Once the nodes in the brain are defined, the connectivity or edge structure of the network can be eastimated. Edges may be based on structural relationships, where for example bundles of axons connect brain regions, or on functional relationships, based on statistical or causal associations between functional signals.

## Structural networks

Structural networks are commonly defined by using the diffusion information available from diffusion MRI. Once the diffusion data has been gathered, major pathways within the brain can be estimated using tractography. Tractography is a method that integrates voxel-wise (3D pixels) fibre orientation estimates. A variety of aspects have been investigated using tractography and diffusion MRI, such as asymmetry of white matter structure and the evolution of the emerging path-ways and overall fibre organisation. It should be noted that, using diffusion MRI, the direction of movement is degenerate with regard to its sign. In simple terms, both going "forward" and "backwards" along the direction the anisotropy is possible.

### Compartment models

In order to infer fibre orientation, one needs to model the diffusion direction in each compartment. That means a model is fitted to the diffusion information, which tries to capture the information of the orientation of the fibres within the voxel. One of the most common models is called "ball and stick", which assumes a combination of the extra-axonal space (in form of a ball) and the intra-axonal space (in form of a stick). Therefore it models the most prominent orientation in the voxel, while accounting for the fact that a voxel is composed of more than just the fibre itself. Multiple sticks might be fitted as well, leading to more than just one principal orientation in the voxel.

### Tractography

Once the compartment models have been fitted to each voxel, pathways can be estimated using streamlines through the vector fields of diffusion directions. This process is called tractography. In the discrete case of voxels, diffusion information is averaged across a voxel and streamlines may be represented by a series of connected voxels. In general one can distinguish between deterministic and probabilistic methods.

In case of**deterministic tractography**, it is common practice to define termination criteria, such as an anisotropy or curvature threshold. Deterministic tractography then uses the voxel wise information to connect the voxels and stops, when any termination criterion is reached. This way the algorithm tries to only account for pathways with a high confidence and discards the rest.

**Probabilistic tractography** on the other hand, tries to handle uncertainties in the individual voxels mathematically by assigning probability density functions to each voxel. The streamlining process is then repeated many times, in order to sample all paths starting from a given voxel. Each path is assigned a compounded probability, thereby describing the uncertainties of individual paths. The benefit of probabilistic tractography algorithms lies in the possibility of "following" streamlines through voxels of low anisotropy. Moreover, this tractography approach is more robust to noise, as wrong pathways usually disperse quickly. Termination criteria can be very lenient or completely erased.

## Functional networks

Functional brain networks investigate the activation patterns of the brain in a time series analysis to describe statistical dependence of regions in the brain. A variety of methods can be used to infer functional connectivity, such as electroencephalography (EEG), magnetoencephalography (MEG) or functional MRI. Two regions in the brain are then assumed to be connected, if there is a statistical relationship between them. However, it is important to distinguish between statistical and causal relationships.

## Edge weights

Networks may have weights assigned to their edges, where the weight corresponds to the strength of the connection. There are a variety of methods with which the weights of an edge in the human brain can be determined, however, there is no consensus on which approach is the most appropriate. Weights in structural brain networks are associated with, for example, fractional anisotropy, fibre count or measurements of myelin content, whereas functional networks may use correlations between regions as weights.

These measurements, however, are difficult to interpret when using probabilistic tractography. Two regions within the brain may be connected by two streamlines inferred from probabilistic tractography with different probabilities. The question is then how the individual streamlines should be weighted, considering that the low probability streamline may be wrong. One way to incorporate the probability information from the probabilistic tractography with biological features, such as anisotropy, was proposed by Robinson and colleagues. Applying probabilistic tractography between two regions in the brain results in a set of sampled voxels. Subsequently each sampled voxel's anisotropy in the direction of the streamline can be weighted by the number of times the voxel was sampled, divided by the total number of samples. The sum of all the measurements in the voxels that are part of a tract can then be assigned as weight of the tract.

In case of

These measurements, however, are difficult to interpret when using probabilistic tractography. Two regions within the brain may be connected by two streamlines inferred from probabilistic tractography with different probabilities. The question is then how the individual streamlines should be weighted, considering that the low probability streamline may be wrong. One way to incorporate the probability information from the probabilistic tractography with biological features, such as anisotropy, was proposed by Robinson and colleagues. Applying probabilistic tractography between two regions in the brain results in a set of sampled voxels. Subsequently each sampled voxel's anisotropy in the direction of the streamline can be weighted by the number of times the voxel was sampled, divided by the total number of samples. The sum of all the measurements in the voxels that are part of a tract can then be assigned as weight of the tract.

© 2016 M.D. Schirmer

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