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Beyond Limit Cycles

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Abstract

Dynamical approaches to biological systems emphasize the qualitative nature of the time-dependent behaviors that can be observed as parameter values are changed. If the important parameters can be identified, then they can be manipulated experimentally. For example, in feedback control, the stability of the fixed point depends on the interplay between the delay τ and the feedback gain. It is surprising that experimentalists did not begin earlier than they did to investigate systematically the behaviors of biological systems as parameters were changed. One barrier was that inexpensive personal computers did not become available until the 1970s. The second barrier was that experimentalists did not have catalogs that documented the types of behaviors that could occur (see, for example, [215, 600]). It can be very difficult at the benchtop to interpret complex time series unless there is some sense of the nature of the phenomena that one is looking for.

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Notes

  1. 1.

    Robert McCredie May, Baron May of Oxford (b. 1936), Australian theoretical ecologist.

  2. 2.

    As in Chapter 5, we use the symbol μ to designate the bifurcation parameter.

  3. 3.

    Mitchell Jay Feigenbaum (b. 1944), American mathematical physicist.

  4. 4.

    Although we have chosen to focus on modeling the human brain to illustrate our points, the same issues arise in other complex biological systems such as the metabolism of a cell, passage from a single cell to tissues and organs, and the behaviors of schools of fish, herds of animals, flocks of birds, and ecosystems.

  5. 5.

    Eugene Izhikevich (b. 1967), Russian-born American mathematical neuroscientist.

  6. 6.

    The FHN equation is the special case for which the w-nullcline is a straight line.

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Authors and Affiliations

  1. W.M. Keck Science Department, The Claremont Colleges, Claremont, CA, USA

    John Milton

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Toru Ohira

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Milton, J., Ohira, T. (2014). Beyond Limit Cycles. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_11

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