Skip to content
Robotic Brain Universe

White matter and tractography

The brain's wiring, how diffusion MRI reveals it, the maths of tensors and networks, and the limits of mapping tracts.

Intermediate · about 8 min · updated 2026-10-02 · awaiting clinical review

See it in 3D:

Illustrative simulation excitatory inhibitory

Association, projection and commissural tracts and the colour code used in this atlas; myelin and conduction speed; small-world brain networks and split-brain research; diffusion MRI, the diffusion tensor, fractional anisotropy and streamline tractography; development and training; multiple sclerosis, disconnection and tractography's false positives; population atlases and the connectome.

Contents
  1. The brain's wiring
  2. What white matter is
  3. Why wiring matters
  4. How white matter is imaged
  5. When white matter changes
  6. When white matter is damaged
  7. The mathematics of diffusion and networks
  8. Technology: atlases and connectomes
  9. Milestones
  10. Frontiers
  11. Check yourself

The brain's wiring

Under the grey cortex lies the white matter: bundles of insulated nerve fibres that connect regions near and far. Its pale colour comes from myelin, the insulation that lets signals jump along an axon and arrive in milliseconds.[1,2]

Diffusion MRI made it possible to follow these bundles in the living brain: by measuring which way water molecules move most easily, it reveals the direction of fibres in every millimetre of the living brain, and tractography joins those directions into pathways. The streamlines in this atlas come from a population-averaged tractography atlas built from 1,065 young adults in the Human Connectome Project.[3,4,5]

Tractography is powerful but fallible: in an international challenge, most algorithms found 90% of the true bundles but also produced many more invalid ones. This reading explains how white matter works, how it is imaged and mathematically described, what happens when it is damaged, and how it is mapped as a network.[6]

What white matter is

Fibre pathways fall into three classic groups. Association tracts link areas within one hemisphere, such as the arcuate fasciculus joining frontal and temporal language areas. Projection tracts connect the cortex with deeper structures and the spinal cord, such as the corticospinal tract. Commissural tracts cross between the hemispheres, the largest being the corpus callosum.[2,7,8]

In the atlas, tracts are coloured by direction using the standard diffusion-imaging convention: red for left–right, green for front–back and blue for up–down.[9]

Key numbers

Young adults whose scans built the tract atlas used here
1,065[5]
Share of tract-to-region connections consistent across individuals
about 85%[10]
Submissions to the 2017 tractography challenge, from 20 groups
96[6]
Scans in a longitudinal study of white matter development (103 people aged 5–32)
221[11]

Why wiring matters

Speed. How fast a myelinated fibre conducts depends on its diameter and on the thickness of its myelin; in fibres of similar shape, speed is nearly proportional to diameter.[1]

Networks. The brain's connections form a network with the features of other complex networks: small-world topology, highly connected hubs and modules. Small-world networks, first described by Watts and Strogatz, are as clustered as a regular lattice yet have short path lengths like a random graph, which speeds signal propagation and synchronisation.[12,13]

Integration. Split-brain research, in which the corpus callosum was cut to treat epilepsy, revealed how specialised each hemisphere is and how the callosum integrates them, including our sense of a unified self.[14]

How white matter is imaged

Diffusion. In a diffusion-weighted MRI scan, magnetic field gradients make the signal fall in proportion to how far water molecules move. In white matter, water moves more easily along fibres than across them; this anisotropy can be measured in any direction.[3,15]

The tensor. Diffusion tensor imaging fits each voxel with a 3 × 3 tensor whose eigenvectors give the tissue's three principal axes and whose eigenvalues give the diffusivity along each. The largest eigenvector points along the main fibre direction; quantities such as mean diffusivity and fractional anisotropy describe the microstructure independent of the head's orientation.[3,16]

Tracking. Tractography follows the main fibre direction from voxel to voxel to reconstruct pathways in three dimensions; Mori and colleagues first validated this against known anatomy in the rat brain.[4]

From MRI to a tract atlasDiffusion-weighted imagessignal loss along many gradientdirectionsModel per voxeltensor: eigenvectors andeigenvaluesMapsfractional anisotropy, colour bydirectionTractographyfollow fibre directions asstreamlinesPopulation atlasaveraged across many peopleConnectomewhich regions each tract reaches
From MRI to a tract atlas. The diffusion signal is modelled in each voxel, the fibre directions are followed as streamlines, and streamlines from many people are averaged into an atlas from which a tract-to-region connectome can be read.[3,4,10,16]
Text version of the diagram
  1. Diffusion-weighted images: signal loss along many gradient directions. Leads to Model per voxel.
  2. Model per voxel: tensor: eigenvectors and eigenvalues. Leads to Maps; Tractography.
  3. Maps: fractional anisotropy, colour by direction.
  4. Tractography: follow fibre directions as streamlines. Leads to Population atlas.
  5. Population atlas: averaged across many people. Leads to Connectome.
  6. Connectome: which regions each tract reaches.

When white matter changes

Growing up. Scanning 103 people aged 5 to 32 at least twice each, Lebel and Beaulieu found that fractional anisotropy rose and mean diffusivity fell in all ten major tracts studied. Projection and commissural tracts mostly matured by late adolescence, but association tracts kept maturing into adulthood.[11]

Learning. After adults trained on a complex visuo-motor skill (juggling), diffusion imaging showed a localised increase in fractional anisotropy in the white matter beneath the intraparietal sulcus, the first evidence of training-related change in white-matter structure in healthy adults.[17]

When white matter is damaged

Multiple sclerosis is an inflammatory disease in which lymphocytes infiltrate the brain and spinal cord and damage myelin and axons; early episodes often recover as inflammation subsides and remyelination occurs, but damage accumulates over time.[18]

Disconnection. Injury along the corticospinal tract above the pyramidal decussation causes weakness on the opposite side, wherever along its course it occurs.[2,8]

Imaging pitfalls. Tractography can mislead: the 2017 challenge showed that tractograms contain many more invalid than valid bundles, and half of the invalid ones occurred systematically across research groups, a fundamental ambiguity of reconstructing tracts from orientation alone.[6]

The mathematics of diffusion and networks

White matter imaging rests on the physics of diffusion and on linear algebra; mapping it as a network uses graph theory.[12,15]

Stejskal–Tanner signal attenuation[15]
S=S0 e−γ2G2δ2(Δ−δ/3) D=S0 e−bDS = S_0\, e^{-\gamma^{2} G^{2} \delta^{2} (\Delta - \delta/3)\, D} = S_0\, e^{-bD}

With a pair of magnetic gradient pulses, the MRI signal falls exponentially with the diffusion coefficient. The product of the gradient terms is the 'b-value': the larger it is, the more the scan is sensitised to diffusion.

Symbols in Stejskal–Tanner signal attenuation
SymbolMeaningUnit
S,S0S, S_0signal with and without diffusion weighting—
γ\gammagyromagnetic ratio of the hydrogen nucleus—
G,δG, \deltagradient strength and pulse duration—
Δ\Deltatime between the two gradient pulsess
DDdiffusion coefficient along the gradient directionmm²/s
Mean diffusivity and fractional anisotropy[16]
λˉ=λ1+λ2+λ33,FA=32  ∑i(λi−λˉ)2∑iλi2\bar\lambda = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}, \qquad \mathrm{FA} = \sqrt{\frac{3}{2}}\;\frac{\sqrt{\sum_i (\lambda_i - \bar\lambda)^2}}{\sqrt{\sum_i \lambda_i^2}}

From the three eigenvalues of the diffusion tensor: mean diffusivity is their average (a third of the tensor's trace); fractional anisotropy runs from 0, when diffusion is equal in all directions, towards 1, when it is confined to one direction, as in a tightly packed fibre bundle.

Symbols in Mean diffusivity and fractional anisotropy
SymbolMeaningUnit
λ1,λ2,λ3\lambda_1, \lambda_2, \lambda_3eigenvalues of the diffusion tensormm²/s
λˉ\bar\lambdamean diffusivitymm²/s
FAFAfractional anisotropy, between 0 and 1—
Streamline tractography[3,4]
dr(s)ds=e1(r(s))\frac{d\mathbf{r}(s)}{ds} = \mathbf{e}_1\big(\mathbf{r}(s)\big)

A streamline is a curve whose tangent everywhere follows the principal eigenvector of the local diffusion tensor; tractography integrates this equation step by step from a seed point.

Symbols in Streamline tractography
SymbolMeaningUnit
r(s)\mathbf{r}(s)position along the streamline at arc length smm
e1\mathbf{e}_1principal eigenvector (main fibre direction) at that position—
Clustering and path length[12,13]
C=1N∑i2 tiki(ki−1),L=1N(N−1)∑i≠jdijC = \frac{1}{N}\sum_i \frac{2\,t_i}{k_i(k_i - 1)}, \qquad L = \frac{1}{N(N-1)}\sum_{i \ne j} d_{ij}

Two numbers that define a small-world network: the clustering coefficient CC (how often a node's neighbours are also connected to each other) is high, as in a lattice, while the characteristic path length LL (average number of steps between nodes) is short, as in a random graph.

Symbols in Clustering and path length
SymbolMeaningUnit
tit_inumber of connections among the neighbours of node i—
kik_inumber of neighbours (degree) of node i—
dijd_{ij}shortest path length between nodes i and j—
NNnumber of nodes—

Technology: atlases and connectomes

The Human Connectome Project collected high-quality MRI, including diffusion imaging, from large numbers of healthy young adults to map human brain connectivity.[21]

Population atlases. Averaging tractography across many people gives atlases in a standard space: a stereotaxic white matter atlas in an ICBM template, and the HCP-1065 tract atlas shown in this model, from which a tract-to-region connectome was derived.[5,10,22]

The connectome. Sporns, Tononi and Kötter argued in 2005 that a connection matrix of the human brain, the 'connectome', would be an indispensable foundation for neuroscience; graph theory now describes such networks in terms of hubs, modules and small-world organisation.[12,23]

Tract families in the atlas[2,8]
FamilyConnectsExample
AssociationAreas within one hemisphereArcuate fasciculus
ProjectionCortex with deep structures and spinal cordCorticospinal tract
CommissuralThe two hemispheresCorpus callosum

Milestones

Seeing the wiring

  1. 1965Stejskal and Tanner measure diffusion with pulsed magnetic field gradients.[15]
  2. 1994Diffusion tensor imaging is introduced.[3]
  3. 1996Rotation-invariant measures such as fractional anisotropy are derived.[16]
  4. 1998Small-world networks are described.[13]
  5. 1999Three-dimensional tractography is validated; a colour code for fibre direction is proposed.[4,9]
  6. 2005The connectome is proposed; split-brain research is reviewed after 45 years.[14,23]
  7. 2008Tractography and stereotaxic white matter atlases are published.[8,22]
  8. 2009Juggling training changes white matter; graph theory of brain networks is reviewed.[12,17]
  9. 2011White matter is shown to keep maturing into adulthood.[11]
  10. 2013The WU-Minn Human Connectome Project is described.[21]
  11. 2017An international challenge exposes tractography's false positives.[6]
  12. 2022A population-based tract-to-region connectome from 1,065 people.[5,10]

Frontiers

The tract-to-region connectome shows that about 85% of its entries are consistent across individuals, while the remaining 15% vary enough to need individual mapping, which matters for planning surgery in a particular patient.[10]

The tractography challenge provides a framework for testing reliability and calls for innovation beyond orientation information alone, since orientation by itself cannot resolve where bundles really run.[6]

Check yourself

Check yourself

  1. Name the three classic families of white matter tracts, with an example of each.
    Show answer

    Association (arcuate fasciculus), projection (corticospinal tract) and commissural (corpus callosum).

  2. What does diffusion tensor imaging actually measure?
    Show answer

    How easily water diffuses in different directions in each voxel; along fibres it moves more easily than across them.

  3. What does fractional anisotropy of 0 mean, and what does a value near 1 mean?
    Show answer

    0 means diffusion is equal in all directions; near 1 means it is confined mainly to one direction.

  4. What colour means front–back in the tract display?
    Show answer

    Green (red is left–right, blue is up–down).

  5. What did the 2017 tractography challenge find?
    Show answer

    Most algorithms found 90% of true bundles but produced many more invalid than valid ones.

  6. Which tracts keep maturing after adolescence?
    Show answer

    Association tracts.

  7. What makes a network 'small-world'?
    Show answer

    High clustering like a lattice combined with short path lengths like a random graph.

Glossary[2,3,4,13,16,23]

White matter
Brain tissue made mainly of myelinated nerve fibres.
Tract
A bundle of nerve fibres running together between brain regions.
Diffusion MRI
MRI sensitised to the random movement of water molecules.
Diffusion tensor
A 3 × 3 matrix describing how diffusion varies with direction in a voxel.
Fractional anisotropy
A measure from 0 to 1 of how directional diffusion is.
Tractography
Reconstructing fibre pathways by following diffusion directions.
Streamline
A curve traced through the fibre-direction field, representing part of a tract.
Connectome
A map of the connections of the brain.
Small-world network
A network that is highly clustered yet has short paths between nodes.
Corpus callosum
The largest commissural tract, joining the two hemispheres.

References

  1. Waxman SG. Determinants of conduction velocity in myelinated nerve fibers. Muscle and Nerve 1980;3(2):141-150. doi:10.1002/mus.880030207
  2. Schmahmann JD, Pandya DN. Fiber Pathways of the Brain. Oxford University Press 2006. doi:10.1093/acprof:oso/9780195104233.001.0001
  3. Basser PJ, Mattiello J, LeBihan D. MR diffusion tensor spectroscopy and imaging. Biophysical Journal 1994;66(1):259-267. doi:10.1016/S0006-3495(94)80775-1
  4. Mori S, Crain BJ, Chacko VP, van Zijl PCM. Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging. Annals of Neurology 1999;45(2):265-269. doi:10.1002/1531-8249(199902)45:2<265::AID-ANA21>3.0.CO;2-3
  5. Yeh F-C. Population-Probability Atlas and Tract-to-Region Connectome (HCP-1065), data release. brain.labsolver.org 2022. https://brain.labsolver.org/hcp_trk_atlas.html
  6. Maier-Hein KH, Neher PF, Houde JC, Côté MA, Garyfallidis E, Zhong J, et al.. The challenge of mapping the human connectome based on diffusion tractography. Nature Communications 2017;8:1349. doi:10.1038/s41467-017-01285-x
  7. Catani M, Jones DK, ffytche DH. Perisylvian language networks of the human brain. Annals of Neurology 2005;57(1):8-16. doi:10.1002/ana.20319
  8. Catani M, Thiebautdeschotten M. A diffusion tensor imaging tractography atlas for virtual in vivo dissections. Cortex 2008;44(8):1105-1132. doi:10.1016/j.cortex.2008.05.004
  9. Pajevic S, Pierpaoli C. Color schemes to represent the orientation of anisotropic tissues from diffusion tensor data: Application to white matter fiber tract mapping in the human brain. Magnetic Resonance in Medicine 1999;42(3):526-540. doi:10.1002/(SICI)1522-2594(199909)42:3<526::AID-MRM15>3.0.CO;2-J
  10. Yeh FC. Population-based tract-to-region connectome of the human brain and its hierarchical topology. Nature Communications 2022;13:4933. doi:10.1038/s41467-022-32595-4
  11. Lebel C, Beaulieu C. Longitudinal development of human brain wiring continues from childhood into adulthood. The Journal of Neuroscience 2011;31(30):10937-10947. doi:10.1523/JNEUROSCI.5302-10.2011
  12. Bullmore E, Sporns O. Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience 2009;10(3):186-198. doi:10.1038/nrn2575
  13. Watts DJ, Strogatz SH. Collective dynamics of 'small-world' networks. Nature 1998;393(6684):440-442. doi:10.1038/30918
  14. Gazzaniga MS. Forty-five years of split-brain research and still going strong. Nature Reviews Neuroscience 2005;6(8):653-659. doi:10.1038/nrn1723
  15. Stejskal EO, Tanner JE. Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. The Journal of Chemical Physics 1965;42(1):288-292. doi:10.1063/1.1695690
  16. Basser PJ, Pierpaoli C. Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance, Series B 1996;111(3):209-219. doi:10.1006/jmrb.1996.0086
  17. Scholz J, Klein MC, Behrens TEJ, Johansen-Berg H. Training induces changes in white-matter architecture. Nature Neuroscience 2009;12(11):1370-1371. doi:10.1038/nn.2412
  18. Compston A, Coles A. Multiple sclerosis. The Lancet 2008;372(9648):1502-1517. doi:10.1016/S0140-6736(08)61620-7
  19. Berman JI, Berger MS, Mukherjee P, Henry RG. Diffusion-tensor imaging-guided tracking of fibers of the pyramidal tract combined with intraoperative cortical stimulation mapping in patients with gliomas. Journal of Neurosurgery 2004;101(1):66-72. doi:10.3171/jns.2004.101.1.0066
  20. Keles GE, Lundin DA, Lamborn KR, Chang EF, Ojemann G, Berger MS. Intraoperative subcortical stimulation mapping for hemispheric perirolandic gliomas located within or adjacent to the descending motor pathways: evaluation of morbidity and assessment of functional outcome in 294 patients. Journal of Neurosurgery 2004;100(3):369-375. doi:10.3171/jns.2004.100.3.0369
  21. Van Essen DC, Smith SM, Barch DM, Behrens TEJ, Yacoub E, Ugurbil K. The WU-Minn Human Connectome Project: An overview. NeuroImage 2013;80:62-79. doi:10.1016/j.neuroimage.2013.05.041
  22. Mori S, Oishi K, Jiang H, Jiang L, Li X, Akhter K, Hua K, Faria AV, Mahmood A, Woods R, Toga AW, Pike GB, Neto PR, Evans A, Zhang J, Huang H, Miller MI, van Zijl P, Mazziotta J. Stereotaxic white matter atlas based on diffusion tensor imaging in an ICBM template. NeuroImage 2008;40(2):570-582. doi:10.1016/j.neuroimage.2007.12.035
  23. Sporns O, Tononi G, Kötter R. The human connectome: a structural description of the human brain. PLoS Computational Biology 2005;1(4):e42. doi:10.1371/journal.pcbi.0010042

Related readings

  • Neurons and electrical signals

    How a nerve cell makes a 100-millivolt spike, the equations that describe it, and why one neuron may be a deep network in disguise.

    Introductory

  • Language and speech

    Where language lives in the brain, why it is not the same as thought, and how AI language models and speech implants are changing the field.

    Intermediate

  • The motor system

    How the brain turns intention into movement, from the redrawn motor homunculus to the laws of reaching and the technology that restores it.

    Intermediate

  • The cerebral cortex and its maps

    What the folded outer layer of the brain is made of, why it folds, and how scientists map it from Brodmann's microscope to petabyte connectomes.

    Introductory

  • How MRI sees the brain

    Nuclear magnetic resonance, relaxation and k-space; structural, diffusion and functional MRI; their pitfalls; and AI, portable and 11.7 T scanners.

    Intermediate

  • How the brain is organised

    A map of the whole brain: its divisions, cells and energy budget, how it develops from a tube, how its network is wired, and how it changes over a lifetime.

    Introductory

Template anatomy for education. Not patient-specific. Not for clinical decision-making.