Modulated escape from a metastable state driven by colored noise.

Many phenomena in nature are described by excitable systems driven by colored noise. The temporal correlations in the fluctuations hinder an analytical treatment. We here present a general method of reduction to a white-noise system, capturing the color of the noise by effective and time-dependent boundary conditions. We apply the formalism to a model of the excitability of neuronal membranes, the leaky integrate-and-fire neuron model, revealing an analytical expression for the linear response of the system valid up to moderate frequencies. The closed form analytical expression enables the characterization of the response properties of such excitable units and the assessment of oscillations emerging in networks thereof.

[1]  Eric Shea-Brown,et al.  Correlation and synchrony transfer in integrate-and-fire neurons: basic properties and consequences for coding. , 2008, Physical review letters.

[2]  Doering,et al.  Bistability driven by weakly colored Gaussian noise: The Fokker-Planck boundary layer and mean first-passage times. , 1987, Physical review letters.

[3]  Brent Doiron,et al.  Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Nicolas Brunel,et al.  Dynamics of the Firing Probability of Noisy Integrate-and-Fire Neurons , 2002, Neural Computation.

[5]  Mantegna,et al.  Experimental investigation of resonant activation , 2000, Physical review letters.

[6]  H. Nyquist.  Thermal Agitation of Electric Charge in Conductors , 1928 .

[7]  H Markram,et al.  Dynamics of population rate codes in ensembles of neocortical neurons. , 2004, Journal of neurophysiology.

[8]  S. Sharma,et al.  The Fokker-Planck Equation , 2010 .

[9]  Tobias C. Potjans,et al.  The Cell-Type Specific Cortical Microcircuit: Relating Structure and Activity in a Full-Scale Spiking Network Model , 2012, Cerebral cortex.

[10]  Doering et al. reply. , 1988, Physical review letters.

[11]  Zhukov,et al.  Dynamics of activated escape and its observation in a semiconductor laser , 2000, Physical review letters.

[12]  U. Frisch,et al.  Intermittency in nonlinear dynamics and singularities at complex times , 1981 .

[13]  Xiao-Jing Wang,et al.  What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance. , 2003, Journal of neurophysiology.

[14]  Eric Shea-Brown,et al.  Impact of Network Structure and Cellular Response on Spike Time Correlations , 2011, PLoS Comput. Biol..

[15]  Bernardo Spagnolo,et al.  Suppression of timing errors in short overdamped josephson junctions. , 2004, Physical review letters.

[16]  Jaime de la Rocha,et al.  Supplementary Information for the article ‘ Correlation between neural spike trains increases with firing rate ’ , 2007 .

[17]  P. Grigolini.  The projection approach to the problem of colored noise , 1986 .

[18]  P. Jung,et al.  Colored Noise in Dynamical Systems , 2007 .

[19]  Abbott,et al.  Asynchronous states in networks of pulse-coupled oscillators. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Benjamin Lindner,et al.  Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity , 2014, Front. Comput. Neurosci..

[21]  J. M. Sancho,et al.  Analytical and numerical studies of multiplicative noise , 1982 .

[22]  Frances S. Chance,et al.  Effects of synaptic noise and filtering on the frequency response of spiking neurons. , 2001, Physical review letters.

[23]  J. Johnson.  Thermal Agitation of Electricity in Conductors , 1928 .

[24]  H. Risken.  Fokker-Planck Equation , 1996 .

[25]  Mantegna,et al.  Noise enhanced stability in an unstable system. , 1996, Physical review letters.

[26]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[27]  Bruce J. West,et al.  Finite correlation time effects in nonequilibrium phase transitions , 1984, Physica A: Statistical Mechanics and its Applications.

[28]  Andreas Schulze-Bonhage,et al.  Movement related activity in the high gamma range of the human EEG , 2008, NeuroImage.

[29]  Farzad Farkhooi,et al.  Renewal Approach to the Analysis of the Asynchronous State for Coupled Noisy Oscillators. , 2015, Physical review letters.

[30]  B. Spagnolo,et al.  Stability measures in metastable states with Gaussian colored noise. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Peter Hänggi,et al.  Ion channel gating: A first-passage time analysis of the Kramers type , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[32]  P. Hagan,et al.  Colored noise and a characteristic level crossing problem , 1998 .

[33]  H. Kramers.  Brownian motion in a field of force and the diffusion model of chemical reactions , 1940 .

[34]  D. Hansel,et al.  How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs , 2003, The Journal of Neuroscience.

[35]  Benjamin Lindner,et al.  Statistical structure of neural spiking under non-Poissonian or other non-white stimulation , 2015, Journal of Computational Neuroscience.

[36]  Nicolas Brunel,et al.  Fast Global Oscillations in Networks of Integrate-and-Fire Neurons with Low Firing Rates , 1999, Neural Computation.

[37]  Virginia Giorno,et al.  AN OUTLINE OF THEORETICAL AND ALGORITHMIC APPROACHES TO FIRST PASSAGE TIME PROBLEMS WITH APPLICATIONS TO BIOLOGICAL MODELING , 1999 .

[38]  Andrey L. Pankratov,et al.  Influence of thermal fluctuations on time characteristics of a single Josephson element with high damping exact solution , 1996 .

[39]  E. Niebur,et al.  Neural Correlates of High-Gamma Oscillations (60–200 Hz) in Macaque Local Field Potentials and Their Potential Implications in Electrocorticography , 2008, The Journal of Neuroscience.

[40]  M. J. Richardson,et al.  Rate response of neurons subject to fast or frozen noise: from stochastic and homogeneous to deterministic and heterogeneous populations. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[42]  Fox,et al.  Functional-calculus approach to stochastic differential equations. , 1986, Physical review. A, General physics.

[43]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[44]  Riccardo Mannella,et al.  Noise in Nonlinear Dynamical Systems , 2009 .

[45]  R. Stein.  Some models of neuronal variability. , 1967, Biophysical journal.

[46]  Frank Moss,et al.  Bistability driven by colored noise : theory and experiment. , 1985 .

[47]  Néstor Parga,et al.  Auto- and crosscorrelograms for the spike response of leaky integrate-and-fire neurons with slow synapses. , 2006, Physical review letters.

[48]  Benjamin Lindner,et al.  Interspike interval statistics of neurons driven by colored noise. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  Comment on "Bistability driven by weakly colored Gaussian noise: The Fokker-Planck boundary layer and mean first-passage times" , 1988, Physical review letters.

[50]  P. Hänggi,et al.  Reaction-rate theory: fifty years after Kramers , 1990 .

[51]  Moritz Helias,et al.  A unified view on weakly correlated recurrent networks , 2013, Front. Comput. Neurosci..

[52]  S. Swain.  Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences , 1984 .

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

[54]  A. Siegert.  On the First Passage Time Probability Problem , 1951 .

[55]  Nicolas Brunel,et al.  Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons , 2000, Journal of Computational Neuroscience.

[56]  L Schimansky-Geier,et al.  Transmission of noise coded versus additive signals through a neuronal ensemble. , 2001, Physical review letters.