Evaluation of stochastic differential equation approximation of ion channel gating models

From the outer ear to the nerve: A complete computer model of the peripheral auditory system

Computer models of the individual components of the peripheral auditory system - the outer, middle, and inner ears and the auditory nerve - have been developed in the past, with varying level of detail, breadth, and faithfulness of the underlying parameters. Building on previous work, we advance the modeling of the ear by presenting a complete, physiologically justified, bottom-up computer model based on up-to-date experimental data that integrates all of these parts together seamlessly. The detailed bottom-up design of the present model allows for the investigation of partial hearing mechanisms and their defects, including genetic, molecular, and microscopic factors. Also, thanks to the completeness of the model, one can study microscopic effects in the context of their implications on hearing as a whole, enabling the correlation with neural recordings and non-invasive psychoacoustic methods. Such a model is instrumental for advancing quantitative understanding of the mechanism of hearing, for investigating various forms of hearing impairment, as well as for devising next generation hearing aids and cochlear implants.

Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor D and suggested a Langevin formulation with noise coefficient matrix S such that SS⊤=D. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this article, we present a natural 14-dimensional dynamics for the HH system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 × 28 noise coefficient matrix S. We show that (1) the corresponding 14D system of ordinary differential equations is consistent with the classical 4D representation of the HH system; (2) the 14D representation leads to a noise coefficient matrix S that can be obtained cheaply on each time step, without requiring a matrix decomposition; (3) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (4) our 14D representation (and those equivalent to it) gives the most accurate interspike interval distribution, not only with respect to moments but under both the L1 and L∞ metric-space norms; and (5) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies S with minimal loss of accuracy under both voltage- and current-clamp conditions.

Strong diffusion formulation of Markov chain ensembles and its optimal weaker reductions

Two self-contained diffusion formulations, in the form of coupled stochastic differential equations, are developed for the temporal evolution of state densities over an ensemble of Markov chains evolving independently under a common transition rate matrix. Our first formulation derives from Kurtz's strong approximation theorem of density-dependent Markov jump processes [Stoch.

PROCESS

Their Appl. 6, 223 (1978)STOPB70304-414910.1016/0304-4149(78)90020-0] and, therefore, strongly converges with an error bound of the order of lnN/N for ensemble size N. The second formulation eliminates some fluctuation variables, and correspondingly some noise terms, within the governing equations of the strong formulation, with the objective of achieving a simpler analytic formulation and a faster computation algorithm when the transition rates are constant or slowly varying. There, the reduction of the structural complexity is optimal in the sense that the elimination of any given set of variables takes place with the lowest attainable increase in the error bound. The resultant formulations are supported by numerical simulations.

Regularity of beating of small clusters of embryonic chick ventricular heart-cells: experiment vs. stochastic single-channel population model

The transmembrane potential is recorded from small isopotential clusters of 2-4 embryonic chick ventricular cells spontaneously generating action potentials. We analyze the cycle-to-cycle fluctuations in the time between successive action potentials (the interbeat interval or IBI). We also convert an existing model of electrical activity in the cluster, which is formulated as a Hodgkin-Huxley-like deterministic system of nonlinear ordinary differential equations describing five individual ionic currents, into a stochastic model consisting of a population of ∼20 000 independently and randomly gating ionic channels, with the randomness being set by a real physical stochastic process (radio static). This stochastic model, implemented using the Clay-DeFelice algorithm, reproduces the fluctuations seen experimentally: e.g., the coefficient of variation (standard deviation/mean) of IBI is 4.3% in the model vs. the 3.9% average value of the 17 clusters studied. The model also replicates all but one of several other quantitative measures of the experimental results, including the power spectrum and correlation integral of the voltage, as well as the histogram, Poincaré plot, serial correlation coefficients, power spectrum, detrended fluctuation analysis, approximate entropy, and sample entropy of IBI. The channel noise from one particular ionic current (I_Ks), which has channel kinetics that are relatively slow compared to that of the other currents, makes the major contribution to the fluctuations in IBI. Reproduction of the experimental coefficient of variation of IBI by adding a Gaussian white noise-current into the deterministic model necessitates using an unrealistically high noise-current amplitude. Indeed, a major implication of the modelling results is that, given the wide range of time-scales over which the various species of channels open and close, only a cell-specific stochastic model that is formulated taking into consideration the widely different ranges in the frequency content of the channel-noise produced by the opening and closing of several different types of channels will be able to reproduce precisely the various effects due to membrane noise seen in a particular electrophysiological preparation.

Channel noise effects on first spike latency of a stochastic Hodgkin-Huxley neuron

While it is widely accepted that information is encoded in neurons via action potentials or spikes, it is far less understood what specific features of spiking contain encoded information. Experimental evidence has suggested that the timing of the first spike may be an energy-efficient coding mechanism that contains more neural information than subsequent spikes. Therefore, the biophysical features of neurons that underlie response latency are of considerable interest. Here we examine the effects of channel noise on the first spike latency of a Hodgkin-Huxley neuron receiving random input from many other neurons. Because the principal feature of a Hodgkin-Huxley neuron is the stochastic opening and closing of channels, the fluctuations in the number of open channels lead to fluctuations in the membrane voltage and modify the timing of the first spike. Our results show that when a neuron has a larger number of channels, (i) the occurrence of the first spike is delayed and (ii) the variation in the first spike timing is greater. We also show that the mean, median, and interquartile range of first spike latency can be accurately predicted from a simple linear regression by knowing only the number of channels in the neuron and the rate at which presynaptic neurons fire, but the standard deviation (i.e., neuronal jitter) cannot be predicted using only this information. We then compare our results to another commonly used stochastic Hodgkin-Huxley model and show that the more commonly used model overstates the first spike latency but can predict the standard deviation of first spike latencies accurately. We end by suggesting a more suitable definition for the neuronal jitter based upon our simulations and comparison of the two models.

Accurate Langevin approaches to simulate Markovian channel dynamics

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

Minimal diffusion formulation of Markov chain ensembles and its application to ion channel clusters

We study ensembles of continuous-time Markov chains evolving independently under a common transition rate matrix in some finite state space. A diffusion approximation, composed of two specifically coupled Ornstein-Uhlenbeck processes in stochastic differential equation representation, is formulated to deduce how the number of chains in a given particular state evolves in time. This particular form of the formulation builds upon a theoretical argument adduced here. The formulation is minimal in the sense that it is always a two-dimensional stochastic process regardless of the state space size or the transition matrix density, and that it requires no matrix square root operations. A set of criteria, put forward here as to be necessarily captured by any consistent approximation scheme, is used together with the master equation to determine uniquely the parameter values and noise variances in the formulation. The model is applied to the gating dynamics in ion channel clusters.

Diffusion approximation-based simulation of stochastic ion channels: which method to use?

To study the effects of stochastic ion channel fluctuations on neural dynamics, several numerical implementation methods have been proposed. Gillespie's method for Markov Chains (MC) simulation is highly accurate, yet it becomes computationally intensive in the regime of a high number of channels. Many recent works aim to speed simulation time using the Langevin-based Diffusion Approximation (DA). Under this common theoretical approach, each implementation differs in how it handles various numerical difficulties—such as bounding of state variables to [0,1]. Here we review and test a set of the most recently published DA implementations (Goldwyn et al., 2011; Linaro et al., 2011; Dangerfield et al., 2012; Orio and Soudry, 2012; Schmandt and Galán, 2012; Güler, 2013; Huang et al., 2013a), comparing all of them in a set of numerical simulations that assess numerical accuracy and computational efficiency on three different models: (1) the original Hodgkin and Huxley model, (2) a model with faster sodium channels, and (3) a multi-compartmental model inspired in granular cells. We conclude that for a low number of channels (usually below 1000 per simulated compartment) one should use MC—which is the fastest and most accurate method. For a high number of channels, we recommend using the method by Orio and Soudry (2012), possibly combined with the method by Schmandt and Galán (2012) for increased speed and slightly reduced accuracy. Consequently, MC modeling may be the best method for detailed multicompartment neuron models—in which a model neuron with many thousands of channels is segmented into many compartments with a few hundred channels.

The ISI distribution of the stochastic Hodgkin-Huxley neuron

The simulation of ion-channel noise has an important role in computational neuroscience. In recent years several approximate methods of carrying out this simulation have been published, based on stochastic differential equations, and all giving slightly different results. The obvious, and essential, question is: which method is the most accurate and which is most computationally efficient? Here we make a contribution to the answer. We compare interspike interval histograms from simulated data using four different approximate stochastic differential equation (SDE) models of the stochastic Hodgkin-Huxley neuron, as well as the exact Markov chain model simulated by the Gillespie algorithm. One of the recent SDE models is the same as the Kurtz approximation first published in 1978. All the models considered give similar ISI histograms over a wide range of deterministic and stochastic input. Three features of these histograms are an initial peak, followed by one or more bumps, and then an exponential tail. We explore how these features depend on deterministic input and on level of channel noise, and explain the results using the stochastic dynamics of the model. We conclude with a rough ranking of the four SDE models with respect to the similarity of their ISI histograms to the histogram of the exact Markov chain model.

Channel-noise-induced stochastic facilitation in an auditory brainstem neuron model

Neuronal membrane potentials fluctuate stochastically due to conductance changes caused by random transitions between the open and closed states of ion channels. Although it has previously been shown that channel noise can nontrivially affect neuronal dynamics, it is unknown whether ion-channel noise is strong enough to act as a noise source for hypothesized noise-enhanced information processing in real neuronal systems, i.e., "stochastic facilitation". Here we demonstrate that biophysical models of channel noise can give rise to two kinds of recently discovered stochastic facilitation effects in a Hodgkin-Huxley-like model of auditory brainstem neurons. The first, known as slope-based stochastic resonance (SBSR), enables phasic neurons to emit action potentials that can encode the slope of inputs that vary slowly relative to key time constants in the model. The second, known as inverse stochastic resonance (ISR), occurs in tonically firing neurons when small levels of noise inhibit tonic firing and replace it with burstlike dynamics. Consistent with previous work, we conclude that channel noise can provide significant variability in firing dynamics, even for large numbers of channels. Moreover, our results show that possible associated computational benefits may occur due to channel noise in neurons of the auditory brainstem. This holds whether the firing dynamics in the model are phasic (SBSR can occur due to channel noise) or tonic (ISR can occur due to channel noise).

An Investigation of the Stochastic Hodgkin-Huxley Models Under Noisy Rate Functions

The effects of ion channel fluctuations on the transmembrane voltage activity are potentially profound in small-size excitable membrane patches. Different groups have extended Hodgkin-Huxley equations into stochastic differential equations to capture the effects of ion channel noise analytically (Fox & Lu, 1994 ; Linaro, Storace, & Giugliano, 2011 ; Güler, 2013 ). Studies have shown that the accuracy of spiking statistics by Fox and Lu's model does not match well with the corresponding statistics from the exact microscopic simulations. The models of both Linaro et al. and Güler, however, were found to produce highly accurate statistics. Here we extend the examination of these models to the case in which the rate functions for the opening and closing of gates are under the influence of noise. For that purpose, the usual rate functions are accompanied additively by Ornstein-Uhlenbeck–type stochastic angular variables. Moreover, we argue that the existence of such noise in the rate functions is a plausible physiological phenomenon for finite-size membranes. It is observed that the presence of noise in the rates is not effective on the degree of inaccuracies within the Fox and Lu model. Güler model's accuracy is found to remain high as in the case of noise free rates. But the performance of Linaro et al.’s model is seen to degrade seriously with the increasing strength of the introduced rate function noise. We attribute this failure of Linaro et al.’s model to the use of the covariance function of open channels at the steady state, in its derivation.

Stochastic Hodgkin-Huxley Equations with Colored Noise Terms in the Conductances

The excitability of cells is facilitated by voltage-gated ion channels. These channels accommodate a multiple number of gates individually. The possible impact of that gate multiplicity on the cell's function, specifically when the membrane area is of limited size, was investigated in the author's prior work (Güler, 2011 ). There, it was found that a nontrivially persistent correlation takes place between the transmembrane voltage fluctuations (also between the fluctuations in the gating variables) and the component of open channel fluctuations attributed to the gate multiplicity. This nontrivial phenomenon was found to be playing a major augmentative role for the elevation of excitability and spontaneous firing in small cells. In addition, the same phenomenon was found to be enhancing spike coherence significantly. Here we extend Fox and Lu's ( 1994 ) stochastic Hodgkin-Huxley equations by incorporating colored noise terms into the conductances there to obtain a formalism capable of capturing the addressed cross-correlations. Statistics of spike generation, spike coherence, firing efficiency, latency, and jitter from the articulated set of equations are found to be highly accurate in comparison with the corresponding statistics from the exact microscopic Markov simulations. This way, it is demonstrated vividly that our formulation overcomes the inherent inadequacy of the Fox and Lu equations. Finally, a recently proposed diffusion approximation method (Linaro, Storace, & Giugliano, 2011 ) is taken into consideration, and a discussion on its character is pursued.

Channel-based Langevin approach for the stochastic Hodgkin-Huxley neuron

Stochasticity in ion channel gating is the major source of intrinsic neuronal noise, which can induce many important effects in neuronal dynamics. Several numerical implementations of the Langevin approach have been proposed to approximate the Markovian dynamics of the Hodgkin-Huxley neuronal model. In this work an improved channel-based Langevin approach is proposed by introducing a truncation procedure to limit the state fractions in the range of [0, 1]. The truncated fractions are put back into the state fractions in the next time step for channel noise calculation. Our simulations show that the bounded Langevin approaches combined with the restored process give better approximations to the statistics of action potentials with the Markovian method. As a result, in our approach the channel state fractions are disturbed by two terms of noise: an uncorrelated Gaussian noise and a time-correlated noise obtained from the truncated fractions. We suggest that the restoration of truncated fractions is a critical process for a bounded Langevin method.

Stochastic-shielding approximation of Markov chains and its application to efficiently simulate random ion-channel gating

Markov chains provide realistic models of numerous stochastic processes in nature. We demonstrate that in any Markov chain, the change in occupation number in state A is correlated to the change in occupation number in state B if and only if A and B are directly connected. This implies that if we are only interested in state A, fluctuations in B may be replaced with their mean if state B is not directly connected to A, which shortens computing time considerably. We show the accuracy and efficacy of our approximation theoretically and in simulations of stochastic ion-channel gating in neurons.

Comparison of continuous and discrete stochastic ion channel models

The stochastic behaviour of ion channels can be described by a discrete model or by an approximate continuous approach. While the discrete approach is exact, it is also less computationally efficient, and so the continuous model is often the method of choice since it allows for incorporation into a multiscale environment. However, in recent years the accuracy of the stochastic continuous approach for calculating statistics of certain quantities in the Hodgkin-Huxley model has come into question. In this paper, we show that by correct formulation of the continuous model, the first two moments in the number of open sodium and potassium channels in the Hodgkin-Huxley model, calculated under voltage clamp conditions using the continuous approach are in good agreement with those obtained from the discrete model.

Simple, fast and accurate implementation of the diffusion approximation algorithm for stochastic ion channels with multiple states

BACKGROUND

The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies.

MAIN CONTRIBUTIONS

We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable--allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when short time steps or low channel numbers were used.

Modeling ion channel dynamics through reflected stochastic differential equations

Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks.

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.

Perturbation analysis of spontaneous action potential initiation by stochastic ion channels

A stochastic interpretation of spontaneous action potential initiation is developed for the Morris-Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently developed quasistationary perturbation method. Using the fact that the activating ionic channel's random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT.

Stochastic amplification of calcium-activated potassium currents in Ca2+ microdomains

Small conductance (SK) calcium-activated potassium channels are found in many tissues throughout the body and open in response to elevations in intracellular calcium. In hippocampal neurons, SK channels are spatially co-localized with L-Type calcium channels. Due to the restriction of calcium transients into microdomains, only a limited number of L-Type Ca(2+) channels can activate SK and, thus, stochastic gating becomes relevant. Using a stochastic model with calcium microdomains, we predict that intracellular Ca(2+) fluctuations resulting from Ca(2+) channel gating can increase SK2 subthreshold activity by 1-2 orders of magnitude. This effectively reduces the value of the Hill coefficient. To explain the underlying mechanism, we show how short, high-amplitude calcium pulses associated with stochastic gating of calcium channels are much more effective at activating SK2 channels than the steady calcium signal produced by a deterministic simulation. This stochastic amplification results from two factors: first, a supralinear rise in the SK2 channel's steady-state activation curve at low calcium levels and, second, a momentary reduction in the channel's time constant during the calcium pulse, causing the channel to approach its steady-state activation value much faster than it decays. Stochastic amplification can potentially explain subthreshold SK2 activation in unified models of both sub- and suprathreshold regimes. Furthermore, we expect it to be a general phenomenon relevant to many proteins that are activated nonlinearly by stochastic ligand release.

Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons

The random transitions of ion channels between conducting and nonconducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling the states of ion channels nonlinearly couples continuous-time Markov chains to a differential equation for voltage. Beginning with the work of R. F. Fox and Y.-N. Lu [Phys. Rev. E 49, 3421 (1994)], there have been attempts to generate simpler models that use stochastic differential equation (SDEs) to approximate the stochastic spiking activity produced by Markov chain models. Recent numerical investigations, however, have raised doubts that SDE models can capture the stochastic dynamics of Markov chain models.We analyze three SDE models that have been proposed as approximations to the Markov chain model: one that describes the states of the ion channels and two that describe the states of the ion channel subunits. We show that the former channel-based approach can capture the distribution of channel noise and its effects on spiking in a Hodgkin-Huxley neuron model to a degree not previously demonstrated, but the latter two subunit-based approaches cannot. Our analysis provides intuitive and mathematical explanations for why this is the case. The temporal correlation in the channel noise is determined by the combinatorics of bundling subunits into channels, but the subunit-based approaches do not correctly account for this structure. Our study confirms and elucidates the findings of previous numerical investigations of subunit-based SDE models. Moreover, it presents evidence that Markov chain models of the nonlinear, stochastic dynamics of neural membranes can be accurately approximated by SDEs. This finding opens a door to future modeling work using SDE techniques to further illuminate the effects of ion channel fluctuations on electrically active cells.

Accurate and fast simulation of channel noise in conductance-based model neurons by diffusion approximation

Stochastic channel gating is the major source of intrinsic neuronal noise whose functional consequences at the microcircuit- and network-levels have been only partly explored. A systematic study of this channel noise in large ensembles of biophysically detailed model neurons calls for the availability of fast numerical methods. In fact, exact techniques employ the microscopic simulation of the random opening and closing of individual ion channels, usually based on Markov models, whose computational loads are prohibitive for next generation massive computer models of the brain. In this work, we operatively define a procedure for translating any Markov model describing voltage- or ligand-gated membrane ion-conductances into an effective stochastic version, whose computer simulation is efficient, without compromising accuracy. Our approximation is based on an improved Langevin-like approach, which employs stochastic differential equations and no Montecarlo methods. As opposed to an earlier proposal recently debated in the literature, our approximation reproduces accurately the statistical properties of the exact microscopic simulations, under a variety of conditions, from spontaneous to evoked response features. In addition, our method is not restricted to the Hodgkin-Huxley sodium and potassium currents and is general for a variety of voltage- and ligand-gated ion currents. As a by-product, the analysis of the properties emerging in exact Markov schemes by standard probability calculus enables us for the first time to analytically identify the sources of inaccuracy of the previous proposal, while providing solid ground for its modification and improvement we present here.

A possible approach to understanding the neuronal bases of the computational properties of the nervous system consists of modelling its basic building blocks, neurons and synapses, and then simulating their collective activity emerging in large networks. In developing such models, a satisfactory description level must be chosen as a compromise between simplicity and faithfulness in reproducing experimental data. Deterministic neuron models – i.e., models that upon repeated simulation with fixed parameter values provide the same results – are usually made up of ordinary differential equations and allow for relatively fast simulation times. By contrast, they do not describe accurately the underlying stochastic response properties arising from the microscopical correlate of neuronal excitability. Stochastic models are usually based on mathematical descriptions of individual ion channels, or on an effective macroscopic account of their random opening and closing. In this contribution we describe a general method to transform any deterministic neuron model into its effective stochastic version that accurately replicates the statistical properties of ion channels random kinetics.

Automaticity in acute ischemia: bifurcation analysis of a human ventricular model

Acute ischemia (restriction in blood supply to part of the heart as a result of myocardial infarction) induces major changes in the electrophysiological properties of the ventricular tissue. Extracellular potassium concentration ([K(o)(+)]) increases in the ischemic zone, leading to an elevation of the resting membrane potential that creates an "injury current" (I(S)) between the infarcted and the healthy zone. In addition, the lack of oxygen impairs the metabolic activity of the myocytes and decreases ATP production, thereby affecting ATP-sensitive potassium channels (I(Katp)). Frequent complications of myocardial infarction are tachycardia, fibrillation, and sudden cardiac death, but the mechanisms underlying their initiation are still debated. One hypothesis is that these arrhythmias may be triggered by abnormal automaticity. We investigated the effect of ischemia on myocyte automaticity by performing a comprehensive bifurcation analysis (fixed points, cycles, and their stability) of a human ventricular myocyte model [K. H. W. J. ten Tusscher and A. V. Panfilov, Am. J. Physiol. Heart Circ. Physiol. 291, H1088 (2006)] as a function of three ischemia-relevant parameters [K(o)(+)], I(S), and I(Katp). In this single-cell model, we found that automatic activity was possible only in the presence of an injury current. Changes in [K(o)(+)] and I(Katp) significantly altered the bifurcation structure of I(S), including the occurrence of early-after depolarization. The results provide a sound basis for studying higher-dimensional tissue structures representing an ischemic heart.

Comparison of Langevin and Markov channel noise models for neuronal signal generation

The stochastic opening and closing of voltage-gated ion channels produce noise in neurons. The effect of this noise on the neuronal performance has been modeled using either an approximate or Langevin model based on stochastic differential equations or an exact model based on a Markov process model of channel gating. Yet whether the Langevin model accurately reproduces the channel noise produced by the Markov model remains unclear. Here we present a comparison between Langevin and Markov models of channel noise in neurons using single compartment Hodgkin-Huxley models containing either Na+ and K+, or only K+ voltage-gated ion channels. The performance of the Langevin and Markov models was quantified over a range of stimulus statistics, membrane areas, and channel numbers. We find that in comparison to the Markov model, the Langevin model underestimates the noise contributed by voltage-gated ion channels, overestimating information rates for both spiking and nonspiking membranes. Even with increasing numbers of channels, the difference between the two models persists. This suggests that the Langevin model may not be suitable for accurately simulating channel noise in neurons, even in simulations with large numbers of ion channels.