1.8 The Physics of Excitable and Active Media

With evolution of living cells and multicellular organisms, novel systems appeared on Earth. Biological systems consist of excitable elements: in addition to being kept far from equilibrium by a flow of energy, some cells can be quickly ‘activated’ with transient release of energy, subsequently returning to a resting state far from equilibrium. They only becoming in excitable again after a ‘refractory period’. Collections of excitable cells form excitable media. Each activated cell (reaction) influences neighbouring cells by a coupling process across some space (diffusion). Thus the activity spreads, generating spatio-temporal patterns ruled by reaction-diffusion equations like the Belousov-Zhabotinsky reaction¹s. 

When elements of excitable media are themselves ‘spontaneously active, they constitute active media. Their activity (signals) couple neighbouring elements and generate spatio-temporal patterns also ruled by the reaction-diffusion equations like the BZ reactions.

Active media are best exemplified as coupling between oscillators. Already in the 1600s, the physicist Christiaan Huygens, the inventor of the pendulum clock, had surmised the principle that guides weakly coupled oscillations by observing that a series of clock pendula attached to a wooden wall become synchronised over time. He realised that the small oscillations imparted by each clock to the incompletely rigid wooden wall acted as a mechanical coupling and entrained the neighbouring clocks². Rows of such oscillating pendula weakly coupled to each other simulate well the general principles of one-dimensional active media. If the coupling between the pendula is strong or if the coupling between oscillators is weak but reciprocal like in Huygens pendula, they will oscillate at a unison. Weaker unidirectional coupling called entrainment, produce delays between oscillations and synchronise less rigidly the oscillations and the process, generates ‘travelling waves’ (propagation).³

The rules that operate in excitable and active media are fundamentally the same as those applying to systems of coupled oscillators⁴. However, the rules of generation and propagation of the waves of activity in excitable and active media are very different from the propagation of waves in classic physics such as ocean waves or electromagnetic waves⁵. 

An electrical model of weakly coupled oscillators was used by Balthasar van der Pol in the 1920s to explain a variety of phenomena including relationships between species, shivering from the cold, menstruation, as well as rhythms of the heart.⁶ The Van der Pol Oscillator was based on an electrical feedback system to translate electrical current into tones producing both regularity and chaos⁷. 

Since Van der Pol’s studies, the occurrence of weakly coupled oscillators in biology has been widely proven⁸. We will see that the concepts behind the Van der Pol weakly coupled oscillators have been extended to the nervous system and any system of excitable cells by Richard FitzHugh and Jinichi Nagumo who recognised their similarities with the Hodgkin and Huxley equations for nerve cells. We will also see how extensive the principles based on weakly coupled oscillating systems can be applied to describe biological phenomena more broadly. Albert-László Barabási fervently concludes in his popular book Linked: The New Science of Networks “These laws, applying equally well to the cell and the ecosystem, demonstrate how unavoidable nature’s laws are and how deeply self-organization shapes the world around us.”

Processes, that occur in such excitable media generate identifiable complex spatio-temporal patterns that obey the following rules:

  :: From their origin travelling waves spread in space in a non-decremental manner. 
  :: The waves of activation in 2 or 3 dimensional systems extend from their origin in circumferential or spiral patterns and become travelling waves.
  :: Because of their refractory period, colliding waves in active media annihilate each other rather than summate. 
  :: The direction and apparent speed of propagation of the waves of activity is determined by the frequency gradient of the oscillators such that higher frequencies drive lower frequency oscillators by entrainment with some delay.
  :: The strength and stability of the coupling between excitable elements determines the degree of stereotypy of the spatio-temporal patterns.

An example of the fundamental rules that control the states of activity of excitable and active media is a forest fire. When two fronts of spreading fires in a forest meet, they stop because there are no more trees to burn behind each fire front. The period between the burned region and regrowth of the forest represents the refractory period before a fire can start again. 

The physical nature of the coupling between the interacting elements in excitable and active media varies enormously in different natural systems. The coupling in BZ chemical reactions is mediated by the spatial diffusion of molecules. In systems of connected pendula, the coupling is some mechanical link between individual pendula. In the electrical oscillators of van der Pol, the coupling occurs via currents passively affecting the neighbouring circuit. We will see below that similar coupling occurs in the excitable membranes of nerve and muscle cells during action potentials. Neural circuits driven by oscillating neurons also behave like coupled oscillators with the spatial coupling represented by nerve fibres projecting some distance and acting on other cells via synaptic transmission.

Further down, I will discuss how coupling between congeneric animals forms dynamic clusters as a kind of super-organism (swarms, flocks, schools, herds, etc), that involves specific sensors in each animal activated by neighbouring animals of the cluster. Similar coupling applies to human societies during social interactions giving rise to diverse cultural phenomena. I also will discuss how the discovery of new phenomena involves the identification of the nature of coupling between them.

As we will see, in all these cases, the spatio-temporal patterns generated by excitable and active media can be portrayed as spatio-temporal maps, which represent geometrical representations of the natural phenomena.

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1. See Section 1.6. ↩︎
2. Christiaan Huygens (1673/1986): The pendulum clock or geometrical demonstrations concerning the motion of pendula as applied to clocks, English translation, Iowa State University Press). ↩︎
3. Arthur T Winfree (2001): The Geometry of Biological Time, Springer-Verlag.  ↩︎
4. See Strogatz SH & Stewart I (1993): Coupled oscillators and biological synchronization. Scientific American 269(6):102-109. ↩︎
5. For more on different types of waves, see https://en.wikipedia.org/wiki/Wave ↩︎
6. Van der Pol, B. (1940): Biological rhythms considered as relaxation oscillations. theory of weakly coupled oscillators Acta Medica Scandinavica 103 (S108), 76–88. ↩︎
7. Click here to download a PDF discussing the development of the Van der Pol oscillator from circuits producing audio effects. ↩︎
8. For example, Glass L & Mackey M (1988): From Clocks to Chaos: The Rhythms of Life Princeton University Press;
   Glass L (2001): Synchronisation and rhythmic processes in physiology. Nature 410, 277-284;
   Winfree AT (1967): Biological Rhythms and the Behavior of Populations of Coupled Oscillators. Journal of Theoretical Biology 16(1) 15-42;
   Elbert T, Ray WJ, Kowalik ZJ, Skinner JE, Graf KE & Birbaumer N (1994): Chaos and physiology: deterministic chaos in excitable cell assemblies. Physiological Reviews 74 (1); 47. ↩︎

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