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Persistent instability in a nonhomogeneous delay differential equation system of the Valsalva maneuver. (English) Zbl 1437.92034

Summary: Delay differential equations are widely used in mathematical modeling to describe physical and biological systems, often inducing 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 (VM). We reduce this model from 5 to 2 states (predicting sympathetic tone and heart rate) and categorize the stability properties of the reduced model using a two-parameter bifurcation analysis of the sympathetic delay (\(D_s\)) and time-scale (\(\tau_s\)). Stability regions in the \(D_s\)\(\tau_s\)-plane for this nonhomogeneous system and its homogeneous analog are classified numerically and analytically, identifying transcritical and Hopf bifurcations. Results show that the Hopf bifurcation remains for both the homogeneous and nonhomogeneous systems, while the nonhomogeneous system stabilizes the transition at the transcritical bifurcation. This analysis was compared with results from blood pressure and heart rate data from three subjects performing the VM: a control subject exhibiting sink behavior, a control subject exhibiting stable focus behavior, and a patient with postural orthostatic tachycardia syndrome (POTS) also exhibiting stable focus behavior. Results suggest that instability caused from overactive sympathetic signaling may result in autonomic dysfunction.

MSC:

92C30 Physiology (general)
92C20 Neural biology
34K18 Bifurcation theory of functional-differential equations
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