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 ${\text{Na}}^{+}$ and ${\text{K}}^{+}$, or only ${\text{K}}^{+}$ 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.

Figure 1

Markov state transitions for the voltage-gated ion channels. (a) Gating scheme for the ${\text{Na}}^{+}$ channel. (b) Gating scheme for the ${\text{K}}^{+}$ channel.

Figure 2

(Color online) Langevin model overestimates the information rates in spiking compartments, irrespective of compartment size. The stimulus cut-off frequency was set to 300 Hz. $\mu$ is the mean and $\sigma$ is the standard deviation of the current injection. Top row: comparison of information rates for models with three different areas [(a) $\left(1\mu {\text{m}}^2\right)$, (b) $\left(10\mu {\text{m}}^2\right)$, and (c) $\left(100\mu {\text{m}}^2\right)$]. Wireframe mesh represents the Langevin implementation of channel noise, while the filled mesh is the Markov implementation. Bottom row: error surfaces between Langevin and Markov representations [(d) $\left(1\mu {\text{m}}^2\right)$, (e) $\left(10\mu {\text{m}}^2\right)$, and (f) $\left(100\mu {\text{m}}^2\right)$].

Figure 3

(Color online) Langevin model underestimates the firing rate and overestimates the information obtained per spike of the spiking compartments, regardless of the size of the compartment. Wireframe mesh represents the Langevin implementation of channel noise, while the filled mesh is the Markov implementation. The stimulus cut-off frequency was set to 300 Hz. $\mu$ is the mean and $\sigma$ is the standard deviation of the current injection. Top row: comparison of firing rate for models with three different areas [(a) $\left(1\mu {\text{m}}^2\right)$, (b) $\left(10\mu {\text{m}}^2\right)$, and (c) $\left(100\mu {\text{m}}^2\right)$]. Firing rates are obtained from trial averaged responses to frozen current injections. Bottom row: comparison of information per spike for models with three different areas [(d) $\left(1\mu {\text{m}}^2\right)$, (e) $\left(10\mu {\text{m}}^2\right)$, and (f) $\left(100\mu {\text{m}}^2\right)$].

Figure 4

(Color online) Comparison of trial averaged, response power spectral density of the voltage traces (with spikes intact) with Langevin and Markov models of channel noise for three different membrane areas [(a) $\left(1\mu {\text{m}}^2\right)$, (b) $\left(10\mu {\text{m}}^2\right)$, and (c) $\left(100\mu {\text{m}}^2\right)$]. Input statistics were sampled from a Gaussian distribution with $\mu =5\mu \text{A}/{\text{cm}}^2$, $\sigma =10\mu \text{A}/{\text{cm}}^2$, and ${\tau }_{correlation}=3.3\text{ms}$.

Figure 5

(Color online) Comparison of trial averaged signal [(a) $\left(1\mu {\text{m}}^2\right)$, (c) $\left(10\mu {\text{m}}^2\right)$, and (e) $\left(100\mu {\text{m}}^2\right)$] and noise [(b) $\left(1\mu {\text{m}}^2\right)$, (d) $\left(10\mu {\text{m}}^2\right)$, and (f) $\left(100\mu {\text{m}}^2\right)$] power spectral density of the voltage traces (with spikes removed) with Langevin and Markov models of channel noise, for three different membrane areas. Input statistics were sampled from a Gaussian distribution with $\mu =5\mu \text{A}/{\text{cm}}^2$, $\sigma =10\mu \text{A}/{\text{cm}}^2$, and ${\tau }_{correlation}=3.3\text{ms}$.

Figure 6

(Color online) Phase plots comparing action potential initiation with Langevin and Markov models of channel noise for compartments with two different areas [(a) $\left(10\mu {\text{m}}^2\right)$ and (b) $\left(100\mu {\text{m}}^2\right)$]. Arrow indicates the action potential initiation zone. Input statistics were sampled from a Gaussian distribution with $\mu =5\mu \text{A}/{\text{cm}}^2$, $\sigma =10\mu \text{A}/{\text{cm}}^2$, and ${\tau }_{correlation}=3.3\text{ms}$.

Figure 7

Comparison of probability density function of Langevin and Markov models of channel noise for spiking compartments. The distance between the distributions is quantified by the Kullback-Leibler (relative entropy) divergence. Input statistics were sampled from a Gaussian distribution with $\mu =5\mu \text{A}/{\text{cm}}^2$, $\sigma =10\mu \text{A}/{\text{cm}}^2$, and ${\tau }_{correlation}=3.3\text{ms}$.

Figure 8

(Color online) Difference between Langevin and Markov models of channel noise with two levels of input noise for a $10\mu {\text{m}}^2$ compartment $\left(f_c=300\text{Hz}\right)$. Filled mesh represents the Markov model; wireframe mesh displays the Langevin model. $\mu$ is the mean and $\sigma$ is the standard deviation of the current injection. (a) Signal-to-noise ratio equals 2; (b) signal-to-noise ratio equals 20.

Figure 9

(Color online) Langevin model overestimates information rate in nonspiking compartments. The stimulus cut-off frequency was set to 300 Hz. $\mu$ is the mean and $\sigma$ is the standard deviation of the current injection. Top row: comparison of information rates for models with three different areas [(a) $\left(1\mu {\text{m}}^2\right)$, (b) $\left(10\mu {\text{m}}^2\right)$, and (c) $\left(100\mu {\text{m}}^2\right)$]. Wireframe mesh represents the Langevin implementation of channel noise, while the filled mesh is the Markov implementation. Bottom row: error surfaces between Langevin and Markov representations [(d) $\left(1\mu {\text{m}}^2\right)$, (e) $\left(10\mu {\text{m}}^2\right)$, and (f) $\left(100\mu {\text{m}}^2\right)$].

Figure 10

(Color online) Comparison of trial averaged signal [(a) $\left(1\mu {\text{m}}^2\right)$, (c) $\left(10\mu {\text{m}}^2\right)$, and (e) $\left(100\mu {\text{m}}^2\right)$] and noise [(b) $\left(1\mu {\text{m}}^2\right)$, (d) $\left(10\mu {\text{m}}^2\right)$, and (f) $\left(100\mu {\text{m}}^2\right)$] power spectral density of the voltage traces with Langevin and Markov models of channel noise, for three different membrane areas. Input statistics were sampled from a Gaussian distribution with $\mu =5\mu \text{A}/{\text{cm}}^2$, $\sigma =10\mu \text{A}/{\text{cm}}^2$, and ${\tau }_{correlation}=3.3\text{ms}$.

Figure 11

Comparison of probability density function of Langevin and Markov models of channel noise for nonspiking compartments. The distance between the distributions is quantified by the Kullback-Leibler (relative entropy) divergence. Input statistics were sampled from a Gaussian distribution with $\mu =5\mu \text{A}/{\text{cm}}^2$, $\sigma =10\mu \text{A}/{\text{cm}}^2$, and ${\tau }_{correlation}=3.3\text{ms}$.