Mapping
Here is a small specification of how chains of layers are defined in PAM.
The order for connections between two layers is:
- Pre-Synaptic layer (Pre)
- arbitrary number of intermediate layers (A_Pre)
- Synaptic layer (S)
- arbitrary number of intermediate layers (A_Post)
- Post-synaptic layer (Post)
Here is a summary of all ways to determine the connectivity and the distances between layers
When two layers have the same internal topology (e.g. identical number and ordering of vertices and definition of quads and triangles), for any point on the first layer its corresponding position on the second layer can be directly computed. This mapping technique is useful whenever topological relations between neurons should be preserved independent from the origin and target location of axons and dendrites in space. The most obvious example for this is the mapping between photoreceptor cells in the retina to V1, where intermediate layers could be used to layout the realistic trajectory of the fibers to the visual cortex. But also the mapping of the dentate gyrus layer on a synaptic layer around CA3 in the hippocampus could make use of topological similarities to constrain the axonal projections along the septo-temporal axis.
Any point p on a layer X is mapped on another layer Y by computing the intersection between the line normal to X through the point p and layer Y. If there is no intersection, there is no connection. This mapping technique can be used when the projection direction of neurons solely depends on the layer it is located in (e.g. cortical layers). Furthermore, this mapping technique can be helpful to selectively map subareas of a layer onto certain target regions (e.g. connections from the lateral and medial entorhinal cortex to different parts of the dentate gyrus, see exemplary demonstration section).
Euclidean mapping computes for a given point p on the first layer the closest point on the next layer. Such a mapping can be useful, when the relative position of neurons on the first layer and its proximity to the target layer determine their entry direction on the target layer. This can be helpful if the curvature of layers in space do not allow a reliable mapping between layers based on normal mapping.
For a given point on the first layer, its corresponding position on the UV-map is calculated. These UV-coordinates are converted to 3d-coordinates for the second layer. This method is useful, when two layers do not have the same mesh topology, but their neurons share the same topological organization. Then, the relation between the first and the second layer is defined in UV-space.
Neurons can be distributed in a 3D volume. In some case you want to mask a part of the volume for mapping. With this mapping techniques each pre- oder post-synaptic neuron is only processed, when it is within this mask. This means, technically, this is not a real mapping but rather a masking-function.
The random mapping maps a point p on one layer to an arbitrary location on the next layer. This mapping is useful when the projection kernels of neurons are well defined while the axonal or dendritic projections through space are randomly distributed across brains.
The distance between a neurons position on various layers can be computed through several methods. The calculation differs slightly between intermediate layers and between intermediate and synaptic layers.
The distance between two points is computed according to the euclidean distance. In real networks this means that the dendrite projects from the soma directly to the synapse
The distance between the two points on both layers is the distance from the first point to the topological corresponding point on the second layer and from their to the synapse. In real neural networks, it means that the dendrite projects first to the topological homologous region on the second layer and from their along the layer to the synapse.
The normal vectir between the first and the second layer is used to determine the projection points. In UVNormal, the normal vector on the second layer is used to determine the projection point on the first layer. Then, the distance between the neuron point and the projection point on the first layer is determined on the UV surface.
The distance between the two points on both layers is the distance from the first point to the closest point on the second layer and from their to the topological corresponding position on the surface. In real neural networks, it means that the dendrite projects first directly to the layer and from their along the layer to the synapse.
Calculates the euclidean distance between the point on the second layer and the first layer. The UV surface on the first layer is used to determine the distance between the two points on the first layer
Starting with the point on the first layer, the closest distance to the second layer is determined. From their the distance on the UV-surface is used to determine the distance to the corresponding point on the second layer.