Serial correlation coefficient as a function of the lag between ISIs.

<p><b>A</b> The case of deterministic adaptation with for different values of the time constant (as indicated in the legend). The theoretical curves, Eq. (9), are depicted by solid lines; the zero baseline is indicated by a dotted line. <b>B</b> The case of stochastic adaptation with for different values of the time constant (as in <b>A</b>). The channel model, Eq. (20), is represented by white symbols, the diffusion approximation (Eq. (1)) is represented by black symbols. The theory based on the colored noise approximation, Eq. (4), is depicted by a solid line. Other parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001026#pcbi-1001026-g002" target="_blank">Fig. 2</a>.</p

Diffusion approximation of adaptation current.

<p><b>A</b> Sample traces of the integrate-and-fire dynamics with two-state adaptation channels, Eq. (20), (35) ( and ). The fraction of open channels (top) exhibits discontinuous jumps with directions that depend on the presence or absence of a spike as given by the activation function (middle panel). <b>B</b> Sample traces of the diffusion model, Eq. (1), with the same 1st and 2nd infinitesimal jump moments of as in the channel model (<b>A</b>). <b>C</b> The fraction of open channels can be split into the deterministic part , Eq. (1b), corresponding to , and an Ornstein-Uhlenbeck process , Eq. (1c), with a correlation time equal to the adaptation time constant (colored noise). The parameters are , , .</p

Comparison of the ISI statistics of the Traub-Miles model – deterministic vs. stochastic adaptation.

<p><b>A</b> Rescaled skewness (Eq. (61)) and <b>B</b> rescaled kurtosis (Eq. (62)) as a function of the coefficient of variation (CV). For stochastic adaptation (Eq. (116), – black circles) the number of channels was varied from to ; for deterministic adaptation (Eq. (115), – gray squares), the noise intensity was varied from to . The corresponding inverse Gaussian statistics (Eq. (64)) is indicated by the dotted line. <b>C, D</b> show the rescaled kurtosis and the serial correlation coefficient (Eq. (63)) at lag 1 as a function of the time scale separation . Stochastic adaptation () and deterministic adaptation () are marked as in <b>A,B</b>. <b>E,F</b> The serial correlation coefficient as a function of the lag for different time constants in as indicated (<b>E</b> deterministic adaptation, <b>F</b> stochastic adaptation; and as in <b>C,D</b>). <b>G</b> The rescaled kurtosis in the mixed case at a fixed amount of white noise () and varying channel numbers . <b>H</b> The corresponding values of the serial correlation coefficient at lag one. The intersection of the curve with the zero line (dotted line) defines the adaptation-noise dominated regime (white region) and the white-noise dominated regime (gray-shaded region). The units of the noise intensities are . For stochastic adaptation . For deterministic adaptation was adjusted to result in a firing rate at around . For the current was . With increasing noise intensity decreased to for .</p

Shape parameters of the ISIH for deterministic and stochastic adaptation.

<p><b>A</b> Rescaled skewness for deterministic adaptation (white squares) and for stochastic adaptation (channel model – white circles, diffusion model – black circles, colored noise approximation – gray circles). Different CVs were obtained by varying or . The dashed line depicts the theoretical curve, Eq. (7), and the solid line depicts the semi-analytical result obtained from the moments of the ISI density, Eq. (69), using numerical integration. <b>B</b> The corresponding plot for the rescaled kurtosis . The adaptation time constant was . All other parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001026#pcbi-1001026-g002" target="_blank">Fig. 2</a>.</p

Serial correlation coefficient at lag 1 as a function of the time scale separation.

<p><b>A</b> Serial correlation coefficient in the case of channels (corresponding to ) for stochastic adaptation (circles, Eq. (3)) and for deterministic adaptation (squares, Eq. (2)). Theoretical curves for stochastic adaptation, Eq. (72), and deterministic adaptation, Eq. (9), are displayed by a dashed line and a solid line, respectively. The zero baseline is indicated by a dotted line. <b>B</b> shows the corresponding plot for channels (corresponding to ) and . The gray-shaded region marks the relevant range for spike-frequency adaptation. was varied by changing at fixed (), all other parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001026#pcbi-1001026-g002" target="_blank">Fig. 2</a>.</p

ISI statistics of the PIF model in the presence of both stochastic adaptation and white noise.

<p>For a fixed level of white noise () the number of adaptation channels was varied. For a small channel population slow channel noise dominates (white region), whereas for a large population the fast fluctuations dominate (gray-shaded region). <b>A</b> Rescaled kurtosis for the channel model (white diamonds) and the diffusion model (black circles). The gray symbols display simulations where the adaptation was replaced by an effective colored noise as before but with the additional white noise input. The case of an inverse Gaussian is indicated by the dotted line. <b>B</b> Corresponding serial correlation coefficient at lag one. The zero line is indicated by the dotted line. The adaptation time constant was chosen as , other parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001026#pcbi-1001026-g002" target="_blank">Fig. 2</a>.</p

ISI histograms of a PIF neuron – theory vs. simulation.

<p><b>A</b> ISI densities in the case of deterministic adaptation () for different noise intensities . Gray bars show the histograms obtained from simulations of Eq. (2); solid lines display the mean-adaptation approximation, Eq. (64) (inverse Gaussian density). <b>B</b> ISI densities in the case of stochastic adaptation () for different as indicated in the panels. The adaptation current was modeled either by the channel model (gray bars), Eq. (20), or by the diffusion model (circles), Eq. (1). The theory, Eq. (69), is displayed as a solid line. Parameters are chosen as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001026#pcbi-1001026-g002" target="_blank">Fig. 2</a>.</p

Illustration of the deterministic dynamics of the adaptive PIF neuron.

<p>The flow in the phase space spanned by possesses a stable limit cycle, which consists of the voltage drift from at to the threshold (A), the following increase of by an amount (B) and the subsequent voltage reset (C). The period of the limit cycle is solely due to the time of the process A, whereas the processes B and C occur instantaneously. A trajectory starting off the limit cycle at reaches the threshold at the time . The deviation from after one period has an absolute magnitude .</p

Shape parameters of the ISIH as a function of the time scale separation.

<p><b>A</b> Rescaled skewness and <b>B</b> rescaled kurtosis for channels (corresponding ) for stochastic adaptation (circles) and for deterministic adaptation (squares). Theory Eq. (113) and (114) is displayed by the solid line, the line is indicated by a dotted line. <b>C</b> and <b>D</b> corresponding plots for channels (corresponding to ) and . was varied by changing at a fixed (), all other parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001026#pcbi-1001026-g002" target="_blank">Fig. 2</a>.</p

Channel model with voltage-dependent transition rates.

<p><b>A</b> In a simple model of a M-type adaptation channel, the slow voltage-gated adaptation current is mediated by a population of slow ion channels that can be either in an open or a closed state. In the activation state (), channels open with rate , in the deactivation state () channels close with rate . <b>B</b> Both transition rates show a sigmoidal dependence on the voltage (Eq. (24) and (27), ).</p