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Multi-scale brain networks

Richard F. Betzel, Danielle S. Bassett

arXiv:1608.08828v2q-bio.NCphysics.soc-ph

TL;DR

Brain networks are organized across spatial, temporal, and topological scales, but many analyses examine only one scale at a time. This review surveys evidence and network-based methods for connecting scales, emphasizing multi-scale community detection and temporal multi-layer analysis.

  • Problem

    Brain-network analyses often focus on a single spatial, temporal, and topological scale, limiting description of the brain’s multi-scale, multi-modal nature.

  • Method

    The review synthesizes empirical evidence and methodological approaches for multi-scale topological, temporal, and spatial brain-network analysis, emphasizing community detection and multi-layer techniques.

  • Results

    The review identifies network-analysis tools and applications that characterize community structure across topological scales, time-resolved connectivity across temporal resolutions, and brain networks across spatial scales.

  • Takeaways & Limitations

    Results at different scales can be both redundant and complementary, supporting multi-scale analysis of brain-network organization.

Abstract

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The network architecture of the human brain has become a feature of increasing interest to the neuroscientific community, largely because of its potential to illuminate human cognition, its variation over development and aging, and its alteration in disease or injury. Traditional tools and approaches to study this architecture have largely focused on single scales -- of topology, time, and space. Expanding beyond this narrow view, we focus this review on pertinent questions and novel methodological advances for the multi-scale brain. We separate our exposition into content related to multi-scale topological structure, multi-scale temporal structure, and multi-scale spatial structure. In each case, we recount empirical evidence for such structures, survey network-based methodological approaches to reveal these structures, and outline current frontiers and open questions. Although predominantly peppered with examples from human neuroimaging, we hope that this account will offer an accessible guide to any neuroscientist aiming to measure, characterize, and understand the full richness of the brain's multiscale network structure -- irrespective of species, imaging modality, or spatial resolution.

I. INTRODUCTION

Brain networks are multi-scale entities spanning spatial, temporal, and topological axes, yet many analyses examine only one point in this space. This review surveys methods and evidence for linking scales, especially across topology and time.

  • Multi-scale brain networks: Brain networks span spatial scales from cells and synapses to regions and fiber tracts, temporal scales from sub-milliseconds to lifespans and evolution, and topological scales from nodes to whole networks.These axes define a three-dimensional space in which brain-network analyses are situated.
  • Multi-scale brain networks: Many brain-network studies focus singularly on one spatial, temporal, and topological scale.The review argues that this narrow focus should be extended by forming bridges between scales.
  • Review scope: The review presents network algorithms for multiple topological scales and multi-layer techniques for temporal resolutions, with emphasis on community detection.It also discusses methodological limitations, best practices, and future directions.
  • Review scope: Brain networks are organized across multiple spatiotemporal scales and can be analyzed from individual nodes to the network as a whole.Figure 1 summarizes the spatial, temporal, and topological dimensions of the multi-scale brain.

II. FUNCTIONAL AND STRUCTURAL BRAIN NETWORKS

Structural and functional brain networks represent different kinds of connectivity in matrices, and graph-theoretic analysis examines their collective architecture. Network properties can be studied locally, globally, or at intermediate mesoscale organization.

  • Network types: Structural connectivity links nodes through physical connections, whereas functional connectivity captures statistical relationships between node activity over time.With MRI, structural connections usually reflect tractography-derived white-matter fibers, while functional connections are often correlations or coherence measures.
  • Network types: Both structural and functional networks are represented by connectivity matrices whose element Aij denotes the connection weight between regions i and j.The passage introduces a common matrix representation for both network types.
  • Network analysis: Graph-theoretic network analysis evaluates the architecture and configuration of connections rather than isolated connection weights.This architecture is treated as relevant to system-level behavior, influential nodes, and robustness or vulnerability.
  • Topological scales: Local measures characterize individual nodes, global measures describe the network collectively, and mesoscale measures characterize clusters such as communities, cores, peripheries, and rich clubs.Degree or strength exemplifies a local measure, while path length exemplifies a global measure.
  • Topological scales: Mesoscale structures occupy a range between local and global extremes and can emerge, persist, and dissolve across multiple topological scales.The review therefore focuses on techniques that detect mesoscale structure over a range of scales.

A. Multi-scale community structure

Mesoscale community structure is not directly evident from complex network topology, so it is detected algorithmically. Modularity maximization compares observed and null-model connectivity, while a resolution parameter enables community detection across scales.

  • A. Multi-scale community structure: Mesoscale structure must be searched for algorithmically because its presence depends on complex patterns of edges among network nodes.Community-detection algorithms differ in how they define communities and identify them.
  • A. Multi-scale community structure: Modularity maximization partitions network nodes into communities by maximizing an objective function that compares observed connections with a specified null model.The comparison evaluates edge weights against the weights expected under that null model.
  • A. Multi-scale community structure: In the modularity formulation, Aij and Pij represent observed and expected connection weights, while σi identifies node i’s community assignment.The resulting optimization estimates a partition of network nodes into communities.
  • A. Multi-scale community structure: Multi-resolution community detection can identify scales of interest by sweeping γ and locating local minima in mean pairwise variation of information.In the synthetic hierarchical network, these minima correspond to scales recovering planted communities.
  • A. Multi-scale community structure: The standard modularity objective has a resolution limit that can make communities below some size mathematically undetectable.A resolution parameter was introduced to address this limitation.
  • A. Multi-scale community structure: The resolution parameter γ acts as a tuning knob: larger values favor smaller communities, whereas smaller values favor larger communities.Sweeping γ can span partitions from one network-wide community to singleton communities; the figure illustrates γ = 1 and γ = 2.5.

1. Multi-scale community structure in the neuroimaging literature

Multi-scale community detection is increasingly used to examine brain-network organization across resolutions, revealing scale-dependent structure and links between topological and spatial organization.

  • Most brain-network studies examine community structure at a single scale or use heuristics such as recursive partitioning and edge thresholding.
  • Multi-scale analyses of resting-state functional connectivity found an interaction between age and resolution: smaller communities become less segregated, whereas larger communities become more segregated across the lifespan.
  • Some studies estimate community structure across resolutions, then select a single partition using a secondary objective function.
  • Anatomical brain-network studies relate community size to spatial radius, suggesting constraints on embedding network architecture within the human skull.

2. Implementation and practical considerations

Multi-scale modularity maximization inherits important methodological problems from ordinary community detection, and varying the resolution parameter can amplify them.

  • Modularity maximization can produce false positives, variable outputs, density-biased communities, and computationally intractable global optimization.
  • Sweeping the resolution parameter γ extends these complications across every analyzed topological scale.
  • A principled approach to multi-scale modularity maximization therefore remains an open practical question.

Selecting the resolution parameter

Selecting resolution values is difficult without prior knowledge of community number or size, so proposed approaches use stability, statistical deviation, or persistence across γ.

  • Without prior knowledge of community number and size, there is no good rationale for preferring one γ value over another, including γ = 1.
  • One approach repeatedly optimizes modularity across γ values and focuses on resolutions where detected partitions are highly similar and minimally variable.
  • Statistical selection can target resolutions where observed community sizes deviate most from chance.
  • Another strategy assumes that meaningful community structure persists across a range of γ values rather than appearing fleetingly.

Consensus community structure and communities of interest

Consensus analysis must address both which resolutions to retain and which detected clusters merit interpretation; related multi-scale structures likewise require continuous rather than binary descriptions.

  • Consensus community structure and communities of interest: Consensus partitions can be selected by maximizing average partition similarity or reclustering node co-occurrence frequencies across an ensemble.
  • Consensus community structure and communities of interest: Modularity assigns every node to a cluster, but hubs and rich clubs may span modules rather than conform to a strict community template.
  • Consensus community structure and communities of interest: A statistical null-model comparison can help identify communities whose modularity contributions are unlikely under chance organization.
  • Consensus community structure and communities of interest: Multiscale tools capture non-random organization from individual nodes to whole networks and may provide a richer basis for relating network structure to cognition and disease.
  • B. Multi-scale rich club and core-periphery organization: Rich clubs are groups of highly connected hubs that are densely interconnected and may integrate otherwise separate modules.
  • B. Multi-scale rich club and core-periphery organization: Binary rich-club and core assignments obscure organization that can persist across multiple topological scales.
  • B. Multi-scale rich club and core-periphery organization: Reporting the range of significant rich clubs and parameterized core-periphery landscapes provides continuous descriptions of alternative hub and core configurations.

C. Multi-scale temporal networks

Brain networks fluctuate across timescales from sub-second dynamics to changes over the lifespan. Multi-layer temporal network models represent topology at different time points as layers and can accommodate virtually any neuroimaging-accessible timescale.

  • Brain-network organization fluctuates across timescales ranging from sub-second intervals to the lifespan.
  • Multi-layer temporal models represent estimates of network topology at different time points as separate layers.
  • The multi-layer framework is mathematically agnostic to the timescales represented by its layers.
  • Consequently, the framework can accommodate virtually any timescale accessible through neuroimaging technologies.

1. Multi-scale, multi-layer network analysis

Multi-layer network analysis extends familiar network measures and community detection to networks represented across layers. Combining this framework with a resolution parameter enables analysis across multiple topological scales, while community structure can be tracked across changing layers.

  • Path length, clustering, and some centrality measures can be calculated on multi-layer networks.
  • Multi-layer, multi-scale community detection is currently the most widely used multi-layer measure in network neuroscience.
  • The resolution parameter γ lets researchers incorporate multiple topological scales into temporal network analyses.
  • Individual networks sharing the same nodes can be combined by adding ordinal or categorical links between corresponding nodes across layers.
  • Community detection on the resulting multi-layer network tracks community formation and dissolution across layers and quantifies node flexibility.

2. Practical considerations

Multi-layer construction requires choices about how layers are linked and weighted. These choices can influence network measurements, and interlayer weights lack a firmly established data-driven selection procedure.

  • Constructing one multi-layer object requires adding artificial links between layers, either manually or through a data-driven procedure.
  • The choice of interlayer-linking strategy can influence the network measurement being made.
  • For temporal networks, ordinal interlayer links are currently standard practice in network neuroscience.
  • Interlayer links are usually assigned a common weight ω and varied over only a narrow range because evidence for choosing among weighting schemes is limited.
  • A principled, data-driven approach for selecting the interlayer weight ω remains desirable.

D. Multi-scale spatial networks

MRI-based brain networks span spatial scales from voxels to whole-brain parcellations, although MRI cannot generally resolve individual cells or neuronal populations. Parcellation choices provide a practical way to examine spatial scale but can alter network topology, requiring reproducibility checks.

  • Functional and diffusion MRI networks span spatial scales from individual voxels to the whole brain.
  • MRI makes it virtually impossible to construct brain networks at finer scales such as individual cells or neuronal populations.
  • Brain parcellation is the most obvious MRI-based way to examine different spatial scales by grouping voxels into regions.
  • Whole-brain parcellations range from approximately 1000 parcels to around 60 parcels.
  • Parcellation choice affects network topology, so results should be checked for qualitative reproducibility across different parcel sets.
  • Parcellation-based spatial analysis can subdivide specific brain areas, including visual-cortex parcels identified from connectivity patterns.

IV. CONCLUSION AND FUTURE DIRECTIONS

Multi-scale network analysis is necessary because different scales can reveal both shared network properties and scale-specific functions. The review therefore argues that network neuroscience needs complementary cross-scale and scale-focused approaches.

  • Network properties can be redundant across scales because similar energetic and spatial constraints shape network structure from cellular levels to brain regions.
  • Network properties can also be complementary, because node and circuit functions and biophysical attributes depend on the scale of construction and analysis.
  • Scale-specific network analysis can provide unique insight into the architecture underpinning functions performed at that scale.
  • Network neuroscience ultimately needs both cross-scale understanding and analyses focused on particular scales.
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