A Novel Use of Pseudospectra in Mathematical Biology: Understanding HPA Axis Sensitivity

TL;DR

This work develops and applies pseudospectral analysis to a nonlinear delay differential equation model of the HPA axis to understand perturbation sensitivity and transient dynamics. By computing three distinct pseudospectra—the time-dependent Jacobian, Floquet pseudospectrum around the limit cycle, and a data-driven Koopman/DMD approach—the authors obtain complementary views on local and global stability, transient growth, and data-driven perturbation behavior. They derive new methods for pseudospectra on Banach spaces and for applying DMD to DDEs, linking mathematical findings to experimental observations such as enhanced responses during upward cortisol slopes. The results support model substantiation by aligning with rat experiments, quantify transient effects via Kreiss constants, and point to pathways for personalized, data-informed modeling in endocrinology.

Abstract

The Hypothalamic-Pituitary-Adrenal (HPA) axis is a major neuroendocrine system, and its dysregulation is implicated in various diseases. This system also presents interesting mathematical challenges for modeling. We consider a nonlinear delay differential equation model and calculate pseudospectra of three different linearizations: a time-dependent Jacobian, linearization around the limit cycle, and dynamic mode decomposition (DMD) analysis of Koopman operators (global linearization). The time-dependent Jacobian provided insight into experimental phenomena, explaining why rats respond differently to perturbations during corticosterone secretion's upward versus downward slopes. We developed new mathematical techniques for the other two linearizations to calculate pseudospectra on Banach spaces and apply DMD to delay differential equations, respectively. These methods helped establish local and global limit cycle stability and study transients. Additionally, we discuss using pseudospectra to substantiate the model in experimental contexts and establish bio-variability via data-driven methods. This work is the first to utilize pseudospectra to explore the HPA axis.

Figures (7)

• Figure 1: A schematic of the HPA axis. The sun represents that CRH is entrained by the day and night cycle, which triggers an enveloping of cortisol and ACTH on a circadian level. On an ultradian level, ACTH triggers cortisol, which then inhibits ACTH. Both operate on regular pulses, the amplitude of which varies according to the other and the wider system. The lightning bolt represents external stressors that cause ACTH and, thereby, cortisol to rise. The dashed line represents the restriction we make to the negative feedback loop between ACTH and cortisol that we consider in this paper.
• Figure 2: Left: Trajectories with 100 randomly selected constant initial conditions (red dots) for $h=7.66$ and limit cycle (green). Right: The limit cycle as $h$ is varied.
• Figure 3: (a) -- (d): Solution trajectories, spectral abscissa, distance to instability, and ratio $-\alpha/d_\tau$. (e) -- (h): Pseudospectra of the linearized system for different $\tau$. We display pseudospectra by plotting contours of $\epsilon$ (colorbars). The eigenvalues are shown in red. The stability line (imaginary axis) is the dotted magenta line.
• Figure 4: Maximum $\alpha$ and index $-\alpha/d\tau$ (over the limit cycle) as $h$ varies.
• Figure 5: Left: The pseudospectrum of $T$ for the default parameter $h=7.66$. The eigenvalues are shown in red, and the stability circle (unit circle) is the dotted magenta circle. Right: The Kreiss constant for different values of $h$ (the dashed line is $h=7.66$).

Theorems & Definitions (1)

• definition thmcounterdefinition: Pseudospectra of Matrices