Introduction

 

Dynamical systems, model structure, and parameters

Many time-dependent processes can be described by a mathematical model if enough information about the systems' structure is available. In a very simple case, the observed system could be a damped harmonic oscillator, e.g. a spring-mass-system. Suppose for a given initial displacement, time-resolved data about the position of the mass is measured and the strength of the damping has to be determined. This could be done by comparing the measurements with the behavior of a suitable mathematical model, in this case consisting of ordinary differential equations (ODE). The model consists of

  1. a structure, and
  2. a set of parameters.

The structure is a differential equation with placeholders D and k instead of concrete values for the parameters characterizing the spring-mass-system (D) and the strength of the damping (k):

The dynamical variable x represents the displacement of the mass from its resting position. Depending on the damping coefficient, qualitatively different trajectories are possible, as shown in the Figure below. A trajectory is a graph showing x depending on the time t. In order to determine the damping strength, the value of this parameter is changed in the mathematical model until the same trajectory is produced as in the real system. This procedure is called optimization, fitting, or parameter calibration.

Successful parameter calibration requires that a suitable model is fitted to the measured data. If e.g. a model for a single pendulum is fitted to measurements from a double pendulum, the fit is probably not very good and the estimated values for the parameters are difficult to interpret.

 

Driving input functions

 

Suppose the spring-mass-system is not attached to a ceiling but a person holds it. If the person first uplifts and then lowers it again, the displacement x will be changed and the dynamical systems has consequently been driven into a different state. The position of the holding hand is not part of the dynamical system, because its location for each time-point is already known when the experiment is planned and will not to be affected by the state of the spring-mass-system. In general driving input functions affect the systems' state and have a known, independent trajectory.

Experiments and observables

 

An experiment consists of

  1. instructions when to measure which system variables, i.e. the sampling and set of so-called observables
  2. if possible initial values, e.g. a known initial displacement, and
  3. given trajectories of all driving input functions, e.g. how the spring holding hand will be moved.

 

Challenges

 

If the treated systems are more complex, the underlying model structure is often unknown. In addition, usually only a subset of system variables can be measured. And since experiments are expensive, the question arises which measurements will provide the most information. This setting leads to several challenges:

  1. Model discrimination: How to distinguish competing model hypotheses?
  2. Parameter non-identifiabilities: Which parameters may compensate each other leading to the same measurements?
  3. Experimental design: What is the next best experiment given the research question?

 

Multi-Experiment Fitting

 

A key approach to address the above challenges is multi-experiment fitting, where a single model is fitted to several experiments at the same time. It is based on the assumption that kinetic system parameters stay unchanged if different initial values for the system variables or different driving input functions are applied. In the mass-spring-system for example, the damping coefficient will be the same independent on the initial displacement of the mass from its resting position.