Modeling Active Non-Markovian Oscillations

Modeling noisy oscillations of active systems is one of the current challenges in physics and biology. Because the physical mechanisms of such processes are often difficult to identify, we propose a linear stochastic model driven by a non-Markovian bistable noise that is capable of generating self-sustained periodic oscillation. We derive analytical predictions for most relevant dynamical and thermodynamic properties of the model. This minimal model turns out to describe accurately bistablelike oscillatory motion of hair bundles in bullfrog sacculus, extracted from experimental data. Based on and in agreement with these data, we estimate the power required to sustain such active oscillations to be of the order of $100k_{B}T$ per oscillation cycle.

Figure 1

(a) Schematic representation of the switching mechanism controlling the dynamics described in Eq. (1): after a time $\tau$ drawn from the distribution ${\psi }_{\pm }(\tau )$ the center $c$ of a harmonic potential $V(x)=(\kappa /2)(x-c{)}^2$ switches from $\pm c_0$ to $\mp c_0$. (b), (e) Realizations of the stochastic driving $c(t)$ (dashed blue line) and of the process $x(t)$ (solid blue line) obtained from a numerical simulation of Eq. (1), for (b) the exponential and (e) the gamma waiting-time probability density function (PDF) plotted, respectively, in (c) and (f). In particular, the exponential distributions have rates $r_{+}=1/7$ and $r_{-}=2/17$, whereas the gamma distributions (see the main text) have shape parameters $k_{+}=15$, $k_{-}=10$, and scale parameters ${\theta }_{+}=7/15$, ${\theta }_{-}=17/20$. (d), (g) Power spectral density $S_x$ (symbols) of $x(t)$ on the doubly logarithmic scale, obtained for two time series of total duration $t=1.5\times {10}^3$ with the same parameters as those in (b) and (d). The dashed lines are given by Eq. (6). The dynamics was simulated with $D=1$, $c_0=5$, $\nu =2.5$, and a time step $\mathrm{\Delta }t={10}^{-3}$.

Figure 2

Stationary probability density ${\rho }^{\mathrm{st}}(x)$ for symmetric exponentially distributed waiting times: numerical simulations (symbols) are compared with the analytical solution in Eq. (3) (dashed lines). The three cases correspond to fixed values of $D=1$ and $\nu =2.5$, but various values of $r$ and $c_0$: blue, $c_0=2$ and $r=5>\nu$; red, $c_0=0.5$ and $r=1.25<\nu$, for which $\chi \simeq 0.31$; green, $c_0=2.5$ and $r=1.25$, for which $\chi \simeq 7.8$. In the latter two cases, $\zeta =1/2$ corresponding to the critical value ${\chi }^{*}(\zeta =1/2)\simeq 1.58$ (see main text). The numerical estimates of ${\rho }^{\mathrm{st}}(x)$ are obtained from $N={10}^4$ simulations of Eq. (1) using Euler’s numerical integration method with time step $\mathrm{\Delta }t=5\times {10}^{-3}$.

Figure 3

Frequencies of the first (blue) and second (red) peaks of the power spectrum $S_x(\omega )$ in Eq. (4), as functions of $k$, for $\theta =1.5$, $D=0.5$, $c_0=1$, and $\nu =2.5$. As $k$ increases above $\simeq 1.5$ (dashed blue vertical line) a first local maximum appears in $S_x(\omega )$ at a typical frequency (blue symbols), well approximated by the blue solid line. As $k$ exceeds $\simeq 14.6$ (dashed red vertical line), a second peak appears at a typical frequency (red symbols), well approximated by the red solid line.

Figure 4

Oscillations of a hair bundle’s tip $x(t)$ modeled by Eq. (1) in the case of symmetric gamma-distributed waiting times: experimental observations and simulations with inferred parameter values. (a) Example segments of experimental and simulated time series. (b) Probability density function (PDF) of $x(t)$. (c) Power spectrum of $x(t)$ with its autocorrelation function (ACF) shown in the inset. (d) Energy dissipated by hair bundles per one cycle in three experimental cases, see Table I in the Supplemental Material [47]. Only data of the experimental case 1 are shown in (a)–(c). Data for all the three cases are reported in the Supplemental Material [47].