Skip to main content
SpringerLink
Log in

Fitness landscapes for coupled map lattices

  • Original Paper
  • Published:
Journal of Biological Physics Aims and scope Submit manuscript

Abstract

Our goal is to match some dynamical aspects of biological systems with that of networks of coupled logistic maps. With these networks we generate sequences of iterates, convert them to symbol sequences by coarse-graining, and count the number of times combinations of symbols occur. Comparison of this with the number of times these combinations occur in experimental data—a sequence of interbeat intervals for example—is a measure of the fitness of each network to describe the target data. The most fit networks provide a cartoon that suggests a decomposition of the experimental data into a component that may be produced by a simple dynamical subsystem, and a residual component, the result of detailed, particular characteristics of the system that generated the target data. In the space of all network parameters, each point corresponds to a particular network. We construct a fitness landscape when we assign a fitness to each point. Because the parameters are distributed continuously over their ranges, and because fitnesses are estimated numerically, any plot of the landscape involves a finite sample of parameter values. We’ll investigate how the local landscape geometry changes when the array of sample parameters is refined, and use the landscape geometry to explore complex relations between local fitness maxima.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)

    Article  Google Scholar 

  2. Wright, S.: The role of mutation, inbreeding, crossbreeding, and selection in evolution. Proc. Sixth Int. Cong. Genetics 1, 356–366 (1932)

    Google Scholar 

  3. Nowak, M.: Evolutionary Dynamics: Exploring the Equations of Life. Harvard University Press (2006)

    Book  Google Scholar 

  4. Waddington, C.: Principles of Development and Differentiation. Macmillian (1966)

  5. Waddington, C.: Strategy of the Genes. A Discussion of Some Aspects of Theoretical Biology. George Allen & Unwin (1957)

  6. Thom, R.: Structural Stability and Morphogenesis. An Outline of a General Theory of Models. Benjamin (1975)

  7. Frauenfelder, H., Sligar, S., Wolynes, P.: The energy landscape and motions of proteins. Science 254, 1589–1603 (1991)

    Article  ADS  Google Scholar 

  8. Frauenfelder, H., et al.: A unified model of protein dynamics. Proc. Nat. Acad. Sci. U.S.A. 106, 5129–5134 (2009)

    Article  ADS  Google Scholar 

  9. Frauenfelder, H., Leeson, D.: The energy landscape in non-biological and biological molecules. Nat. Struct. Biol. 5, 757–759 (1998)

    Article  Google Scholar 

  10. Frauenfelder, H.: Energy landscape and dynamics of biomolecules. J. Biological Physics 31, 413–416 (2005)

    Article  Google Scholar 

  11. Jeffrey, H.: Chaos game representation of gene structure. Nucl. Acid Res. 18, 2163–2170 (1990)

    Article  Google Scholar 

  12. Jeffrey, H.: Chaos game visualization of sequences. Comput. Graph. 16, 25–33 (1992)

    Article  Google Scholar 

  13. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press (1995)

    Book  Google Scholar 

  14. Kitchens, B.: Symbolic Dynamics: One-Sided, Two-Sided, and Countable State Markov Shifts. Springer (1998)

  15. Blanchard, F., Maass. A., Nogueira, A.: Topics in Symbolic Dynamics and Applications. Cambridge University Press (2000)

    Book  Google Scholar 

  16. Hutchinson, J.: Fractals and self-similarity. Indiana Univ. J. Math. 30, 713–747 (1981)

    Article  MathSciNet  Google Scholar 

  17. Barnsley, M.: Fractals Everywhere. 2nd ed., Academic Press (1993)

  18. Stewart, I.: Order within the Chaos Game? Dynamics Newsletter 3, 4–9 (1989)

    Google Scholar 

  19. May, R.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)

    Article  ADS  Google Scholar 

  20. Devaney, R.: An Introduction to Chaotic Dynamical Systems. 2nd ed., Addison-Wesley (1989)

  21. Rasband, N.: Chaotic Dynamics of Nonlinear Systems. Wiley-Interscience (1990)

  22. Hindmarsh, J., Rose, M.: A model of neuronal bursting using using three coupled first order differential equations. Proc. Roy. Soc. London. Biol. Sci. 221, 87–102 (1984)

    Article  ADS  Google Scholar 

  23. Storace, M., Linaro, D., de Lange, E.: The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos 18, 033128 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  24. Sorkin, G.: Efficient simulated annealing on fractal energy landscapes. Algorithmica 6, 367–418 (1991)

    Article  MathSciNet  Google Scholar 

  25. Adami, C.: Introduction to Artificial Life. Springer (1997)

  26. Weinberger, E., Stadler, P.: Why some fitness landscapes are fractal. J. Theoret. Biol. 163, 255–275 (1993)

    Article  ADS  Google Scholar 

  27. Bahar, S.: The Essential Tension: Competition, Cooperation and Multilevel Selection in Evolution. Springer Nature (2018)

    Book  Google Scholar 

  28. Strogatz, S.: Sync: The Emerging Science of Spontaneous Order. Hyperion Books (2003)

  29. http://physionet.org/challenge/2002/event-2.shtml#downloading-the-challenge-dataset Accessed 10 Feb 2013 (2013)

  30. Wessel, N., et al.: Classifying simulated and physiological heart rate variability signals. Computers in Cardiology 29, 133–135 (2002)

    Article  Google Scholar 

  31. Accardo, A., et al.: Use of the fractal dimension for the analysis of electroencephalographic time series. Biol. Cybernetics 77, 339–350 (1997)

    Article  Google Scholar 

  32. Kulish, V., Sourin, A., Sourina, O.: Human encephalograms seen as fractal time series: Mathematical analysis and visualization. Comput. Biol. Med. 36, 291–302 (2006)

    Article  Google Scholar 

  33. Schalk, G., et al.: BCI2000: A general-purpose brain-computer interface (BCI) system. IEEE Trans. Biomed. Eng. 51, 1034–1043 (2004)

    Article  Google Scholar 

  34. Frame, M., Urry, A.: Fractal Worlds: Grown, Built, and Imagined. Yale University Press (2016)

    MATH  Google Scholar 

  35. Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications. 3rd ed., Wiley (2014)

  36. Thew, N.: A Dictionary of Driven Iterated Function Systems to Characterize Chaotic Time Series. Yale University Applied Mathematics Senior Thesis (2014)

Download references

Acknowledgements

This paper is an elaboration of the work in the first author’s Applied Mathematics Yale senior thesis [36]. In early stages of this project we benefitted from many thoughtful and energetic discussions with Rachel Lawrence and Hannah Otis. Comments by a very thoughtful anonymous referee led to improvements in the readability of this paper.

Funding

This research was supported by grant funding for the first author from the Science and Technology Research Scholars (STARS) II program at Yale University.

Author information

Authors and Affiliations

  1. School of Graduate Medical Education, Mayo Clinic, Rochester, MN, USA

    Noelle Driver

  2. Mathematics Department, Yale University, New Haven, CT, USA

    Michael Frame

Authors
  1. Noelle Driver
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Frame
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Frame.

Ethics declarations

Conflicts of interest

We have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the Topical Collection: The Revolutionary Impact of Landscapes in Biology

Guest Editors: Robert Austin, Shyamsunder Erramilli, Sonya Bahar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Driver, N., Frame, M. Fitness landscapes for coupled map lattices. J Biol Phys 47, 215–235 (2021). https://doi.org/10.1007/s10867-021-09577-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10867-021-09577-6

Keywords

Search

Navigation