We consider a generic biological process described by a dynamical system, subject to an input signal with a high-frequency periodic component. The rapid oscillations of the input signal induce inherently multiscale dynamics, motivating order-reduction techniques. It is intuitive that the system behaviour is well approximated by its response to the averaged input signal. However, changes to the high-frequency component that preserve the average signal are beyond the reach of such intuitive reasoning. In this study, we explore system response under the influence of such an input signal by exploiting the timescale separation between high-frequency input variations and system response time. Employing the asymptotic method of multiple scales, we establish that, in some circumstances, the intuitive approach is simply the leading-order asymptotic contribution. We focus on higher-order corrections that capture the response to the details of the high-frequency component beyond its average. This approach achieves a reduction in system complexity while providing valuable insight into the structure of the response to the oscillations. We develop the general theory for nonlinear systems, while highlighting the important case of systems affine in the state and the input signal, presenting examples of both discrete and continuum state spaces. Importantly, this class of systems encompasses biochemical reaction networks described by the chemical master equation and its continuum approximations. Finally, we apply the framework to a nonlinear system describing mRNA translation and protein expression previously studied in the literature. The analysis shines new light on several aspects of the system quantification and both extends and simplifies results previously obtained.

我们考虑由动态系统描述的通用生物过程，受具有高频周期分量的输入信号的影响。输入信号的快速振荡会引起固有的多尺度动态，从而激发降阶技术。直观的是，系统行为可以通过其对平均输入信号的响应来很好地近似。然而，保持平均信号的高频分量的变化超出了这种直观推理的范围。在这项研究中，我们通过利用高频输入变化和系统响应时间之间的时间尺度分离来探索在这种输入信号影响下的系统响应。采用多尺度的渐近方法，我们确定，在某些情况下，直观的方法只是领先的渐近贡献。我们专注于捕获对超出其平均值的高频分量细节的响应的高阶校正。这种方法降低了系统复杂性，同时提供了对振荡响应结构的宝贵见解。我们开发了非线性系统的一般理论，同时强调了状态和输入信号中系统仿射的重要情况，并提供了离散和连续状态空间的示例。重要的是，这类系统包含由化学主方程及其连续近似描述的生化反应网络。最后，我们将该框架应用于描述先前在文献中研究的 mRNA 翻译和蛋白质表达的非线性系统。