HHS Public Access Author manuscript Author Manuscript
Signif (Oxf). Author manuscript; available in PMC 2016 July 18. Published in final edited form as: Signif (Oxf). 2015 August ; 12(4): 34–39. doi:10.1111/j.1740-9713.2015.00843.x.
The brain science interface Sean Simpson, Jonathan Burdette, and Paul Laurienti
Abstract
Author Manuscript
The human brain consists of billions of neurons clustered into many different regions, each of which work together to give rise to complex human behaviour. But understanding how these regions are networked together – and the effect trauma and disease might have on network functions – requires greater statistical input. Sean L. Simpson, Jonathan H. Burdette and Paul J. Laurienti discuss neuroscience past and present, and how statistics can help its future Understanding the human brain remains the Holy Grail in biomedical science, and arguably in all of the sciences. Our brains represent the most complex systems in the world (and, some contend, the universe), comprising nearly 100 billion neurons with septillions of possible connections between them.
Author Manuscript
The structure of these connections gives rise to an efficient hierarchical system capable of consciousness, as well as complex thoughts, feelings and behaviours that make us uniquely human. The history of our attempts to understand this organ of intelligence has followed an evolutionary process replete with regression, progression and many “false starts”. However, the overall trend appears to be moving from simplistic ideas about brain function to more appropriately holistic views about how system-level properties allow the emergence of complex behaviours. The newest branch on the neuroscientific evolutionary tree is brain network analysis, which aims to characterise the systemic structure of the brain and how this structure relates to various conditions (e.g., disease) and behaviours. Understanding the relationship between brain structure – composed of the interactions between brain regions – and brain function necessitates statistical methodologies that account for data complexity. That is, the statistical evaluation of systemic properties of brain networks requires tools that can capture the complex interactions present in networks.
Author Manuscript
Despite this, most current brain network studies employ rudimentary statistical approaches. There is a pressing need for further statistical development in order to engender powerful analytical tools that will leverage the wealth of data present in brain networks and aid in our understanding of normal and abnormal brain function.
A very brief history of neuroscience Neuroscience has experienced a meteoric rise in importance during human history. Ancient Egyptians (eighteenth century BC) considered the brain to be “cranial stuffing”, yet still they provided the first written account of the organ (Figure 1), describing its anatomy and the cerebrospinal fluid.
Simpson et al.
Page 2
Author Manuscript
The status of the brain was elevated by the Pythagorean Alcmaeon of Croton, Hippocrates and Plato (sixth to fourth centuries BC) who considered it to be the seat of the mind and intelligence. However, it received a demotion from Aristotle, who believed that the brain merely served as a radiator to cool the real location of the mind, the heart. The brain experienced a resurgence in importance during the Renaissance and Age of Enlightenment. Andreas Vesalius, Leonardo da Vinci and Thomas Willis further detailed its structural characteristics and connection to the rest of the nervous system. Descartes studied its physiology and proposed the theory of dualism, believing that the mind and brain (body) were separate entities connected via the pineal gland.
Author Manuscript
The modern era has ushered in a few (pseudo)scientific false starts, a number of neuroscientific and neurotechnological advances, and scientific and philosophical debates regarding brain function. But it was the confluence of the invention of the microscope and development of a staining procedure by Camillo Golgi in the late 1890s that gave birth to the “localist tradition” – the view that neurons and brain areas have specific functions. This shifted the focus of study from the brain itself to single neurons within the brain, as well as brain regions, and localism continues to be a dominant theory today. Zip forward a hundred years and a second technological transformation occurred when Seiji Ogawa invented blood-oxygen-level dependent (BOLD) functional magnetic resonance imaging (fMRI), a technology that has revolutionised our ability to study brain function. Within the localist paradigm, activation analyses – focused on which areas of the brain “light up” during specific tasks – have flourished. However, ambiguous findings and disparate brain region activity during many cognitive tasks yielded evidence contradicting the idea of one-to-one mapping of brain regions and cognitive functions.
Author Manuscript
This has set the stage for the recent emergence of functional connectivity and network analyses. Functional connectivity analysis focuses on the communication between individual brain regions. But it is network analysis – the focus of this article – that gives weight to the holist viewpoint of the brain as an integrated system. Network analysis allows the quantification of the connectivity between all brain regions and the construction of an interconnected representation of the brain.1
Author Manuscript
The advantage of this network perspective of the brain is that interactions between brain areas become paramount. When disrupting connections in a network, the resultant changes in system behaviour may be seen in distant portions of the system due to altered information flow. Damage to one part of the brain can therefore have dramatic effects on activity levels in other brain regions if the damaged area is a critical input into those other regions. Statistical methodologies to examine the systemic features of brain networks and how these features are related to complex behaviours and diseases are just beginning to be developed.
(Brain) network science Network analyses provide a unique perspective on the brain that cannot be obtained using traditional localist methods. Studies designed to localise the focus of a particular cognitive function do not evaluate how interacting brain regions may contribute to that function.
Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 3
Author Manuscript
Rather, they identify the areas that are active during task performance and not active when the task is not being performed. Network analyses, however, allow for the evaluation of how brain regions interact to produce cognitive and/or behavioural outcomes. Thus, the interactions that culminate in the activation of a brain region during a task are identified in addition to the area itself. This is a more complete view of brain function as it is sensitive to changes in the organisation of connectivity between regions that may be directly active during a cognitive process as well as those regions that may contribute to or modulate the activity of the activated regions.
Author Manuscript
Consider Alzheimer’s disease, for example. An area of the brain known as the precuneus has been heavily implicated in the disease, but this theory has been difficult to reconcile with the predominant clinical symptom of Alzheimer’s: memory dysfunction, a cognitive process associated with the hippocampi. However, recent network analyses have discovered that the precuneus is anatomically and physiologically a central hub – a highly connected area in the brain. Damage to a central network hub like the precuneus can reverberate throughout the system and alter processing in many areas of the brain, including the hippocampus. Both structural (DTI, sMRI) and functional (BOLD fMRI, EEG, MEG) studies have provided useful insight on Alzheimer’s and other diseases, including Parkinson’s and schizophrenia. Here we focus on fMRI network analyses, given their dominance in the field.
Author Manuscript
Functional magnetic resonance imaging measures localised brain activity by capturing changes in blood flow and oxygenation that occur as brain activity levels change. These measurements are recorded across a series of scans from cubic subdivisions of the brain roughly a few millimetres in size, called voxels. Functional brain network analysis quantifies the similarities between all time-series pairs from these n voxels, or between averaged timeseries pairs across voxels within specified regions for coarser representations, creating an interconnected representation of the brain (a brain network). The resulting n × n connection matrix is commonly thresholded to retain strong connections while removing weaker ones that are thought to be spurious associations. A schematic exhibiting this network generation process is presented in Figure 2.
Properties of the brain
Author Manuscript
Early network studies revealed that a healthy brain exhibits “small-world” properties, namely that the brain is well wired for both functional segregation (local communication) and integration (global communication). Conceptually, the brain’s efficient long- and shortdistance communication properties are analogous to those of a modified game of Telephone in a large sports arena. In this context, the goal of the game is to pass a message around to the other side of the stadium via neighbour-to-neighbour (local) transmission. With this approach, the message will take a long time to get around and, as is known by those who played this game as children, the message will often be quite distorted. However, by introducing just a few cell phones on either side of the stadium, the message can be quickly transmitted across the stadium (global) with high fidelity.
Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 4
Author Manuscript
Our brains operate in a similar “small-world” fashion by efficiently relaying information both locally (functional segregation) and globally (functional integration). Functional segregation network metrics characterise the brain’s local communication ability by quantifying the presence of densely interconnected groups of brain regions which allow for segregated neural processing (regional specificity). The clustering coefficient (C, Figure 3a) and its scaled analogue, local efficiency (Eloc) (Table 1), serve as the two most commonly used measures of regional specificity.
Author Manuscript
Functional integration network metrics, meanwhile, characterise the brain’s global communication ability by quantifying the presence of statistical dependencies between distributed brain regions, indicating the capacity for rapid information transfer (distributive processing). The characteristic path length (L, Figure 3b) and its scaled analogue, global efficiency (Eglob) (Table 1), serve as the two most commonly used measures of distributive processing. Other commonly assessed brain network properties include: resilience, the capacity of localised brain injury or degeneration to affect overall brain capabilities; graph centrality and information flow, the relative importance of a given node (region) in a brain network for the transfer of information; and community structure, the brain’s ability to subdivide into interconnected communities (modules) that allow an efficient division of labour. The community structure of the brain can be likened to a city, with neighbourhoods serving as the units (communities) of efficiency while still being interconnected with and contributing to the functioning of the city as a whole. A change in this structure (e.g., the merging of two “incompatible” neighbourhoods) may affect the functioning of the entire system.
Author Manuscript
Many algorithmic methods exist for detecting community structure (with none being uniformly best) – although modularity maximisation, which aims to optimise the modularity statistic (Figure 4, Table 1), serves as most common in the brain network literature. Recent algorithmic developments also allow identifying (potentially) overlapping communities since many brain areas likely belong to multiple communities simultaneously given that they can perform several roles.
Author Manuscript
Despite these advancements in brain network description, much work remains in refining and statistically assessing these network properties and determining the best approach to estimating the underlying brain networks from which these properties are extracted. For example, real networks likely contain negative feedback loops, having a positive connection in one direction and negative connection in the other, but current network generation methods are unable to capture this. Additionally, current metric assessments ignore the fact that the underlying network is estimated and thus that these metrics are estimates themselves with certain probability distributions. Making formal inferential decisions about network metric values and gaining a better understanding of topological variability in normal and abnormal brain function requires assessment of the estimation error resulting from the estimation of the underlying brain networks. This area of network error propagation provides extremely fertile ground for Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 5
Author Manuscript
statistical research. To our knowledge, there is currently no active research in this area for functional brain networks other than that of Balachandran et al. who quantified the propagation of error from network estimation to subgraph counts.3
Fusing statistics and network science Accurately and precisely drawing inferential insight into the link between brain network properties and brain (dys)function requires some statistical gaps to be addressed. While recent univariate developments have proven useful, gleaning deeper insights into normal and abnormal changes in complex functional organisation demands methods that leverage the wealth of data present in an entire brain network.1
Author Manuscript
This systemic organisation confers much of our brain’s functional abilities as functional connections may be lost due to an adverse health condition, but compensatory connections may develop as a result in order to maintain organisational consistency and functional performance. Thus, gaining insight into this organisation requires a multivariate modelling framework that allows assessing the effects of endogenous (network measures) and exogenous (demographics, disease status, etc.) variables of interest on the overall network structure. We have made strides in developing such a framework both with exponential random graph models and mixed models, but more work is needed.4–6 Moreover, extending such frameworks to the longitudinal context will allow us to explain how dynamic network changes relate to normal and abnormal brain function over time.
Author Manuscript
The movement of neuroscience from simplistic ideas about brain function to holistic views about how systemic properties underpin the emergence of complex behaviours promises to take us a step closer to understanding who we are and what makes us unique. The future of brain network science remains promising as long as we take a conscientious approach to developing methods that appropriately account for data complexity. More generally, complexity-based neuroscience – which subsumes network-based analyses – represents a new paradigm aimed at quantifying the complex patterns inherent in physiological systems. This systems-based approach represents the frontier in neuroscience, statistics, and the sciences more generally.
Author Manuscript
Incorporating innovative methodologies within this paradigm will aid in revolutionising our understanding of the human brain. Given the nascency of the field and the variety of scientific questions of interest, it is unlikely that an optimal analysis method exists. Rather than being “locked in” to what have emerged as standard brain network science techniques, this generation of scientists has to be more flexible in thought and open to newer, often better, ways of analysing and using the richness of this complex data. A multi-faceted suite of complementary approaches situated at the interface of statistical, network and brain science will likely be needed to move the field forward.
Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 6
Author Manuscript
Acknowledgments This work is supported by the National Institute of Biomedical Imaging and Bioengineering (NIBIB) (K25 EB012236-01A1) and the Wake Forest Older Americans Independence Center (P30 21332). We thank past and present members of the Laboratory for Complex Brain Networks for some of the figures used in this article.
References
Author Manuscript
1. Simpson SL, Bowman FD, Laurienti PJ. Analyzing complex functional brain networks: fusing statistics and network science to understand the brain. Statistics Surveys. 2013; 7:1–36. [PubMed: 25309643] 2. Telesford QK, Simpson SL, Burdette JH, Hayasaka S, Laurienti PJ. The brain as a complex system: Using network science as a tool for understanding the brain. Brain Connectivity. 2011; 1:295–308. [PubMed: 22432419] 3. Balachandran P, Kolaczyk E, Viles W. On the propagation of low-rate measurement error to subgraph counts in large, sparse networks. arXiv:1409.5640. 2014 [math.ST]. 4. Simpson SL, Hayasaka S, Laurienti PJ. Exponential random graph modeling for complex brain networks. PLoS One. 2011; 6:e20039. 5. Simpson SL, Laurienti PJ. A two-part mixed-effects modeling framework for analyzing whole-brain network data. NeuroImage. 2015; 113:310–319. 6. Simpson SL, Moussa MN, Laurienti PJ. An exponential random graph modeling approach to creating group-based representative whole-brain connectivity networks. NeuroImage. 2012; 60:1117–1126.
Biographies
Author Manuscript
Sean Simpson is an associate professor of biostatistical sciences and member of the Laboratory for Complex Brain Networks (LCBN) at Wake Forest School of Medicine with a focus on fusing multivariate statistical methods with network science for brain network analysis Jonathan Burdette is a professor of neuroradiology and member of the LCBN at Wake Forest School of Medicine, and he studies brain–body interactions as a complex system Paul Laurienti is a professor of radiology and director of the LCBN at Wake Forest School of Medicine. He is developing methods to understand the brain as a complex, integrated system
Author Manuscript Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 7
Author Manuscript Figure 1.
Hieroglyphic for “brain”
Author Manuscript Author Manuscript Author Manuscript Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 8
Author Manuscript Author Manuscript
Figure 2.
Schematic for generating brain networks from fMRI time series data (partially recreated from Simpson et al.1). Functional connectivity between brain areas is estimated based on time-series pairs to produce a connection matrix. A threshold is commonly applied to the matrix to remove negative and/or “weak” connections
Author Manuscript Author Manuscript Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
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Author Manuscript Author Manuscript
Figure 3.
(a) The clustering coefficient (C) for the source region (node) equals the proportion of its connections that are also connected with each other. (b) The characteristic path length (L) for a pair of regions equals the shortest distance (smallest number of edges that must be traversed) to get from the one region to the other. The overall C and L for the network are then the averages of these values across all regions and region pairs, respectively
Author Manuscript Author Manuscript Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
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Author Manuscript Author Manuscript Author Manuscript
Figure 4.
Author Manuscript
Modularity analysis (reproduced from Telesford et al.2). Depending on the level where the subdivisions are made (dashed line), the number of communities can change. (A) In this example network, the optimal division, as indicated by maximum modularity, yields four communities (indicated by the red dashed line). Shifting this line up or down (indicated by the dashed line with an arrow) produces a lower modularity value that yields a suboptimal community structure. (B) As the line shifts higher, fewer communities are formed (approaching a single community comprising all nodes). (C) As the line shifts lower, more communities are formed (approaching every node being in their own community)
Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
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Table 1
Author Manuscript
Network metric definitions Metric
Definition
Local efficiency
where Eloc,i is the local efficiency of node i, N is the set of all nodes in the network, n is the number of nodes, aij is an indicator function for the existence of an edge between nodes i and j, ki is the degree of node i, and djh(Ni) is the shortest path between nodes j and h that contains only neighbours (connected nodes) of node i. Eloc ranges from 0 to 1, with larger values representing more functional segregation. Global efficiency
where Eglob,i is the global efficiency of node i, dij is the shortest path between nodes i and j, and N and n are defined as before. Like Eloc, Eglob also ranges from 0 to 1, with larger values representing more functional integration.
Author Manuscript
Modularity
where M indexes the set of non-overlapping modules from the fully subdivided network, and euv denotes the proportion of all edges that connect nodes in module u with those in module v.
Author Manuscript Author Manuscript Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Signif (Oxf). Author manuscript; available in PMC 2016 July 18. Published in final edited form as: Signif (Oxf). 2015 August ; 12(4): 34–39. doi:10.1111/j.1740-9713.2015.00843.x.
The brain science interface Sean Simpson, Jonathan Burdette, and Paul Laurienti
Abstract
Author Manuscript
The human brain consists of billions of neurons clustered into many different regions, each of which work together to give rise to complex human behaviour. But understanding how these regions are networked together – and the effect trauma and disease might have on network functions – requires greater statistical input. Sean L. Simpson, Jonathan H. Burdette and Paul J. Laurienti discuss neuroscience past and present, and how statistics can help its future Understanding the human brain remains the Holy Grail in biomedical science, and arguably in all of the sciences. Our brains represent the most complex systems in the world (and, some contend, the universe), comprising nearly 100 billion neurons with septillions of possible connections between them.
Author Manuscript
The structure of these connections gives rise to an efficient hierarchical system capable of consciousness, as well as complex thoughts, feelings and behaviours that make us uniquely human. The history of our attempts to understand this organ of intelligence has followed an evolutionary process replete with regression, progression and many “false starts”. However, the overall trend appears to be moving from simplistic ideas about brain function to more appropriately holistic views about how system-level properties allow the emergence of complex behaviours. The newest branch on the neuroscientific evolutionary tree is brain network analysis, which aims to characterise the systemic structure of the brain and how this structure relates to various conditions (e.g., disease) and behaviours. Understanding the relationship between brain structure – composed of the interactions between brain regions – and brain function necessitates statistical methodologies that account for data complexity. That is, the statistical evaluation of systemic properties of brain networks requires tools that can capture the complex interactions present in networks.
Author Manuscript
Despite this, most current brain network studies employ rudimentary statistical approaches. There is a pressing need for further statistical development in order to engender powerful analytical tools that will leverage the wealth of data present in brain networks and aid in our understanding of normal and abnormal brain function.
A very brief history of neuroscience Neuroscience has experienced a meteoric rise in importance during human history. Ancient Egyptians (eighteenth century BC) considered the brain to be “cranial stuffing”, yet still they provided the first written account of the organ (Figure 1), describing its anatomy and the cerebrospinal fluid.
Simpson et al.
Page 2
Author Manuscript
The status of the brain was elevated by the Pythagorean Alcmaeon of Croton, Hippocrates and Plato (sixth to fourth centuries BC) who considered it to be the seat of the mind and intelligence. However, it received a demotion from Aristotle, who believed that the brain merely served as a radiator to cool the real location of the mind, the heart. The brain experienced a resurgence in importance during the Renaissance and Age of Enlightenment. Andreas Vesalius, Leonardo da Vinci and Thomas Willis further detailed its structural characteristics and connection to the rest of the nervous system. Descartes studied its physiology and proposed the theory of dualism, believing that the mind and brain (body) were separate entities connected via the pineal gland.
Author Manuscript
The modern era has ushered in a few (pseudo)scientific false starts, a number of neuroscientific and neurotechnological advances, and scientific and philosophical debates regarding brain function. But it was the confluence of the invention of the microscope and development of a staining procedure by Camillo Golgi in the late 1890s that gave birth to the “localist tradition” – the view that neurons and brain areas have specific functions. This shifted the focus of study from the brain itself to single neurons within the brain, as well as brain regions, and localism continues to be a dominant theory today. Zip forward a hundred years and a second technological transformation occurred when Seiji Ogawa invented blood-oxygen-level dependent (BOLD) functional magnetic resonance imaging (fMRI), a technology that has revolutionised our ability to study brain function. Within the localist paradigm, activation analyses – focused on which areas of the brain “light up” during specific tasks – have flourished. However, ambiguous findings and disparate brain region activity during many cognitive tasks yielded evidence contradicting the idea of one-to-one mapping of brain regions and cognitive functions.
Author Manuscript
This has set the stage for the recent emergence of functional connectivity and network analyses. Functional connectivity analysis focuses on the communication between individual brain regions. But it is network analysis – the focus of this article – that gives weight to the holist viewpoint of the brain as an integrated system. Network analysis allows the quantification of the connectivity between all brain regions and the construction of an interconnected representation of the brain.1
Author Manuscript
The advantage of this network perspective of the brain is that interactions between brain areas become paramount. When disrupting connections in a network, the resultant changes in system behaviour may be seen in distant portions of the system due to altered information flow. Damage to one part of the brain can therefore have dramatic effects on activity levels in other brain regions if the damaged area is a critical input into those other regions. Statistical methodologies to examine the systemic features of brain networks and how these features are related to complex behaviours and diseases are just beginning to be developed.
(Brain) network science Network analyses provide a unique perspective on the brain that cannot be obtained using traditional localist methods. Studies designed to localise the focus of a particular cognitive function do not evaluate how interacting brain regions may contribute to that function.
Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 3
Author Manuscript
Rather, they identify the areas that are active during task performance and not active when the task is not being performed. Network analyses, however, allow for the evaluation of how brain regions interact to produce cognitive and/or behavioural outcomes. Thus, the interactions that culminate in the activation of a brain region during a task are identified in addition to the area itself. This is a more complete view of brain function as it is sensitive to changes in the organisation of connectivity between regions that may be directly active during a cognitive process as well as those regions that may contribute to or modulate the activity of the activated regions.
Author Manuscript
Consider Alzheimer’s disease, for example. An area of the brain known as the precuneus has been heavily implicated in the disease, but this theory has been difficult to reconcile with the predominant clinical symptom of Alzheimer’s: memory dysfunction, a cognitive process associated with the hippocampi. However, recent network analyses have discovered that the precuneus is anatomically and physiologically a central hub – a highly connected area in the brain. Damage to a central network hub like the precuneus can reverberate throughout the system and alter processing in many areas of the brain, including the hippocampus. Both structural (DTI, sMRI) and functional (BOLD fMRI, EEG, MEG) studies have provided useful insight on Alzheimer’s and other diseases, including Parkinson’s and schizophrenia. Here we focus on fMRI network analyses, given their dominance in the field.
Author Manuscript
Functional magnetic resonance imaging measures localised brain activity by capturing changes in blood flow and oxygenation that occur as brain activity levels change. These measurements are recorded across a series of scans from cubic subdivisions of the brain roughly a few millimetres in size, called voxels. Functional brain network analysis quantifies the similarities between all time-series pairs from these n voxels, or between averaged timeseries pairs across voxels within specified regions for coarser representations, creating an interconnected representation of the brain (a brain network). The resulting n × n connection matrix is commonly thresholded to retain strong connections while removing weaker ones that are thought to be spurious associations. A schematic exhibiting this network generation process is presented in Figure 2.
Properties of the brain
Author Manuscript
Early network studies revealed that a healthy brain exhibits “small-world” properties, namely that the brain is well wired for both functional segregation (local communication) and integration (global communication). Conceptually, the brain’s efficient long- and shortdistance communication properties are analogous to those of a modified game of Telephone in a large sports arena. In this context, the goal of the game is to pass a message around to the other side of the stadium via neighbour-to-neighbour (local) transmission. With this approach, the message will take a long time to get around and, as is known by those who played this game as children, the message will often be quite distorted. However, by introducing just a few cell phones on either side of the stadium, the message can be quickly transmitted across the stadium (global) with high fidelity.
Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 4
Author Manuscript
Our brains operate in a similar “small-world” fashion by efficiently relaying information both locally (functional segregation) and globally (functional integration). Functional segregation network metrics characterise the brain’s local communication ability by quantifying the presence of densely interconnected groups of brain regions which allow for segregated neural processing (regional specificity). The clustering coefficient (C, Figure 3a) and its scaled analogue, local efficiency (Eloc) (Table 1), serve as the two most commonly used measures of regional specificity.
Author Manuscript
Functional integration network metrics, meanwhile, characterise the brain’s global communication ability by quantifying the presence of statistical dependencies between distributed brain regions, indicating the capacity for rapid information transfer (distributive processing). The characteristic path length (L, Figure 3b) and its scaled analogue, global efficiency (Eglob) (Table 1), serve as the two most commonly used measures of distributive processing. Other commonly assessed brain network properties include: resilience, the capacity of localised brain injury or degeneration to affect overall brain capabilities; graph centrality and information flow, the relative importance of a given node (region) in a brain network for the transfer of information; and community structure, the brain’s ability to subdivide into interconnected communities (modules) that allow an efficient division of labour. The community structure of the brain can be likened to a city, with neighbourhoods serving as the units (communities) of efficiency while still being interconnected with and contributing to the functioning of the city as a whole. A change in this structure (e.g., the merging of two “incompatible” neighbourhoods) may affect the functioning of the entire system.
Author Manuscript
Many algorithmic methods exist for detecting community structure (with none being uniformly best) – although modularity maximisation, which aims to optimise the modularity statistic (Figure 4, Table 1), serves as most common in the brain network literature. Recent algorithmic developments also allow identifying (potentially) overlapping communities since many brain areas likely belong to multiple communities simultaneously given that they can perform several roles.
Author Manuscript
Despite these advancements in brain network description, much work remains in refining and statistically assessing these network properties and determining the best approach to estimating the underlying brain networks from which these properties are extracted. For example, real networks likely contain negative feedback loops, having a positive connection in one direction and negative connection in the other, but current network generation methods are unable to capture this. Additionally, current metric assessments ignore the fact that the underlying network is estimated and thus that these metrics are estimates themselves with certain probability distributions. Making formal inferential decisions about network metric values and gaining a better understanding of topological variability in normal and abnormal brain function requires assessment of the estimation error resulting from the estimation of the underlying brain networks. This area of network error propagation provides extremely fertile ground for Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 5
Author Manuscript
statistical research. To our knowledge, there is currently no active research in this area for functional brain networks other than that of Balachandran et al. who quantified the propagation of error from network estimation to subgraph counts.3
Fusing statistics and network science Accurately and precisely drawing inferential insight into the link between brain network properties and brain (dys)function requires some statistical gaps to be addressed. While recent univariate developments have proven useful, gleaning deeper insights into normal and abnormal changes in complex functional organisation demands methods that leverage the wealth of data present in an entire brain network.1
Author Manuscript
This systemic organisation confers much of our brain’s functional abilities as functional connections may be lost due to an adverse health condition, but compensatory connections may develop as a result in order to maintain organisational consistency and functional performance. Thus, gaining insight into this organisation requires a multivariate modelling framework that allows assessing the effects of endogenous (network measures) and exogenous (demographics, disease status, etc.) variables of interest on the overall network structure. We have made strides in developing such a framework both with exponential random graph models and mixed models, but more work is needed.4–6 Moreover, extending such frameworks to the longitudinal context will allow us to explain how dynamic network changes relate to normal and abnormal brain function over time.
Author Manuscript
The movement of neuroscience from simplistic ideas about brain function to holistic views about how systemic properties underpin the emergence of complex behaviours promises to take us a step closer to understanding who we are and what makes us unique. The future of brain network science remains promising as long as we take a conscientious approach to developing methods that appropriately account for data complexity. More generally, complexity-based neuroscience – which subsumes network-based analyses – represents a new paradigm aimed at quantifying the complex patterns inherent in physiological systems. This systems-based approach represents the frontier in neuroscience, statistics, and the sciences more generally.
Author Manuscript
Incorporating innovative methodologies within this paradigm will aid in revolutionising our understanding of the human brain. Given the nascency of the field and the variety of scientific questions of interest, it is unlikely that an optimal analysis method exists. Rather than being “locked in” to what have emerged as standard brain network science techniques, this generation of scientists has to be more flexible in thought and open to newer, often better, ways of analysing and using the richness of this complex data. A multi-faceted suite of complementary approaches situated at the interface of statistical, network and brain science will likely be needed to move the field forward.
Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 6
Author Manuscript
Acknowledgments This work is supported by the National Institute of Biomedical Imaging and Bioengineering (NIBIB) (K25 EB012236-01A1) and the Wake Forest Older Americans Independence Center (P30 21332). We thank past and present members of the Laboratory for Complex Brain Networks for some of the figures used in this article.
References
Author Manuscript
1. Simpson SL, Bowman FD, Laurienti PJ. Analyzing complex functional brain networks: fusing statistics and network science to understand the brain. Statistics Surveys. 2013; 7:1–36. [PubMed: 25309643] 2. Telesford QK, Simpson SL, Burdette JH, Hayasaka S, Laurienti PJ. The brain as a complex system: Using network science as a tool for understanding the brain. Brain Connectivity. 2011; 1:295–308. [PubMed: 22432419] 3. Balachandran P, Kolaczyk E, Viles W. On the propagation of low-rate measurement error to subgraph counts in large, sparse networks. arXiv:1409.5640. 2014 [math.ST]. 4. Simpson SL, Hayasaka S, Laurienti PJ. Exponential random graph modeling for complex brain networks. PLoS One. 2011; 6:e20039. 5. Simpson SL, Laurienti PJ. A two-part mixed-effects modeling framework for analyzing whole-brain network data. NeuroImage. 2015; 113:310–319. 6. Simpson SL, Moussa MN, Laurienti PJ. An exponential random graph modeling approach to creating group-based representative whole-brain connectivity networks. NeuroImage. 2012; 60:1117–1126.
Biographies
Author Manuscript
Sean Simpson is an associate professor of biostatistical sciences and member of the Laboratory for Complex Brain Networks (LCBN) at Wake Forest School of Medicine with a focus on fusing multivariate statistical methods with network science for brain network analysis Jonathan Burdette is a professor of neuroradiology and member of the LCBN at Wake Forest School of Medicine, and he studies brain–body interactions as a complex system Paul Laurienti is a professor of radiology and director of the LCBN at Wake Forest School of Medicine. He is developing methods to understand the brain as a complex, integrated system
Author Manuscript Signif (Oxf). Author manuscript; available in PMC 2016 July 18.
Simpson et al.
Page 7
Author Manuscript Figure 1.
Hieroglyphic for “brain”
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Figure 2.
Schematic for generating brain networks from fMRI time series data (partially recreated from Simpson et al.1). Functional connectivity between brain areas is estimated based on time-series pairs to produce a connection matrix. A threshold is commonly applied to the matrix to remove negative and/or “weak” connections
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Figure 3.
(a) The clustering coefficient (C) for the source region (node) equals the proportion of its connections that are also connected with each other. (b) The characteristic path length (L) for a pair of regions equals the shortest distance (smallest number of edges that must be traversed) to get from the one region to the other. The overall C and L for the network are then the averages of these values across all regions and region pairs, respectively
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Figure 4.
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Modularity analysis (reproduced from Telesford et al.2). Depending on the level where the subdivisions are made (dashed line), the number of communities can change. (A) In this example network, the optimal division, as indicated by maximum modularity, yields four communities (indicated by the red dashed line). Shifting this line up or down (indicated by the dashed line with an arrow) produces a lower modularity value that yields a suboptimal community structure. (B) As the line shifts higher, fewer communities are formed (approaching a single community comprising all nodes). (C) As the line shifts lower, more communities are formed (approaching every node being in their own community)
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Table 1
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Network metric definitions Metric
Definition
Local efficiency
where Eloc,i is the local efficiency of node i, N is the set of all nodes in the network, n is the number of nodes, aij is an indicator function for the existence of an edge between nodes i and j, ki is the degree of node i, and djh(Ni) is the shortest path between nodes j and h that contains only neighbours (connected nodes) of node i. Eloc ranges from 0 to 1, with larger values representing more functional segregation. Global efficiency
where Eglob,i is the global efficiency of node i, dij is the shortest path between nodes i and j, and N and n are defined as before. Like Eloc, Eglob also ranges from 0 to 1, with larger values representing more functional integration.
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Modularity
where M indexes the set of non-overlapping modules from the fully subdivided network, and euv denotes the proportion of all edges that connect nodes in module u with those in module v.
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