Stochastically Gating Ion Channels Enable Patterned Spike Firing through Activity-Dependent Modulation of Spike Probability

The transformation of synaptic input into patterns of spike output is a fundamental operation that is determined by the particular complement of ion channels that a neuron expresses. Although it is well established that individual ion channel proteins make stochastic transitions between conducting and non-conducting states, most models of synaptic integration are deterministic, and relatively little is known about the functional consequences of interactions between stochastically gating ion channels. Here, we show that a model of stellate neurons from layer II of the medial entorhinal cortex implemented with either stochastic or deterministically gating ion channels can reproduce the resting membrane properties of stellate neurons, but only the stochastic version of the model can fully account for perithreshold membrane potential fluctuations and clustered patterns of spike output that are recorded from stellate neurons during depolarized states. We demonstrate that the stochastic model implements an example of a general mechanism for patterning of neuronal output through activity-dependent changes in the probability of spike firing. Unlike deterministic mechanisms that generate spike patterns through slow changes in the state of model parameters, this general stochastic mechanism does not require retention of information beyond the duration of a single spike and its associated afterhyperpolarization. Instead, clustered patterns of spikes emerge in the stochastic model of stellate neurons as a result of a transient increase in firing probability driven by activation of HCN channels during recovery from the spike afterhyperpolarization. Using this model, we infer conditions in which stochastic ion channel gating may influence firing patterns in vivo and predict consequences of modifications of HCN channel function for in vivo firing patterns.     

Effects of stochastic channel gating and distribution on the cardiac action potential

Ion channels exhibit stochastic conformational changes determining their gating behavior. In addition, the process of protein turnover leads to a natural variability of the number of membrane and gap junctional channels. Nevertheless, in computational models, these two aspects are scarcely considered and their impacts are largely unknown. We investigated the effects of stochastic current fluctuations and channel distributions on action potential duration (APD), intercellular conduction delays (ICDs) and conduction blocks using a modified ventricular cell model (Rudy et al.) with Markovian formulations of the principal ion currents (to simulate their stochastic open-close gating behavior) and with channel counts drawn from Poisson distributions (to simulate their natural variability). In single cells, APD variability (coefficient of variation: 1.6% at BCL=1000 ms) was essentially caused by stochastic channel gating of I_Ks, persistent I_Na and I_Ca,L. In cell strands, ICD variability induced by stochastic channel gating and Poissonian channel distributions was low under normal conditions. Nonetheless, at low intercellular coupling levels, Poissonian gap junctional channel distribution resulted in a large ICD variability (coefficient of variation >20%), highly heterogeneous conduction patterns and conduction blocks. Therefore, the stochastic behavior of current fluctuations and channel distributions can contribute to the heterogeneity of conduction patterns and to conduction block, as observed previously in experiments in cardiac tissue with altered intercellular coupling.     

Modelling animal growth in random environments: An application using nonparametric estimation

We study a stochastic differential equation growth model to describe individual growth in random environments. In particular, in this paper, we discuss the estimation of the drift and the diffusion coefficients using nonparametric methods for the case of nonequidistant data for several trajectories. We illustrate the methodology by using bovine growth data. Our goal is to assess: (i) if the parametric models (with specific functional forms for the drift and the diffusion coefficients) previously used by us to describe the evolution of bovine weight were adequate choices; (ii) whether some alternative specific parameterized functional forms of these coefficients might be suggested for further parametric analysis of this data.     

Stochastic Ion Channel Gating in Dendritic Neurons: Morphology Dependence and Probabilistic Synaptic Activation of Dendritic Spikes

Neuronal activity is mediated through changes in the probability of stochastic transitions between open and closed states of ion channels. While differences in morphology define neuronal cell types and may underlie neurological disorders, very little is known about influences of stochastic ion channel gating in neurons with complex morphology. We introduce and validate new computational tools that enable efficient generation and simulation of models containing stochastic ion channels distributed across dendritic and axonal membranes. Comparison of five morphologically distinct neuronal cell types reveals that when all simulated neurons contain identical densities of stochastic ion channels, the amplitude of stochastic membrane potential fluctuations differs between cell types and depends on sub-cellular location. For typical neurons, the amplitude of membrane potential fluctuations depends on channel kinetics as well as open probability. Using a detailed model of a hippocampal CA1 pyramidal neuron, we show that when intrinsic ion channels gate stochastically, the probability of initiation of dendritic or somatic spikes by dendritic synaptic input varies continuously between zero and one, whereas when ion channels gate deterministically, the probability is either zero or one. At physiological firing rates, stochastic gating of dendritic ion channels almost completely accounts for probabilistic somatic and dendritic spikes generated by the fully stochastic model. These results suggest that the consequences of stochastic ion channel gating differ globally between neuronal cell-types and locally between neuronal compartments. Whereas dendritic neurons are often assumed to behave deterministically, our simulations suggest that a direct consequence of stochastic gating of intrinsic ion channels is that spike output may instead be a probabilistic function of patterns of synaptic input to dendrites.     

On parameter estimation in population models

We describe methods for estimating the parameters of Markovian population processes in continuous time, thus increasing their utility in modelling real biological systems. A general approach, applicable to any finite-state continuous-time Markovian model, is presented, and this is specialised to a computationally more efficient method applicable to a class of models called density-dependent Markov population processes. We illustrate the versatility of both approaches by estimating the parameters of the stochastic SIS logistic model from simulated data. This model is also fitted to data from a population of Bay checkerspot butterfly (Euphydryas editha bayensis), allowing us to assess the viability of this population.     

Stochastic properties of Ca(2+) release of inositol 1,4,5-trisphosphate receptor clusters

Intracellular Ca(2+) release is controlled by inositol 1,4,5-trisphosphate (IP(3)) receptors or ryanodine receptors. These receptors are typically distributed in clusters with several or tens of channels. The random opening and closing of these channels introduces stochasticity into the elementary calcium release mechanism. Stochastic release events have been experimentally observed in a variety of cell types and have been termed sparks and puffs. We put forward a stochastic version of the Li-Rinzel model (the deactivation binding process is described by a Markovian scheme) and a computationally more efficient Langevin approach to model the stochastic Ca(2+) oscillation of single clusters. Statistical properties such as Ca(2+) puff amplitudes, lifetimes, and interpuff intervals are studied with both models and compared with experimental observations. For clusters with tens of channels, a simply decaying amplitude distribution is typically observed at low IP(3) concentration, while a single peak distribution appears at high IP(3) concentration.     

MCMC estimation of Markov models for ion channels

Ion channels are characterized by inherently stochastic behavior which can be represented by continuous-time Markov models (CTMM). Although methods for collecting data from single ion channels are available, translating a time series of open and closed channels to a CTMM remains a challenge. Bayesian statistics combined with Markov chain Monte Carlo (MCMC) sampling provide means for estimating the rate constants of a CTMM directly from single channel data. In this article, different approaches for the MCMC sampling of Markov models are combined. This method, new to our knowledge, detects overparameterizations and gives more accurate results than existing MCMC methods. It shows similar performance as QuB-MIL, which indicates that it also compares well with maximum likelihood estimators. Data collected from an inositol trisphosphate receptor is used to demonstrate how the best model for a given data set can be found in practice.     

Parameterization for In-Silico Modeling of Ion Channel Interactions with Drugs

Since the first Hodgkin and Huxley ion channel model was described in the 1950s, there has been an explosion in mathematical models to describe ion channel function. As experimental data has become richer, models have concomitantly been improved to better represent ion channel kinetic processes, although these improvements have generally resulted in more model complexity and an increase in the number of parameters necessary to populate the models. Models have also been developed to explicitly model drug interactions with ion channels. Recent models of drug-channel interactions account for the discrete kinetics of drug interaction with distinct ion channel state conformations, as it has become clear that such interactions underlie complex emergent kinetics such as use-dependent block. Here, we describe an approach for developing a model for ion channel drug interactions. The method describes the process of extracting rate constants from experimental electrophysiological function data to use as initial conditions for the model parameters. We then describe implementation of a parameter optimization method to refine the model rate constants describing ion channel drug kinetics. The algorithm takes advantage of readily available parallel computing tools to speed up the optimization. Finally, we describe some potential applications of the platform including the potential for gaining fundamental mechanistic insights into ion channel function and applications to in silico drug screening and development.     

The effects of large and small-scale random events on the synchrony of metapopulation dynamics: a theoretical analysis

We present an analysis of the conditions under which migration and global random factors may determine large scale synchrony in the dynamics of spatially structured populations. We derive an analytic approximation which describes how the desynchronizing influence of local environmental stochasticity combines with the synchronizing influences of larger scale environmental stochastic variation and migration to determine population cross correlation coefficients. Despite the simplifications made by this analysis, computer simulations show that the behaviour of more complicated models is well described by our approximation over considerable regions of parameter space. We conclude that population synchrony is largely determined by the coefficients of variation (CVs) of the local and larger scale stochastic processes, and that migration alone is only likely to maintain population synchrony when the CV of the local stochastic process is very small.     

Effective diffusion coefficients of K+ and Cl- ions in ion channel models.

We have used molecular dynamics simulations, corresponding to a total simulation time of 11 ns, to investigate the effective short-time local diffusion coefficient of potassium and chloride ions in a series of model ion channels. These models, which include channels formed by the fungal peptide alamethicin, by a synthetic leucine-serine peptide, and by the pore-lining M2 helix bundle of the nicotinic acetylcholine receptor, have a range of different secondary structures, diameters and hydrophobicities. We find that the diffusion coefficients of both ions are appreciably reduced in the narrower channels, the extent of the reduction being similar for both the anionic and cationic species. This suggests that a difference in mobility cannot be the source of the ion selectivity exhibited by some of the channels (for example, the leucine-serine peptide). We find no evidence for a reduction in mobility of either ion in the nAChR model. These results are broadly in line with a previous similar study of Na+ ions, and may be useful in Poisson-Nernst-Planck, Eyring rate theory or Brownian dynamics calculations of channel conductance.     

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.     

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.     

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.     

Na channel kinetics: developing models from non-stationary current fluctuations by analytic methods

In a previous study, we analyzed Na current fluctuations in myelinated axons from Xenopus laevis under voltage clamp conditions. The statistical properties were analyzed in terms of covariance functions for consecutive time intervals of varying duration during the pulse step. The underlying channel kinetics was analyzed by performing stochastic simulations of published Na channel models and calculating corresponding covariance functions. None of the models explained the fluctuation results. We therefore developed a novel minimal Na channel model that satisfactorily described the results. In the present paper, we extend the analysis and specify the possible models explaining the experimental data by using analytical methods. We derive general relations between the experimental data, including the covariance functions, and the rate constants of specific one-open-state models. A general feature of these models is that they comprise an inactivation step from the first closed state and a relatively low backward rate from the open state. This is in accordance the minimal model inferred from numerical stochastic calculations in the previous study.     

Two-category model of task allocation with application to ant societies

In many network models of interacting units such as cells or insects, the coupling coefficients between units are independent of the state of the units. Here we analyze the temporal behavior of units that can switch between two 'category' states according to rules that involve category-dependent coupling coefficients. The behaviors of the category populations resulting from the asynchronous random updating of units are first classified according to the signs of the coupling coefficients using numerical simulations. They range from isolated fixed points to lines of fixed points and stochastic attractors. These behaviors are then explained analytically using iterated function systems and birth-death jump processes. The main inspiration for our work comes from studies of non-hierarchical task allocation in, e.g., harvester ant colonies where temporal fluctuations in the numbers of ants engaged in various tasks occur as circumstances require and depend on interactions between ants. We identify interaction types that produce quick recovery from perturbations to an asymptotic behavior whose characteristics are function of the coupling coefficients between ants as well as between ants and their environment. We also compute analytically the probability density of the population numbers, and show that perturbations in our model decay twice as fast as in a model with random switching dynamics. A subset of the interaction types between ants yields intrinsic stochastic asymptotic behaviors which could account for some of the experimentally observed fluctuations. Such noisy trajectories are shown to be random walks with state-dependent biases in the 'category population' phase space. With an external stimulus, the parameters of the category-switching rules become time-dependent. Depending on the growth rate of the stimulus in comparison to its population-dependent decay rate, the dynamics may qualitatively differ from the case without stimulus. Our simple two-category model provides a framework for understanding the rich variety of behaviors in network dynamics with state-dependent coupling coefficients, and especially in task allocation processes with many tasks.     

A Stochastic Model of Conductance Transitions in Voltage-Gated IonChannels

We present a statistical physics model to describe the stochastic behaviorof ion transport and channel transitions under an applied membrane voltage.To get pertinent ideas we apply our general theoretical scheme to ananalytically tractable model of the channel with a deep binding site whichinteracts with the permeant ions electrostatically. It is found that theinteraction is modulated by the average ionic occupancy in the bindingsite, which is enhanced by the membrane voltage increases. Above acritical voltage, the interaction gives rise to a emergence of a newconducting state along with shift of S4 charge residues in the channel.This exploratory study calls for further investigations to correlate thecomplex transition behaviors with a variety of ion channels, withparameters in the model, potential energy parameters, voltage, and ionicconcentration.     

Internal motions in proteins and gating kinetics of ionic channels.

Single-channel current recordings have revealed a complex kinetic behavior of ionic channels. Many channels exhibit closed-time distributions in which long waiting times occur with a much higher frequency than predicted by a simple exponential decay function. In this paper a model for opening-closing transitions that accounts for internal motions in the protein matrix is discussed. The model is based on the notion that the transition between a conductive and a nonconductive state of the channel represents a local process in the protein, such as the movement of a small segment of a peptide chain or the rotation of a single amino-acid residue. When the blocking group moves into the ion pathway, a structural defect is created consisting in a region of loose packing and/or poor hydrogen bonding. By rearrangements of neighboring groups, the defect may migrate within the protein matrix, carrying out a kind of random walk. Once the defect has moved away from the site where it was formed, a transition back to the open state of the channel is possible only when the defect has returned by chance to the original position. The kinetic properties of this model are analyzed by stochastic simulation of defect diffusion in a small domain of the protein. With a suitable choice of domain size and diffusion rate, the model is found to predict closed-time distributions that agree with experimental observations.     

Dynamic properties of Na+ ions in models of ion channels: a molecular dynamics study.

We present simulation results for the effective diffusion coefficients of a sodium ion in a series of model ion channels of different diameters and hydrophobicities, including models of alamethicin, a leucine-serine peptide, and the M2 helix bundle of the nicotinic acetylcholine receptor. The diffusion coefficient, which in the simulations has a value of 0.15(2) A2ps-1 in bulk water, is found to be reduced to as little as 0.02(1) A2ps-1 in the narrower channels, and to about 0.10(5) A2ps-1 in wider channels such as the nicotinic acetylcholine receptor. It is anticipated that this work will be useful in connection with calculations of channel conductivity using such techniques as the Poisson-Nernst-Planck equation, Eyring rate theory, or Brownian dynamics.     

Application of oscillating potentials to the Shaker potassium channel

Nonequilibrium response spectroscopy (NRS) has been proposed recently to complement standard electrophysiological techniques used to investigate ion channels. It involves application of rapidly oscillating potentials that drive the ion channel ensemble far from equilibrium. It is argued that new, so far undiscovered features of ion channel gating kinetics may become apparent under such nonequilibrium conditions. In this paper we explore the possibility of using regular, sinusoidal voltages with the NRS protocols to facilitate Markov model selection for ion channels. As a test case we consider the Shaker potassium channel for which various Markov models have been proposed recently. We concentrate on certain classes of such models and show that while some models might be virtually indistinguishable using standard methods, they show marked differences when driven with an oscillating voltage. Model currents are compared to experimental data obtained for the Shaker K+ channel expressed in mammalian cells (tsA 201).