Figure 1

A plot of $x_n$ and $y_n$ as a function of iteration number $n$ for the Rulkov model with $\alpha =2.02$ and $\beta =\sigma ={10}^{-3}$. The noise terms have been set to zero.Reuse & Permissions

Figure 2

Rulkov model trajectories $x_n$ and $y_n$ as a function of iteration number $n$ for $\alpha =1.99$ and $\beta =\sigma ={10}^{-3}$. The trajectories approach the fixed point, whose coordinates are $x^{*}=-1$ and $y^{*}=-1-\alpha ∕2$. The noise terms have been set to zero.Reuse & Permissions

Figure 3

Noise-induced pulses in the Rulkov model for $\alpha =1.99,\beta =\sigma ={10}^{-3},D_x=1.6\times {10}^{-5}$, and $D_y=0$, after transients have died out. The upper trace is $x_n$ and the lower trace is $y_n$ as a function of iteration number $n$.Reuse & Permissions

Figure 4

Noise-induced pulses in the Rulkov model for $\alpha =1.99,\beta =\sigma ={10}^{-3},D_x=9\times {10}^{-4}$, and $D_y=0$.Reuse & Permissions

Figure 5

The regularity $R$ for the Rulkov model, Eqs. (1, 2), plotted as a function of the logarithm of the noise variance. The squares (on the left) are data for $D_x=0$. The circles (on the right) are data for $D_y=0$. Open symbols: $\alpha =1.99$. Black symbols: $\alpha =1.95$. Gray symbols: $\alpha =1.91$. Here, we use $\beta =\sigma ={10}^{-3}$. The uncertainty bars indicate the standard deviation of the mean over ten noise realizations and are, in most cases, smaller than the plotted symbols.Reuse & Permissions

Figure 6

The regularity $R$ for the Rulkov model, Eqs. (1, 2), as a function of ${\log }_{10}{D}_x$ and ${\log }_{10}{D}_y$ with $\alpha =1.99,\beta =\sigma =0.001$.Reuse & Permissions

Figure 7

The logarithm of the regularity $R$ plotted as a function of ${\log }_{10}D$ for $\beta =\sigma ={10}^{-4}$ (open symbols) and $\beta =\sigma ={10}^{-2}$ (solid symbols) for $\alpha =1.99$. For each data set, on the left (squares), noise is added to $y$ alone. On the right (circles) noise is added to $x$ alone. The uncertainty bars indicate the standard deviation of the mean over ten noise realizations. The solid curves are the results of the Fokker-Planck analysis described in Sec. 4.Reuse & Permissions

Figure 8

Rulkov model state-space trajectories (heavy symbols) for noise-induced pulses with $D_x={10}^{-4}$ and $D_y=0$. The $x$ nullcline is indicated by the thin curve. Here we use $\alpha =1.99,\beta =\sigma =0.001$. The state-space locations of the activation, pulse, and recovery phases are indicated. The larger circles indicate the three fixed points of Eq. (1) with $y$ viewed as a fixed parameter.Reuse & Permissions

Figure 9

The results of the Fokker-Planck analysis described in the text (solid curves) and the simulation data for the Rulkov model with $\beta =\sigma ={10}^{-3}$. Open symbols, $\alpha =1.99$; solid symbols, $\alpha =1.91$. On the left $D_x=0$ (squares); on the right $D_y=0$ (circles).Reuse & Permissions

Figure 10

The results of the Fokker-Planck analysis of the Rulkov model stochastic coherence with noise sources present in both dynamical variables. $\alpha =1.99$ and $\beta =\sigma ={10}^{-3}$. The regularity $R$ is plotted as a function of ${\log }_{10}{D}_x$ and ${\log }_{10}{D}_y$. Note the general agreement with the simulation results presented in Fig. 6.Reuse & Permissions

Figure 11

A plot of the regularity of the noise-induced pulses for the FitzHugh Nagumo model. $\epsilon =0.001,\gamma =1.5$. Solid symbols, $b=0.4812$; open symbols, $b=0.53$; squares, $D_{\upsilon }=0$; circles, $D_w=0$. The uncertainty bars indicate the standard deviation of the mean for ten noise realizations, each consisting of a sequence of 200 time units. (Each pulse has a duration of about 1.5 time units.) The solid curves are the results of the Fokker-Planck analysis described in the text.Reuse & Permissions

Figure 12

$\upsilon$ nullcline (cubic curve) and $w$ nullcline (straight line) for the FitzHugh-Nagumo model superposed on several noise-induced pulse trajectories (small circles) for $b=0.4812,\gamma =1.5$, and $\epsilon ={10}^{-3}$. The activation, pulse, and recovery phases are indicated. The coordinates of the $\upsilon$ nullcline local minimum are $({\upsilon }_A,w_A)$ and those of the local maximum are $({\upsilon }_P,w_P)$. The three fixed points for the no-noise version of Eq. (40) are indicated by the larger, filled-in circles.Reuse & Permissions

Figure 13

A plot of the escape region for the no-noise Rulkov model for $\alpha =1.99$ and $\beta =\sigma =0.001$. Initial points below the curve lead to trajectories that stay in the neighborhood of the fixed point, located at (0,0) in this plot. (The steps in the curve are due to finite sampling of the initial conditions.) Initial points above the curve lead to trajectories that make a large excursion through state space before returning to the fixed point. The cross, centered on the fixed point, indicates the standard deviation of the noise intensity when the regularity has its maximum value for these parameter values. The filled circle and square indicate the escape points for the one-dimensional models. The short vertical and horizontal lines indicate the range of escape points used in the data fits.Reuse & Permissions

Figure 14

A sketch of the potential function $U\left(y\right)$ indicating the absorbing boundary at $a$, the reflecting boundary at $b$, and the fixed point at $y^{*}$. $w$ is the injection point.Reuse & Permissions