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A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves

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Abstract

Homeostasis occurs in a control system when a quantity remains approximately constant as a parameter, representing an external perturbation, varies over some range. Golubitsky and Stewart (J Math Biol 74(1–2):387–407, 2017) developed a notion of infinitesimal homeostasis for equilibrium systems using singularity theory. Rhythmic physiological systems (breathing, locomotion, feeding) maintain homeostasis through control of large-amplitude limit cycles rather than equilibrium points. Here we take an initial step to study (infinitesimal) homeostasis for limit-cycle systems in terms of the average of a quantity taken around the limit cycle. We apply the “infinitesimal shape response curve” (iSRC) introduced by Wang et al. (SIAM J Appl Dyn Syst 82(7):1–43, 2021) to study infinitesimal homeostasis for limit-cycle systems in terms of the mean value of a quantity of interest, averaged around the limit cycle. Using the iSRC, which captures the linearized shape displacement of an oscillator upon a static perturbation, we provide a formula for the derivative of the averaged quantity with respect to the control parameter. Our expression allows one to identify homeostasis points for limit cycle systems in the averaging sense. We demonstrate in the Hodgkin–Huxley model and in a metabolic regulatory network model that the iSRC-based method provides an accurate representation of the sensitivity of averaged quantities.

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Notes

  1. “all the vital mechanisms, however varied they may be, have only one object, that of preserving constant the conditions of life in the internal environment”—quoted in translation by Cannon (1929).

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Acknowledgements

PT thanks the Oberlin College Department of Mathematics for research support.

Funding

This work was supported in part by National Institutes of Health BRAIN Initiative Grant R01 NS118606, and NSF Grant DMS-2052109.

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Authors and Affiliations

  1. Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH, 44106, USA

    Zhuojun Yu & Peter J. Thomas

  2. Department of Biology, Case Western Reserve University, Cleveland, OH, 44106, USA

    Peter J. Thomas

  3. Department of Cognitive Science, Case Western Reserve University, Cleveland, OH, 44106, USA

    Peter J. Thomas

  4. Department of Computer and Data Science, Case Western Reserve University, Cleveland, OH, 44106, USA

    Peter J. Thomas

  5. Department of Electrical, Control and Systems Engineering, Case Western Reserve University, Cleveland, OH, 44106, USA

    Peter J. Thomas

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  1. Zhuojun Yu
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Contributions

ZY and PJT designed the study. ZY and PJT derived the results. ZY performed the simulations. ZY and PJT wrote the manuscript.

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Correspondence to Zhuojun Yu.

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The authors declare no competing interests.

Code availability (software application or custom code)

Custom code written in Matlab is available at https://github.com/zhuojunyu-appliedmath/Homeostasis-iSRC.

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A Hodgkin–Huxley equations

A Hodgkin–Huxley equations

In the Hodgkin–Huxley system (20), the activation/inactivation rate functions are

$$\begin{aligned} \alpha _n(V)&=0.01\frac{10-V}{\exp {(\frac{10-V}{10})}-1},&\beta _n(V)&=0.125\exp {\left( \frac{-V}{80}\right) },\\ \alpha _m(V)&=0.1\frac{25-V}{\exp {(\frac{25-V}{10})}-1},&\beta _m(V)&=4\exp {\left( \frac{-V}{18}\right) },\\ \alpha _h(V)&=0.07\exp {\left( \frac{-V}{20}\right) },&\beta _h(V)&=\frac{1}{\exp {(\frac{30-V}{10})}+1}. \end{aligned}$$

Parameter values for the system are listed in Table 1.

Table 1 Parameter values for HH system
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Yu, Z., Thomas, P.J. A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves. J. Math. Biol. 84, 24 (2022). https://doi.org/10.1007/s00285-022-01724-4

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