Modulating frequency and amplitude of discrete-time biological oscillations

Modulating precisely frequency and amplitude of oscillations is of great significance for sustaining proper functionality of biological systems. Dynamical systems together with bifurcation theories have been proven effective for quantifying these two properties of biological oscillations and to design feasible state-feedback modulators, especially for continuous-time systems. However, a framework for modulating the frequency and the amplitudes of different variables in discrete-time systems is still lacking, although such systems are sometimes more suitable for modeling generation-to-generation or stage-wise biological processes. A corresponding theoretical foundation is therefore needed for the inverse design of feasible modulators targeting prescribed oscillatory properties. To fill the gap, we study the problem of frequency and amplitude modulations of discrete-time oscillations in this work. We consider the oscillations arising from the Neimark-Sacker (NS) bifurcation and formulate precisely the modulation issues. Specifically, we discuss the difference between continuous- and discrete-time oscillations. Because the latter may not be exactly periodic, we establish the problem as modulating the rotational angle and the amplitudes of the invariant closed curve emerging from the NS bifurcation rather than the classical frequency and amplitudes of the discrete-time solution itself. Based on this formulation, we then obtain the analytical approximations of the two properties in terms of system and modulation parameters by computing the normal form of the NS bifurcation. The ratio of the amplitude of different variables is also approximated. In light of these approximations, we successfully design either linear or nonlinear feedback modulators to achieve modulations of rotational angle and amplitude of discrete-time biological oscillations. The efficacy of the proposed framework is validated in representative examples including a biochemical oscillator and a neuronal network exhibiting synchronous dynamics. Our work therefore provides a theoretical foundation of frequency and amplitude modulations in discrete-time biological oscillations.