Double-wavelet approach to study frequency and amplitude modulation in renal autoregulation

Biological time series often display complex oscillations with several interacting rhythmic components. Renal autoregulation, for instance, involves at least two separate mechanisms both of which can produce oscillatory variations in the pressures and flows of the individual nephrons. Using double-wavelet analysis we propose a method to examine how the instantaneous frequency and amplitude of a fast mode is modulated by the presence of a slower mode. Our method is applied both to experimental data from normotensive and hypertensive rats showing different oscillatory patterns and to simulation results obtained from a physiologically based model of the nephron pressure and flow control. We reveal a nonlinear interaction between the two mechanisms that regulate the renal blood flow in the form of frequency and amplitude modulation of the myogenic oscillations.

Figure 1

Experimental recording of the proximal tubular pressure in a single nephron of a rat kidney (a) and results of its wavelet analysis: the dynamics of all local maxima of the energy density (b), the scalogram (c) and the extracted frequencies of rhythmic components (d). The search for $f_{\text{slow}}$ and $f_{\text{fast}}$ was performed in the ranges 0.02–0.07 and $0.1–0.25\mathrm{Hz}$, respectively. The frequency step in the wavelet transform was chosen to be $0.001\mathrm{Hz}$.

Figure 2

Results of the double-wavelet analysis of the time series shown in Fig. 1a. (a) and (b) correspond to the cases of frequency and amplitude modulation, respectively. Black circles mark $f_{\text{slow}}\left(t\right)$ and white circles indicate the instantaneous modulation frequency.

Figure 3

Numerical analysis of the mathematical model (3). Responses to a parameter change of the tubular pressure and modulation properties in the periodic (a,b) and chaotic (c,d) regimes, respectively. Solid curves in (b,d) correspond to the instantaneous frequency of the slow mode. The instantaneous modulation frequencies are presented by dashed curves (the case of amplitude modulation) and by dotted curves (the case of frequency modulation). The parameters for the wavelet analysis are the same as in Fig. 1.

Figure 4

Analysis of the tubular pressure data for a hypertensive rat. (a) is the original time series. (b) demonstrates the instantaneous frequency of the fast mode $f_{\text{fast}}\left(t\right)$. (c) and (d) show the instantaneous frequency of the slow mode (black circles) and the modulation frequency (white circles) for the frequency and amplitude modulation of the fast mode, respectively.

Figure 5

Analysis of blood flow data from the nephron of a normotensive rat. (a) is the original time series. (b) illustrates the instantaneous frequency $f_{\text{fast}}\left(t\right)$. (c) and (d) show $f_{\text{slow}}\left(t\right)$ (black circles) and the modulation frequency (white circles) for the frequency and amplitude modulation of the fast mode, respectively.

Figure 6

Analysis of blood flow data from a nephron of a hypertensive rat (a) exhibiting chaotic dynamics. The instantaneous frequency $f_{\text{fast}}\left(t\right)$ is shown in (b). (c) frequency modulation and (d) amplitude modulation. The extracted modulation frequencies (white circles) correspond to the frequencies of the slow mode (black circles).