Analysis of responses of channel noise models for a fixed voltage trajectory.

<p>(A) Voltage trace obtained from the Markov chain model with no current input, 6,000 channels and 1,800 channels. Dynamics are characterized by a prolonged subthreshold period followed by a spontaneous, channel noise-induced spike at . (B) Means of fraction of open and channels for the voltage trace shown in (A), as computed from Equations 10 and 11. (C) Variance in the fraction of open channels. (D) Variance in the fraction of open channels. Left insets in (C and D) show magnified views of the period preceding the spike. Right inset in (C) shows magnified view during the spike. For (C and D), exact variances (black) were computed from Equation 12 and Equation 13 and all other variances were estimated from 5,000 repeated simulations of the channel noise models.</p

Markov chain kinetic models of the Na⁺ channel in squid giant axon.

<p>(A) Kinetic scheme for the classical HH model of the Na channel. (B) Kinetic scheme for the Vandenberg and Bezanilla model of the Na channel. Arrows are labeled with transition rates that are functions of voltage, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002247#pcbi.1002247-Hodgkin1" target="_blank">[20]</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002247#pcbi.1002247-Clay1" target="_blank">[41]</a> for further details. The open states are those in the bottom right: (3,1) in (A) and in (B).</p

Classification of channel noise models.

<p>A summary of the three classes of channel noise models that we discuss in this review, and how they differ from the deterministic HH equations, which have no noise.</p

ISI statistics for DC input.

<p>(A) Mean of ISIs for a membrane area of channels). (B) CV of ISIs for same membrane area as (A). 500 spikes were used to estimate the mean and variance, and error bars indicate standard error in the mean for ten repeated measurements for all models except the Markov chain model, for which only four repeated measurements were used.</p

Inhibition is subtractive for large A-channel conductance or weak synaptic excitation.

<p><b>A, B</b>: Firing rates computed from simulations with inhibition (<i>g</i>_{<i>Syn</i>,<i>I</i>} = 1, <i>r</i>_<i>I</i> = 50 Hz, abscissa) plotted as a function of firing rates computed from simulations without inhibition (<i>g</i>_{<i>Syn</i>,<i>I</i>} = 0, ordinate). In <b>A</b>: Three values of A-channel conductance are compared (<i>g</i>_<i>A</i> = 20, 30, 40) with synaptic excitation strength fixed at <i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.5. Inhibition is subtractive for large <i>g</i>_<i>A</i> evident in the rightward shift of the threshold-linear relationship between firing rates for <i>g</i>_<i>A</i> = 40. In <b>B</b>: Three values of synaptic excitation strength are compared (<i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.4, 0.5, 0.7) with A-channel conductance fixed at <i>g</i>_<i>A</i> = 30. Inhibition is subtractive for weaker excitation, evident in the rightward shift of the threshold-linear relationship between firing rates for <i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.4.</p

Boundary between subtractive and divisive inhibition in (<i>g</i>_{<i>Syn</i>,<i>E</i>}, <i>g</i>_<i>A</i>) parameter space.

<p><b>A, B</b>: For each parameter set, we fit threshold-linear functions to characterize the relationship between output firing rates in the presence and absence of inhibition. Dots in each panel identify the smallest value of <i>g</i>_<i>A</i> (for a given parameter set) at which inhibition is subtractive. In <b>A</b>: We vary inhibition strength (<i>g</i>_{<i>Syn</i>,<i>I</i>} = 0.5, 1, 2) and keep inhibition rate fixed at 50 Hz. In <b>B</b>: We vary inhibition rate (<i>r</i>_<i>I</i> = 30, 50, 70 Hz) and keep inhibition strength fixed at <i>g</i>_{<i>Syn</i>,<i>I</i>} = 1. The values of <i>g</i>_<i>A</i> that define the boundary between subtractive and divisive inhibition decrease with increases in either inhibition parameter (<i>g</i>_{<i>Syn</i>,<i>I</i>} or <i>r</i>_<i>I</i>).</p

Examples of divisive and subtractive effects of inhibition in the one-compartment model.

<p><b>A, B</b>: Output firing rates as a function of excitatory input rate, computed from simulations without inhibition (empty circles, <i>g</i>_{<i>Syn</i>,<i>I</i>} = 0) and with inhibition (filled circles, <i>g</i>_{<i>Syn</i>,<i>I</i>} = 1 and <i>r</i>_<i>I</i> = 50 Hz). Excitatory synaptic strength is <i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.5. In <b>A</b>: Divisive rescaling of the input/output relation with <i>g</i>_<i>A</i> = 20. In <b>B</b>: Subtractive shifting of the input/output relation with <i>g</i>_<i>A</i> = 40. <b>C</b>: Data from <b>A</b> and <b>B</b> are replotted with output firing rates in the absence of inhibition on the ordinate and output firing rates in the presence of inhibition on the abscissa. Threshold-linear functions are fit to simulation data (black lines). Rightward shift of threshold-linear function for <i>g</i>_<i>A</i> = 40 is characteristic of subtractive inhibition.</p

Dependence of the <i>V</i>-nullcline on A: <i>g</i>_<i>A</i>, B: <i>b</i>, C: <i>s</i>_<i>I</i> and D: <i>s</i>_<i>E</i>.

<p>Default values of the parameters are <i>g</i>_<i>A</i> = 20, <i>b</i> = .5, <i>s</i>_<i>I</i> = .5 and <i>s</i>_<i>E</i> = 1. Moreover, <i>g</i>_{<i>Syn</i>,<i>E</i>} = 3 and <i>g</i>_{<i>Syn</i>,<i>I</i>} = 5. Thin blue line is <i>n</i>_∞(<i>V</i>), the <i>n</i>-nullcline.</p

Response to an excitatory input.

<p><b>A</b>. The neuron will or will not fire an action potential if, at the time of the excitatory input, it lies below or above the left knee of the <i>s</i>_<i>E</i> = 1 cubic, respectively. <b>B</b>. The neuron cannot respond with an action potential if the left knee of the <i>s</i>_<i>E</i> = 1 cubic lies below the <i>n</i> = 0 axis.</p

The slope of the input/output firing rate curves at <i>r</i>_<i>E</i> = 0 computed from both the theoretical prediction Eq 15 and simulations of the full model.

<p>The slope of the input/output firing rate curves at <i>r</i>_<i>E</i> = 0 computed from both the theoretical prediction <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006292#pcbi.1006292.e054" target="_blank">Eq 15</a> and simulations of the full model.</p