Persistent instability in a nonhomogeneous delay differential equation system of the Valsalva maneuver

TL;DR

This paper investigates how delays in a nonlinear neurological control model affect stability during the Valsalva maneuver, revealing unstable modes and bifurcations that relate to POTS patient symptoms.

Contribution

It introduces a detailed bifurcation analysis of a nonhomogeneous delay differential equation model of the autonomic nervous system during the Valsalva maneuver, linking model dynamics to clinical observations.

Findings

Unstable modes linked to parameter interactions between sympathetic delay and time-scale.

Discrepancies between solutions are due to stabilizing forcing functions.

Presence of Hopf bifurcation suggests oscillatory behavior in the system.

Abstract

Delay differential equations (DDEs) are widely used in mathematical modeling to describe physical and biological systems. Delays can impact model dynamics, resulting in oscillatory behavior. In physiological systems, this instability may signify (i) an attempt to return to homeostasis or (ii) system dysfunction. In this study, we analyze a nonlinear, nonautonomous, nonhomogeneous open-loop neurological control model describing the autonomic nervous system response to the Valsalva maneuver. Unstable modes have been identified as a result of parameter interactions between the sympathetic delay and time-scale. In a two-parameter bifurcation analysis, we examine both the homogeneous and nonhomogeneous systems. Discrepancies between solutions result from the presence of the forcing functions which stabilize the system. We use analytical methods to determine stability regions for the homogeneous system, identifying transcendental relationships between the parameters. We also use computational methods to determine stability regions for the nonhomogeneous system. The presence of a Hopf bifurcation within the system is discussed and solution types from the sink and stable focus regions are compared to two control patients and a patient with postural orthostatic tachycardia syndrome (POTS). The model and its analysis support the current clinical hypotheses that patients suffering from POTS experience altered nervous system activity.