Mathematical representation of the structure of neuron-glia networks

Abstract

Network representations of the nervous system have been useful for the understanding of brain phenomena such as perception, motor coordination, and memory. Although brains are composed of both neurons and glial cells, neuron-glial networks have been little studied so far. Given the emergent role of glial cells in information transmission in the brain, we developed a mathematical representation for neuron-glial networks ($\Upsilon$-graph). We also defined the concepts of isomorphisms, unnested form (multidigraph) and matrix equation for $\Upsilon$-graphs. Although we found several network motives where the isomorphism between unnested forms does not guarantees the isomorphism between their respective $\Upsilon$-graphs, we found that if the matrix equations satisfy some conditions, the unnested forms isomorphism guarantees the isomorphism between $\Upsilon$-graphs. Finally, we introduced a novel approach to modeling the network shape. Our work presents a mathematical framework for working with neuron-glia networks.