Mathematical representation of the structure of neuron-glia networks

TL;DR

This paper introduces a mathematical framework for representing neuron-glia networks, enabling better understanding of their structure and potential functions in brain processes.

Contribution

The authors developed a novel mathematical representation called the $$-graph for neuron-glia networks, including concepts like isomorphisms and matrix equations, advancing network modeling.

Findings

Identified conditions where unnested form isomorphism guarantees $$-graph isomorphism.

Defined new concepts such as isomorphisms and matrix equations for $$-graphs.

Proposed a new approach to modeling the shape 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.