Biological version of Braess' paradox arising from perturbed homeostasis

Braess' paradox is an observation in traffic networks in which changes to the structure of the network—such as the addition of a road—which are intended to enhance flow can often instead have the paradoxical effect of reducing flow through the network. Versions of Braess' paradox have subsequently been observed in many other network types. Homeostasis is a seemingly unrelated biological concept in which interacting regulatory mechanisms work in concert to regulate a particular system output such that it is relatively insensitive to changes in input. A classic example is the broad range of ambient temperatures (input) over which mammalian body temperature (output) is close to constant. In this work, we propose a connection between these concepts using the mathematical formulation of infinitesimal homeostasis and argue that perturbations of homeostatic systems in disease may often lead to observations of paradoxical behavior via the universal unfoldings of infinitesimal homeostasis. We illustrate the concept with an example system drawn from the study of the pathophysiology and treatment of asthma; as a network flow model, this exhibits a form of paradoxical behavior akin to Braess' paradox which we argue arises from perturbation of homeostasis due to disease.

Figure 1

Schematic illustration of homeostasis showing relative stability with near constant function (output) over a wider range of stimulus (input). Dashed line indicates the extension for “negative” stimulus.

Figure 2

Unfolding of infinitesimal homeostasis (black, chair homeostasis $Y={\widehat{p}}^3$; blue, $\widehat{q}=+\delta$ perturbation; red, $\widehat{q}=-\delta$ perturbation). Note that the $\widehat{q}=-\delta$ perturbation (red curve) exhibits local extrema while the others are monotonic.

Figure 3

Schematic illustration of transitions between flow configurations expected in the lung (black); possible unfolding with local extrema (gray, dashed).

Figure 4

Input-output map for a fatal asthma simulated lung; input is the ASM fraction remaining after treatment (in the treated airways), while output is the respiratory resistance (at 85% of maximal ASM activation). Region of paradoxical response is highlighted (a). Panels (b) through (f) show flow configurations at selected points (red circles in panel (a), ordered left to right). The color bar indicates flow normalized to nominal [35]. Note in particular that panels (d) and (e) straddle the paradoxical response, with panel (e) exhibiting recruitment in the upper lobe and a visually more homogeneous flow pattern.

Figure 5

Input-output maps (left); comparison between fatal asthma (upper panels) and nonasthma (lower panels). As in Fig. 4, input is the ASM fraction remaining after treatment (in the treated airways), while output is the respiratory resistance (at 85% of maximal ASM activation). Histograms of the instantaneous slope are shown in the right-hand panels (log scale). Fatal asthma maps (upper panels) exhibit nonmonotonic, paradoxical behavior, with significant occurrence of negative slope. Nonasthma maps (lower panels) are predominantly homeostatic and/or monotonic, with slope exclusively near zero or positive. Ten simulations are shown for each group.