title: A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding.
abstract: 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 Hc = 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 Hc = 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.
authors.affiliation: Laboratory of Mathematical Physics, The Rockefeller University, New York, NY 10065, USA.
authors.forename: Thibaud
authors.initials: T
authors.lastname: Taillefumier
authors.forename: Marcelo O
authors.initials: MO
authors.lastname: Magnasco
mesh_terms: D003198:Computer Simulation; D004058:Diffusion; D008954:Models, Biological; D009474:Neurons / Q000378:metabolism* / Q000502:physiology; D013269:Stochastic Processes; D009435:Synaptic Transmission / Q000502:physiology*; D013997:Time Factors
references.citation: Bull Math Biol. 2008 Jul;70(5):1425-49
references.pmid: 18506541
references.citation: J Neurophysiol. 2001 Oct;86(4):1858-68
references.pmid: 11600645
references.citation: Nat Rev Neurosci. 2003 Sep;4(9):739-51
references.pmid: 12951566
references.citation: Proc Natl Acad Sci U S A. 2002 Mar 19;99(6):3552-6
references.pmid: 11891285
references.citation: Proc Natl Acad Sci U S A. 2000 May 9;97(10):5557-61
references.pmid: 10792057
references.citation: Biol Cybern. 2000 Oct;83(4):327-40
references.pmid: 11039698
references.citation: Nature. 2007 Nov 1;450(7166):77-80
references.pmid: 17972880
references.citation: J Neurosci. 2006 Nov 22;26(47):12206-18
references.pmid: 17122045
references.citation: J Chem Phys. 2011 Dec 28;135(24):245102
references.pmid: 22225187
references.citation: Neural Comput. 2011 Feb;23(2):421-34
references.pmid: 21105825
references.citation: Phys Rev Lett. 2004 Mar 19;92(11):118701
references.pmid: 15089179
references.citation: Nat Rev Neurosci. 2008 Feb;9(2):97-107
references.pmid: 18200026
references.citation: Neural Comput. 2003 Feb;15(2):279-308
references.pmid: 12590808
references.citation: Neural Comput. 2003 Aug;15(8):1789-807
references.pmid: 14511513
references.citation: Science. 1995 Jun 9;268(5216):1503-6
references.pmid: 7770778
references.citation: J Neurosci. 2004 Feb 4;24(5):1159-72
references.pmid: 14762134
references.citation: Network. 2001 May;12(2):215-33
references.pmid: 11405423
references.citation: Neural Comput. 2003 Aug;15(8):1715-49
references.pmid: 14511510
references.citation: J Dairy Sci. 2005 Feb;88(2):834-41
references.pmid: 15653551
references.citation: Kybernetik. 1971 Jun;8(6):214-23
references.pmid: 5090384
references.citation: Neural Comput. 2011 Sep;23(9):2169-208
references.pmid: 21671786
references.citation: Cereb Cortex. 2003 Nov;13(11):1257-69
references.pmid: 14576217
references.citation: Neural Comput. 2010 Oct;22(10):2558-85
references.pmid: 20608869
references.citation: Theor Popul Biol. 1972 Dec;3(4):406-33
references.pmid: 4667096