Large-scale white matter pathways crisscrossing the cortex create a complex pattern of connectivity that underlies human cognitive function. Generative mechanisms for this architecture have been difficult to identify in part because little is known in general about mechanistic drivers of structured networks. Here we contrast network properties derived from diffusion spectrum imaging data of the human brain with 13 synthetic network models chosen to probe the roles of physical network embedding and temporal network growth. We characterize both the empirical and synthetic networks using familiar graph metrics, but presented here in a more complete statistical form, as scatter plots and distributions, to reveal the full range of variability of each measure across scales in the network. We focus specifically on the degree distribution, degree assortativity, hierarchy, topological Rentian scaling, and topological fractal scaling—in addition to several summary statistics, including the mean clustering coefficient, the shortest path-length, and the network diameter. The models are investigated in a progressive, branching sequence, aimed at capturing different elements thought to be important in the brain, and range from simple random and regular networks, to models that incorporate specific growth rules and constraints. We find that synthetic models that constrain the network nodes to be physically embedded in anatomical brain regions tend to produce distributions that are most similar to the corresponding measurements for the brain. We also find that network models hardcoded to display one network property (e. g. , assortativity) do not in general simultaneously display a second (e. g. , hierarchy). This relative independence of network properties suggests that multiple neurobiological mechanisms might be at play in the development of human brain network architecture. Together, the network models that we develop and employ provide a potentially useful starting point for the statistical inference of brain network structure from neuroimaging data.
Increasing resolution of noninvasive neuroimaging methods for quantifying structural brain organization in humans has inspired a great deal of theoretical activity [1]–[4], aimed at developing methods to understand, diagnose, and predict aspects of human development and behavior based on underlying organizational principles deduced from these measurements [5]–[7]. Ultimately, the brain is a network, composed of neuronal cell bodies residing in cortical grey matter regions, joined by axons, protected by myelin. Diffusion-weighted magnetic resonance imaging methods trace these white matter connections, based on the diffusion of water molecules through the axonal fiber bundles. While resolution has not reached the level of individual neurons and axons, these methods lead to reliable estimates of the density of connections between regions and fiber path lengths. The result is a weighted adjacency matrix, with a size and complexity that increases with the resolution of the measurements [8], [9]. The immense complexity of this data makes it difficult to directly deduce the underlying mechanisms that may lead to fundamental patterns of organization and development in the brain [10]. As a result, comparison studies with synthetic network models, employing quantitative graph statistics to reduce the data to a smaller number of diagnostics, have provided valuable insights [11]–[15]. These models and statistics provide a vehicle to compare neuroimaging data with corresponding measurements for well-characterized network null models. However, the methods are still in development [16]–[18], and vulnerable to the loss of critical information through oversimplification of complex, structured data sets, by restricting comparisons to coarse measurements that ignore variability [10], [19], [20]. Two critical questions motivate development of network methodologies for the brain. The first question focuses on predictive statistics: Are there graph metrics that may ultimately be useful in parsing individual differences and diagnosing diseases? Comparing empirical brain data to benchmark null models can establish the statistical significance of a topological property [21]–[23], and normalizing a topological property by its null model surrogate can be a useful preprocessing step prior to the determination of statistical differences in brain network structure between groups [16]. The second question focuses on network characteristics from a fundamental, development and evolutionary perspective: What organizational principles underlie growth in the human brain? Here comparing empirical brain data to simplified model networks that have been created to capture some aspect of, for example, neurodevelopmental growth rules [24], neuronal functions [11], or physiological constraints [25] may aid in developing a mechanistic understanding of the brain' s network architecture (e. g. , [26]–[28]). Both efforts require a basic understanding of the topological similarities and differences between synthetic networks and empirical data. In this paper, we perform a sequence of detailed, topological comparisons between empirical brain networks obtained from diffusion imaging data and 13 synthetic network models (see Table 1). The models are investigated in a tree-like branching order, beginning with the simplest, random or regular graphs, and progressively adding complexity and constraints (see Figure 1). The objective of this investigation is to determine, in a controlled, synthetic setting, the impact of network properties on the topological measurements. Our goal is not to create a definitive network model of the brain, but to gain an intuition for structural drivers of network statistics and to create a battery of null models to be used in statistical comparisons of brain networks. At the coarsest level in the model hierarchy, we distinguish between synthetic networks that are constructed purely based on rules for connectivity between nodes (non-embedded), and those that constrain nodes to reside in anatomical brain regions (embedded) (see Figure 1). While non-embedded models are frequently used for statistical inference, recent evidence has suggested that physical, embedding constraints may have important implications for the topology of the brain' s large-scale anatomical connectivity [2], [8], [22], [26]–[29]. By examining both non-embedded and embedded models, we hope our results will help to guide the use, development, and understanding of more biologically realistic models for both statistical and mechanistic purposes [23], [30]. A second important classification of the synthetic models in our study separates those obtained from static ensembles with fixed statistical properties and those generated using mechanistic growth rules (see Figure 1). While algorithms for generating networks based on static sampling and growth rules ultimately both produce ensembles of fixed graphs for our comparison with data, additional constraints imposed by underlying growth rules may facilitate understanding of mechanisms for development and evolution in the brain as well as other biological and technological networks. To compare the models with brain data, we employ a particular subset of the many network diagnostics that have been proposed as measures of network topology [31], specifically chosen to highlight the regional variability and multiscale nature of network architecture. Many network diagnostics can be described as summary diagnostics, in which a property of the network organization is reduced to a single diagnostic number. Examples include average path length and average clustering coefficient. However, the comparison of summary diagnostics between real and model networks can be difficult to interpret [32] because they often hide the granularity at which biological interpretations can be made. To maximize the potential for a mechanistic understanding, we instead study diagnostics that provide distributions, visualized and analyzed by two-dimensional curves or scatter plots where the regional variability of network structure is readily apparent. The following four diagnostic relationships are obtained from a distribution of values over network nodes or topological scales: hierarchy [33], degree assortativity [34], topological Rentian scaling [35], [36], and the topological fractal dimension [37]. Each of these inherently relational properties has previously been investigated in the context of anatomical brain networks in humans [28], [38], [39]. In this paper, we use them to examine the differences between empirically derived anatomical brain networks and synthetic network models.
We utilize previously published diffusion spectrum imaging data [39] to examine the network structure of anatomical connectivity between cortical regions in the human brain. In this data, the direct pathways between N = 998 cortical regions of interest are estimated using deterministic white matter tractography in 5 healthy human participants [39]. This procedure results in an N×N weighted undirected adjacency matrix W representing the network, with elements Wij indicating the (normalized) number of streamlines connecting region i to region j (see Figure 2). The organization of white matter tracts can be examined at two distinct levels of detail: topological and weighted. Studies of the topological organization of brain anatomy focus on understanding the presence or absence of white matter tracts between regions [26]–[28], while studies of the weighted organization focus on understanding the strength of white matter connectivity between those regions. In this paper, we explore the topological organization of white matter connectivity between cortical regions. In future work we plan to build additional constraints into our models that will enable a comparison of model and empirical weighted networks. To study topological organization, we construct the binary adjacency matrix A in which the element Aij is equal to 1 if the employed tractography algorithm identifies any tracts (of any strength) linking region i with region j (i. e. ,). In this data [39], the adjacency matrix A is relatively sparse, resulting in a network density of, where is the total number of connections present. This estimate of brain network sparsity is consistent with estimates extracted from other similar data sets of comparable network size [8], [40]. Given the potential variability in the topological organization of networks extracted from different individuals [8], [41]–[44], we report results for one individual in the main manuscript and describe the consistency of these results across subjects in the Supplementary Materials. We also briefly note that while extremely rich, this data set also has its limitations. In particular, the development of high resolution imaging methods and robust tractography algorithms to resolve crossing fibers are fast-evolving areas of research. Novel imaging techniques have for example recently identified the existence of 90-degree turns in white matter tracts [45], a biological marker that we are not sensitive to in our data. We measure four network properties including degree assortativity, hierarchy, Rentian scaling, and topological fractal dimension as well as several summary diagnostics, as reported in Table 2. All computational and basic statistical operations (such as t-tests and correlations) were implemented using MATLAB (2009b, The MathWorks Inc. , Natick, MA) software. Graph diagnostics were estimated using a combination of in-house software, the Brain Connectivity Toolbox [60], and the MATLAB Boost Graph Library (http: //www. stanford. edu/~dgleich/programs/). To perform the recursive topological partitioning employed in the examination of topological Rentian scaling, we used the software tool hMETIS [61]. Several of the network models that we investigate include one or more tunable parameters affecting the details of the generated graphs. These include the Barabási-Albert, affinity, and hybrid distance growth models. To compare these network models to the data, we optimized parameter values to minimize the difference between the model network and the empirical brain network. Specifically, we used the Nelder-Mead simplex method, which is a derivative-free optimization method, that minimizes the value of a difference metric between the two networks. We chose to let be the sum of the absolute relative difference of nine of the network characteristics reported in Table 2 (clustering coefficient C, path length P, diameter D, degree assortativity r, hierarchical parameter β, topological Rentian exponent pT, topological fractal dimension dB, modularity Q, and number of communities #com). Alternative choices for the difference metric could weight some network characteristics to a greater or lesser degree than others. However, because we do not a priori have a rubric by which to determine the biological relevance of a single network diagnostic in comparison to others, we chose not to utilize such a weighting scheme.
We begin by comparing the network organization of the brain' s anatomical connectivity with that of 8 network models whose structure is not a priori constrained to accommodate a physical embedding of the nodes in cortical areas. (In the next subsection, we will examine 5 embedded network models.) The non-embedded network models include an Erdös-Rényi graph, a configuration model with the same degree distribution as the empirical network, a ring lattice graph, a modular small-world graph, a fractal hierarchical graph, a Gaussian drop-off graph, a Barabási-Albert graph, and an affinity graph (see Figure 2 for associated example adjacency matrices for these graphs and Table 1 for abbreviations of model names). These models range from disordered to ordered (e. g. , the Erdös-Rényi and regular lattice models) with a range of mesoscale organization for intermediate cases (e. g. , modular small-world and fractal hierarchical models) which influence the network diagnostics, and (dis) similarities to corresponding measurements for the brain. The non-embedded models described in the previous section necessarily ignore a fundamental property of the brain: its embedding in physical space. Spatial constraints likely play an important role in determining the topological properties of brain graphs [22], [26]–[29]. In this section, we explore the topological properties of spatially embedded graphs in which the probability of connecting any two nodes in the network depends on the Euclidean distance between them [79]. We explore the same topological diagnostics as we did in the previous section: degree distribution, assortativity, hierarchy, and diagnostics estimating the topological dimension of the network. As a whole, we find that spatially embedded models capture more topological features of the empirical networks than models that lack the physical embedding constraint. To clarify the distinction between embedded and non-embedded network models, it is necessary to highlight the differences between topological and physical notions of space. Many topological models (such as the Barabási-Albert model) are often described in ways that utilize notions of “local” connections. However, this concept of locality is present in a purely topological sense and not in a geographical sense. Topological models such as the Barabási-Albert model are not derived from spatial embeddings in or and therefore the nodes of these networks do not have spatial positions and the edges of these networks do not have physical lengths. The nonequivalence of topological and geographic structure is illustrated by the fact that a network topology (e. g. , BA) can either remain non-embedded or can be embedded into Euclidean space (e. g. ,) in many different ways: in some embeddings, the topological distance between nodes could correlate with the physical distance between nodes, but in other embeddings one need not observe such a correlation. While the previous section described topological and non-embedded models, in this section we focus on networks that have been embedded into Euclidean space.
While the details of this set of analyses are of course important, we can also propose a set of integrated insights into the biological underpinnings of structural brain network organization based on the collective results extracted from these models. First, the fact that models hard-coded to display one topological property are unlikely to also display a second topological property suggests that the processes of neurodevelopment have not been selected by evolutionary drivers to optimize a single topological variable. Such a suggestion is intuitively plausible: What mechanisms exist to isolate and optimize single topological properties in the complicated cellular milieu of a developing organism? Evidence from evolution and development instead suggest that the neuronal systems in living organisms are constrained by energy and metabolic concerns [83]. While energetic concerns may subsequently translate into constraints on topological network architectures [12], [25], [28], topological features are unlikely to be the singular driving mechanism of evolution. Supposing that energetic concerns play a role in guiding network connectivity in large-scale brain structure, how might these concerns manifest themselves in the observed network organization of a single organism at a single point in time? One possibility is that such constraints would impact on the physical length of connections since long connections are arguably more costly to both develop and maintain [12], [25], [28]. Consistent with this possibility, we observe that models that penalize physical length of connections (embedded models) tend to be more similar to the empirical data than models that hard-code specific topological properties (non-embedded models). This gross result, robust to individual variation in different model parameters, supports the view that biological physics may be a more fundamental driver of structural brain architecture than network topology. However, we also note that simple distance models remain unable to capture all of the intricacies of the observed network architecture. While there is certainly room to construct more complicated physical models, it is also arguable that additional biophysical constraints are playing a secondary but influential role. A key feature of networked neuronal systems is their development over time, which displays complicated maturation-dependent trajectories [84]–[86]. It is therefore intuitively plausible that growth processes pose unique constraints on network development that cannot be captured by static physical distances alone. Indeed, we observe that 2 of the 3 models that display most similarity to the empirical network structure are growing models (the affinity model and the hybrid distance growth model, which we define for the first time in this paper), suggesting that principles underlying the time evolution of network structures is critical. If true, this result uncovers a major gap in current network models of neuronal systems: namely, a sequence of models of increasing complexity that account for both physical constraints and growth processes on final (adult) network architecture. We speculate that such models, which obey principles of both physics and time, will be best able to capture observed empirical brain network structure. Model interpretations aside, it is important to emphasize that this work has a complementary purpose: to provide researchers with mathematical null models to inform statistical inference. The pragmatic uses of these models fall under two broad categories: (i) the use of a single model and (ii) the use of the full model battery. Single models can be used to address the question “How different are my two sets of networks in property y beyond that expected by their differences in property x. ” For example, one might have a group of networks from a clinical population and a group of networks from a control population. The two groups might differ in both their degree distribution and their clustering coefficient. However, one would like to test whether their difference in clustering coefficient is more than expected given their difference in degree distribution. That is, one would like to isolate the independent contribution of one network parameter to the phenotype of the disease. The statistical test one could then employ is to compare the clustering coefficient of the empirical networks in one group (normalized by the clustering coefficient of the associated configuration models, which control for degree distribution) to the clustering coefficient of the empirical networks in the other group (again normalized by the clustering coefficient of the associated configuration models). Such a test directly determines whether the clustering coefficient is more different between the two groups than expected given the differences in their degree distributions. While we have used the clustering coefficient and degree distribution for simplicities sake in this argument, all other (potentially more complicated) pairs of properties can be examined similarly (e. g. , hierarchical structure, preferential attachment, modular structure, wiring properties, etc.). In addition to single models, model batteries can be used to probe more general questions of group differences between sets of networks, for example from clinical and control populations. In some group comparisons, it is possible to observe marginally significant group differences in many network properties but to not observe any single network property that is affected drastically in isolation. In such cases, it is useful to report a comprehensive statistical test that encompasses these findings, rather than report a series of separate t-tests. In this context, model batteries can be extremely useful because they can provide response functions (such as the summed relative difference from data, illustrated for a single individual in Figure 11A) that indicate the differences between the data and the model battery. Different individuals can have different response functions (as illustrated in Figure 11B), as can different groups. To directly compare these functions between groups, one can use a branch of statistics known as functional data analysis (for a relevant textbook see [87] and for an application in network neuroimaging see [88]). Model batteries therefore complement network diagnostics in providing measurable statistics that can be used to identify subtle differences in network architecture between groups. In the following sections we discuss the details of each model more fully and relate our results to prior work. We conclude with a description of model interpretations, future directions, and methodological limitations. We probe non-embedded models with differing amounts and types of structure. While the Erdös-Rényi model provides an important benchmark with a random topology, it bears little resemblance to the brain network. Although a homogeneous random distribution of links has been suggested to characterize the small-scale structure of neuron-to-neuron connections [89], [90], the large-scale structure of human and animal brains instead displays heterogeneous connectivity [67]. Perhaps one of the simplest measures of this heterogeneity is found in the degree distribution, which displays a predominance of low degree nodes and a long tail of high degree nodes. In comparing the degree distribution of the brain to that obtained from a BA model, it is clear that this tail, however, is not well-fit by a power-law, a finding consistent with previous reports in brain anatomy [21], [38] and function [15], [91]. However, by matching the empirical data, for example using a configuration model with the same degree distribution, we note that we do not automatically uncover higher order structures like assortativity, suggesting that the degree distribution provides only limited insight into the forces constraining brain network development. Several decades ago, neuroanatomists observed that the pattern of connections in several animal brains displayed a combination of both densely clustered areas and long range projects between distant areas [92]–[95]. The regular lattice and Gaussian drop-off models are able to capture these densely connected structures but fail to capture the extent of long-range connectivity observed in the brain. The small-world modular and fractal hierarchical models contain both properties: dense local connectivity and long-range interactions. The fractal hierarchical model has the added benefit of containing nested structures, which have been implicated in the heterogeneity of neuronal ensemble activity [11] and in the separation and integration of information processing across multiple frequency bands [96]. Moreover, hierarchical modular structure has been identified in organization of white matter streamlines in human diffusion weighted imaging data [8], [72], [74] and implicated in neurobiological phenomena [11], [75], [76]. None of the non-embedded models discussed earlier in this section simultaneously provide a heterogeneous degree distribution, degree assortativity, hierarchical topology, and realistic topological dimensions. Such a “No Free Lunch” rule is perhaps unsurprising, in that a network that is developed to directly obtain one property typically fails to also display a second property. This result suggests that the topological properties that we explore here are in some sense independent from one another. It is, however, important to clarify that the interpretation of our findings in light of the observed correlations between network diagnostic values themselves, estimated over different networks or models (see previous literature, e. g. , [97], [98], and results for the current data presented in Figure S4 in the Supplementary Materials), that suggest the need for methods to identify distinguishing properties among networks [56], [99]. The two sets of observations can be brought together by realizing that while classes of networks (e. g. , brain networks) might display correlated network diagnostics values, these relationships need not be expected theoretically from any randomly chosen set of networks. Indeed, networks can be segregated into families based on the profile of interdependence between network diagnostic values [100]. Finally, in our affinity model, we hard-code both degree assortativity and a continuous hierarchical topology, rather than the discrete hierarchy employed in nested models like the fractal hierarchical model examined here. Interestingly, however, and in contrast to the other non-embedded models, we simultaneously obtain a heterogeneous degree distribution, and similar estimates of the topological dimension. This model fits multiple properties of brain networks that were not explicitly included in the construction of the network model, but are nevertheless a consequence of a three-parameter fit in the specific affinity model selected. The affinity model therefore serves as a promising candidate as both a generative model and statistical null model of brain organization. In an effort to include additional biological constraints, we also explore several models that employ information regarding either the physical placement of network nodes or that place constraints on the Euclidean lengths of network edges. In general, this set of networks outperforms most of the non-embedded network models that we studied, demonstrating that the brain is highly spatially organized and supporting the notion that physical constraints might play important roles in brain network development and structure [8], [25]–[29], [90], [101], [102]. It is important to preface the discussion of our results by mentioning the fact that the properties of empirically derived brain networks display a heterogeneity that could at least in part stem from the peculiar physical properties of the organ. Brains are symmetric objects, with the two hemispheres being connected with one another via tracts in the corpus callosum and via subcortical structures. This separation allows for a very different topology within a hemisphere than between hemispheres. Moreover, cortical areas (gray matter) form a shell around the outer edges of the brain while their connections (white matter) compose the inner volume. Finally, brain areas are inherently heterogeneous in physical volume, making their distances from one another far from homogeneous. While the morphology of the brain constrains its potential topological properties, evidence also suggests that the lengths of tracts connecting brain areas follow a heavy tailed distribution, with short tracts being relatively common and long tracts being relatively rare [26], [27]. These findings are in concert with the idea that energy efficiency—to develop, maintain, and use neuronal wiring—remains a critical factor in brain evolution and development [29], [103]. In this study, we begin with a random geometric model, whose nodes are placed uniformly at random in a volume but whose edges selectively link nodes that are nearby in physical space. In light of the simplicity of this model, it is somewhat surprising that we obtain such good agreement with the empirical degree distribution, the presence of assortativity, and the presence of a hierarchical topology. In the minimally wired graph we employ a similar connection rule but also fix node placement to be identical to that in the empirical brain network, following previous studies [28]. However, neither of these two models are able to capture the extent of long-distance connections observed in the empirical data. By employing the distance drop-off model, we can fix a connection probability that varies with distance, rather than simply a connection threshold. This connection probability, however, is not enough to provide a realistically broad degree distribution. Our distance drop-off growth model combines the strengths of each of these models by laying down a set of seed edges uniformly at random in a volume and then iteratively adding edges between pairs of nodes according to a probability that falls off with inter-node distance. The resulting degree distribution and assortativity properties are the best match to the empirical data of the models that we studied. A hybrid between the minimally wired model and the distance drop-off growth model does not perform significantly better in matching these properties and shows a hierarchical structure that is more pronounced than the data. Importantly, the embedded network models examined here are purposely simplistic. While arbitrarily more complex models could be constructed, our goal was to isolate individual drivers of topology and probe their relationship to observed network diagnostics. Other studies of interest in relation to these findings include those that explore the effects of geometric folding [90], radial surface architectures [102], and the effects of wiring minimization on functional networks [25]. While the construction of network models is genuinely critical in providing null tests for statistical inference of brain structure from data, this avenue of research also has the potential to provide key insights into the neurobiological mechanisms of brain development and function if performed with appropriate caution. In light of this second use, we note that several of the network models discussed in this paper employ rules that are reminiscent of—or even directly inspired by—known biological phenomena. For example, physical models that place constraints on the length of connections in Euclidean space are consistent with the known distribution of connection lengths in the brain and the modern understanding of metabolic constraints on the development, maintenance, and use of long wires [26]–[29], [101], [103]. However, even topological constraints that link nodes that have similar sets of neighbors can be interpreted as favoring links between neurons or regions that share similar excitatory input [25]. As an example, our affinity model hard-codes two inter-node relationships. First, nodes with a similar degree are more likely to be connected to one another by an edge, leading to degree assortativity throughout the network. This behavior can be thought of as a mathematical representation of the intuitive principle of spatial homophily: large neurons with expansive projections (e. g. , pyramidal or basket cells) are more likely to connect to one another because they densely innervate tissue over large distances. Network assortativity can also stem from the temporal homophily that occurs during development: neurons that migrate over longer distances during development are more likely to come into contact with—and therefore generate a synapse with—one another than neurons that migrate over shorter distances. The second topological relationship hard-coded into the affinity model is the prevalence of clustering in local neighborhoods, a property consistent with physical constraints on network development. As neurons develop, it is intuitively more likely for them to create synapses with neighboring neurons than non-neighboring neurons, thereby closing topological loops in close geographic proximity. While we have only provided a few examples here, links between topological rules and biological phenomena provide potentially critical neurophysiological context for the development and assessment of synthetic network models. The perspective that we have taken in choosing synthetic network models is one of parsimonious pragmatism. We seek to identify models with simplistic construction rules or growth mechanisms to isolate topological (non-embedded) and physical (embedded) drivers of network topology. One alternative perspective would be to begin with a certain graph topology (for example, an Erdős-Rényi graph), and iteratively rewire edges to maximize or minimize a network diagnostic or set of network diagnostics [25]. However, this approach requires prior hypotheses about which network diagnostics are most relevant for brain network development, a choice that is complicated by the observed correlations between such diagnostics [97]. Another approach is to employ exponential random graph models [16], [19], [104], which provide a means to generate ensembles of networks with a given set of network properties but do not provide a means to isolate mechanistic drivers of those network properties. A third approach is to construct a mechanistic model based on particle-particle collisions, which might serve as a physical analogy to the biological phenomena of neuronal migration through chemical gradients [105], [106]. In each of these cases, a perennial question remains: at what spatial scale should we construct these models to gain the most insight into the relevant biology? Important future directions could include the development of multiscale growth models, enabling us to bridge the scales between neuronal mechanisms and large-scale structure. There remain important limitations to our work. In particular, we have focused on understanding the (binary) topology of brain network architecture rather than its weighted connection strengths. Our choice was informed by three factors: 1) An understanding of the relationship between synthetic network models and brain network topology could be useful for informing a similar investigation into network geometry, 2) In these particular networks, node degree (binary) and node strength (weighted by the number of streamlines) are strongly correlated (Pearson' s correlation coefficient,) and therefore topology serves as a proxy for weighted connectivity, and 3) The choice of how to weight the edges in an anatomical network derived from diffusion imaging is an open one [107], and therefore investigations independent of these choices are particularly useful. Network models constitute necessarily simplified representations of often very complex systems. The 13 synthetic network models we study in this work could be extended to include additional physical features of the human brain. For example, a key constraint on brain morphology and connectivity lies in the organ' s bilateral symmetry. This symmetry in brain structure is evident in the distribution of anatomical connectivity in the brain networks examined in this study: pairs of homologous regions are more than 3 times more likely to be connected to one another than pairs of non-homologous regions. As described in [39], each of the 998 regions used in the parcellation is affiliated with one of 66 anatomical parcels defined based on surface reconstruction performed in Freesurfer. We calculated the average density of connections between all of the regions in one anatomical parcel and all of the regions in another anatomical parcel. In this way, we obtain a pairwise density of connectivity between all 66 anatomical parcels. The average density of connections between homologous regions is 15. 22% and the average density of connections between non-homologous regions is 4. 05%. The topological ramifications of this symmetry are not well understood. Moreover, in simple network models, emphasis is placed on characterizing the patterns of network edges while the characteristics of individual nodes (apart from their connectivity) are examined to a lesser degree [108]. The development of more complicated models that account for feature vectors of brain region properties could provide additional insights into neurophysiological phenomena. Indeed, quantifying the relationship between a brain region' s connectivity and its functional or anatomical properties is a critical goal of network neuroscience. Initial forays into this area have demonstrated that topological properties of a brain region (node degree) can be linked to neurophysiological properties (prevalence of amyloid-beta deposition) [109], suggesting the utility of network approaches in providing mechanistic hypotheses regarding disease attributes. In this paper, we have examined the mechanistic drivers of network topologies by employing and developing a range of synthetic network models governed by both topological (non-embedded) and physical (embedded) rules and comparing them to empirically derived brain networks. These tools may prove useful in the statistical inference of anatomical brain network structure from neuroimaging data. Future efforts can further build on these findings to identify neurobiologically relevant mechanisms for healthy brain architecture and its alteration in disease states.
White matter tracts crisscrossing the human cortex are linked in a complex pattern that constrains human thought and behavior. Why the human brain displays the complex pattern that it does is a fascinating open question. Progress in uncovering generative mechanisms for this architecture requires greater knowledge about mechanistic drivers of anatomical networks. Here we contrast network properties derived from images of the human brain with 13 synthetic network models investigated in a progressive, branching sequence, chosen to probe the roles of physical embedding and temporal growth. We characterize both the empirical and synthetic networks using network diagnostics presented here in statistical form, as scatter plots and distributions, to reveal the full range of variability of each measure. We find that synthetic models that constrain the network nodes to be physically embedded in anatomical brain regions tend to produce distributions that are most similar to the corresponding measurements for the brain. We also find that network models hardcoded to display one network property do not in general simultaneously display a second, suggesting that multiple neurobiological mechanisms drive human brain network development. The network models that we develop and employ enable statistical inference of brain network structure from neuroimaging data.
statistical mechanics
nervous system
applied mathematics
neuroscience
systems science
mathematics
statistics (mathematics)
computational neuroscience
neuroimaging
complex systems
computer and information sciences
connectomics
systems biology
physics
neuroanatomy
anatomy
interdisciplinary physics
biology and life sciences
physical sciences
computational biology
statistical methods
Resolving Structural Variability in Network Models and the Brain