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

Joshua H. Goldwyn, Nikita S. Imennov, Michael Famulare, Eric Shea-Brown

arXiv:1009.4172v1q-bio.NCq-bio.QM

TL;DR

Markov-chain models capture channel noise but are computationally expensive, and previous work raised doubts about whether SDE approximations preserve their stochastic behavior. This paper analyzes channel-based and subunit-based SDEs and finds that channel-based models can approximate Markov-chain channel noise and spiking, whereas subunit-based models cannot. The analysis links the difference to how subunit combinations shape temporal noise correlations.

  • Problem

    Markov-chain channel-noise models are computationally expensive, while the accuracy of SDE approximations to their stochastic behavior has been questioned.

    Source: §I · pp. 1–2

  • Method

    The paper analyzes and simulates channel-based and subunit-based SDE models against Markov-chain models, including their channel-noise statistics and spiking behavior.

    Source: §I · p. 2, §1 · p. 4

  • Results

    The channel SDE reproduces Markov-chain spiking statistics much better than subunit SDEs, and temporal correlations in channel noise are critical to spike timing.

    Source: C. Simulation Results: Interspike… · p. 12, §V · p. 14

  • Takeaways & Limitations

    Accurate SDE approximations require modeling the entire channel structure because it shapes channel-noise history and temporal correlations.

    Source: §V · pp. 13–14

Abstract

from arXiv · show

The random transitions of ion channels between conducting and non-conducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling fluctuations in the states of ion channels uses continuous-time Markov chains nonlinearly coupled to a differential equation for voltage. Beginning with the work of Fox and Lu, 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 preserve 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 effect 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, and the subunit-based approaches do not correctly account for this structure. Our study therefore confirms and elucidates the findings of previous numerical investigations of subunit-based SDE models. Moreover, it presents the first 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.

I. INTRODUCTION

Channel noise arises from random channel transitions and can affect neuronal dynamics, but the commonly used SDE approximations have been questioned. This paper compares channel- and subunit-based formulations, analyzing how channel structure affects their ability to reproduce Markov-chain behavior.

II. CONDUCTANCE MODELS BASED ON ION CHANNELS AND THEIR SUBUNITS

  • The Hodgkin–Huxley model couples ion-specific conductances to a membrane current-balance equation; conductances depend on the proportion of channels that are open.Channel states reflect configurations of subunits, also called particles or gates.Source: §I · p. 1

I. INTRODUCTION

  • Channel noise can alter firing threshold, spike timing, interspike-interval statistics, stochastic resonance, and synaptic integration.It may also contribute to nervous-system variability that constrains motor and sensory fidelity and neuron miniaturization.Source: §I · p. 1

A. Markov Chain Ion Channel Model

  • In the Markov-chain model, each channel occupies a discrete state in a continuous-time chain, with transition rates determined by its current state and membrane voltage.The K+ channel has four subunits and conducts only when all are open; its population conductance depends on the fraction of open channels.Source: §I · p. 2, A. Markov Chain Ion Channel Model · p. 3

I. INTRODUCTION

  • Prior simulations found that the widely used SDE approach does not accurately reproduce Markov-chain stochastic responses, despite being orders of magnitude faster to simulate.The approach adds noise terms to equations for gating variables.Source: §I · p. 2
  • The paper compares three proposed SDEs and uses analysis and simulations to show that channel-based SDEs can approximate Markov-chain behavior, whereas subunit-based SDEs cannot.The authors attribute the difference to temporal noise correlations shaped by channel structure.Source: §I · p. 2

B. Deterministic Conductance Models

The deterministic conductance description is equivalent to the classical Hodgkin–Huxley formulation in the infinite-channel limit. For finite channel populations, a system-size expansion yields a channel-based SDE whose variables represent channel states.

B. Deterministic Conductance Models

  • With infinitely many independent channels, the open-channel fraction becomes deterministic and equals the probability that an individual channel is open.For K+ conductance, this gives gK = ḡKp4, equivalent to the classical gK = ḡKn4 formulation.Source: B. Deterministic Conductance Models · p. 3

1. Channel SDE Model

  • For finite channel populations, the system-size expansion derives a stochastic channel-state model from the master equation.Its state variables xi represent the proportions of channels with i open subunits, which fluctuate across realizations.Source: §1 · p. 4
  • The channel SDE uses channel-state proportions and was analyzed under voltage clamp and simulated to test whether it reproduces Markov-chain stochastic properties.Its construction distinguishes closed channel states, increasing system dimensionality relative to classical Hodgkin–Huxley equations.Source: §1 · p. 4

2. Subunit SDE Models

Subunit-based SDEs construct channel conductance from noisy subunit populations rather than modeling channel states directly. The resulting aggregation changes channel-noise statistics, while channel-based modeling preserves the relevant structure.

2. Subunit SDE Models

  • The subunit SDE avoids modeling distinguishable closed-channel states, but builds conductance from noisy subunit variables.The paper’s two-subunit illustration contrasts grouping subunits into channels with averaging across subunit classes before estimating channel conductance.Source: §2 · p. 4, D. The Distinction Between Subunit and… · p. 5, Figure 1 · p. 5
  • The independent-subunit model multiplies four independent subunit-population realizations, while the identical-subunit model raises one realization to the fourth power.Both converge to deterministic Hodgkin–Huxley behavior with infinitely many K+ channels.Source: §2 · p. 4

D. The Distinction Between Subunit and Channel Models

  • Channel-based SDEs can approximate Markov-chain channel noise and spiking statistics, whereas subunit-based SDEs cannot.The distinction is whether channel states are modeled before conductance is computed or subunit states are modeled first and then combined.Source: D. The Distinction Between Subunit and… · p. 5
  • Subunit averaging has the same expected open-channel fraction as channel averaging, but its variance differs because it averages over N^2 subunit pairings rather than N channels.The independent-class subunit variance is always below the Markov-chain variance when subunit open probability is in [0, 1].Source: D. The Distinction Between Subunit and… · pp. 5–6
  • Identical subunit classes produce twice the variance of the independent-subunit model, and their variance may be either below or above the Markov-chain variance.These discrepancies persist for finite channel populations but vanish as channel number tends to infinity.Source: D. The Distinction Between Subunit and… · p. 6

III. VOLTAGE CLAMP ANALYSIS OF STOCHASTIC MODELS

The voltage-clamp analysis characterizes the stationary fraction of open channels at fixed membrane potential. For statistically identical, independent channels, its distribution follows from the single-channel open probability and the channel population size.

A. Stationary Distribution

  • The analysis fixes membrane voltage and characterizes the probability distribution of the fraction of open channels.For K+ channels, the stationary probability of an open channel is the probability that all four subunits are open.Source: A. Stationary Distribution · p. 6, §1 · p. 6

1. Markov Chain Model

  • For identical independent channels under voltage clamp, the open-channel count is binomial, so the fraction’s mean is independent of channel number and its variance scales as 1/N.Figure 2 reports means and standard deviations versus clamp voltage for Na+ and K+ channels; the channel-SDE analytical results match the Markov-chain results.Source: §1 · p. 7, Figure 2 · p. 7

2. Channel SDE Model

The channel-based SDE reproduces the Markov chain model’s limiting stationary distribution, while the two subunit-based SDEs misrepresent channel-noise distributions and variance. The subunit models can match the mean, but their distributions and noise estimates diverge from the Markov chain model.

2. Channel SDE Model

  • The channel SDE analysis uses an equilibrium noise approximation, replacing diffusion-matrix state variables with their mean equilibrium values.For voltage clamp, the paper reports this approximation holds to O(1/N^2); the analysis does not constrain channel proportions to [0, 1].Source: §2 · p. 7
  • At large channel counts, the channel SDE’s stationary Gaussian matches the Markov chain model’s binomial limit in mean and variance.This is the paper’s first confirmation that the channel SDE approximates the Markov chain model.Source: §2 · p. 7

3. Subunit SDE Models

  • The independent-subunit model’s open-channel distribution is a product of four independent Gaussian subunit variables and does not approach a Gaussian at large channel counts.The authors therefore characterize it as fundamentally incompatible with the Markov chain model.Source: §3 · p. 8
  • The identical-subunit model’s open-channel distribution is a single subunit Gaussian raised to the fourth power and likewise does not approach a Gaussian.The authors also characterize this distribution as fundamentally incompatible with the Markov chain model.Source: §3 · p. 8
  • The two subunit approaches match the Markov chain model’s mean fraction of open channels at large channel counts, but describe its variance poorly.The identical-subunit model drastically underestimates channel noise for subthreshold voltages; both subunit models underestimate Na+ channel noise near the resting potential.Source: §3 · p. 8

B. Autocorrelation in Voltage Clamp

The section derives stationary voltage-clamp autocorrelations for the Markov chain model and shows that channel structure sets their temporal scales. Na+ correlations decay much faster than K+ correlations.

B. Autocorrelation in Voltage Clamp

  • The autocorrelation is treated as stationary, and the section defines it for the time series of the fraction of open channels under voltage clamp.The analysis assumes R(t) does not depend on the initial time.Source: B. Autocorrelation in Voltage Clamp · p. 8

1. Markov Chain Model

  • The Markov chain autocorrelation is obtained from the probability that an individual channel is open at both the initial and later times, using its master equation.The calculation simplifies because the channels are statistically identical and independent, and the voltage-clamp master equation is linear with constant coefficients.Source: §1 · p. 8
  • At 0 mV, Na+ channel autocorrelations decay within the first millisecond, whereas K+ channel autocorrelations persist for nearly 10 ms.The comparison is shown for a 10 µm2 membrane area, with 600 Na+ and 180 K+ channels.Source: §1 · p. 9, Figure 3 · p. 9
  • Channel structure sets the temporal profile of Markov chain channel noise: K+ channels have four time scales, while Na+ channels have seven.These arise from the five possible K+ channel configurations and eight possible Na+ channel configurations, respectively.Source: §1 · p. 9

2. Channel SDE Model

The channel SDE reproduces the Markov chain model’s voltage-clamp autocorrelation, but the subunit-based models retain channel-noise correlations for too long. Their leading-order autocorrelations lack the faster K+ channel time scales.

2. Channel SDE Model

  • The channel SDE autocorrelation is identical to the Markov chain model’s autocorrelation.The channel SDE calculation uses the same approximations that yield a multivariable Ornstein-Uhlenbeck process.Source: §2 · p. 9

3. Subunit SDE Models

  • The subunit-model autocorrelations are calculated from Ornstein-Uhlenbeck solutions, combining four independent solutions for InS and raising one solution to the fourth power for IdS.The calculations use Ito calculus and normalize by the corresponding voltage-clamp mean and variance.Source: §3 · p. 9
  • To leading order in 1/N, both subunit-based models lack the faster K+ channel time scales, so their channel-noise autocorrelations decay more slowly than the Markov chain and channel SDE autocorrelations.The longer decay is reported for both Na+ and K+ channels at 0 mV.Source: §3 · p. 9

IV. NUMERICAL SIMULATIONS

The simulations compare the Markov chain model with three SDE models across membrane areas and test voltage-clamp statistics and interspike intervals. The channel SDE is implemented with unconstrained state variables and an equilibrium-value approximation in its diffusion matrices.

IV. NUMERICAL SIMULATIONS

  • Simulations test voltage-clamp statistics and interspike intervals across membrane areas of 1, 10, and 100 µm2, with corresponding channel counts listed in Table II.The study uses Hodgkin-Huxley parameter values from Table I, with resting potential shifted to 0 mV.Source: §IV · p. 9, Table II · p. 10, Table I · p. 10

A. Methods

  • The simulations use a 0.01 ms time step and define a spike as exceeding 60 mV after remaining below that threshold for the previous 2 ms.Gaussian random numbers are generated from Mersenne Twister uniform draws using the Box-Muller method.Source: A. Methods · p. 10
  • The stochastic stimuli combine a DC current with Gaussian white noise, and the reported spike-train measures are the mean and coefficient of variation of the first 2,000 interspike intervals.The simulations vary membrane area, DC input current, and input-noise level.Source: A. Methods · p. 10

1. Markov Chain Model

  • The Markov chain model tracks channel states, with Na+ channels represented by three m subunits and one h subunit in an eight-state chain; channel counts are simulated with Gillespie’s algorithm.The fraction of open Na+ and K+ channels is used to update voltage with forward Euler.Source: §1 · p. 10

2. Channel SDE Model (IdS)

  • The channel SDE is a 12-equation system whose state variables represent proportions of channels in each subunit configuration; an exact reduction uses the sum-to-one relations.The original 14-dimensional system is reduced to 12 dimensions.Source: §2 · pp. 10–11
  • Unconstrained channel-proportion variables can leave their admissible surfaces, making states biologically meaningless and diffusion matrices not guaranteed to have real square roots.The simulations use equilibrium values in the diffusion matrices; comparison with bounded SDE solutions is left for future work.Source: §2 · p. 11

3. Subunit SDE Models

The independent-subunit and identical-subunit SDEs differ in their assumptions about subunit correlations. In voltage clamp, the channel SDE matches the Markov chain model closely, while subunit models show a high-voltage discrepancy in mean open Na+ channels.

3. Subunit SDE Models

  • Independent-subunit SDEs use multiple independent realizations, whereas identical-subunit SDEs assume perfect correlation and raise one subunit solution to the appropriate power.The gating variables represent proportions of open subunits and are constrained to remain between zero and one.Source: §3 · p. 11

B. Simulation Results: Voltage Clamp

  • In voltage clamp, simulated mean and standard deviation values generally match analytical calculations, and MC and channel-model results are virtually indistinguishable.The comparison uses a 10 µm2 membrane patch with voltage held constant.Source: B. Simulation Results: Voltage Clamp · pp. 11–12
  • The only numerical-versus-analytical deviation occurs in subunit-model mean open Na+ channels at high voltage, where a Gaussian approximation permits negative subunit values.Because simulated gating variables are bounded, the Gaussian treatment underestimates the mean open fraction; this discrepancy decreases as channel number increases.Source: B. Simulation Results: Voltage Clamp · p. 12

C. Simulation Results: Interspike Intervals

The channel SDE produces spiking statistics close to the Markov chain model, whereas subunit SDEs generally produce longer interspike intervals. Channel-model accuracy persists across tested stimuli, with some discrepancies depending on membrane area and input conditions.

C. Simulation Results: Interspike Intervals

  • MC and channel models show close quantitative agreement in mean ISI and CV, while subunit models produce dramatically longer ISIs, especially at low stimulus levels and larger membrane areas.At high current levels, all models have similar mean ISI and CV values.Source: C. Simulation Results: Interspike… · p. 12
  • Subunit SDE mean ISIs exceed MC values across stimulus conditions and membrane areas, consistent with smaller conductance variance and reduced spike rates at low stimulus levels.The smaller variance in open K+ and Na+ channel proportions was also found in the voltage-clamp analysis.Source: C. Simulation Results: Interspike… · p. 12
  • The channel SDE can approximate the MC model quantitatively even with small channel numbers, and equilibrium-noise and boundary treatments do not substantially degrade accuracy across stimuli.The system size expansion is formally valid only in the infinite-channel limit.Source: C. Simulation Results: Interspike… · p. 12
  • Channel-model mean ISIs tend to be longer at 1 and 10 µm2, but are shorter at 100 µm2 under weak or absent input; boundary treatment and equilibrium-noise approximation may contribute.The authors state that further investigation of the approximation is needed.Source: C. Simulation Results: Interspike… · p. 12
  • With noisy stimuli, MC and channel-SDE ISI statistics remain quantitatively similar, while stimulus fluctuations elicit spikes even at low or zero DC current.The differences in mean ISIs become less apparent, and the equilibrium noise approximation does not break down with rapidly fluctuating input.Source: C. Simulation Results: Interspike… · p. 12

V. DISCUSSION

The channel-based SDE can approximate Markov-chain channel noise and spiking, while subunit-based SDEs do not generally preserve the channel’s stochastic structure. The discussion connects this difference to temporal correlations and notes computational costs and model-scope limits.

V. DISCUSSION

  • Subunit-based SDEs produce weaker fluctuations and lower firing rates than the Markov-chain model, and these discrepancies persist as channel numbers increase.The literature also reports less spike variability but higher information transmission for this approach.Source: §V · p. 13
  • The channel-based SDE reproduces Markov-chain channel-noise effects in spiking even with relatively small channel populations, provided its system-size expansion is properly formulated.For constant and noisy inputs, its interspike-interval statistics were generally in close quantitative agreement with the Markov-chain model.Source: §V · pp. 13–14
  • The channel SDE requires about 25 times the computational time of the identical-subunit model, but becomes faster than the Markov-chain simulation above approximately 1200 Na+ and 360 K+ channels.Its computation time does not depend on channel count, whereas the reported Gillespie implementation took 12 times longer when channel counts increased tenfold.Source: §V · p. 14
  • Matching stationary means and variances by adjusting subunit noise is insufficient to match Markov-chain spiking statistics.The subunit models cannot reproduce the Markov-chain model’s multiple-timescale autocorrelation, while the second quasistationary model matches that correlation in voltage clamp.Source: §V · p. 14
  • Temporal correlations in channel noise critically affect spike timing, so a valid model must represent the whole channel’s dynamics, not only individual subunit kinetics.For channels with a single subunit, the channel- and subunit-based SDEs are mathematically identical, explaining why the subunit model can approximate the Markov-chain model in that case.Source: §V · p. 14
  • The authors caution that their models are not complete descriptions of channel noise and may need revision as membrane structure and dynamics are better understood.Their stated guiding principle is that approximations should incorporate channel states to capture noise state- and history-dependence.Source: §V · pp. 14–15

Appendix A: Matrices used in numerical simulations of the channel SDE model

This appendix specifies the state vectors and matrices used to simulate the channel SDE, then justifies an equilibrium-noise approximation under voltage clamp.

Appendix A: Matrices used in numerical simulations of the channel SDE model

  • The simulation matrices are defined for K+ and Na+ channel-state vectors, with channel counts denoted by N_K and N_Na and diffusion matrices represented through their square roots.The appendix gives the state vectors, transition matrices, diffusion matrices, and diagonal elements used in those matrices.Source: Appendix A · p. 16

a. Voltage Clamp

  • Under voltage clamp, a small-noise expansion shows that the equilibrium-noise approximation is an O(1/N) approximation to the long-time behavior of the subunit and channel SDEs.The derivation fixes membrane potential so the transition rates are constant, then replaces the deterministic leading-order term with its equilibrium value.Source: a. Voltage Clamp · pp. 16–17

b. Time-dependent voltage

The equilibrium-noise approximation reproduces spiking statistics accurately despite the lack of a generic separation between voltage and gating-variable relaxation times. The appendix contrasts a variance-matching quasistationary model with a second model designed to match channel autocorrelation, means, and variances.

b. Time-dependent voltage

  • Despite lacking a generic separation between voltage and gating-variable relaxation times, the equilibrium-noise approximation reproduces spiking statistics with high accuracy.The stated time-scale separation does not appear to be generic for Hodgkin-Huxley models.Source: b. Time-dependent voltage · p. 17

Appendix C: Quasistationary Models

  • The first quasistationary model adjusts subunit-SDE noise to match stationary open-channel variance, but does not satisfactorily approximate the Markov-chain model.The approach modifies the potassium subunit noise and also changes the sodium m-noise while leaving h-noise unchanged.Source: Appendix C · p. 17
  • The second quasistationary model is designed to match Markov-chain open-channel autocorrelation functions, means, and variances in voltage clamp.Its conductance combines deterministic Hodgkin-Huxley gating with a colored Gaussian process.Source: Appendix C · p. 17
  • The second model uses a non-Markovian Gaussian process to represent four distinct autocorrelation time scales in the open potassium-channel proportion.The process is constructed from the channel autocorrelation function.Source: Appendix C · p. 17
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