General principles of the synthesis algorithm

Computing the source and target points of an axon

The long-range axon synthesis algorithm starts by collecting the given morphologies from the given cell collection file and the morphology directory. Then the starting point of each axon are computed. These starting points are called the source points of the axons and can be given explicitly using the axon grafting points file (see also the related CLI argument).

The next step is to associate a source population to each long-range axon. These populations can also be arbitrarily associated to each axon using the axon grafting points file. If they are not given, the brain region of the atlas in which the cell is located is computed. Then a source population is randomly picked among the ones associated to this brain region (the probabilities are given in the population_probabilities.json file of the input folder).

After that, the target populations of each axon are computed. The probabilities of each target population depending on the source population are given in the projection_probabilities.json file of the input folder).

Once the target populations have been chosen, a voxell is randomly chosen in the brain region associated to each target population. Then a random shift is applied inside this voxel. This final location is called a target point of the axon.

Here is a schema of the data workflow to generate the source and target points used for synthesis:

digraph { { atlas [label="atlas"] soma_center [label="soma center"] axon_grafting_pts [label="axon grafting points", style=dotted] pop_probs [label="population probabilities"] source_pop [label="source population"] source_pt [label="source point", shape=box, style=rounded, color=blue] proj_probs [label="projection probabilities"] target_pops [label="target populations"] target_pts [label="target points", shape=box, style=rounded, color=blue] synthesis [label="Synthesis", shape=box, color=red] } soma_center -> source_pop; atlas -> source_pop; pop_probs -> source_pop; soma_center -> source_pt; axon_grafting_pts -> source_pt [style=dotted]; axon_grafting_pts -> source_pop [style=dotted]; source_pop -> target_pops; proj_probs -> target_pops; target_pops -> target_pts; atlas -> target_pts; source_pt -> synthesis target_pts -> synthesis }

Synthesizing the main trunk of an axon

The main trunk of the axon is the part the connects the source point to all the target points. The synthesis of this main trunk will be performed using a Steiner Tree algorithm. This algorithm is NP-complete, which means that is can be very long to compute a solution. In our case, we suppose that a reasonable approximation is sufficient (the code pcst_fast will be used for this task).

In order to use this approximation, the first step is to transform the problem in the Euclidean space into a graph problem. To do this, some intermediate points are created between the source and target points in order to refine the future graph. Then, for the same reason, some random points are created in the 3D bounding box containing the source and target points. After that, the Voronoï points of all the previous points are created, in order to add points with more realistic angles because the Voronoï points have angle properties close to the ones of the Steiner points. Once all the points are created, a triangulation is computed to create the edges that connect them.

The edges used to compute the Steiner Tree must have positive weights. So the first step is to consider that each edge has a weight equal to its length in the Euclidean space. Then several optional penalties can be applied to these weights in order to guide the Steiner Tree algorithm. Here are the possible penalties:

  • a penalty to increase the weights if the edge is not in the radial direction of the source point. This is to ensure that the main trunk does not deviate too much from the target point directions.

  • a penalty to increase the weights if the edge direction is not perpendicular to the depths fields direction. This is to ensure that the main trunk follows the shape of the atlas layers when it is possible.

  • a penalty to decrease the weights if the edge is located in a set of specific brain regions. This is to guide the main trunk in the preferred regions for axonal development, like the fiber tracts for example.

Once the edges all have a weight, the Steiner Tree algorithm is computed in order to selected the edges that connect the source point to all the target points using the shortest possible paths.

_images/graph_creation_solution.png

The final step to build the main trunk is to perform a guided random walk that follows the edges selected by the Steiner Tree algorithm. The random walk is tinkered in a way that ensures that the final result of this step is a main trunk whose morphometrics are realistic.

Considering preferred regions

It is possible to make the main trunk prefer some regions during the Steiner Tree process. To do this, the preferred regions are modeled by attractor points positioned in space. These attractors are used to update the weights of the edges that are used by the Steiner Tree algorithm to compute the solution which minimizes the total weight of the edges selected in the solution. In this case, the edge weights will decrease as their distance to the attractor decreases, making them more likely to be selected by the Stein Tree algorithm.

_images/graph_creation_solution_preferred_regions.png

Synthesizing the tufts of an axon

Once the main trunk of the axon has been synthesized, a tuft is grown from each target point. These tufts aims at connecting the axon to the dentrites surrounding the target points. The generation of these tufts has twow steps:

  • a barcode representing the future tuft is randomly chosen among the possible ones, based one the target population (these barcodes are given in the Clustering/tuft_properties.json file of the input folder).

  • the tuft is grown based on this barcode using the NeuroTS code.