A phase transition in the first passage of a Brownian process through a fluctuating boundary: implications for neural coding

Finding the first time a fluctuating quantity reaches a given boundary is a deceptively simple-looking problem of vast practical importance in physics, biology, chemistry, neuroscience, economics and industry. Problems in which the bound to be traversed is itself a fluctuating function of time include widely studied settings in neural coding, such as neuronal integrators with irregular inputs and internal noise. We show that the probability p(t) that a Gauss-Markov process will first exceed the boundary at time t suffers a phase transition as a function of the roughness of the boundary, as measured by its H\"older exponent H, with critical value Hc = 1/2. For smoother boundaries, H > 1/2, the probability density is a continuous func- tion of time. For rougher boundaries, H < 1/2, the probability is concentrated on a Cantor-like set of zero measure: the probability density becomes divergent, almost everywhere either zero or infin- ity. The critical point Hc = 1/2 corresponds to a widely-studied case in the theory of neural coding, where the external input integrated by a model neuron is a white-noise process, such as uncorrelated but precisely balanced excitatory and inhibitory inputs. We argue this transition corresponds to a sharp boundary between rate codes, in which the neural firing probability varies smoothly, and temporal codes, in which the neuron fires at sharply-defined times regardless of the intensity of internal noise.