Nonautonomous Dynamical Systems in the Life Sciences

Nonautonomous dynamics describes the qualitative behavior of evolutionary differential and difference equations, whose right-hand side is explicitly time-dependent. Over recent years, the theory of such systems has developed into a highly active field related to, yet recognizably distinct from that of classical autonomous dynamical systems. This development was motivated by problems of applied mathematics, in particular in the life sciences where genuinely nonautonomous systems abound. In this survey, we introduce basic concepts and tools for appropriate nonautonomous dynamical systems and apply them to various representative biological models.

This work introduces a notion of random dynamical systems with inputs, providing several basic definitions and results on equilibria and convergence. It also presents a “converging input to converging state” (“CICS”) result, a concept that plays a key role in the analysis of stability of feedback interconnections, for monotone systems.

An important feature of many physiological systems is that they evolve on multiple scales. From a mathematical point of view, these systems are modeled as singular perturbation problems. It is the interplay of the dynamics on different temporal and spatial scales that creates complicated patterns and rhythms. Many important physiological functions are linked to time-dependent changes in the forcing which leads to nonautonomous behaviour of the cells under consideration. Transient dynamics observed in models of excitability are a prime example.Recent developments in canard theory have provided a new direction for understanding these transient dynamics. The key observation is that canards are still well defined in nonautonomous multiple scales dynamical systems, while equilibria of an autonomous system do, in general, not persist in the corresponding driven, nonautonomous system. Thus canards have the potential to significantly shape the nature of solutions in nonautonomous multiple scales systems. In the context of neuronal excitability, we identify canards of folded saddle type as firing threshold manifolds. It is remarkable that dynamic information such as the temporal evolution of an external drive is encoded in the location of an invariant manifold—the canard.

If a network of cells is repeatedly driven by the same sustained, complex signal, will it give the same response each time? A system whose response is reproducible across repeated trials is said to be reliable. Reliability is of interest in, e.g., computational neuroscience because the degree to which a neuronal network is reliable constrains its ability to encode information via precise temporal patterns of spikes. This chapter reviews a body of work aimed at discovering network conditions and dynamical mechanisms that can affect the reliability of a network. A number of results are surveyed here, including a general condition for reliability and studies of specific mechanisms for reliable and unreliable behavior in concrete models. This work relies on qualitative arguments using random dynamical systems theory, in combination with systematic numerical simulations.

First, we introduce nonautonomous oscillator—a self-sustained oscillator subject to external perturbation and then expand our formalism to two and many coupled oscillators. Then, we elaborate the Kuramoto model of ensembles of coupled oscillators and generalise it for time-varying couplings. Using the recently introduced Ott-Antonsen ansatz we show that such ensembles of oscillators can be solved analytically. This opens up a whole new area where one can model a virtual physiological human by networks of networks of nonautonomous oscillators. We then briefly discuss current methods to treat the coupled nonautonomous oscillators in an inverse problem and argue that they are usually considered as stochastic processes rather than deterministic. We now point to novel methods suitable for reconstructing nonautonomous dynamics and the recently expanded Bayesian method in particular. We illustrate our new results by presenting data from a real living system by studying time-dependent coupling functions between the cardiac and respiratory rhythms and their change with age. We show that the well known reduction of the variability of cardiac instantaneous frequency is mainly on account of reduced influence of the respiration to the heart and moreover the reduced variability of this influence. In other words, we have shown that the cardiac function becomes more autonomous with age, pointing out that nonautonomicity and the ability to maintain stability far from thermodynamic equilibrium are essential for life.

One of the key aspects in the study of cellular communication is understanding how cells receive a continuous input and transform it into a discrete, all-or-none output. Such so-called ultrasensitive dose responses can also be used in a variety of other contexts, from the efficient transport of oxygen in the blood to the regulation of the cell cycle and gene expression. This chapter provides a self contained mathematical review of the most important molecular models of ultrasensitivity in the literature, with an emphasis on mechanisms involving multisite modifications. The models described include two deeply influential systems based on allosteric behavior, the MWC and the KNF models. Also included is a description of more recent work by the author and colleagues of novel mechanisms using alternative hypotheses to create ultrasensitive behavior.

Mathematical modeling plays an important and increasing role in drug development. The objective of this chapter is to present the concept of pharmacokinetic (PK) and pharmacodynamic (PD) modeling applied in the pharmaceutical industry. We will introduce typically PK and PD models and present the underlying pharmacological and biological interpretation. It turns out that any PKPD model is a nonautonomous dynamical system driven by the drug concentration. We state a theoretical result describing the general relationship between two widely used models, namely, transit compartments and lifespan models. Further, we develop a PKPD model for tumor growth and anticancer effects based on the present model figures and apply the model to measured data.

Chronic infection with hepatitis C or hepatitis B virus are important world-wide health problems leading to long-term damage of the liver. There are, however, treatment options which can lead to viral eradication in hepatitis C or long-term viral suppression in hepatitis B in some patients. Nevertheless, there is still room for improvement. Mathematical compartment models based on ordinary differential equation systems have successfully been applied to improve antiviral treatment. Here, we illustrate how mathematical and statistical analysis of such models influenced clinical research and give an overview on the most important models for hepatitis C and hepatitis B viral kinetics.

Stochastic differential equations (SDEs) provide an appropriate framework for modeling biomedical problems, since they allow detailed a priori biochemical knowledge to be accounted for and at the same time are able to describe the noise in the systems under investigation and in the data without excessively complicating the settings. We present three application paradigms related to an intracellular signaling pathway, to radio-oncological treatments, and to cell dispersal.