Exported oscillator competition: A concept to analyze complex rhythms

We study the interaction between two independent nonlinear oscillators competing through a neutral excitable element. The first oscillator, completely deterministic, acts as a normal pacemaker sending pulses to the neutral element which fires when it is excited by these pulses. The second oscillator, endowed with some randomness, though unable to make the excitable element to beat, leads to the occasional suppression of its firing. The missing beats or errors are registered and their statistics analyzed in terms of the noise intensity and the periods of both oscillators. This study is inspired in some complex rhythms such as a particular class of heart arrhythmia.

Figure 1

Nullclines (dashed lines) and $v\text{−}u$ trajectories (solid lines) of oscillatory $I=0.68$ (a) and excitable $I=0.6$ (b) element.

Figure 2

Period of firing for an oscillatory element as a function of the parameter $I$, corresponding to the deterministic case $(D=0)$ and parameter $ϵ=0.1$. Insets: Dynamical evolution for $I=0.68$ (top) and 0.8 (bottom).

Figure 3

Normalized variance $R$ and averaged period $T$ (inset) as a function of the noise intensity. They correspond to parameters $I=0.68$ (solid lines) and 0.8 (dotted lines).

Figure 4

Schematic diagram of the proposed coupling to simulate the considered oscillator competition.

Figure 5

(a) Train of pulses $U$ for parameters: $\gamma =0.2$, $\alpha =0.1$, $I_S=0.82(T_S\approx 4)$, $I=0.6$ (excitable), $I_E=0.68(T_E\approx 11)$, ${ϵ}_S={ϵ}_E=0.1$, $ϵ=0.01$, and $D={10}^{-6}$. (b) Ectopic input $\left(u_E\right)$ for the record in (a). Normal input $\left(u_S\right)$ is also shown (dotted line).

Figure 6

Normalized $V\text{−}V$ interval histogram and NIB histogram for noise intensities: $D={10}^{-6}$ (left column) and $D={10}^{-5}$ (right column). Parameters $I_S$ and $I_E$ have been chosen to have $T_S=3.6$ and $T_E=10.8$. Other parameters are those of Fig. 5.

Figure 7

Sequence of pulses corresponding to the cases of Fig. 6. $D={10}^{-6}$ (upper panel) and $D={10}^{-5}$ (lower panel). NIB sequences are pointed out.

Figure 8

Normalized NIB histograms for noise intensity $D={10}^{-6}$ and ratio $T_E∕T_S=2$, 2.5 and 3, from left to right.

Figure 9

Relative error versus period ratio $T_E∕T_S$ for $T_E=10.8$ (circles), 9.2 (triangles), and 7.2 (squares). $T_S$ is changing from 2.8 to 5, approximately. $D={10}^{-6}$ and other parameters are those of previous figures. Dotted line represents the limit condition $n_V∕N=T_S∕T_E$.

Figure 10

Relative error versus period ratio $T_E∕T_S$ for $T_E=11$, $D={10}^{-5}$, $\gamma =0.2$, and the coupling parameter $\alpha$ going from 0.08 (bottom curve) to 1.8 (top curve) every 0.02. Dotted line represents the limit condition $n_V∕N=T_S∕T_E$. Inset: Time $\tau$ as a function of $\alpha$.