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Homeostasis, singularities, and networks
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Golubitsky, Martin and Stewart, Ian (2017) Homeostasis, singularities, and networks. Journal of Mathematical Biology, 74 (1-2). pp. 387-407. doi:10.1007/s00285-016-1024-2 ISSN 0303-6812.
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Official URL: http://dx.doi.org/10.1007/s00285-016-1024-2
Abstract
Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter λλ varies over some interval. We discuss two main aspects of homeostasis, both related to the effect of coordinate changes on the input–output map. The first is a reformulation of homeostasis in the context of singularity theory, achieved by replacing ‘approximately constant over an interval’ by ‘zero derivative of the output with respect to the input at a point’. Unfolding theory then classifies all small perturbations of the input–output function. In particular, the ‘chair’ singularity, which is especially important in applications, is discussed in detail. Its normal form and universal unfolding λ3+aλλ3+aλ is derived and the region of approximate homeostasis is deduced. The results are motivated by data on thermoregulation in two species of opossum and the spiny rat. We give a formula for finding chair points in mathematical models by implicit differentiation and apply it to a model of lateral inhibition. The second asks when homeostasis is invariant under appropriate coordinate changes. This is false in general, but for network dynamics there is a natural class of coordinate changes: those that preserve the network structure. We characterize those nodes of a given network for which homeostasis is invariant under such changes. This characterization is determined combinatorially by the network topology.
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Journal of Mathematical Biology | ||||||||
Publisher: | Springer | ||||||||
ISSN: | 0303-6812 | ||||||||
Official Date: | January 2017 | ||||||||
Dates: |
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Volume: | 74 | ||||||||
Number: | 1-2 | ||||||||
Page Range: | pp. 387-407 | ||||||||
DOI: | 10.1007/s00285-016-1024-2 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access |
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