"A few open mathematical questions in neuroscience" Thibaud Taillefumier

First, I will address the problem of neural coding, which aims at deciphering the behaviorally-relevant patterns of neural activity. I will discuss the distinction between rate coding and temporal coding for idealized neurons called integrate-and-fire neurons. This distinction will lead us to consider unsolved questions about the regularity of the distribution of first-passage times and about the synchronous/asynchronous behavior of neural network dynamics.

Second, I will explore the efficient coding hypothesis, which posits that sensory systems allocate their limited range of responses to optimally encode relevant information about the environment. I will quantify the information encoding capabilities of distinct rate-coding neural populations, each with their own tuning curves, in response to a one-dimensional stimulus. Even in this simple setting, resolving optimal trade-offs between tuning specialization and population redundancy remains a theoretical challenge.