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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 industrial engineering. Problems in which the bound to be traversed is itself a fluctuating function of time include widely studied problems 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 Hölder exponent H. The critical value occurs when the roughness of the boundary equals the roughness of the process, so for diffusive processes the critical value is H c = 1/2. For smoother boundaries, H > 1/2, the probability density is a continuous function 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 infinity. The critical point H c = 1/2 corresponds to a widely studied case in the theory of neural coding, in which the external input integrated by a model neuron is a white-noise process, as in the case of uncorrelated but precisely balanced excitatory and inhibitory inputs. We argue that 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.first-passage time | neural code | random walk A Brownian process W(t) that starts at t = 0 from W(t = 0) = 0 will fluctuate up and down, eventually crossing the value 1 infinitely many times: for any given realization of the process W, there will be infinitely many different values of t for which W(t) = 1. Finding the very first such time,known as the "first passage" of the process through the boundary B = 1, is easier said than done, one of those classical problems whose concise statements conceal their difficulty (1-4). For general fluctuating random processes, the first-passage time problem is both extremely difficult (5-9) and highly relevant, due to its manifold practical applications: it models phenomena as diverse as the onset of chemical reactions (10-14), transitions of macromolecular assemblies (15-19), time-to-failure of a device (20)(21)(22), accumulation of evidence in neural decision-making circuits (23), the "gambler's ruin" problem in game theory (24), species extinction probabilities in ecology (25), survival probabilities of patients and disease progression (26-28), triggering of orders in the stock market (29-31), and firing of neural action potentials (32-37).Much attention has been devoted to two extensions of this basic problem. One is the first passage through a stationary boundary within a complex spatial geometry, such as diffusion in porous media or complex networks. These models are used to describe foragi...

Natural auditory scenes possess highly structured statistical regularities, which are dictated by the physics of sound production in nature, such as scale-invariance. We recently identified that natural water sounds exhibit a particular type of scale invariance, in which the temporal modulation within spectral bands scales with the centre frequency of the band. Here, we tested how neurons in the mammalian primary auditory cortex encode sounds that exhibit this property, but differ in their statistical parameters. The stimuli varied in spectro-temporal density and cyclo-temporal statistics over several orders of magnitude, corresponding to a range of water-like percepts, from pattering of rain to a slow stream. We recorded neuronal activity in the primary auditory cortex of awake rats presented with these stimuli. The responses of the majority of individual neurons were selective for a subset of stimuli with specific statistics. However, as a neuronal population, the responses were remarkably stable over large changes in stimulus statistics, exhibiting a similar range in firing rate, response strength, variability and information rate, and only minor variation in receptive field parameters. This pattern of neuronal responses suggests a potentially general principle for cortical encoding of complex acoustic scenes: while individual cortical neurons exhibit selectivity for specific statistical features, a neuronal population preserves a constant response structure across a broad range of statistical parameters.

Neural computations emerge from myriads of neuronal interactions occurring in intricate spiking networks. Due to the inherent complexity of neural models, relating the spiking activity of a network to its structure requires simplifying assumptions, such as considering models in the thermodynamic mean-field limit. In the thermodynamic mean-field limit, an infinite number of neurons interact via vanishingly small interactions, thereby erasing the finite size of interactions.To better capture the finite-size effects of interactions, we propose to analyze the activity of neural networks in the replica-mean-field limit. Replica-mean-field models are made of infinitely many replicas which interact according to the same basic structure as that of the finite network of interest.Here, we analytically characterize the stationary dynamics of an intensity-based neural network with spiking reset and heterogeneous excitatory synapses in the replica-mean-field limit. Specifically, we functionally characterize the stationary dynamics of these limit networks via ordinary differential equations derived from the Poisson Hypothesis of queuing theory. We then reduce this functional characterization to a system of self-consistency equations specifying the stationary neuronal firing rates. Of general applicability, our approach combines rate-conservation principles from point-process theory and analytical considerations from generating-function methods. We validate our approach by demonstrating numerically that replica-mean-field models better capture the dynamics of neural networks with large, sparse connections than their thermodynamic counterparts. Finally, we explain that improved performance by analyzing the neuronal rate-transfer functions, which saturate due to finite-size effects in the replica-mean-field limit.