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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Science. Author manuscript; available in PMC 2013 September 15.
Published in final edited form as:
PMCID: PMC3773464

The Geometric Structure of the Brain Fiber Pathways


The structure of the brain as a product of morphogenesis is difficult to reconcile with the observed complexity of cerebral connectivity. We therefore analyzed relationships of adjacency and crossing between cerebral fiber pathways in four nonhuman primate species and in humans by using diffusion magnetic resonance imaging. The cerebral fiber pathways formed a rectilinear three-dimensional grid continuous with the three principal axes of development. Cortico-cortical pathways formed parallel sheets of interwoven paths in the longitudinal and medio-lateral axes, in which major pathways were local condensations. Cross-species homology was strong and showed emergence of complex gyral connectivity by continuous elaboration of this grid structure. This architecture naturally supports functional spatio-temporal coherence, developmental path-finding, and incremental rewiring with correlated adaptation of structure and function in cerebral plasticity and evolution.

The organizing principles of cerebral connectivity remain unclear. In the brainstem and spinal cord, fiber pathways are organized as parallel families derived from the three principal axes of embryonic development: the rostro-caudal, the medio-lateral (or proximo-distal), and the dorso-ventral (16). In the forebrain of advanced species, however, corresponding patterns of connectivity have yet to be established. Many studies of evolution, development, and gene expression point to a geometric organization of cerebral fiber pathways similar to that of the brainstem (35, 79), and functional studies (1013) also suggest that connectivity is geometrically organized. Several leading theories of cerebral function (1417) propose geometric organization at multiple scales. However, high-resolution studies of cerebral connectivity with tract tracers have given only limited evidence of geometric organization (1012, 18, 19).

A challenge in the investigation of cerebral structure and connectivity can be traced to the common occurrence of distinct pathways within the same small volumes of tissue, or “path crossing.” Crossing is a pervasive feature of brain structure and may be essential for efficient connectivity (20, 21). Owing to crossing, the mapping of connectivity must untangle pathways from cellular to macroscopic scales simultaneously (22, 23). This was accomplished with tract tracers methods, which are considered a gold standard (18, 19). Tracer studies inject compounds into the live brain and allow them to disperse by means of axonal transport, marking individual axons over large distances. However, these can map only a small fraction of the pathways in any single brain and are not feasible in humans. Thus, the discovery and analysis of the structural relationships between pathways—and their context within cerebral connectivity—has remained challenging (18, 19, 24).

To address these limitations, methods have been developed to map the fiber pathways of the brain through use of diffusion magnetic resonance imaging (MRI). Diffusion MRI creates multidimensional contrast that is representative of the distribution of fiber orientations at each location in the tissue (21). Though of lower resolution than tract tracing, diffusion MRI is noninvasive, applicable to humans and synoptic, and able to map the connectional anatomy of a single brain in its entirety, including spatial correlations between pathways. These correlations represent the mesoscale structure of connectivity, within the scope of which are the questions of whether cerebral pathways are discrete versus continuous and the detailed character of the spatial organization of connectivity.

To probe the spatial relations between the pathways of the brain, we analyzed path-adjacency and path-crossing in four nonhuman primate species and in humans. Diffusion spectrum MRI (DSI) was acquired in whole-brain specimens ex vivo in rhesus, owl monkey, marmoset, and the prosimian galago, and in vivo in subjects (515 directions; pathways were computed with deterministic streamline integration) (21). To demonstrate pathways' structural relationships, we augmented interactive software to compute for any path the set of all paths with which it shares one or more voxels, termed its path neighborhood.

The character of a typical cerebral path neighborhood is illustrated in Fig. 1. In Fig. 1A, an association pathway—rhesus monkey superior longitudinal fasciculus-3 (SLF3)—is identified by its termination within the parietal lobe; this is the MRI analog of a tracer injection (25). So defined, this pathway appeared as an almost isolated structure. In Fig. 1B, the path neighborhood of SLF3 was identified and was astonishingly simple. It entirely consists of a single curved two-dimensional (2D) sheet of paths, all mutually parallel, transversely oriented, and all crossing SLF3 at nearly right angles.

Fig. 1
Neighborhood structure of cerebral pathways. (A) MRI analog of tracer injection. A major association pathway of the rhesus forebrain SLF3 (blue) was identified by its termination in a selected region (yellow sphere in the parietal lobule). A co-terminus ...

To investigate neighborhood structure independent of path identifications, we adopted the following procedure: select a small region, identify its incident paths, and compute the paths incident on these paths, their neighborhood. This was performed in the rhesus frontal lobe, as illustrated in Fig. 1, C to E. The path neighborhood comprises two sets—transverse callosal paths and longitudinal paths of the cingulum and SLF1—that crossed like the warp and weft of a fabric as a near-orthogonal grid. Thus, these paths formed a single biaxial system. This pattern was typical of cerebral white matter.

The 3D structure of cerebral pathways was demonstrated through analysis of several path neighborhoods in a single region. Deep white matter of the occipital lobe—the sagittal stratum—of the rhesus monkey is shown in Fig. 2. Here, four seed regions were selected whose incident paths and their neighborhoods formed the edges and faces of a curved rectangular box, demonstrating that parallel grid structure of pathways was not limited to particular 2D surfaces but extended throughout entire 3D volumes. Paths in the third nearly orthogonal axis were also seen and no diagonal paths observed. Similar 3D organization was demonstrated in the highly curved midline system [supporting online material (SOM) text and fig. S1].

Fig. 2
Grid structure of cerebral pathways. The grid structure of the pathways of the sagittal stratum in the rhesus occipital lobe was demonstrated by means of neighborhood analysis. (A and B) Four seed regions were selected (spheres are indicated as superficial ...

In all present studies, cerebral path crossings formed well-defined 2D sheets. For example, in Fig.2B paths crossed (arrows) so as to form closed quadrilaterals that are elements of well-defined 2D surfaces. Geometrically, this configuration is highly exceptional (26, 27). Just as it is exceedingly unlikely that several points fall on a straight line, it is similarly improbable that two families of curves in 3D have any 2D surfaces in common. This sheet structure was found throughout cerebral white matter and in all species, orientations, and curvatures. Moreover, no brain pathways were observed without sheet structure. Further, because the processes of diffusion encoding, reconstruction, and tractography are purely local, limited to single or to adjacent voxels, whereas the spatial correlations entailed in this pattern were longrange and nonlinear, this structure could not be attributed to technical artifacts related to the imaging of diffusion (SOM text and fig. S2). A histological counterpart of the crossing structure of fiber pathways was observed with confocal microscopy: a tissue section through the rhesus monkey sagittal stratum cut parallel to the sheets of Fig. 2, A to C, that showed axons interweaving and crossing in two axes, which was confirmed by the orientation distribution of their spatial autocorrelation (28).

Grid structure of cerebral pathways was pervasive, coherent, and continuous with the three principal axes of development. In each region in Fig. 3A, continuous grid structure is demonstrated across several regions in the frontal lobes in rhesus monkey, including left arcuate sulcus, midline callosal region, and right central sulcus. The grid orientations in central sulcus were not parallel to gyral topography but were oblique, following spiral trajectories. This contrasted with the temporal lobes, where path and gyral orientations were closely aligned (SOM text and fig. S3A).

Fig. 3
Continuous grid structure of rhesus frontal lobes. (A) The grid structure of subcortical pathways was continuous within and between path neighborhoods. Three neighborhoods were constructed of the left arcuate and right central sulci and the midline. (Insets) ...

In rhesus deep white matter, the major frontal association pathways were parallel elements of a single grid (Fig. 3B, SOM text, and fig. S4). These pathways were not highly distinct but continuous with their neighbors via sparse interposed parallel paths. This was consistent in analysis at multiple depths (SOM text and fig. S4). Thus, major longitudinal pathways of the frontal lobe as well as longitudinal U-fiber pathways could be considered local condensations of a single system that spans the frontal lobe. The grid structure of pathways of the cerebral mantle was continuous with those of the limbic system and basal ganglia (SOM text and fig. S3B).

To investigate how pathways may change course, we analyzed a known turn in a major cerebral pathway identified in the tracer studies of Schmahmann and Pandya in rhesus monkey (17). Tracer and DSI showed bifurcation of a frontal projection pathway into transverse and dorsoventral components. In lieu of diagonals, pathways were found to bifurcate and turn between axes (SOM text and fig. S5).

To investigate grid continuity in the cerebral hemisphere, widely separated path neighborhoods were constructed in owl monkey. Grid structure was maintained at all scales, from the single voxel, to the lobe, to the hemisphere (SOM text and fig. S6). Continuity of grid structure between superficial gyral and deep cerebral pathways was demonstrated through analysis of neighborhoods at sequential depths (SOM text and fig. S4). Grid structure in all three orthogonal axes was observed in the centrum semiovale, including the longitudinal and transverse paths and dorso-ventral projection paths. This was observed in the rhesus and in the human, in vivo and ex vivo (SOM text and fig. S7). Further validation of grid structure of cerebral pathways was obtained with an alternative MRI contrast mechanism of circular diffusion contrast (SOM text and fig. S8) (29), and the recognized triaxial structure of pathways of the brainstem was shown with present methods (SOM text and fig. S9).

Strong homology of deep cerebral grid structure was found across all species studied (Fig. 4). These included the grid systems of the callosum, sagittal stratum, and supra-Sylvian pathways, as well as the crossing of the fornix and anterior commissure in all species studied ex vivo at high resolution. In the rhesus monkey, central and subcortical grid structures (Fig. 4D), including those of the major frontal sulci (principal, arcuate, central), fit together continuously like a jigsaw puzzle (SOM text and figs. S3A, S4, and S6, owl monkey). Thus, we hypothesize that the complex connectivity of the cerebral mantle represents a continuous elaboration of the simpler core.

Fig. 4
Homologous cerebral grid structure in (A) galago, (B) marmoset, (C) owl monkey, (D) rhesus monkey, and (E) human, left lateral views. Homologous grid structures including those of the corpus callosum/cingulum bundle (CC/CB); sagittal stratum and supra-Sylvian ...

We have found that the fiber pathways of the forebrain are organized as a highly curved 3D grid derived from the principal axes of development. This structure has a natural interpretation. By the Frobenius theorem, any three families of curves in 3D mutually cross in sheets if and only if they represent the gradients of three corresponding scalar functions (26, 27). Accordingly we hypothesize that the pathways of the brain follow a base-plan established by the three chemotactic gradients of early embryogenesis (30). Thus, the pathways of the mature brain presents an image of these three primordial gradients, plastically deformed by development. This could be tested by obtaining diffusion MRI during cerebral embryogenesis and assessing the derivation of pathways from the primordial directions. Grid structure should restrict and simplify axonal path-finding compared with models that allow less constrained and less correlated connectivity within and between cerebral areas. If grid structure guides connectivity similar to the lane markers in a highway, then navigation would be reduced from a general 3D problem to a far simpler question of when to exit. In the brain, fibers growing in any axis would have a choice at each moment of just the four orthogonal directions perpendicular to their course. Grid structure would increase the efficacy of path orientation as a mechanism of axonal path-finding (31, 32). Simultaneously, this structure supports incremental modification of connectivity by geometric modification within broad continuous families of parallel paths. Thus, the grid organization of cerebral pathways may represent a “default connectivity,” on which adaptation of structure and function can both occur incrementally in evolution and development, plasticity, and function.

A grid organization of cerebral pathways has been suggested in several contexts. Katz et al. (8) hypothesized that a large-scale substrate of grid organization is applicable to forebrain development. Checkerboard organization of cortical Brodmann fields has been noted in visual areas of the temporal lobe (1) and frontal motor areas (3) in monkeys. In humans, Badre and D'Esposito (13) have suggested hierarchic organization of the frontal cortex along a rostro-caudal axis.

The correlation between grid and topographic orientations observed in the rhesus temporal lobe supports Van Essen's hypothesis relating cerebral folding to fiber tension (33); however, the variable relation of fiber structure and cortical folding observed in the frontal lobe merits further investigation. More recently, Clochoux et al. have derived from its patterns of folding a spherical coordinate system for the longitude and latitude of the human cerebral cortex (34). This coordinate system seems generally congruent with that implied by the present grid structure of pathways, and investigation of their relationship may shed light on the linkage in structure and development between cerebral gray matter and white matter.

Greater differentiation between pathways was observed in the more complex rhesus brain than in simpler species, suggesting that the evolutionary emergence of discrete pathways parallels the increasing cerebral complexity in the primate lineage. Functionally, pervasive cerebral organization with parallel paths and similar lengths would naturally support neural coding via spatial and temporal coherence (16).

The grid structure of cerebral pathways has implications for brain mapping: It suggests a simplifying framework and natural coordinate system (35) for the description of brain structure, its pathways, and connectivity; simplifies and constrains models of cerebral white matter; and indicates that topographic organization is characteristic of cerebral connectivity and not limited to a few major pathways. Of concern, present findings suggest that existing MRI tractography may underestimate sharp turning (36). It also provides a means to validate MRI tractography through consistency with grid structure.

Supplementary Material

Supplemental material

Materials and Methods

SOM Text

Figs. S1 to S9


The authors thank L. C. Abbate, J. W. Belliveau, D. A. Feinberg, B. R. Rosen, S. A. Wedeen, and M. W. Weiner for reviewing this manuscript; J. D. Schmahmann, B. I. Shraiman, R. Turner, H. E. Stanley, and T. J. Brady for their comments; K. Mansfield of the New England Primate Center and L. Worthylake of the Oregon National Primate Research Center for primate specimens; C. Devitt for myelin stains; and M. P. Frosch for human specimens. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Science Foundation, the National Institute Of Mental Health, or the National Institutes of Health. This work is directly supported by grants NSF PHY-0855161, NIH R01-MH652456, P41 RR-023953, P41 RR-14075, and The Human Connectome Project U01 MH093765. V.J.W. designed the methods of MRI acquisition, reconstruction, and analysis; acquired and analyzed the data; discovered the grid structure; and wrote the paper. W.-Y.I.T. collaborated in the development of theoretical, imaging, and anatomic ideas and methods. G.D. implemented MRI acquisition techniques, including hardware and pulse sequences; optimized protocols; and acquired ex vivo MRI. R.W. implemented analysis algorithms and created their user interface. J.H.K. and D.L.R. obtained and prepared specimens, participated in analyses, and contributed to this manuscript. F.M. optimized and obtained immunofluorescent microscopy. P.H. provided in vivo human studies. All authors discussed the results and commented on the manuscript. No author has a major competing interest to declare. All human studies were obtained after signed informed consent, with review and approval by the Institutional Review Board. Tissue specimens were studied as discarded materials, with review and approval by the Institutional Subcommittee on Animal Care. Human tissue specimens were studied as deidentified discarded materials, with review and approval by the Institutional Review Board.

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