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The what and where of adding channel noise to the Hodgkin-Huxley equations
Joshua H. Goldwyn, Eric Shea-Brown
TL;DR
The paper addresses how to represent stochastic ion-channel activity within the simpler Hodgkin-Huxley framework rather than using higher-dimensional kinetic equations. It reviews noise formulations and finds that accurate approximations emerge when fluctuations are introduced through conductances or fractions of open channels, while voltage-clamp-based methods have a dynamic-voltage limitation.
Problem
It has remained unclear which noise terms can incorporate channel noise accurately into the canonical Hodgkin-Huxley equations.
Method
The review compares stochastic Hodgkin-Huxley formulations with Markov-chain channel models, focusing on subunit noise and conductance-noise approaches.
Results
Conductance-noise models preserve the Hodgkin-Huxley structure and accurately approximate Markov-chain channel fluctuations, whereas subunit and voltage-clamp approaches can misestimate fluctuations.
Takeaways & Limitations
Adding noise through conductance terms or fractions of open channels provides the preferred route for incorporating channel noise into Hodgkin-Huxley equations.
Takeaways & Limitations
Voltage-clamp-based conductance-noise methods assume voltage-clamp statistics remain valid with a freely evolving membrane potential and may be less reliable during rapid action-potential voltage changes.
Abstract
from arXiv · showhide
One of the most celebrated successes in computational biology is the Hodgkin-Huxley framework for modeling electrically active cells. This framework, expressed through a set of differential equations, 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. Latter 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. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the Hodgkin-Huxley equations. 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 Hodgkin-Huxley equations. 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.
Introduction
Channel noise is a widespread source of stochasticity in electrically active cells, yet its incorporation into Hodgkin-Huxley equations has lacked clear, reliable options. The review examines simpler noise-augmented formulations and emphasizes that accurate conductance-noise methods depend critically on where noise terms are placed.
- Motivation: Channel noise can influence information processing, spike timing, firing irregularity, subthreshold dynamics, and action-potential initiation and propagation.It has been studied across auditory nerve, entorhinal cortex, cerebellar, hippocampal, and other cellular systems.
- Problem: It remains unclear how best to include channel noise in the canonical Hodgkin-Huxley equations.The central question is whether a few added noise terms can provide a simpler alternative to the higher-dimensional kinetic description.
- Reference model: The direct reference model treats each channel as an independently transitioning discrete-state process, yielding a voltage-dependent continuous-time Markov chain.For finite channel numbers, the Markov process is simulated with a Gillespie-type algorithm.
- Approach: Noise-augmented Hodgkin-Huxley equations are attractive because they could preserve conceptual and computational simplicity while linking channel fluctuations to established HH dynamics and geometry.Fox and Lu initiated this approach by deriving candidate stochastic differential equations through a system-size expansion of the Markov chain.
- Contribution: Recent methods can accurately reproduce channel fluctuations, but their success depends critically on placing noise terms as conductances in the Hodgkin-Huxley equations.The review unifies methods that provide accurate approximations to Markov-chain channel-noise models.
Stochastic versions of the Hodgkin-Huxley Equations
Stochastic HH models add channel fluctuations while preserving links to deterministic gating dynamics, but their accuracy depends critically on where noise enters the equations. Conductance-based approaches model open-channel fluctuations directly and can match Markov-chain statistics, whereas subunit and voltage-clamp-based approximations have documented limitations.
- HH baseline: The deterministic HH model uses V, m, h, and n to represent membrane voltage and aggregated fractions of open subunits regulating Na+ and K+ conductances.The combinations m3h and n4 represent mean fractions of open Na+ and K+ channels in the infinite-channel limit.
- Current noise: Current noise adds a fluctuating term directly to dV/dt, but its intensity lacks a principled determination and may not reflect voltage- or subunit-dependent channel activity.The approach can nevertheless be justified empirically in some settings.
- Subunit noise: Subunit noise perturbs aggregated open-subunit fractions, but finite-channel fluctuations in m3h and n4 need not match fluctuations in membrane-wide open-channel fractions.Because specific packages of subunits determine channel states, aggregated subunit transitions have different statistics from individual-channel transitions.
- Subunit noise: Subunit-noise approximations produce weaker conductance and voltage fluctuations, lower firing rates, and less spike variability than Markov-chain models.They also yield higher information transmission rates and inaccurate stationary open-channel distributions, with errors persisting as channel numbers increase.
- Conductance noise: Voltage-clamp-based quasistationary and effective models use voltage-dependent conductance fluctuation statistics, but applying them during spiking requires assuming their voltage-clamp dynamics remain valid.This assumption may be more plausible when voltage changes slowly than fluctuation correlation times and less reliable during rapid action potentials.
- Conductance noise: Fox and Lu’s conductance-noise formulation models fractions of channels in specific configurations and decomposes them into deterministic HH-compatible and fluctuation components.Its high-dimensional SDEs shape open-channel fluctuations without modifying the deterministic HH structure.
Comparing stochastic versions of the Hodgkin-Huxley equations: simulations
Simulations compare simplified channel-noise models against the Markov chain reference using channel-fraction statistics and interspike-interval measures. Fox and Lu’s conductance-noise model most consistently reproduces the reference, whereas the subunit model shows systematic variance and ISI discrepancies.
- Simulation design: The simulations compare Markov chain, subunit, voltage-clamp conductance, current-noise, and Fox–Lu conductance-noise models using channel statistics and spike-train ISIs.Comparisons use fixed voltage trajectories and constant-current inputs, with Euler–Maruyama for SDEs and a Gillespie-type algorithm for channel kinetics.
- Fixed-voltage trajectory: All models closely reproduce mean open-channel fractions for the fixed voltage trajectory, so those results are not plotted.
- Fixed-voltage trajectory: Fox and Lu’s conductance-noise model accurately captures Na+ variance and best matches K+ variance, while subunit and voltage-clamp models misestimate fluctuations.The voltage-clamp model fails to track Na+ variance during the rapid spike, while the subunit model underestimates subthreshold variance and overestimates spike variance.
- Spike-train statistics: For DC input, all models except the subunit model reproduce mean ISIs reasonably accurately, though current-noise and voltage-clamp methods show slight discrepancies.ISI means and coefficients of variation were estimated from repeated simulations at membrane areas of 10µm2 and 100µm2.
- Spike-train statistics: Across tested conditions, Fox and Lu’s conductance-noise model generates ISI statistics most similar to the Markov chain model.
Discussion
The review argues that stochastic HH models have regained credibility, especially when channel fluctuations are added at the conductance level, while noting that Markov chains may not remain the definitive benchmark.
- Recent analysis and testing have renewed confidence that SDE versions of the HH equations can model stochastic ion-channel activity.The authors connect this framework to questions about spike timing, reliability, propagation, and other neural dynamics.
- Conductance noise, equivalently fluctuations in open-channel fractions, is presented as the preferred way to add channel noise while preserving HH structure and approximating Markov-chain models.The Fox–Lu high-dimensional SDEs can be decomposed into the classical deterministic HH equations plus a channel-noise fluctuation component.
- The review’s simulations and discussion distinguish derived models tied to ion-channel conformational states from empirical models constructed from observable quantities.Empirical conductance fluctuations could be fitted to or validated against electrophysiological measurements without relying on a Markov-chain model.
- Markov-chain models serve as the usual validation reference, but critiques and alternative mathematical models mean they are not guaranteed to remain the gold standard.This motivates continued development of empirically grounded approaches to conductance fluctuations.
- The reviewed stochastic approaches extend the Hodgkin–Huxley conductance-based framework and can leverage existing theoretical and numerical tools for SDEs.