Understanding the relationship between the dynamical rules of a single neuron and the collective behavior of a neural network remains one of the most broad and complex problems in theoretical neuroscience. One approach to this problem is the conception of functional, rather than physical, geometries.
A torus (i.e. donut) is a morphism of the periodic 2-D metric space that constitutes an abstract space for the neural network depicted here. The mapping of neurons onto the donut is functional, not physical, because each neuron is placed closer to neurons with which it is synaptically connected, not necessarily to neurons that are nearby in the brain.
The colors depicted here correspond to the voltage of the neuron at a given abstract point. This functional donut mapping allows us to recognize an emergent global organization, which arises despite mere local connectivity between neurons.
Initializing in a random state (top left), we see seemingly chaotic activity wave behavior (top right), and eventually diverse instances of globally ordered and stable wave-like phenomena (bottom four) All six simulations shown here have identical parameters; they differ only in random initialization voltages. I originally developed this software to model rat hippocampus field CA3, which may allow the rat to understand its location in space as it moves through multiple environments. The network topology is maintained, but some parameters in the depicted simulations have been altered to values unusable for position integration. I would like to acknowledge Professor Carlos Brody (Neuroscience) for his guidance.