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

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 twoparameter 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.

Global sensitivity analysis informed model reduction and selection applied to a Valsalva maneuver model

“…As shown in our previous work [36], τ s , appearing in equation ( 9), is nonlinearly related to the delay D s and this relationship can lead to instability. Therefore, we restricted the upper and lower bounds of τ s in this study to ±50% rather than use the entire physiological range (mean ± 2 SD) determined in [35] to ensure the parameter space Ω p did not intersect an oscillatory regime.…”

“…Parameters calculated from the data have bounds set to the mean value ± 2 standard deviations (SD). An exception is the parameter τ s , which interacts with the delay D s and can cause instability [36]. To remain in the stable region, τ s is varied ± 50% of its nominal value.…”

“…The necessity of this parameter has been supported in previous studies [20,35], but the aim of the model reduction in the next section explores whether it is structurally necessary to include both baroreceptor regions. The sympathetic time-scale τ s has been found in our previous work to be nonlinearly related to the delay D s [36]. In this study, we have held D s constant at 3 s and have only varied τ s within parameter bounds that maintain stable model behavior (Table 1).…”