Non-linear dynamics of biological systems

Abstract

All biological systems can be classified as open, dissipative and non-linear. This review introduces the most typical phenomena associated with non-linearity, dissipation and openness in biological systems. Namely, damped oscillations, self-oscillations, synchronisation, chaotic and noise-induced oscillations are explained, and illustrated by examples from various biological systems. The link is made between the experimentally registered activities and their mathematical counterparts with the help of the qualitative theory of ordinary differential equations. We introduce the ideas of non-linear and oscillatory thinking proposed by Mandelstam in the 1930s, and show how they can be applied to biological systems. An emphasis is made on the intuitive explanation of mathematical concepts rather than mathematical rigour.

Keywords:

• non-linear thinking
• dynamics
• biology
• self-organisation
• oscillation

Acknowledgements

The author is grateful to Dr Alexander Balanov for his helpful comments on the draft of this paper and for help in preparing some figures; to Christopher Marsden for assistance in editing the text of the manuscript; and to Mario Markus and Lewi Stone for providing pictures illustrating their research.

Notes

 1. In biological systems the parameters are often not constant, e.g. the external temperature might vary, or adrenalin level might float. But we can assume that these parameters obey some other evolution equations with constant parameters (and possibly noise), which are coupled to the given biological subsystem. If necessary, we can assume a hierarchy of such systems. Thus, a biological system with changing parameters can still be described in terms of a dynamical system with constant parameters.

 2. However, in principle, we can include more variables than necessary, provided that any new variable is not a linear combination of others, see below. In that case the dimension of the system is the total number of variables considered.

 3. The most popular examples are ordinary differential equations of the first and higher orders, discrete-time equations (or maps), partial differential equations, integral equations, delay differential equations, and stochastic differential equations.

 4. A function is called smooth if it is infinitely many times differentiable with respect to all of its arguments.

 5. In fact, this can serve an illustration of Euler method to numerically solve systems of ordinary differential equations.

 6. Understanding of a mathematically rigorous definition of an invariant manifold requires specialist training in mathematics and can be found e.g. in [119]. Here we assume that the reader does not necessarily have such mathematical background, and present a relaxed idea of a manifold in almost layman terms. We only mention the useful properties of the manifolds that are relevant to the problems considered.

 7. Although of course always approximate to a lesser or a greater extent.

 8. A similar, but mathematically more rigorous and detailed, discussion can be found in [120].

 9. Self-oscillations will be discussed in Section 5.

10. Strictly speaking, energy of conservative systems can be rigorously introduced only for Hamiltonian systems, i.e. the ones that can be rewritten in a special mathematical form [16]. However, it helps to speak about the energy of a conservative system to convey the general idea.

11. A heart contains a large group of different pacemaker cells that jointly regulate the heart rhythm.

12. These are the most popular routes, which are most often observed in various experiments. There is a number of less popular routes predicted by mathematicians.

13. ‘All animals are equal, but some animals are more equal than others’ from Animal Farm by George Orwell.