Searching bifurcations in high-dimensional parameter space via a feedback loop breaking approach

📝 Abstract

Bifurcations leading to complex dynamical behaviour of non-linear systems are often encountered when the characteristics of feedback circuits in the system are varied. In systems with many unknown or varying parameters, it is an interesting, but difficult problem to find parameter values for which specific bifurcations occur. In this paper, we develop a loop breaking approach to evaluate the influence of parameter values on feedback circuit characteristics. This approach allows a theoretical classification of feedback circuit characteristics related to possible bifurcations in the system. Based on the theoretical results, a numerical algorithm for bifurcation search in a possibly high-dimensional parameter space is developed. The application of the proposed algorithm is illustrated by searching for a Hopf bifurcation in a model of the mitogen activated protein kinase (MAPK) cascade, which is a classical example for biochemical signal transduction.

💡 Analysis

Bifurcations leading to complex dynamical behaviour of non-linear systems are often encountered when the characteristics of feedback circuits in the system are varied. In systems with many unknown or varying parameters, it is an interesting, but difficult problem to find parameter values for which specific bifurcations occur. In this paper, we develop a loop breaking approach to evaluate the influence of parameter values on feedback circuit characteristics. This approach allows a theoretical classification of feedback circuit characteristics related to possible bifurcations in the system. Based on the theoretical results, a numerical algorithm for bifurcation search in a possibly high-dimensional parameter space is developed. The application of the proposed algorithm is illustrated by searching for a Hopf bifurcation in a model of the mitogen activated protein kinase (MAPK) cascade, which is a classical example for biochemical signal transduction.

📄 Content

A frequent challenge in the analysis of non-linear dynamical systems is to find parameter values for which the system undergoes changes in its dynamical behaviour. Such changes are directly related to the emergence of complex dynamical behaviour.

Standard cases of complex dynamical behaviour are multistability, i.e. the existence of several stable steady states, limit cycle oscillations, and non-periodic oscillations.

Feedback circuits are the major structural feature in the emergence of complex dynamical behaviour. In particular, it can be shown that a positive feedback circuit in the system is required for multistationarity [Kaufman et al., 2007], whereas a negative circuit is typically required for limit cycle oscillations [Snoussi, 1998]. This importance of feedback circuits makes control theory a natural tool for the analysis of complex dynamical behaviour.

Yet, the main properties of a system’s qualitative dynamical behaviour are the location and stability of equilibrium points. Knowledge of these is often also useful when analysing complex dynamical behaviour. It is well known from dynamical systems theory that two stable equilibrium points are separated by an invariant repellor, which contains an unstable equilibrium point in most cases. Similarly, stable limit cycle oscillations usually coexist with an unstable equilibrium point. Also transient behaviour is often governed by the attraction to and repulsion from equilibrium points. Thus a convenient first step when studying the qualitative behaviour of a dynamical system is to look at stability properties of equilibrium points.

A classical tool for analysing the influence of parameter values on the location and stability of equilibrium points is bifurcation analysis. Bifurcation analysis is done routinely with numerical continuation methods for one adjustable bifurcation parameter [Kuznetsov, 1995]. Methods for numerical bifurcation analysis in several parameters are now being developed [Henderson, 2007, Stiefs et al., 2008], but due to practical considerations, they remain limited to two or three adjustable bifurcation parameters.

The challenge to find parameter values for bifurcations is of particular relevance in the area of biological systems. The main reasons for this are that biological function is often based on complex dynamical behaviour, and that parameters can vary within a large range due to environmental or internal conditions.

There are many examples where complex dynamical behaviour of a non-linear biological system can directly be related to biological function. Some examples from the specific area of biochemical signal transduction within living cells are given by bistability in the mitogen activated protein kinase (MAPK) pathway to induce developmental processes [Ferrell and Xiong, 2001], rapid activation of caspases upon an over-threshold stimulus in programmed cell death [Eissing et al., 2004], and sustained oscillations in circadian clocks [Leloup and Goldbeter, 2003].

Systems for biochemical signal transduction are usually modelled with non-linear ordinary differential equations (ODEs). Many models of biochemical systems contain a high number of model parameters, usually even more parameters than state variables. A major problem in understanding biochemical systems is that most of these parameters are not very well known from measurements, and that they often vary significantly due to internal or environmental conditions of the cell. Thus analysing the influence of uncertain or varying parameters on stability properties is a fundamental issue towards understanding dynamical behaviour of biochemical systems. Moreover, to avoid overlooking relevant effects it is necessary to consider simultaneous changes in all adjustable parameters [Stelling et al., 2004, Kim et al., 2006].

The requirement of looking at simultaneous changes in several parameters makes the application of classical continuation methods problematic, as these require to define a line in parameter space along which equilibrium points are tracked. A good choice of this line is essential to obtain meaningful results, yet this choice is often done by intuitive understanding of the system in the better case or iterative trials in the worse.

Often only a single parameter is varied at a time, but then again the choice of the parameter to vary is not trivial and needs to be done for example via sensitivity considerations.

In this paper, we present a new method to locate points in a possibly high-dimensional parameter space for a change in stability properties of equilibrium points, often hinting to either emergence or loss of complex dynamical behaviour. The method is based on considering the dynamical system as a closed loop feedback system. It is then possible to study properties of the original system in terms of an adequately defined open loop system. If the open loop system is well chosen, then its dynamical behaviour is much simpler than that of the closed l

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