In recent years, there has been increasing interest in the simulation of stochastic channel dynamics, i.e., the random opening and closing of ion channels. The stochastic Hodgkin– Huxley (HH) neuron^[1] can be accurately simulated with the Markov method or Gillespie method.^{[2– 5]} However, the computational cost of these Markov-based methods becomes very high with an increase in the number of channels. Thus, various Langevin approaches have been suggested wherein the channel variables are modulated by Gaussian noise.^{[6– 11]}

Among these approaches, Fox and Lu suggested a simple gate-based Langevin approach for a stochastic HH model.^[6] With such an approach, the stochastic dynamics of Na⁺ and K⁺ channels, consisting of several gates to control the channel opening and closing, can be calculated with the gate open fractions disturbed by Gaussian noise. The noise strength is determined by the channel number and fraction of open gates. Because of its simplicity, such a gate-based Langevin approach has been applied extensively to simulate stochastic HH channel dynamics and other channel systems.^{[12– 15]}

However, in the past few years, increasing evidence has indicated apparent inaccuracies in gate-based Langevin approaches.^{[16– 19]} Mino et al.^[16] first reported that the firing efficiency and spiking latency of the Fox– Lu gate-based approach for a neuron with 1000 Na⁺ channels responding to a pulse current differed from those for the Markov method. They suggested that, because the generation of action potentials in the HH model is highly dependent on the number of open Na⁺ channels, the number of open channels in the approach should be rounded down to the nearest integer to obtain better statistical results. Later, Bruce^[17] argued that a better treatment would be to round up the number of open Na⁺ channels to its nearest integer. Compared to rounding to the nearest integer, the rounding-down method always underestimates the Na⁺ current. Thus, statistically, the rounding-down treatment may produce decreased firing efficiency and increased spike latency.

Because each channel consists of several gates, Shuai and Jung^[18] suggested two different considerations for the gate-based approach. In detail, each K⁺ channel has 4 n-gates and each Na⁺ channel has an h-gate and 3 m-gates.^[1] For the same type of gate, the gates can be either identical (and thus be disturbed by identical Gaussian noise) or independent (and thus be disturbed by different Gaussian noises). Later, Goldwyn et al.^[8] pointed out that although the consideration of independent gates is more biologically realistic, the identical gate approach derives better statistics for action potentials than the independent gate approach for the stochastic HH neuronal model. However, Sengupta et al.^[7] showed that the identical gate approach still underestimates the channel noise, resulting in overestimation of information rates with 6000 Na⁺ and 1800 K⁺ channels.

Huang et al.^[19] proposed the rescaled gate-based approach to address the localized intracellular Ca²⁺ signals (Ca²⁺ puffs) that are released from a cluster of Ca²⁺ channels in the ER membrane. It has been shown that, by properly introducing a factor to rescale the gate noise strength, the modified Langevin approach can better simulate stochastic calcium channel dynamics. Compared to the Ca²⁺ channels in ER membrane where only one type of gate is discussed with stochastic dynamics, the Na⁺ and K⁺ channels in the HH neuron present a more complicated situation because there are three different types of gates (i.e., 4 n-gates, 1 h-gate, and 3 m-gates). Thus, it is interesting to consider whether the gate noise can be rescaled to improve the gate-based Langevin approach for stochastic HH neuronal dynamics. In this study, we compare the identical gate Langevin approach and independent gate Langevin approach with rescaled-strength of Gaussian noise for the stochastic HH neuronal model. We show that the rescaled Langevin approach can greatly improve the statistical results.

The neuron membrane voltage for the deterministic HH model is given by the following equation^[1]

with V representing the membrane potential and C representing the membrane capacitance. The currents of Na⁺ channels and K⁺ channels and the leakage are given by I_Na = g_Nahm³(V − E_Na), I_K = g_Kn⁴(V − E_K), and I_L = g_L(V − E_L), respectively. The equation for the gating variable x (with n for K⁺ channels and m and h for Na⁺ channels) is

All of the model parameters have been described previously.^{[1, 11]} The open/close two-state Markov process for each single gate (n, m, and h gates) has also been previously described^{[4, 11]} and is considered to be a standard Markov method for stochastic channel dynamics in the paper.

There are two ways to consider the Gaussian noise for the gate variables. The first one is to assume identical gate dynamics with noise:^{[6, 8, 18]}

where ζ (t) is the uncorrelated Gaussian white noise with zero mean and unit variance, and N_x represents the number of corresponding Na⁺ or K⁺ channels. The term G is a time-dependent noise strength with

The second way is to assume the independent gate dynamics for each gate. As a result, the channel currents are rewritten as I_Na = g_Nahm₁m₂m₃(V − E_Na) and I_K = g_Kn₁n₂n₃n₄(V − E_K), with

where x_i represents h, m₁, m₂, m₃, n₁, n₂, n₃, and n₄. In the simulation, we keep the boundary limitation of 0 ≤ x ≤ 1 by simply requiring x = 0 or 1 once it is out of [0, 1].

In the original Fox– Li Langevin approach, λ _x = 1 (i.e., λ _h = λ _m = λ _n = 1).^[6] However, we introduce the rescale factor λ _x to better simulate the strength of Gaussian noise on gate dynamics in the Langevin approach. We suggest that a set of rescale factors {λ _x} can be optimally selected to produce better statistical results for action potentials of a stochastic HH neuron.

Gate-based Langevin approaches have been widely applied for the study of stochastic channel dynamics after Fox and Lu proposed such approaches to simulate the stochastic HH neuronal model in 1994.^[6] It has been realized that these simple approaches actually suggest an over-simplified noise approximation for stochastic channel simulation, leading to qualitatively correct but quantitatively incorrect conclusions. Thus, an interesting question is how to construct an improved Langevin approach to better approximate the Markovian channel dynamics.

In this paper, we indicate that the Gaussian noise suggested by the original identical and independent gate approaches is typically small. Thus, we propose to rescale the strength of the Gaussian noise for the gate-based Langevin approaches for the stochastic HH neuron model. With a larger Gaussian noise, the rescaled Langevin approaches can generate a strong fluctuation for gate variables, resulting in a larger probability at the zero open fraction, as given by the Markov method. With a better approximation of open fractions for K⁺ and Na⁺ channels, the rescaled Langevin approaches can achieve better statistical results for fluctuating membrane voltage. Our simulation results show that the rescaled identical gate approach agrees better with the Markov method regarding the calculation of mean membrane voltage and mean inter-spike interval, whereas the rescaled independent gate approach agrees better with the Markov method for the calculation of the bifurcation diagram of the averaged maximal and minimal voltages and the mean spike amplitude.

Because the rescale parameters for Langevin noise are not universal, the rescaled Langevin approach is an ad-hoc approach that calls for preliminary simulations to determine the empirical scaling factors. Stochastic channel dynamics are found widely in neuron dynamics and intracellular calcium signaling systems.^[15] Considering the simplicity of the gate-based Langevin approach, we believe that the rescaled Langevin approach may be applicable in many stochastic channel systems.