Asymptotic Formulas for Extreme Statistics of Escape Times in 1, 2 and 3-Dimensions

The first of N identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target sets the time scale of activation, which in general is much faster than the arrival to the target of a single trajectory only. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number N of i.i.d. Brownian trajectories to a small target in 1, 2 and 3-dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular biology.

[1]  D. Holcman,et al.  Estimating the synaptic current in a multiconductance AMPA receptor model. , 2010, Biophysical journal.

[2]  Zeev Schuss,et al.  On Recovering the Shape of a Domain from the Trace of the Heat Kernel , 2005, SIAM J. Appl. Math..

[3]  S B Yuste,et al.  Order statistics for d-dimensional diffusion processes. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  E. Pugh,et al.  Calcium Feedback to cGMP Synthesis Strongly Attenuates Single-Photon Responses Driven by Long Rhodopsin Lifetimes , 2013, Neuron.

[5]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[6]  B. Hille,et al.  Ionic channels of excitable membranes , 2001 .

[7]  G Bel,et al.  Crowding effects in non-equilibrium transport through nano-channels , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[8]  David Holcman,et al.  Survival probability of diffusion with trapping in cellular neurobiology. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Z. Schuss,et al.  Stochastic Narrow Escape in Molecular and Cellular Biology: Analysis and Applications , 2015 .

[10]  Alberto Rosso,et al.  Maximum of N independent Brownian walkers till the first exit from the half-space , 2010, 1004.5042.

[11]  Z. Schuss.  Theory and Applications of Stochastic Processes: An Analytical Approach , 2009 .

[12]  Z. Schuss,et al.  Diffusion escape through a cluster of small absorbing windows , 2008 .

[13]  Tom Chou,et al.  First Passage Problems in Biology , 2014, 1408.4518.

[14]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[15]  David Holcman,et al.  The Narrow Escape Problem , 2014, SIAM Rev..

[16]  Arnab K. Pal,et al.  Extreme value statistics of correlated random variables , 2014, 1406.6768.

[17]  Ralf Metzler,et al.  First passage time of N excluded-volume particles on a line. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Marie E Burns,et al.  Calcium feedback to cGMP synthesis strongly attenuates single-photon responses driven by long rhodopsin lifetimes. , 2012, Neuron.

[19]  W. Chen,et al.  The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model , 2010, SIAM J. Appl. Dyn. Syst..

[20]  David Holcman,et al.  Calcium Dynamics in Neuronal Microdomains: Modeling, Stochastic Simulations, and Data Analysis , 2014, Encyclopedia of Computational Neuroscience.

[21]  S. Redner,et al.  First invader dynamics in diffusion-controlled absorption , 2014, 1404.2181.

[22]  Y. C. Verdière,et al.  Spectre du laplacien et longueurs des géodésiques périodiques. II , 1973 .

[23]  D Holcman,et al.  Calcium dynamics in dendritic spines and spine motility. , 2004, Biophysical journal.

[24]  Gordon L. Fain,et al.  Detection of single photons by toad and mouse rods , 2013, Proceedings of the National Academy of Sciences.

[25]  Z. Schuss,et al.  The narrow escape problem for diffusion in cellular microdomains , 2007, Proceedings of the National Academy of Sciences.

[26]  Gregory Schehr,et al.  Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces , 2012, 1203.1658.

[27]  M Segal,et al.  Geometry of dendritic spines affects calcium dynamics in hippocampal neurons: theory and experiments. , 1999, Journal of neurophysiology.

[28]  Z. Schuss,et al.  Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane , 2004 .

[29]  Armin Huber,et al.  The latency of the light response is modulated by the phosphorylation state of Drosophila TRP at a specific site , 2017, Channels.

[30]  Katja Lindenberg,et al.  Order statistics for first passage times in one-dimensional diffusion processes , 1996 .

[31]  N. Rouach,et al.  Synapse Geometry and Receptor Dynamics Modulate Synaptic Strength , 2011, PloS one.

[32]  Zeev Schuss,et al.  Brownian Dynamics at Boundaries and Interfaces , 2013 .

[33]  Z. Schuss,et al.  Narrow Escape, Part III: Non-Smooth Domains and Riemann Surfaces , 2006 .