The What and Where of Adding Channel Noise to the Hodgkin-Huxley Equations

Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage—and the highly nonlinear impact of that voltage back on the currents themselves—into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.

[1]  B. Mandelbrot,et al.  RANDOM WALK MODELS FOR THE SPIKE ACTIVITY OF A SINGLE NEURON. , 1964, Biophysical journal.

[2]  L. Pinneo On noise in the nervous system. , 1966, Psychological review.

[3]  D. Gillespie Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[4]  L. Walløe,et al.  Firing behaviour in a stochastic nerve membrane model based upon the Hodgkin-Huxley equations. , 1979, Acta physiologica Scandinavica.

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

[6]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[7]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[8]  J. E. Glynn,et al.  Numerical Recipes: The Art of Scientific Computing , 1989 .

[9]  G. Ermentrout,et al.  Analysis of neural excitability and oscillations , 1989 .

[10]  Huxley Af,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve. 1952. , 1990 .

[11]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1990 .

[12]  F Bezanilla,et al.  A sodium channel gating model based on single channel, macroscopic ionic, and gating currents in the squid giant axon. , 1991, Biophysical journal.

[13]  Fox,et al.  Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  J. Rinzel,et al.  Equations for InsP3 receptor-mediated [Ca2+]i oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. , 1994, Journal of theoretical biology.

[15]  Carson C. Chow,et al.  Spontaneous action potentials due to channel fluctuations. , 1996, Biophysical journal.

[16]  J. Ruppersberg Ion Channels in Excitable Membranes , 1996 .

[17]  R. Morris Foundations of cellular neurophysiology , 1996 .

[18]  R. Fox Stochastic versions of the Hodgkin-Huxley equations. , 1997, Biophysical journal.

[19]  Idan Segev,et al.  Ion Channel Stochasticity May Be Critical in Determining the Reliability and Precision of Spike Timing , 1998, Neural Computation.

[20]  A. Alonso,et al.  Noise from voltage-gated ion channels may influence neuronal dynamics in the entorhinal cortex. , 1998, Journal of neurophysiology.

[21]  J. R. Clay,et al.  Excitability of the squid giant axon revisited. , 1998, Journal of neurophysiology.

[22]  R O Edeson,et al.  Evidence for cooperativity between nicotinic acetylcholine receptors in patch clamp records. , 2000, Biophysical journal.

[23]  J. White,et al.  Channel noise in neurons , 2000, Trends in Neurosciences.

[24]  A. Sherman,et al.  Channel sharing in pancreatic beta -cells revisited: enhancement of emergent bursting by noise. , 2000, Journal of theoretical biology.

[25]  M B Jackson,et al.  Single‐Channel Recording , 1998, Current protocols in neuroscience.

[26]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[27]  L S Liebovitch,et al.  Fractal methods to analyze ion channel kinetics. , 2001, Methods.

[28]  Peter Dayan,et al.  Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems , 2001 .

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

[30]  I. Goychuk,et al.  Stochastic resonance as a collective property of ion channel assemblies , 2001, physics/0106036.

[31]  P. Jung,et al.  Optimal intracellular calcium signaling. , 2002, Physical review letters.

[32]  José M. Casado,et al.  Synchronization of two Hodgkin–Huxley neurons due to internal noise , 2003 .

[33]  Peter F. Rowat,et al.  State-Dependent Effects of Na Channel Noise on Neuronal Burst Generation , 2004, Journal of Computational Neuroscience.

[34]  Zhonghuai Hou,et al.  Double-system-size resonance for spiking activity of coupled Hodgkin-Huxley neurons. , 2004, Chemphyschem : a European journal of chemical physics and physical chemistry.

[35]  Nicholas T. Carnevale,et al.  ModelDB: A Database to Support Computational Neuroscience , 2004, Journal of Computational Neuroscience.

[36]  Jay T. Rubinstein,et al.  Comparison of Algorithms for the Simulation of Action Potentials with Stochastic Sodium Channels , 2002, Annals of Biomedical Engineering.

[37]  Shangyou Zeng,et al.  Mechanism for neuronal spike generation by small and large ion channel clusters. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Apostolos P. Georgopoulos,et al.  Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems.Computational Neuroscience.ByPeter Dayanand, L F Abbott.Cambridge (Massachusetts): MIT Press. $50.00. xv + 460 p; ill.; index. ISBN: 0–262–04199–5. 2001. , 2004 .

[39]  Mahmut Ozer,et al.  Effect of channel noise on the time-course of recovery from inactivation of sodium channels , 2005 .

[40]  John R Clay,et al.  Axonal excitability revisited. , 2005, Progress in biophysics and molecular biology.

[41]  D. Koh,et al.  How noise and coupling induce bursting action potentials in pancreatic {beta}-cells. , 2005, Biophysical journal.

[42]  Ian C. Bruce,et al.  Implementation Issues in Approximate Methods for Stochastic Hodgkin–Huxley Models , 2007, Annals of Biomedical Engineering.

[43]  Tim Austin,et al.  The emergence of the deterministic Hodgkin--Huxley equations as a limit from the underlying stochastic ion-channel mechanism , 2006, math/0609068.

[44]  S. W. Jones,et al.  Are rate constants constant? , 2006, The Journal of physiology.

[45]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[46]  A. Aldo Faisal,et al.  Stochastic Simulations on the Reliability of Action Potential Propagation in Thin Axons , 2007, PLoS Comput. Biol..

[47]  Peter F. Rowat,et al.  Interspike Interval Statistics in the Stochastic Hodgkin-Huxley Model: Coexistence of Gamma Frequency Bursts and Highly Irregular Firing , 2007, Neural Computation.

[48]  Olli Yli-Harja,et al.  Stochastic Differential Equation Model for Cerebellar Granule Cell Excitability , 2008, PLoS Comput. Biol..

[49]  Svetlana Postnova,et al.  Propagation effects of current and conductance noise in a model neuron with subthreshold oscillations. , 2008, Mathematical biosciences.

[50]  A. Aldo Faisal,et al.  Stochastic Simulation of Neurons, Axons, and Action Potentials , 2009 .

[51]  Gabriel J. Lord,et al.  Stochastic Methods in Neuroscience , 2009 .

[52]  James P Keener,et al.  Invariant manifold reductions for Markovian ion channel dynamics , 2009, Journal of mathematical biology.

[53]  Paul J. Abbas,et al.  The Dependence of Auditory Nerve Rate Adaptation on Electric Stimulus Parameters, Electrode Position, and Fiber Diameter: A Computer Model Study , 2010, Journal of the Association for Research in Otolaryngology.

[54]  Hilary DeRemigio,et al.  Markov Chain Models of Ion Channels and Calcium Release Sites , 2009 .

[55]  Ian C. Bruce,et al.  Evaluation of Stochastic Differential Equation Approximation of Ion Channel Gating Models , 2009, Annals of Biomedical Engineering.

[56]  Jay T. Rubinstein,et al.  Stochastic Population Model for Electrical Stimulation of the Auditory Nerve , 2009, IEEE Transactions on Biomedical Engineering.

[57]  Hasan Alzubaidi,et al.  Numerical Simulations of SDEs and SPDEs From Neural Systems Using SDELab , 2009 .

[58]  J. Weiss,et al.  Irregularly appearing early afterdepolarizations in cardiac myocytes: random fluctuations or dynamical chaos? , 2010, Biophysical journal.

[59]  Pierre Giraud,et al.  Spike-Time Precision and Network Synchrony Are Controlled by the Homeostatic Regulation of the D-Type Potassium Current , 2010, The Journal of Neuroscience.

[60]  M. Dawkins The Noisy Brain: Stochastic Dynamics as a Principle of Brain Function The Noisy Brain: Stochastic Dynamics as a Principle of Brain Function. By Edmund T. Rolls & Gustavo Deco. Oxford: Oxford University Press (2010). Pp. 310. Price £37.95 hardback. , 2010, Animal Behaviour.

[61]  J E Niven,et al.  Comparison of Langevin and Markov channel noise models for neuronal signal generation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  K. Pakdaman,et al.  Fluid limit theorems for stochastic hybrid systems with application to neuron models , 2010, Advances in Applied Probability.

[63]  Cian O'Donnell,et al.  Stochastic Ion Channel Gating in Dendritic Neurons: Morphology Dependence and Probabilistic Synaptic Activation of Dendritic Spikes , 2010, PLoS Comput. Biol..

[64]  Joshua H. Goldwyn,et al.  Publisher's Note: Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons [Phys. Rev. E 83, 041908 (2011)] , 2011 .

[65]  Michele Giugliano,et al.  Accurate and Fast Simulation of Channel Noise in Conductance-Based Model Neurons by Diffusion Approximation , 2011, PLoS Comput. Biol..

[66]  Eric Shea-Brown,et al.  Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.