Modulating Biological Rhythms: A Noncomputational Strategy Harnessing Nonlinearity and Decoupling Frequency and Amplitude

Understanding and achieving concurrent modulation of amplitude and frequency, particularly adjusting one quantity and simultaneously sustaining the other at an invariant level, are of paramount importance for complex biophysical systems, including the signal pathway where different frequency indicates different upstream signal yielding a certain downstream physiological function while different amplitude further determines different efficacy of a downstream output. However, such modulators with clearly described and universally valid mechanisms are still lacking. Here, we rigorously propose an easy-to-use control strategy containing only one or two steps, leveraging the nonlinearity in the modulated systems to decouple frequency and amplitude in a noncomputational manner. The strategy’s efficacy is demonstrated using representative biochemical systems and, thus, it could be potentially applicable to modulating rhythms in experiments of biochemistry and synthetic biology.

Figure 1

(a) Achieving IAM or IFM by a linear controller requires computing the normal form. (b) Achieving IAM and IFM needs, respectively, one and two steps without computation by harnessing nonlinearity (zigzag) in the controller.

Figure 2

IAM of system (5) achieved by (a) the linear controller $\mathit{u}=\mathit{F}\mathit{y}$, and (b), the nonlinear controller $c(Y-Y\mathrm{ss}{)}^2$ (blue: amplitude; red: frequency). Depictions are the numerical results (solid) and the theoretical predictions obtained by computing the normal form (light dashed). The circles represent the simulated results using the Hill-type function, i.e., $y_j/(y_j+k)\approx y_j$ [25], and the results obtained by using $u(Y)$.

Figure 3

(a),(b) Variations of the amplitude and the frequency of the oscillation in system (5) under the control strategy harnessing nonlinearity. Dashed lines represent isoamplitude and isofrequency contours. A two-step controller for IFM is highlighted. (c) Three oscillations correspond to the highlighted circles.

Figure 4

(a) IAM in the AC-DC model achieved by intervening the nonlinear term [$\beta =g_2^{(300)}$] (solid: numerical result; dashed: theoretical prediction). Under certain intensity, the amplitude (blue) is greatly varied whereas the frequency (red) remains static. (b) Bifurcation diagram with (red) and without (blue) control. The steady state (SS) undergoes the Hopf bifurcation at $S^{*}\approx 0.1589$. An oscillation arises when $S>S^{*}$, whose amplitude is altered by the nonlinear controller with ${\mathrm{\Gamma }}_{\beta }=-1$ (SM S.5). (c) The evolutions of (blue: original; red: controlled) oscillations suggesting that the amplitudes of $y$ and $z$ are also suppressed (dashed cycles: $S=0.2$).