Stochastic entrainment of a stochastic oscillator

In this work, we consider a stochastic oscillator described by a discrete-state continuous-time Markov chain, in which the states are arranged in a circle, and there is a constant probability per unit time of jumping from one state to the next in a specified direction around the circle. At each of a sequence of equally spaced times, the oscillator has a specified probability of being reset to a particular state. The focus of this work is the entrainment of the oscillator by this periodic but stochastic stimulus. We consider a distinguished limit, in which (i) the number of states of the oscillator approaches infinity, as does the probability per unit time of jumping from one state to the next, so that the natural mean period of the oscillator remains constant, (ii) the resetting probability approaches zero, and (iii) the period of the resetting signal approaches a multiple, by a ratio of small integers, of the natural mean period of the oscillator. In this distinguished limit, we use analytic and numerical methods to study the extent to which entrainment occurs.

Figure 1

An $n$-state oscillator with transition rate $\frac{n}{T}$ and resetting probability $p$.

Figure 2

Comparison of the fraction of systems in state $j$ at time $t$ and the probability $P_j\left(t\right)$ of the oscillator being in state $j$ at time $t=0, T, S^{+}$, and $2S^{-}$. We simulate 80 000 oscillators with $n=20, T=50$ s, $S=100.25$ s, and $p=0.2$. The bars represent the fractions, and the curve is the plot of $P_j$. Here $P_j\left(t\right)$ is the numerical solution of Eqs (1)–(3). The total time simulated is $2S$. Agreement is enforced at $t=0$ by setting $P_j\left(0\right)$ equal to the actual fraction of systems that were started in state $j$.

Figure 3

Probability density functions of the period of the oscillator with $n=1$, 4, 16, and 64 states.

Figure 4

Comparison of the solution by discrete Fourier transform and the approximate expression given in Eq. (33) at times $t=0, \frac{T}{2}, T$, and $S$. The oscillator has 150 states with $\alpha =20, \beta =0.1, m=2$, and $T=50$ s. The bars represent the exact solution $P_j$ obtained by discrete Fourier transform, and the curve is the plot of $\frac{1}{n}\rho (\frac{j}{n},t)$.

Figure 5

${\rho }_0^{\text{max}}$ for $\beta \in [-10,10]$ and $\frac{m}{l}$ with both $m$ and $l$ ranging from 1 to 4.