Neuron PRM : A Framework for Constructing Cortical Networks

Algorithms & Applications Group
Neuron PRM : A Framework for Constructing Cortical Networks

supported by NSF, Dept. of Education, Texas Higher Education Coordinating Board
Marco Morales, Nancy M. Amato
Project Alumni: Jyh-Ming Lien

The brain has extraordinary computational power to represent and interpret complex natural environments. The topology and geometry of the structures in the brain have a decissive role in these natural computations. NeuronPRM is a framework to construct a 3D model of a cortical network. using probabilistic roadmap methods. Although not the usual motion planning problem, our objective of building a network that encodes the pathways of the cortical network is analogous to the PRM objective of constructing roadmaps that contain a representative sample of feasible paths. We represent the network as a large-scale directed graph, and use L-systems and statistics data to `grow' neurons that are morphologically indistinguishable from real neurons. Our ultimate goal is to map and understand the connectivity and geometry of the cortical network.

Sample of synthetic neurons in NeuronPRM

NeuronPRM Architechture

We begin our modeling by partitioning the cerebral cortex into a set of finite elements (FEs). Each FE has a set of geometrical models of reconstructed neurons which are used to generate additional anatomically realistic synthetic neurons. The cortical network is completed by making connections or synapses between the neurons. The construction process borrows from PRM (Probabilistic Roadmap) techniques by using simple, local techniques to understand a much larger unknown space viewed as a robotics problem. Here, the workspace is enclosed inside the cortical model, and, within this space, points (neurons) are generated and connected. Each point is a different configuration of a tree-like robot. Unlike a traditional PRM, the tree-like robots have the ability to change shape (i.e., connectivity and number of links), so each roadmap node, like a reconfigurable robot, will represent not only a transformation of the robot's joints but also its topology (joint connectivity). In short, we are trying to solve a robotics problem with an infinite numbers of degrees of freedom.

Neuron Model Each neuron is represented as a tree of neural components. The first components are the soma and a set of arbors. Each arbor has as children a set of segments who in turn have children that are either junctions (when they branch) or termination marks (when they are terminal segments). Each segment has a list of micro-segments which may or may not have a spine where synapses take place.

Neuron-PRM (N-PRM) samples graph nodes and connects nodes using local information. Each vertex in the graph defines a configuration of a neuron and each directed edge represents an abstract connection or synapse between neurons. Here, we define an abstract synapse as a set of real synapses from segments of one neuron to another. Since actual synapses can be quite complex, these abstract synapses help us not only maintain data more easily but also provide a hierarchical representation for searching the cortical network. Following figure shows the result of N-PRM in abstract level.

Mouse Cortex Left: empt cortex with FE decomposition. Center: Filled with abstract neurons and connections Right: A closer view.

Statistically each neuron has thousands of synapses. An adult human being has more than one hundred billion neurons and even the cortex of the mouse has more than seventeen billion neurons. Based on current technology, it is not feasible to compute and store such a huge amount of synaptic connections between all possible pairs of neurons. The traditional PRM connection strategy of computing all pairwise distances and attempting to connect the k-closest nodes is not feasible for roadmaps with millions or billions of nodes, each of which consists of thousands of segments. To deal with such large numbers of complex nodes, we define simple distance metrics to reject neuron pairs and find potential synapses very quickly. most neuron pairs are quickly rejected by a filtering test which checks for intersection of their bounding volumes. Thus, detailed distance computations between the neuronal segments will only be performed for those neuron pairs that pass the bounding volume test. Three bounding volume tests are implemented. They are bounding sphere, bounding box, bounding convex hull.

Connections in FE Left: An Finite Element extracted from mouse cortex filled with abstract neural network. Center: Same network as left, but neurons are represented as bounding box. Right: This shows abstract connection between sythetic neurons.

After constructing the roadmaps with abstract synaptic connections, we applied the simple synapse discovery algorithm to identify real synapses. The running time for discovery is linear in the number of edges of the input roadmap, so the convex hull is the fastest.

We determined that the approximate convex hull bounding volume was the fastest overall (considering both the cost of the abstract and the actual synapse discovery phases); its drawback is the increased storage requirements for the convex hull as compared to the bounding sphere or box.

Bounding Volumes Left: A Granule neuron generated from L-System. Right: Bounding box and Convex Hull are generated from this neuron.

Papers

Neuron PRM: A Framework for Constructing Cortical Networks, Jyh-Ming Lien, Marco Morales, Nancy M. Amato, Neurocomputing, 52-54(28):191-197, Jun 2003. Also, Technical Report, TR01-002, Parasol Laboratory, Department of Computer Science, Texas A&M University, Oct 2001.
Journal(ps, pdf, abstract) Technical Report(ps, pdf, abstract)

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