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Detection of Nonlinearity in Cardiovascular Variability Signals using Cyclostationary Analysis

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Abstract

A novel approach for detection of polynomial nonlinearity in the neuro-cardiovascular system based on cyclostationary analysis is presented. Metronome breathing is employed to provide a sinusoidal input to the neuro-cardiovascular system in which Heart Rate Variability (HRV) and Blood Pressure Variation (BPV) are considered as its outputs. The presence of new harmonics of the main respiratory rate in the HRV and BPV is investigated by using the concept of (self) phase and (self) frequency coupling. It is shown that a second order polynomial nonlinear system is actually involved in producing the HRV and BPV. The strength of this nonlinearity decreases with increasing the breathing rate.

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Notes

  1. For real signals C 1y_o (α) is symmetric with respect to α, therefore only α ≥  0 is needed for the analysis.

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Authors and Affiliations

  1. Division of Medical Devices, University of Ottawa Heart Institute, 40 Ruskin St., Ottawa, ON, Canada, K1Y 4W7

    Saeid Seydnejad

  2. Division of Medical Devices, University of Ottawa Heart Institute, 1370 Chicory Place, Ottawa, ON, Canada, K1J 1E7

    Saeid Seydnejad

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  1. Saeid Seydnejad
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Correspondence to Saeid Seydnejad.

Appendix A

Appendix A

The amplitude of the input sinusoids in Eq. (18) are written in the form of a modulated signal composed of a constant term and a time varying part, w 1(t) = W 1 + Δw 1(t), w 2(t) = W 2 + Δw 2(t). In order for the inequality (19) to be valid we should have,

$$ \frac{C_{1y} (2f_0)}{C_{1y} (f_0)} > \frac{C_{1x} (2f_0)}{C_{1x} (f_0)} \Rightarrow \frac{\frac{a}{2}m_{w_2} +\frac{b}{4}m_{w_1^2}} {\frac{a}{2}m_{w_1} +\frac{b}{2}m_{w_1 w_2}} > \frac{m_{w_2}}{m_{w_1}} \Rightarrow \frac{b}{4}m_{w_1} m_{w_1^2} > \frac{b}{2}m_{w_1 w_2} m_{w_2} $$
(20)

Hence,

$$ \begin{array}{l} E\left. \left\{ W_1 \left( 1+\frac{\Delta w_1 (t)}{W_1} \right) \right. \right\} E\left. \left\{ W_1^2\left( 1+\frac{\Delta w_1 (t)}{W_1} \right)^2 \right. \right\} > 2E\left. \left\{ W_1 \left( 1+\frac{\Delta w_1 (t)}{W_1} \right) W_2 \left( 1+\frac{\Delta w_2 (t)}{W_2} \right) \right. \right\} E\left. \left\{ W_2 \left( 1+\frac{\Delta w_2 (t)}{W_2} \right) \right. \right\} \\ \Rightarrow W_1^3E\left. \left\{ 1+\frac{\Delta w_1 (t)}{W_1} \right. \right\} E\left. \left\{ 1+\frac{2\Delta w_1 (t)}{W_1} \right. \right\} > 2W_1 W_2^2E\left. \left\{ 1+\frac{\Delta w_1 (t)}{W_1}+\frac{\Delta w_2 (t)}{W_2} \right. \right\} E\left. \left\{ 1+\frac{\Delta w_2 (t)}{W_2} \right. \right\} \end{array} $$
(21)

Note that we have ignored the second order terms compared to the first order terms (e.g. Δw 21 (t) << Δw 1 (t)) in the above derivation. The worst case condition for the above inequality to be valid occurs when Δw 1(t) = 0 and Δw 2(t) = W 2 then Eq. (21) becomes,

$$ W_1^3 > 8W_1 W_2^2 \Rightarrow W_1 > \sqrt 8 \,W_2 \Rightarrow W_1 > 2.828 W_2 \Rightarrow W_1 > 3 W_2 $$
(22)

This means that in the worst case condition (19) remains valid as long as the magnitude of the second order harmonic is three times smaller than the magnitude of the main component.

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Seydnejad, S. Detection of Nonlinearity in Cardiovascular Variability Signals using Cyclostationary Analysis. Ann Biomed Eng 35, 744–754 (2007). https://doi.org/10.1007/s10439-007-9281-2

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