input:

A preprocessed rsfMRI data array D (number of voxel n_v x number of frames n_f x number of scans n_s)

A reference pattern array REF (n_v x number of reference patterns n_pat)

A list of transition paths P = [pi], where pi is a vector of reference pattern labels (from 1 to n_pat).

A time delay array T (n_pat x n_pat). T[i][j] represents averaged time delay between reference pattern i and j.

A constant padding number n_pad

Output:

Occurrence ratios or for each path in P

for pi in P:

l ← transition step of pi

count ← 0

% Get time delays of the 1st principal lag thread in pi

if l = 2:

td (l x 1) ← Algorithm 1(pi, T)

else:

td [0, T[pi[1]], pi[2]]

% Calculate correlations between the patterns and the regressed rsfMRI frames and get occurrence counts

for s = 1…n_s: % loop through scans

for f = 1…n_f – n_pad: % loop through frames

for m = 1…l: % loop through RSFC patterns in the path pi

d_reg regress out RSFC patterns 1..m-1, m+1..l if they are positively correlated with RSFC pattern m from D[:, f+td[m], s]

r0[m], p0[m] ← correlation(REF[:,pi[m]], d_reg)

if all elements in r0 >threshold:

count ← count + 1

or[i] ← count/(n_f-n_pad)/n_s

TEXTBOX 3.

Algorithm 3: Recursively searching for multi-step transitions

Input:

A preprocessed rsfMRI data array D (number of voxel n_v x number of frames n_f x number of scans n_s)

A reference pattern array REF (n_v x number of reference patterns n_pat)

A time delay array T (n_pat x n_pat). T[i][j] represents averaged time delay between reference pattern i and j.

Null distribution of occurrence ratios Hor[l] for l-step transitions. l = 1…7

Non-null distribution of sum of squares of time delay error percentages (SSEP) of l-step transitions HSSEP[l]. l = 1…7

Output:

A transition path list P

Initialize p_val, path_set, and P with empty lists

% Get significant 1-step transition paths

for i = 1…n_pat:

for j = 1…n_pat:

path ← [i,j]

or ← Algorithm 2(path)

Get the p value of or with Hor[1] and append it to p_val

Include path into path_set

p_val ← FDR_correct(p_val)

path_set ← path_set[p_val < 0.05]

Include path_set into P

% Get significant multi-step transition paths

path_set ← all 2-step transitions composed of 1-step transitions in P

n_step ← 2

while path_set is not empty:

p_val ← Get the p values of SSEP of paths in path_set

path_set ← path_set[p_val > 0.1]

ors ← Algorithm 2(path_set) % calculate occurrence ratios

p_val ← Get the p values of or with Hor[nstep]

p_val ← FDR_correct(p_val)

path_set ← path_set[p_val < 0.05]

Include path_set into P

path_set ← all (n_step+1)-step transitions composed of n_step-step transitions in P

n_step ← n_step+1