The human connectome undergoes complex transformations across the lifespan, and it can be mathematically modeled as a complex network by graph theoretical analysis. Imaging connectomics based on graph theory has become an effective and unique methodological framework for studying functional connectivity patterns of the developing and aging brain [1
]. Normal brain development is characterized by continuous and significant network evolution through infancy, childhood, and adolescence, following specific maturational patterns. Normal aging is related to some resting state brain networks disruption, which are associated with certain cognitive decline. Specifically, a connectome (i.e., a large-scale resting-state functional connectivity network, RSFC) can be mapped by taking each parcellation unit of the whole-brain as nodes and estimating inter-unit correlations (i.e., functional connectivity, FC) in spontaneous blood oxygenation level dependent (BOLD) fluctuations as edges. Combined with graph theoretical analysis, this study aimed to explore typical age-related architecture changes of the human brain, which is a prerequisite for studying neuropsychiatric diseases such as autism, depression, schizophrenia, Alzheimer's disease, and Parkinson’s disease.
Many connectomics studies using network science and graph theory have shown age-related changes of FC patterns. FC patterns between/within resting-state subsystems (a set of brain regions that exhibit coherent activity in a task-free state [2
] and exhibit consistent spatial topographic patterns across the cerebral cortex [3
]) have been investigated with age [5
], and FC patterns for brain regions had also been examined across the lifespan [6
]. In addition, several whole-brain FC pattern studies on RSFC networks have focused either on early or late age-related changes [3
], while other studies have focused on specific subsystems involving default networks [14
], and cognitive-control [18
]. Moreover, findings on large-scale RSFC network properties have revealed that modularity and local efficiency decreased over aging [8
]. In contrast, another study reported that modularity was similar in older and younger participants [24
]. Furthermore, studies of connectomics on global efficiency have not reached a consensus [8
]. These studies focused on studying the FC pattern or some network properties in the developmental stage or across the lifespan. Little attention has been focused on the changes to whole-brain centralities with age.
Nodal connectivity or centralities account for a given region’s relationship with the entire functional connectome and not just its relationship to specified regions (seed-based analysis), or to brain subsystems (independent component analysis). As such, centrality measures allow us to capture the given region’s importance in the functional connectome. Prior studies of nodal connectivity or centralities have focused primarily on identifying “cortical hubs” [8
] and finding rich-clubs [21
] in healthy adulthood participants. A few researchers have reported that the degree distribution in the normal adult human brain conformed to an exponentially truncated power law [25
]. Recent neuroscience studies have employed eigenvector centrality (EC) mapping for measuring FC in both diseases [34
] and task states of brain activity [26
]; however, these studies have not explored EC pattern changes of the whole-brain RSFC network across the lifespan in healthy individuals.
The network eigen-entropy (NEE) was defined based on EC and the Shannon entropy. In a complex system, it characterizes the orderness change of a system indicating the state or the development direction of a system. EC defined in this way assigned each node a centrality that depended on both the quantity and the quality of its connections. It integrated the centralities of a node’s neighbors and each node was ranked by its EC scores in a network. Therefore, EC was extended to capture the global features of the graph [36
]. The energy probability (EP) of each node was derived from EC scores in a graph network. The resulting set of energy probabilities generated a frequency distribution. The EP can be regarded as a random variable that follows this frequency distribution, and the distribution measured the functional connectome heterogeneity [36
]. If the nodes’ function in the network is not specialized, and the communication of information is not preferential, the EP distribution is uniform, and the network is homogenous. However, if the nodes’ function is specialized, and the communication of information is preferential, the EP distribution is distributed, and the network is heterogeneous. Collectively, the more distributed the EP histogram of a system is, the lower the entropy is and vice versa. Thus, the NEE allows us to analyze the organizational mechanisms of the functional connectome development.
We initiated this study under the assumption that the orderness of the functional connectome was related to the age of the brain.
In this study, we calculated the whole-brain NEE from the EP of parcellation units of the cerebral cortex. By systematically investigating the development and aging of human connectome from 7 to 85 years, we found that the whole-brain NEE showed a non-linear decrease with age. The EP distribution within the whole-brain shifted from concentration to wide dispersion with age at group level. Moreover, the NEE values of subsystems showed non-linear or linear decreases with age. Interestingly, the NEE values of the sensory motor area showed negative quadratic changes with age, while the NEE of limbic system showed positive quadratic changes with age, and the other five subsystems showed linear decreases. Furthermore, brain regions with significant EP changing from flourishing (7–20 years) to the youth period (23–38 years) were mainly located in the right prefrontal cortex and basal ganglia, the areas involved in emotion regulation and executive function in coordination with the action of the sensory system, thus implying that self-awareness and voluntary control performance significantly changed during neurodevelopment. Those from the youth period through to middle (40–59 years) age were located in the mesial temporal lobe and caudate (which are associated with long-term memory), implying that the memory of the human brain begins to decline with age during this period. These results may be relevant for understanding normal developmental and aging changes in neural circuits that underlie age–related variation in cognition and behavior. These novel findings are discussed below in detail.
4.1. The Orderness of the Functional Connectome Measured by the Network Eigen-Entropy
Previous studies on structural and functional connectomics have revealed that the human brain has a small-world structure [33
], which possesses a greater clustering coefficient than the random network and lower characteristic path length than the regular network. Unlike the properties of the small-world structure, whole-brain NEE characterizes the orderness of network structure from a spatial perspective based on the EC and statistical entropy. As EC conveys the global structural information of the network, the uniformity of the EC distribution reflects the degree of complex network heterogeneity. So, NEE can also capture the heterogeneity of the network. The entropy of a single discrete random variable is a measure of its average uncertainty, characterizing the degree of underlying randomness of the random variable. The orderness/certainty implied by NEE is dependent on the similarity of the intrinsic structure on multi-levels in a network. The more uniform energy distribution, the greater the entropy, the more uncertainty in the system; on the other hand, the more orderness/certainty in the system.
On the application of entropy in the human brain, there are several studies related to our study. Yao and his colleagues [51
] regarded the correlation coefficients between brain regions as a random variable, and applied relative entropy to measure the variability in the distribution of correlation coefficients. The results demonstrated that the relative entropy increased with age. These results investigated whether the FC changed with age. Jia and his colleagues [51
] regarded the dynamic correlation coefficients between brain regions as a random variable, and applied sample entropy to measure the complexity of dynamic FC, while the results demonstrated that the sample entropy reduced with age. Mcintosh and his colleagues [52
] used multiscale entropy (MSE) to measure brain signal variability through a collection of the electroencephalography (EEG) and magnetoenphalography (MEG) signals in tasks. MSE characterized the temporal complexity of the brain signal, while NEE depicted the spatial complexity of the interaction between brain regions signal activities across the whole brain. However, these entropy analyses cannot capture the orderness of the brain network structure from a spatial perspective.
Through the analysis of the Mann–Whitney rank-sum test, we found that the mean NEE value of the RSFC networks was lower than that of random networks and their variance box distribution differed by two magnitudes (Figure 6
d). As the connections between nodes in the random network were random, the EC distribution was uniform, and the intrinsic architecture of the functional brain network was regular relative to the random network. The degree of orderness of the functional brain network was higher than the random network, and the NEE could robustly distinguish functional brain networks from random networks.
4.2. The Orderness Variability of the Whole-Brain with Age
To specifically assess the association of the whole-brain NEE with age, we divided the fMRI dataset I into four age groups. In the analysis of different aged individuals in dataset I, we found that at the group level, the EP distribution of the RSFC networks shifted from concentration to dispersion (Figure 1
). Through a statistical analysis of whole-brain NEE of different aged individuals, we found that the whole-brain NEE was robust when compared among the flourishing, youth period and middle age groups. Furthermore, the whole-brain NEE was significantly different among the entropies of the three sample groups, and further analysis over continuous aging showed that an approximate U-shaped trajectory of the whole-brain NEE fitted. The distribution of EP in the middle age group was more widely spread than that of the flourishing and youth periods. This led to the middle age group having a smaller NEE than that of the youth period group, which had an even smaller NEE than that of the flourishing. This revealed that there was a higher level of heterogeneity of functional connectome during middle age group than the youth period, and the youth group had a higher level of heterogeneity of functional connectome than that of the flourishing. This implied that the orderness/certainty of functional connectome enhanced rapidly until middle age. In contrast, the EP distribution was partially adjusted from middle age to old age. This led to no significant change of NEE between middle age and old age (61–85 years), and these findings may be associated with the degree of functional connectome adjustment with age. This finding verified the opinion in What is Life
written by Schrodinger [53
] that life continues to draw on negative entropy from the environment. The finding also was consistent with the evolution theory of the life system proposed by Darwin that the development of the organism and the highly complex brain is a process of diminishing entropy. These results may partly reveal the architectural transition of the human brain shift from a relatively anatomically driven configuration to a functional well-organized configuration with age. Furthermore, the current findings seem consistent with the notion of free energy minimization in the brain, as the entropy-like term (NEE) did indeed decrease with aging subjects [54
]. Considering the notion of entropy in the context of free energy minimization may extend the value of this study to the broader community.
4.3. The Orderness Variability of Functional Subsystems with Age
In addition to determining the NEE of the whole brain, we determined the NEE of seven intra-subsystems of the brain across a lifespan. The subsystem NEE exhibited the FC pattern within this brain functional system, which indicated the orderness variability of the specific brain functional system. We observed distinct trajectories of these subsystem NEEs with age. Concretely, the NEE of limbic system decreased quickly over the early part of the lifespan (peak age of about 60 years); thereafter it showed a slightly increasing trend. This meant that the orderness of the limbic system was enhanced before 60 years, and weakened afterwards. The limbic system, as a subcortical cortex controlling human emotions and spiritual activities, showed a consistent trajectory with the whole-brain NEE. In contrast, the NEE of the sensory motor area remained at a high level before the age of 50, indicating that it maintained high homogeneity before the age of 50. After 50 years of age, its NEE decreased, which may be associated with the increases in the inter-connections and EP, making the heterogeneity of the sensory motor area and information exchange between the sensory motor area and other subsystems increase. The EP in the sensory motor area was slightly higher than other subsystems, which may be related to the earlier maturation of the sensory motor subsystem [10
], while the heteromodal areas of the association cortex governing higher cognitive function were still developmental [12
]. The finding implied that the orderness/certainty of all subsystems were the same as the orderness/certainty of whole-brain.
The energy probabilities of part brain regions of the ventral attention subnetwork and sensory motor area were apparently higher than other networks in the youth period (Group II), while the energy probabilities of the default network, frontoparietal network and limbic subsystem decreased. This finding may imply that youth is a prominent period of change to the subsystems’ heterogeneity. The energy probabilities of the sensory motor network and part of the ventral attention network brain regions were highest in the whole-brain in the middle age (Group III). However, the energy probabilities of the visual network declined, while the energy probabilities of the default, frontoparietal, and limbic networks decreased further. This result may be consistent with the further enhanced heterogeneity of whole-brain functional subsystems [60
]. The values of EP in the brain regions of the frontoparietal network increased slightly. And the energy probabilities of part brain regions of the default network decreased further in old age (Group IV). In summary, complicated changes to subsystems were not synchronous across the lifespan, which revealed the compensatory function among functional brain areas during normal aging [66
], which may partly reflect the human brain plasticity.
4.4. Brain Regions with Significant Energy Probablity Changing
Through the Mann–Whitney rank-sum test on whole-brain EPs between Group II (youth period) and Group I (flourishing), we found that the significantly changed regions were mainly located in the right prefrontal cortex and basal ganglia, which are involved in emotion regulation and executive function in coordination with the action of the sensory system, implying that self-awareness and voluntary control performance significantly changed during neurodevelopment [69
]. In addition, with the exception of the energy probabilities of the right median cingulate gyrus and right superior frontal gyrus which increased with age, other regions decreased with age. In contrast, a small set of significantly different regions was revealed between Group III (middle age) and Group II (youth period), which were less than that between Group II and Group I, and belonged to subcortical regions. These regions are mainly involved in long-term memory, implying that the memory of the human brain begins to decline with age during this period. Furthermore, we found that the brain regions where EP changes only comprised a small part of the whole brain and these changed brain regions were not synchronized. Between Groups III and IV, no brain region with significant EP changes was detected.
4.5. Limitations and Future Directions
It is worth noting that there were several issues to be considered when interpreting our findings. First, two strategies are commonly used for thresholding an FC matrix to format a graph adjacency matrix: (1) the correlation criterion using a fixed correlation value for all subjects, and (2) edges density criterion using a fixed density value for all subjects. In this study, we used a constant significance criterion. Although age and edges density were unrelated (r = −0.08, p = 0.35), more consistent results might be obtained if graph analyses could be conducted on density criterion. This issue should be studied carefully in future. Second, we regressed out the global signal to partly reduce physiological and other global noise. Third, it was a challenge to map the brain’s parcellation-based functional connectome appropriately and precisely. In the study, we used a random-generated high-resolution template and explored the scale effect of a parcellation scheme on whole-brain NEE values. To eliminate the effect of templates on estimating brain system orderness, we will consider establishing voxel-wise RSFC networks in future work. Fourth, we attempted to explore the age-related differences of functional connectome over a continuous age range that covered both development and aging. However, the analyzed samples were not perfectly distributed across the entire lifespan as the number of young adults was greater than the number of older people. We utilized a quadratic model to explore the age changes across the human lifespan. The incomplete distribution of ages in our sample may have affected parametric curve fitting. Fifth, the RSFC network we constructed was binary and undirected; however, we know that an effective connection between two brain regions indicated a causal relationship, which contained more precise information. Furthermore, weighted connections indicated the strength of the information communication between two brain regions, so in future, we will also consider establishing weighted directed RSFC networks. Sixth, the present work discussed the orderness variability of functional connectome with normal developmental and aging, while structural and functional connectomics were of mutual interaction. Brain structural connectome supported and constrained functional connectome, meanwhile brain network organization refined brain structure gradually. We will consider discussing the orderness variability of structural connectomics with age in further works to reveal the consistency and discrepancy between them.
There were some disadvantages of NEE. It can describe the integral topological properties of the system, but it cannot locate local development of the system. In addition, the model used in this study is relatively simple and may impede its generalization and application to broader areas. In the future, we will expand to encompass more graph theoretic formalisms, and pursue more accurate and detailed models. The whole-brain NEE discussed in this study was estimated only for healthy individuals. We know that there is abnormal functional connectivity specific to some RSFC subsystems in brain diseases (e.g., depression, schizophrenia, autism, Alzheimer's disease) [72
]. So, we would examine whether the NEE of the whole brain or specific subsystems could be used to predict the course of brain diseases and to evaluate treatment effects at a mesoscale level, shedding light on the pathophysiology of brain diseases in the future.