布朗过程首次通过涨落边界时的相变：对神经编码的启示
A phase transition in the first passage of a Brownian process through a fluctuating boundary: implications for neural coding
摘要：
寻找一个波动量第一次到达给定边界的时间，是一个看似简单的问题，在物理学、生物学、化学、神经科学、经济学和工业中具有巨大的实际重要性。要遍历的边界本身是时间的波动函数的问题包括神经编码中广泛研究的设置，例如具有不规则输入和内部噪声的神经元积分器。我们证明了高斯-马尔可夫过程在时间t首先超过边界的概率p（t）作为边界粗糙度的函数发生相变，由其Hölder指数H测量，临界值Hc = 1/2。对于较平滑的边界，H > 1/2，概率密度是时间的连续函数。对于较粗糙的边界，H < 1/2，概率集中在一个类似康托的零测度集上：概率密度变得发散，几乎处处为零或无穷大。临界点Hc = 1/2对应于神经编码理论中广泛研究的情况，其中模型神经元整合的外部输入是白噪声过程，例如不相关但精确平衡的兴奋性和抑制性输入。我们认为，这种过渡对应于一个尖锐的边界之间的速率代码，其中的神经放电概率变化平稳，和时间的代码，其中的神经元火灾在sharp定义的时间，无论内部噪声的强度。
英文摘要：
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 Hölder 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.
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