Comparison between the responses obtained by the deterministic and stochastic models.

<p>The length of each trace is 0.4 seconds. In A) the depolarizing current is below firing threshold (<i>I_app</i> = 11 pA) and in B) just above firing threshold (<i>I_app</i> = 12 pA). In C), the depolarizing current is considerably larger (<i>I_app = </i>29 pA). The deterministic model (right panels) does not reproduce the experimentally observed irregularity in firing. The responses simulated by the stochastic model (left panels), on the contrary, very closely resemble the experimentally obtained irregularities (for more details, see the Electroresponsiveness Obtained by the Stochastic Model Only section). For the stochastic traces σ = 0.5. For each value of the depolarizing current, <i>I_app</i>, traces from three independent simulations of the stochastic model are shown to illustrate the variability of firing.</p

Quantitative analysis of the interspike intervals of the stochastic model.

<p>Firing is simulated for 50 seconds with three different values of depolarizing current pulses, <i>I_app</i>, and the parameter σ. The chosen levels for depolarizing current pulses are: <b>i)</b><i>I_app</i> = 11 pA (below firing threshold, <i>I_th</i>), <b>ii)</b><i>I_app</i> = 12 pA (just above the firing threshold), and <b>iii)</b><i>I_app</i> = 29 pA (a considerably larger stimulus). From each trace the mean, standard deviation (std) and the coefficient of variation (CV) of the interspike intervals are calculated. Seconds are used as units for the mean and standard deviation; coefficient of variation is dimensionless. Same simulated data is used as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000004#pcbi-1000004-g008" target="_blank">Figure 8</a>.</p

Exploring the intrinsic burst generation with the stochastic granule cell model.

<p>A small depolarizing current pulse (shown by a rectangular bar at the bottom of the figure) below firing threshold is injected into the cell soma. The bursts are evoked by random changes of σ between the values σ = 0.3 and σ = 1.1 (i.e., during a burst the value of parameter σ is increased to 1.1 otherwise it being 0.3). For illustrative purposes the trace with two bursts of action potentials is shown here (compare also with <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000004#pcbi-1000004-g005" target="_blank">Figure 5</a>). A fast afterhyperpolarization (fAHP), an afterdepolarization (ADP), and a slow afterhyperpolarization (sAHP) are indicated by arrows.</p

Forward and backward rate functions for different ion channel types in the stochastic model (see Equation 8).

<p>Forward and backward rate functions for different ion channel types in the stochastic model (see Equation 8).</p

Exploring the variability in spike timing.

<p>Three sets of ten realizations of firing are simulated with the stochastic granule cell model using a depolarizing current pulse just above the firing threshold (<i>I_app</i> = 12 pA). The length of each trace is 0.1 seconds. In the upper panel σ = 0.1, in the middle panel σ = 0.3, and in the lower panel σ = 0.5. The main variability does not arise only from the timing of the first action potential, but there is significant variability also after the first spike.</p

Frequency-current (<i>f-I</i>) curve of the stochastic granule cell model presented as a box and whisker plot.

<p>Depolarizing current pulses from 0 pA to 45 pA are used. For each value of depolarizing current we simulated 50 realizations, each 50 seconds long. Median, upper and lower quartiles, and the maximal and minimal firing frequencies are given for each depolarizing current pulse; outliers are marked with+symbol. Spontaneous activity is observed at low firing frequencies with depolarizing currents below 11 pA which is the firing threshold of the model. The <i>f-I</i> curve of the stochastic model is linear up to a frequency of 125 Hz, after which it shows saturation. For every realization σ = 0.5.</p

Dynamic behavior of the stochastic granule cell model simulated for a longer duration (15 seconds).

<p>A small depolarizing current below firing threshold is applied throughout the simulation, similarly as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000004#pcbi-1000004-g004" target="_blank">Figure 4</a>. Bursts and occasional spontaneous firing can be observed. Bursts are evoked by random changes of σ between the values σ = 0.3 and σ = 1.1 (during a burst the value of parameter σ is increased to 1.1 otherwise it being 0.3). This 15-second simulation also provides evidence that stable solutions are obtained when bursts are evoked.</p

Parameter values used in both stochastic and deterministic simulations.

<p>See the sections Deterministic Model and Complete Stochastic Model for more details on ion channel types and the description of the complete mathematical model.</p

Gating variables for K_A channel activation and inactivation processes.

<p>The stochastic model is simulated for 0.5 seconds and depolarized from 0.15 seconds to 0.35 seconds. The value of the parameter σ is set to 0.15 and all other parameters are fixed as explained in the text.</p

Example of stable, long-lasting, continuous simulation with irregular firing using the stochastic model.

<p>A depolarizing current pulse just above the firing threshold is given. A simulation of 5 seconds is shown to provide evidence that stable solutions are obtained with stochastic differential equations and Brownian motion. The simulation time of this trace with time-step of 10⁻⁵ seconds is ca. 15 seconds. For this simulation σ = 0.5.</p