GEOMETRIC REALIZATION OF SPARSE NEURAL CODES
Aleina Wachtel, Mohamed Omar, Nora Youngs, R. Amzi Jeffs, Natchanon Suaysom.
Harvey Mudd College, Claremont, CA.
Understanding how a neural code stores information is one of the most pressing problems in neuroscience. It is vital to study these neural codes from a mathematical perspective to gain a better understanding of the brain. The combinatorial information in a neural code can allow us to determine if the code reflects the firing behavior of neurons with convex receptive fields. Thus, we seek to determine which neural codes can be realized as convex open sets in R2. We restrict to codes showing sparse behavior, specifically codes in which no more than 2 neurons fire simultaneously, to reduce the intractability of determining which neural codes can be so realized. Using geometric properties of neural realizations, we find necessary and sufficient conditions for realizability of 2-sparse codes based on the associated co-firing graph. This yields a complete characterization of which 2-sparse codes can be realized in R3. We also exhibit several classes of codes that are always realizable in R2, and give a class of graphs never realizable in R2. With a characterization of 2-sparse neural codes, we can better comprehend the manner in which biologically viable codes are spatially represented in the brain.