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Conceived and designed the experiments: PH MK XG. Performed the experiments: PH MK XG. Analyzed the data: PH MK XG PT VW RM JT. Contributed reagents/materials/analysis tools: PH. Wrote the paper: PH MK.

The authors have declared that no competing interests exist.

Understanding the large-scale structural network formed by neurons is a major challenge in system neuroscience. A detailed connectivity map covering the entire brain would therefore be of great value. Based on diffusion MRI, we propose an efficient methodology to generate large, comprehensive and individual white matter connectional datasets of the living or dead, human or animal brain. This non-invasive tool enables us to study the basic and potentially complex network properties of the entire brain. For two human subjects we find that their individual brain networks have an exponential node degree distribution and that their global organization is in the form of a small world.

Biological neuronal networks, and in particular the human brain, are remarkable natural systems capable of complicated patterns of behavior. This capability seems possible due to the combination of an enormous computational capacity given by a huge amount of neurons, and a highly evolved communication network ^{11} nodes and 10^{16} edges is not only impossible to obtain with current techniques, but it also would carry a great deal of information that is irrelevant from the global organization point of view. We must therefore resort to a different level of granularity, where a node represents thousands or millions of neurons grouped together. Unfortunately, such available graphs are today limited to small post-mortem datasets (only 50–70 nodes) of rat

Crick and Jones stated that

As an illustration of our approach, we analyze the basic brain graph properties of two healthy volunteers. In particular, we study a number of distributions derived for nodes (e.g., degree, strength) and edges (weight, length). We also answer some questions related to the topology, e.g., “Is the brain network a small world?”. With technology improvements, finer resolution and a better Signal-to Noise Ratio (SNR), or a deeper analysis of the network, more complex and accurate network characteristics will be accessible, thus potentially contributing to the answers of some key questions in neuroscience.

The path from diffusion MRI to a graph mapping brain connectivity is a four step process: (1) diffusion MRI acquisition, (2) white matter tractography, (3) white matter-gray matter interface partition into Regions Of Interest (ROIs) and (4) network construction. We present a general scheme of our methodology in

First, Diffusion Spectrum MRI (DSI) is performed on a subject or sample. This acquisition provides a 3D diffusion function at every location in the brain. This data set is called a

We use Diffusion Spectrum Imaging (DSI)

In the present experiment, after having obtained the informed consent of two healthy volunteers, two data sets are acquired at 3T with an Achieva (Philips, Einthoven, The Netherlands) MRI scanner using a diffusion weighted spin echo EPI technique ^{2} ^{2} and the slice thickness 3 mm, which yields a voxel size of 2×2×3 mm^{3}. The classical DSI scheme we use goes as follows: diffusion-weighted images covering the whole brain are acquired for 515 different values of diffusion sensitizing gradient intensity and direction (i.e., different q-vectors) _{x}_{y}_{z}_{x}_{y}_{z}^{2}dρ_{i}

In

A) The result of the “diffusion MRI acquisition” step. In every voxel of a coronal slide the Orientation Density Function (ODF) captures locally for every direction the diffusion “intensity”. B) Zoom in the centrum semi-ovale C) Each ODF is replaced by a set of vectors defining its local maxima. D) Fibers are computed following the local diffusion maxima; they are uniformly initiated over the whole brain white matter. See also

Tractography is a post-processing method that based on the diffusion map, constructs 3-dimensional curves of maximal diffusion coherence. These curves, called

Detection of the directions of maximum diffusion. At each voxel, we define a set of directions of maximum diffusion as local maxima of _{i}_{j}_{i}_{j}_{i}

Fiber computation. We initiate the same number of fibers for every direction of maximum diffusion in every white matter voxel. For example, in a voxel with 2 directions, we initiate 30 fibers along each direction, total 60. The starting points are chosen at random within the voxel. Next, from each such point we begin growing a fiber in two opposite directions with a fixed step of 1 mm. On entering a new voxel, the fiber growth continues along the direction of the vector _{j}

The goal of the third step is to partition the WGM interface in a number of areas that we call Regions Of Interest (ROIs). In this step we use exclusively the 3D mask of the brain WGM interface (i.e., the cortex and the thalamus for simplicity). The ROIs should be compact and of similar surface (counted in the number of voxels), which is a non-trivial task to achieve for the complex, strongly folded shape of the brain. For instance, a straightforward approach would be to partition this interface according to some 3D regular lattice

Finally, in the fourth step, we combine the output of steps two and three and create the graph of brain connectivity. Every ROI constructed in step three becomes a node in the graph. We denote by ROI(_{e}_{e}_{e}_{f∈Fe}l_{f∈Fe}

ROI size is a parameter of our methodology. On the one hand, a natural lower limit for this size is one voxel of the WGM interface. However, we prefer to combine at least several voxels into one ROI to be sure to have a representative number of fibers connecting this ROI to the rest of the brain. On the other hand, taking ROIs that are too large results in a network of insufficient resolution and of trivially small size. In our simulations we set the ROI size to between 8 and 64 voxels of WGM interface. This results in a weighted network of between 500 to 4000 nodes representing small areas of WGM interface between ∼250 mm^{2} (64 voxels/ROI) and ∼30 mm^{2} (8 voxels/ROI), respectively. This graph has between 25'000 to 100'000 edges that represent axonal bundles of millimetric or centimetric diameter. For simplicity, in the remainder of this text we analyze graphs of about 1'000 nodes. In particular, |_{1}_{1}_{2}_{2}

Once the network constructed, several graph statistics characterizing the architecture of the network can be computed and examined.

We first turn our attention to the nodes of our graph. A basic characteristic of a node _{e:v∈e}w

A closer look at node degrees suggests that, from a developmental and energy optimization point of view, hubs do not seem to be favored. This was suggested by

The edge length

(a) The distribution of edge lengths

Having examined separately the distributions of nodes and edges, we now discuss the topology of the graph itself. An interesting question one can ask is: “Is the brain a small world?”. The more formal definition of a small world graph involves two metrics, _{v∈V}c

There are two issues that we have to address before we attempt to decide if our graph

We present the results in

The edges are chosen from the set of all edges

In order to test the robustness of our methodology and because of uncertainty about the ideal number of nodes for the presented methodology, we measured the brain network at 4 different node resolutions on data set 1 (see

While basic connectivity parameters differed slightly for data sets 1 and 2 (see

Data Set 1 | Data Set 2 | |

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The question of investigating structural network deteriorations in diseased populations like schizophrenics or demented patients is challenging and should be addressed in the future

Although our methodology yields promising results, we need to keep in mind that there are some steps prone to various kinds of noise and distortions whose effect is difficult to evaluate. First of all, we work at a given level of granularity. The spatial and angular resolution of our diffusion MRI experiment is limited, which makes it difficult to tell much about submillimetric fiber tracts and crossing axonal bundles separated with angles smaller than 20°. The ROIs have a given size, which automatically groups tens of thousands of neurons into a single node. Noise is also introduced during the MRI acquisition, and the tractography algorithm might not perfectly model the relationship between water diffusion and axonal orientation. Although all these points are constantly being improved, there will always remain a huge discrepancy between our constructed graph and the real neuronal network made of 10^{11} neurons and several orders more connections.

Nevertheless, diffusion MRI tractography is a widely used and accepted method to map axonal bundle trajectories. Furthermore it was validated experimentally to large extent in the case of DSI. First,

White matter–gray matter interfaces: Magenta = posterior part of the thalamus, blue and gray = right and left V1, orange = V2, cyan = V3, red = V5. Fibers: yellow = homotopic V1, red = V1–V2, white = V2–V3, green = V2–V5, blue = lateral geniculate body–V1. See also

In this article we have proposed a methodology for mapping networks of structural connectivity in the brain. Our approach is non-invasive, efficient, individual and of relatively high-resolution. For the first time we can globally characterize brain connectivity with individual tract properties or network statistics in an individual living subject. Based on the analysis of two healthy subjects we found that the graph of the human brain is a small world, but not a scale-free network. Large new areas of application are foreseen; in basic neuroscience our technique may contribute to the discovery of the general principles regulating communication, evolution and development of the brain; in clinical neuroscience it may shed new light into diseases of disorders that involve disruptions of anatomical brain connectivity.

The number of edges in the resulting graph as a function of the number of fibers connecting two points in the gray-white mater interface. The straight line represents the y = x relation.

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Histograms of ROI sizes for the number of ROIs ranging from N = 506 to 4052 in subject 1. One voxel translates to about 4 mm^{2}.

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The results generated for all four considered scales in subject 1. The symbols in the last two rows are (as in the main paper): blue circles-“Top-weight edges”, red triangles-“Random fibers”, and black disks-“Random graph”.

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Comparison of edge weights inside the visual system with the rest of the brain. Each box plot represents all edge weights in the brain of similar white matter length. The big black dot represents the weight of the considered connection, namely V1-V2, V2-V3, V2-V5, as well as the connections between the lateral geniculate body and V1 (LGB-V1), and between left and right V1 areas (V1^{left}-V1^{right}). Each connection is compared with the other connections in the brain of same white matter length as short connections are usually denser that long ones. The considered connections in the visual system are largely above their respective medians (horizontal line in within each box, whiskers represent 5th and 95th quantiles).

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Whole brain tractography result in subject 1.

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Partition of the white-gray matter interface in approximately 1000 ROIs.

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Connections between different visual areas.

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