The data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database¹ (Jack et al., 2008; Jagust et al., 2010), which are used to measure the progression of mild cognitive impairment (MCI) and early AD. The rs-fMRI data set was acquired from ADNI-2 in this study. Specifically, the 3.0T Philips MRI scanner was used to scan the brain of the subjects, and a high-resolution rs-fMRI three-dimensional image was acquired by an echo plane imaging (EPI) sequence. The acquisition parameters are as follows: echo time (TE) = 30 ms, turning angle = 80°, slice thickness = 3.3 mm, slice number = 48, matrix size = 64 × 64, repetition time (TR) = 3,000 ms, and time point volume = 140.

In this study, we used the commonly available rs-fMRI data of the ADNI database and screened the subjects’ age, education level, and gender. Thus, there were a total of 68 AD and NC studies. Then, spm12² and DPARSF toolboxes³ (Yan and Zang, 2010) were applied for rs-fMRI data preprocessing. The specific steps are as follows.

Firstly, for each subject, the first 10 time points of rs-fMRI images were discarded to improve the signal-to-noise ratio and achieve signal stability. The remaining 130 time points were used for further analysis.

Secondly, the time correction was applied according to the acquisition time difference between the scanning layers. The slice at TR/2 time point was used as the reference scanning layer, and other scanning layers were aligned to it to ensure that the acquisition of each slice corresponds to the same time point.

Further, a spatial correction was applied. Each subject’s functional images were aligned to the Montreal Institute of Neurology (MNI) space according to its structural image (T1 image).

Then, spatial smoothing with a Gaussian kernel of 4 × 4 × 4 mm full width at half maximum, linear trend removal, band-pass filtering (0.01–0.08 Hz), and global signal regression were carried out in turn.

Finally, a covariate regression analysis was used to eliminate the signal fluctuation caused by a cerebrospinal fluid signal, white matter signal, and head movement.

As there is still a controversy on whether to eliminate the whole brain signal, this study did not regress the whole brain signal. At the end of the experiment, the data of head displacement more than 1 mm or of head-motion rotation more than 1° were excluded.

We constructed the two modes of brain network, i.e., spatial network and spatiotemporal network. A spatial network was established according to the brain partition template on spatially anatomical labeling, while a spatiotemporal network was constructed by sliding a time series window.

In this study, we propose a novel method based on persistent homology to detect the differences of dynamic functional networks between AD and NC groups. However, the mostly applied strategy in this area is the state extraction of dynamic functional connectivity through k-means clustering (Preti et al., 2017). To verify the effectiveness of our proposed method, we also use the traditional method to conduct experiments on spatiotemporal networks. Specifically, we used k-means clustering (de Vos et al., 2018) to capture the state changes of functional connectivity in a dynamic time series, and then analyzed the differences between the groups of all patients with AD and NC subjects in the identical states of dynamic functional connectivity. The k-means clustering method uses a distance function to cluster data into different clusters in an iterative manner. It clustered the time-varying matrices of all sliding windows of all individuals into k clustering states. The similarity within the cluster is very high, while the similarity between the clusters is extremely low. In this study, the k-means algorithm was applied to a series of time-varying matrices according to its Manhattan distance (Allen et al., 2014).

We further observed each clustering state from two aspects, window ratio and average residence time using the DynamicBC toolbox (Liao et al., 2014). Specifically, the window ratio is calculated as the proportion of staying in a given state window, and the average residence time represents the number of consecutive windows belonging to a state.

In this study, network measurement methods based on graph theory and persistent homology were used to measure brain functional organization.

A two-sample t-test was performed on the brain network properties based on graph theory and persistent homology, as well as clustering states. p < 0.05 indicates that the difference between the two groups is statistically significant. We used Matlab (R2017a) for statistical test.

The experimental participants were obtained from ADNI-2.⁵ Details of demographic characteristics are shown in Table 1. There were 31 AD subjects aged 60–90 years old in rs-fMRI data set. To conduct a comparative study with the patient group, 37 matched NC subjects were selected in this study. Diagnostic classification was made by ADNI investigators using the established criteria. The Clinical Dementia Rating (CDR) global score of any AD patient was no less than 1, while that of NC subject was 0. There are no significant differences in age, education, and gender between the groups. However, there is a significant difference in Mini-Mental Status Examination (MMSE) scores (Folstein et al., 1975). In addition, the displacement of head movement in any subject’s image is less than 1 mm, and the rotation angle in any direction is less than 1°.

In this study, the properties based on graph theory and persistent homology methods were used for the whole brain network analysis using AAL atlas. Their statistical differences between groups are shown in Table 2 using a two-sample t-test.

In the network measurement using graph theory-based properties, the network radius showed significant differences between patients with AD and subjects with NC (AD: 6.17 ± 2.63; NC: 5.97 ± 2.64, p = 0.0493) in a spatial network. However, other network properties based on the graph theory could not detect any significant differences between the groups either in a spatial or spatiotemporal network.

Then, in the network measurement using persistent homology-based properties, the SIP index showed very significant differences between the groups in a spatial network (AD: − 0.67 ± 0.13; NC: − 0.75 ± 0.10, p = 0.008 < 0.01), so does the BNP index (AD: − 257.16 ± 43.18; NC: − 283.12 ± 56.82, p = 0.041 < 0.05). Moreover, in the spatiotemporal network measurement, the group difference of BNP is more significant (AD: − 718.53 ± 84.92; NC: − 793.13 ± 109.96, p = 0.0030, much less than 0.041) than in a spatial network, and so is SIP (AD: − 0.52 ± 0.11; NC: − 0.59 ± 0.09, p = 0.002 < 0.008).

To sum up, the differences of persistent homology indicators between AD and NC are significantly greater than those of graph theory indicators. Moreover, their group differences in a spatiotemporal network are more significant than those in a spatial network. It suggests that a brain spatiotemporal network analysis based on persistent homology is more likely to explore potential biomarkers.

Default mode network is a functional subnetwork with the strongest variability in the study of AD, which contains the most obvious areas of brain atrophy. In this study, 26 areas (Vriend et al., 2018) in AAL were identified as the ROIs of DMN. The statistical differences of network properties between the groups in the DMN atlas are shown in Table 3.

In the study of DMN based on the graph theory method, there is no significant difference in the small-world attribute in the spatial DMN (p > 0.05), but it shows a very significant difference in the spatiotemporal network (AD: 1.03 ± 0.02; NC: 1.02 ± 0.01, p = 0.007), suggesting a small-world attribute could detect subtle time variability in the spatiotemporal DMN. In addition, it can be seen that the eigenvector centrality has very significant differences between the groups in both the spatial network (AD: 0.1046 ± 0.01; NC: 0.1047 ± 0.01, p = 0.009) and spatiotemporal network (AD: 0.1946 ± 0.01; NC: 0.1950 ± 0.01, p = 0.017), while there is no difference in the whole brain study.

Then, in the study of DMN based on persistent homology, both SIP and BNP detected significant differences between the groups in spatial and spatiotemporal networks. Especially in the spatiotemporal DMN, the SIP (AD: − 0.69 ± 0.12; NC: − 1.46 ± 0.10, p = 0.003) and BNP (AD: − 171.41 ± 32.24; NC: − 198.84 ± 34.08, p = 0.001) have very significant group differences.

In general, the spatiotemporal DMN properties based on persistent homology performed the best. Research on the persistent homology of spatiotemporal DMN is more likely to distinguish AD from NC, and it is more likely to explore the potential biomarkers for AD imaging.

To determine the optimal number of clusters k, we took k from 2 to 9 and repeated the test 100 times for each value. The experimental result is shown in Figure 3, where the solid point represents the best number of clusters. Here, the most suitable number of clusters was k = 2. The Silhouette score and Calinski–Harabasz index were used to evaluate the effectiveness of the clustering results. We further check the optimal number of clusters using a fivefold cross-validation. The average distance of test sets was calculated while selecting different cluster numbers, as shown in Figure 4. The error distance reached the minimum when the cluster number was 2, suggesting that the optimal value of k was 2 again. Thus, the AD and NC subjects’ varying-time matrices of sliding windows were clustered into two highly structured functional connectivity states separately.

We calculated a specific functional connectivity matrix (i.e., cluster center) for all windows in each state, and each element in the matrix was the median value of the corresponding element belonging to a state in all window matrices. The cluster center diagrams of the whole brain network and DMN are shown in Figures 5, 6, respectively, where maps correspond to the number of windows in a state, and the color bar represents the Pearson correlation coefficient value. It can be seen that state A is a stronger connection state, and state B is a sparser one. In the whole brain network clustering (see Figure 5), the window ratio of patients with AD in state A is reduced by about 5% compared with NC subjects, while it increases accordingly in state B. Similarly, in the DMN clustering (see Figure 6), compared with NC subjects, the window ratio of patients with AD in state B (AD: 0.67 ± 0.05; NC: 0.47 ± 0.06, p < 0.05) is significantly higher.

Further, the results of average residence time are shown in Figures 7, 8. It is calculated by averaging the number of consecutive windows before changing from one to another state. The number of consecutive windows of AD in state B is relatively large, either in the whole brain network (see Figure 7, AD: 40 ± 8 windows; NC: 27 ± 5 windows, p < 0.05), or in DMN (see Figure 8, AD: 47 ± 8; NC: 28 ± 6, p < 0.05).

In summarily, the clustering results of the DMN subnetwork were similar to those of the whole brain network, and the difference between the AD and NC group was more significant. The brain network of patients with AD is more likely to stay in state B, a weak connection state.

This paper studies the differences of spatiotemporal network dynamics between patients with AD and NCs in the whole brain network and its subnetwork DMN. There are three main findings in this study.

First, the spatiotemporal dynamics method combining a sliding window and persistent homology performs better than existing methods in the AD brain network analysis, and the persistent homology-based measures can be used as a potential biomarker. In our experiment, most of the network properties were more powerful in the spatiotemporal brain network than in the spatial brain network, particularly two persistent homology-based properties. Both SIP and BNP detected significant group differences (p < 0.01) in a spatiotemporal network after introducing time-varying information, in both the whole brain network and DMN subnetwork. In addition, in the DMN, the small-world attribute of a spatiotemporal network (p < 0.01) is significantly different from that of a spatial network (p > 0.05). In general, the brain spatiotemporal network could provide a more subtle temporal variability than a spatial network.

Second, in the study on the AD brain network, the DMN is a more suitable network than the whole brain network. The difference between groups of most network properties in DMN was more apparent than that in the whole brain network. In the DMN measurement using the graph theory method, the eigenvector centrality detected very significant differences between the groups in both spatial and spatiotemporal networks, while there was no difference in the whole brain study. Similarly, there was no significant difference in the small-world attribute in the whole brain network, but there was a very significant difference in the spatiotemporal DMN. Overall, the DMN is more likely to distinguish AD from NC, and it is more likely to explore the potential biomarkers for AD imaging.

Finally, the clustering results show that AD tends to be in the weak connection state. The window ratio and the average residence time in the weak connection state were relatively higher for patients with AD than those of NC. It may be due to the fact that patients with AD have not established a firm connection.

As the persistent homology-based features are monotonically decreasing convergence functions according to filtration values, both the network properties BNP and SIP can be considered as the information diffusion rate or convergence rate in the process of reaching a fully connected component. In this study, in both the whole brain network and DMN, the values of the BNP group and the SIP group showed the same pattern of | AD| < | NC|, with a significance in the level p < 0.01 based on a two-sample t-test. The fact that the BNP and SIP curves of the AD group are not as steep as those of the NC group may mean that the information diffusion speed in the AD group is slow. Because the persistent homology-based features measure the topology of the whole brain network, it is reasonable to speculate that the slow convergence rate is caused by the reduction of functional integration of the whole brain, which may further lead to cognitive defects in patients with AD.

Further, we apply the traditional method of cluster analysis to verify the abovementioned assumption. The spatiotemporal networks of AD and NC individuals were clustered into two states. One shows the pattern with a stronger connectivity in the brain region, and another presents the pattern with less connection definitions and a lower connectivity. Moreover, the brain network of patients with AD were more likely to stay in a weak connection state. This finding confirms the abovementioned assumption about the weak integration ability of AD brain network. Therefore, our proposed method based on persistent homology provides a novel viewpoint of the overall tissue injury and the interruption of neuronal integration in patients with AD dementia.

We proposed a novel method to construct and measure a spatiotemporal network based on persistent homology (i.e., spatiotemporal network + persistent homology-based properties). We compared our experimental results with two dynamic spatiotemporal methods and two static spatial methods for a functional connectivity analysis, including spatiotemporal network + graph theory-based properties, spatiotemporal network + k-means clustering, spatial network + persistent homology-based properties, and spatial network + graph theory-based properties. Because the strength of functional connectivity varies in the range of seconds to minutes, it is important to analyze the differences between different time periods, forming temporal dynamics. Only a spatiotemporal brain network is able to capture the inner dynamic nature of connectivity alterations, therefore the dynamic spatiotemporal methods are demonstrated to be more powerful to distinguish the AD group from the NC group than the static spatial methods without time-varying information, as shown in Tables 2, 3.

Further, in dynamic spatiotemporal network analyses, after sliding time windows, the state extraction through k-means clustering and the network topology measurement have been the two most widely applied strategies (Preti et al., 2017). The former technique enables to detect differences between AD and NC groups based on the dynamical occurrence and connectivity strength of connectivity states. Then, the latter technique allows to refine the meaning of the observed spatial differences across groups using some network metrics. With graph theory-based network properties, various non-trivial topological features, including small-world organization, modular structure, and highly connected hubs, have been observed to be disrupted in patients with AD (Sporns, 2018; Kuang et al., 2019a). Moreover, the persistent homology-based network properties allow to address dynamic abnormalities at a more global level, where the evolution of the global pattern of connectivity contributed by all scales is probed.

In summary, the dynamic spatiotemporal network analysis provides more completed functional alterations in connectivity, which cannot be depicted by a stationary analysis. Such kinds of technique have a greater potential for clinical applications in AD.

Although the promising results were obtained by applying the two suggested network properties SIP and BNP based on persistent homology to discriminate spatiotemporal networks between AD and NC groups, there are some important caveats. First, most network properties showed a better statistical power in spatiotemporal networks in ADNI data set. To validate the robustness of the proposed method, we will further study on other independent data sets. Then, current research is limited to the analysis of the rs-fMRI data collected in a single time period and does not verify multiple time periods. In the future, we will study the longitudinal trajectories (Dautricourt et al., 2021) of functional brain dynamics over multiple time periods. Last but not least, although this article interprets the temporal dynamics of the functional brain network well and has obtained good experimental results, a correlation analysis with behavioral data should be performed in the future.