A Resilient, Low-Frequency, Small-World Human Brain Functional Network with Highly Connected Association Cortical Hubs
Open accessAbstract
Small-world properties have been demonstrated for many complex networks. Here, we applied the discrete wavelet transform to functional magnetic resonance imaging (fMRI) time series, acquired from healthy volunteers in the resting state, to estimate frequency-dependent correlation matrices characterizing functional connectivity between 90 cortical and subcortical regions. After thresholding the wavelet correlation matrices to create undirected graphs of brain functional networks, we found a small-world topology of sparse connections most salient in the low-frequency interval 0.03–0.06 Hz. Global mean path length (2.49) was approximately equivalent to a comparable random network, whereas clustering (0.53) was two times greater; similar parameters have been reported for the network of anatomical connections in the macaque cortex. The human functional network was dominated by a neocortical core of highly connected hubs and had an exponentially truncated power law degree distribution. Hubs included recently evolved regions of the heteromodal association cortex, with long-distance connections to other regions, and more cliquishly connected regions of the unimodal association and primary cortices; paralimbic and limbic regions were topologically more peripheral. The network was more resilient to targeted attack on its hubs than a comparable scale-free network, but about equally resilient to random error. We conclude that correlated, low-frequency oscillations in human fMRI data have a small-world architecture that probably reflects underlying anatomical connectivity of the cortex. Because the major hubs of this network are critical for cognition, its slow dynamics could provide a physiological substrate for segregated and distributed information processing.
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High-resolution tractography of the white matter pathways shows small-world network properties, consistent with efficient allocation of neuronal traffic flow.
“High-resolution tractography of the white matter pathways shows small-world network properties, consistent with efficient allocation of neuronal traffic flow (Achard et al., 2006).”
From Global Workspace Dynamics: Cortical “Binding and Propagation” Enables Conscious Contents (2013) · cites this paper as Achard et al., 2006
The Abstract (page not provided) reports small-world properties in a network derived from resting-state fMRI, not high-resolution white-matter tractography. In “Anatomy of human and primate brain small-world networks” (page not provided), the authors propose future tractography as a way to test whether the functional network reflects a small-world anatomical architecture. The paper links small-world organization to integrated processing, but does not demonstrate efficient neuronal traffic flow through mapped white-matter pathways. (LLM, conf=0.88)
Experimental — accuracy not yet measuredBrain networks do not demonstrate scale-free properties in their degree distribution, but rather have an exponentially truncated power law.
“However, it appears that brain networks do not demonstrate scale-free properties in their degree distribution, but rather have an exponentially truncated power law. 2”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [2]
In “Why the human brain network is not scale-free,” the authors report that their resting-state, regional-resolution network did not have a power-law degree distribution; an exponentially truncated power law fit best. The claim generalizes this finding to brain networks overall, although the same section notes a previous activated-fMRI study at voxel resolution that reported a scale-free topology. (LLM, conf=0.86)
Experimental — accuracy not yet measuredMany real-world networks exhibit a power-law degree distribution, although often the distribution is capped due to physical restraints, in which case an exponentially truncated power law degree distribution is appropriate.
“Many realworld networks have been found to exhibit this property, although often the distribution is capped due to physical restraints (e.g., the finite size of a brain), in which case an exponentially truncated power law degree distribution is appropriate.2”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [2]
The Introduction describes power-law and exponentially truncated degree distributions in real-world networks. In “Why the human brain network is not scale-free,” the authors report that a truncated power law best fits the brain network and suggest that physical constraints may explain such cutoffs. They do not identify the finite size of a brain as the constraint, and they present the causal explanation as likely rather than established. (LLM, conf=0.82)
Experimental — accuracy not yet measuredSparse small-world attributes are robust in that they are reproducible across imaging modalities, disease states, and species.
“Sparse small-world attributes are robust in that they are reproducible across imaging modalities (diffusion MRI, 45,46 resting fMRI, 2,78,102 structural MRI, 48 EEG, 82 MEG, 90 and PET 30,79 ), disease states, and species (such as the cat, macaque, and nematode Caenorhabditis elegans).20”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [2]
The Introduction (page not provided) reports small-world properties in C. elegans, cat and monkey networks, as well as evidence from human fMRI and neurophysiological data. It does not establish reproducibility across the full list of imaging modalities or across disease states; the Sample section describes healthy volunteers without neurologic or psychiatric disorders. (LLM, conf=0.85)
Experimental — accuracy not yet measuredAn exponentially truncated power law degree distribution confers robustness to random error but leaves the network vulnerable to specific removal of hubs.
“An exponentially truncated power law degree distribution has been found to confer robustness to random error but also leaves the network vulnerable to specific removal of hubs. 2”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [2]
The ‘Degree distribution and hubs’ section identifies an exponentially truncated power law as the best-fitting distribution, and ‘Resilience of brain functional network’ reports resilience to random error. Targeted hub removal did cause damage, but the paper says the brain network was considerably more resilient to targeted attack than a scale-free network; the claim overstates its vulnerability and attributes that pattern to the degree distribution more directly than the paper does. (LLM, conf=0.86)
Experimental — accuracy not yet measured
Translating connectivity patterns to physical space was originally performed in a 2D plane, where lines denoted connections between regions in physical space.
“Translating connectivity patterns to physical space was originally performed in a 2D plane (a), where lines were used to denote connections between regions in physical space.2,78”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [2]
The “Anatomical and topological visualization of brain functional networks” section says the authors placed regional nodes in the y-z plane of Talairach space and drew lines between connected nodes; Figure 3 likewise describes a sagittal view. This supports the 2D depiction and lines, but the paper does not establish that this was the original or first way connectivity patterns were translated into physical space. Page numbers are not provided in our copy's passages. (LLM, conf=0.84)
Experimental — accuracy not yet measuredThe power-law fit is believed to reflect the brain network's hub organization wherein the majority of information processing is handled by a small number of core regions.
“The power-law fit is believed to reflect the brain network’s hub organization wherein the majority of information processing is handled by a small number of core regions (Achard et al., 2006).”
From Functional and structural connectome properties in the 5XFAD transgenic mouse model of Alzheimer’s disease (2018) · cites this paper as Achard et al., 2006
The “Degree distribution and hubs” section identifies highly connected association-cortical hubs, supporting the claim's hub-organization component. But it finds that an exponentially truncated power law fits better than a pure power law, and the paper does not establish that a small number of regions handle most information processing (page numbers unavailable in our copy). (LLM, conf=0.88)
Experimental — accuracy not yet measured
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- Authors
- Achard, S., Salvador, R., Whitcher, B., Suckling, J., & Bullmore, E.
- Affiliations
- University of Cambridge (GB)
- GlaxoSmithKline (United Kingdom) (GB)
- Venue
- The Journal of Neuroscience
- Publisher
- Society for Neuroscience
- Published
- 2006-01-04
- Volume
- 26
- Issue
- 1
- Pages
- 63-72
- ISSN
- 0270-6474 (ISSN-L)
- 1529-2401
- PMC
- PMC6674299
- OpenAlex
- W1987924998
- Cited by
- 2,393
- References
- 4
- Citations by year
- 2026: 45 · 2025: 61 · 2024: 75 · 2023: 93 · 2022: 99 · 2021: 133 · 2020: 111 · 2019: 145 · 2018: 149 · 2017: 175 · 2016: 194 · 2015: 165 · 2014: 158 · 2013: 180 · 2012: 152
- FWCI
- 33.20
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Locations
- Journal of Neuroscience (publishedVersion, cc-by)
- PubMed Central (publishedVersion)
- PubMed (publishedVersion)
- Funders
- Florida Atlantic University
- Wellcome Trust
- Universidad del Atlántico
- Medical Research Council
- National Institute of Mental Health
- Topics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Health Sciences › Medicine › Radiology, Nuclear Medicine and Imaging › Advanced Neuroimaging Techniques and Applications
- Keywords
- small-world networks, functional brain networks, resting-state functional connectivity, association cortex, network hubs, network resilience, low-frequency oscillations, discrete wavelet transform, degree distribution, information processing
- MeSH terms
- Adult, Biological Clocks, Brain Mapping, Cerebral Cortex, Female, Humans, Magnetic Resonance Imaging, Male, Nerve Net, Neural Pathways
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