Skip to main content

Using transition systems to describe and predict the behaviour of structured excitable media


I show how transition systems can be applied to the naturally concurrent behaviour of excitable media. I consider structured excitable media, in which excitations are constrained to propagate only in defined narrow channels, and cannot propagate elsewhere. I define a type of transition system that can be used to describe the complete set of behaviours exhibited by simple structures. The composition rules that result from this definition can be used to automatically deduce the behaviour of more complex structures composed from simpler structures. Several examples illustrate the method, and a software implementation is provided.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32


  • Adamatzky A (eds) (2002) Collision based computing. Springer, London

    MATH  Google Scholar 

  • Adamatzky A (2007) Physarum machines: encapsulating reaction-diffusion to compute spanning tree. Naturwissenschaften 94:975–980

    Article  Google Scholar 

  • Adamatzky A (2010) Physarum machines: computers from slime mould. World Scientific Series on Nonlinear Science, Series A. 74. World Scientific, Singapore, ISBN:978-981-4327-58-9

  • Adamatzky A, De Lacy Costello B, Shirakawa T (2008) Universal computation with limited resources: Belousov-Zhabotinsky and Physarum computers. Int J Bifurcation Chaos 18(8):2373–2389

    Article  MathSciNet  Google Scholar 

  • De Lacy Costello B, Adamatzky A, Jahan I, Zhang L (2011) Towards constructing one-bit binary adder in excitable chemical medium. Chem Phys 381(1–3):88–99

    Article  Google Scholar 

  • Gáspár V, Bazsa G, Beck MT (1983) The influence of visible light on the Belousov Zhabotinskii oscillating reactions applying different catalysts. Z Phys Chem (Leipzig) 264(1):43–48

    Google Scholar 

  • Gorecki J, Gorecka JN, Igarashi Y (2009) Information processing with structured excitable medium. Nat Comput 8:473–492

    Article  MathSciNet  MATH  Google Scholar 

  • Halvorsrud R, Wagner G (1998) Growth patterns of the slime mold Physarum on a nonuniform substrate. Phys Rev E 57(1):941–948

    Article  Google Scholar 

  • Harding SL, Miller JF (2005) Evolution in materio: a real-time robot controller in liquid crystal. In: Proceedings NASA/DoD conference on evolvable hardware. IEEE Press, Washington, DC, USA, pp 229–238

  • Harel D, Pnueli A (1985) On the development of reactive systems. In: Apt KR (eds) Logics and models of concurrent systems. NATO ASI series, vol F-13. Springer, New York, pp 477–498

    Chapter  Google Scholar 

  • Igarashi Y, Górecki J, Górecka JN (2008) One dimensional signal diodes constructed with excitable chemical system. Acta Phys Polon 39(5):1187–1197

    Google Scholar 

  • Jones J (2010) The emergence and dynamical evolution of complex transport networks from simple low-level behaviours. Int J Unconv Comput 6(2):125–144

    Google Scholar 

  • Kuhnert L, Agladze KI, Krinsky VI (1989) Image processing using light-sensitive chemical waves. Nature 337(6204):244–247

    Article  Google Scholar 

  • Milner R (1989) Communication and concurrency. Prentice Hall, London

    MATH  Google Scholar 

  • Motoike I, Yoshikawa K (1999) Information operations with an excitable field. Phys Rev E 59:5354–5360

    Article  Google Scholar 

  • Nakagaki T, Kobayashi R, Nishiura Y, Ueda T (2004) Obtaining multiple separate food sources: behavioural intelligence in the Physarum plasmodium. Proc R Soc B Biol Sci 271(1554):2305–2310

    Article  Google Scholar 

  • O’Keefe S (2009) Implementation of logical operations on a domino substrate. Int J Unconv Comput 5(2):115–128

    Google Scholar 

  • Sparsø J, Furber S (eds) (2001) Principles of asynchronous circuit design—a systems perspective. Kluwer, Boston

    Google Scholar 

  • Steinbock O, Kettunen P, Showalter K (1996) Chemical wave logic gates. J Phys Chem 100:18970–18975

    Article  Google Scholar 

  • Stepney S (2008) The neglected pillar of material computation. Physica D 237(9):1157–1164

    Article  MathSciNet  MATH  Google Scholar 

  • Stevens WM (2008) Logic circuits in a system of repelling particles. Int J Unconv Comput 4(1):61–77

    Google Scholar 

  • Stevens WM (2012) Computing with planar toppling domino arrangements. Nat Comput. doi:10.1007/s11047-012-9341-x

  • Tero A, Ryo K, Toshiyuki N (2007) A mathematical model for adaptive transport network in path finding by true slime mold. J Theor Biol 244(4):553–564

    Article  Google Scholar 

  • Toth R, Stone C, Adamatzky A, De Lacy Costello B, Bull L (2009) Experimental validation of binary collisions between wave fragments in the photosensitive Belousov-Zhabotinsky reaction. Chaos Solitons Fractals 41:1605–1615

    Article  Google Scholar 

  • Tsuda S, Zauner K, Gunji Y (2006) Robot control with biological cells. Biosystems 87(2–3):215–223

    Google Scholar 

  • Wagon S, Pontarelli A, Briggs W, Becker S (2005) The dynamics of falling dominoes. The UMAP J 26(1):35–47

    Google Scholar 

  • Wang X, Kwiatkowska M (2007) On process-algebraic verification of asynchronous circuits. Fundam Inform 80(1–3):283–310

    MathSciNet  MATH  Google Scholar 

  • Winskel G, Nielsen M (1995) Models for concurrency, In: Handbook of logic in computer science, vol 4. Oxford University Press, Oxford, pp 1–20

    Google Scholar 

Download references


The work was partially supported by Leverhulme Trust grant F/00577/1. Thanks to Andrew Adamatzky for useful comments on an earlier draft of this paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to William M. Stevens.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Stevens, W.M. Using transition systems to describe and predict the behaviour of structured excitable media. Nat Comput 12, 393–410 (2013).

Download citation

  • Published:

  • Issue Date:

  • DOI: