Conclusion
A. Summary points regarding physiological response systems
This chapter has addressed some of the major issues involved in the use of time series analyses to detect and quantify periodicities in physiological response systems. The points described will be reviewed and we will conclude with a strategy to quantify periodic rhythms which attempts to overcome some of the problems inherent in psychophysiological research.
1. Physiological data may be described conveniently as time series. The hardware that psychophysiologists commonly use, the polygraph and the computer readily lend themselves to the production of time series data. The polygraph provides a continuous analog signal indexed by time. The computer with an analog-to-digital converter transforms the analog physiological signal into a digital representation at equally spaced intervals. This produces a discrete time series which is a manifestation of a continuous time series. The normal collection of physiological data satisfies many of the sampling requirements for time series analyses. With minor modifications inter-event series, such as heart period, may conform to these sampling requirements.
2. Many physiological response systems exhibit periodicities. These periodic processes tend to be of great interest to psychophysiologists because physiological rhythmicity in general is a manifestation of neural modulation of peripheral processes.
3. The accurate quantification of periodic physiological activity is difficult because physiological data do not conform to the underlying assumptions necessary for most time series analyses. Physiological data are not stationary and the non-stationary components must be removed prior to the analyses of periodic activity.
4. Periodic physiological activity is generally not sinusoidal and can not be fit by one sine wave at the frequency of the periodic activity. Therefore, periodic physiological processes have higher frequency components. These higher frequency components are often observed at integer multiples of the dominant frequency and are harmonics. These harmonics do not indicate that the high frequency activity represents unique physiological activity. Rather, the harmonics reflect the non- sinusoidal nature of the main periodic component.
5. Many periodicities are manifested in more than one physiological system. For example, respiratory activity is manifested in heart period, blood pressure, vasomotor, and skin potential patterns. Thus, the problems described above in the quantification of heart period oscillations will be similar to those encountered in other physiological response systems.
6. Measurement of the covariation of periodic processes in pairs of physiological response systems can be evaluated with cross-spectral analysis. Since physiological systems do not produce a constant periodicity, a weighted coherence can be calculated which expresses the shared variance of two processes across a band of frequencies.
B. Quantification strategies
As we have argued in the above sections, the major problem in quantifying the periodic components of physiological activity is that physiological time series are non-stationary and the periodic components are not sinusoidal. The application of spectral methods to physiological response patterns is extremely limited because spectral methods were developed to work with stationary times series. Moreover, the spectral decomposition theory assumes that the sinusoidal components are statistically independent. In our examples, it has become clear that since periodic physiological processes are not sinusoidal, the spectral decomposition of physiological processes often produces harmonic variances which are highly correlated with the dominant "fundamental" frequency.
Adherence to the following guidelines will minimize some of problems described above.
1. The sampling rate must be fast enough to prevent aliasing. Aliasing may be avoided by sampling faster than twice the frequency of the highest frequency potentially present in the signal.
2. The non-stationary component of the process must be removed before the periodic components can be evaluated. This may be accomplished by the moving polynomial filter procedure which successfully removes aperiodic components from the data set. Other methods of detrending such as the successive difference and linear regression methods do not function as well as the moving Polynomial filter procedure.
3. It is necessary to know the transfer function of a detrending or filtering method. As discussed earlier, some filtering procedures do not produce the assumed fidelity within the frequency band of interest. Moreover, procedures, such as the moving polynomial filter, can be designed to reject different low frequency band widths. Selection of a detrending and filtering method is a function of the frequency band of interest.
4. To accurately quantify various periodic components in the same time series, multiple filtering strategies have to progress from low to higher frequencies. Because periodic physiological activity is not sinusoidal, the variance of a periodic physiological process is distributed not only on the fundamental frequency of the process, but also at whole number integers of the fundamental frequency (i.e., harmonics). These harmonic variances distort the estimates of faster periodic processes. By developing moving polynomial filters which fit into the non-sinusoidal characteristics of slow periodic physiological activity, it becomes possible to more accurately quantify the higher frequency physiological activity. For example, in studying the multiple rhythms in the heart period process, a moving polynomial filter passing frequencies above 0.06 Hz might be used to study the Traube-Hering-Mayer wave and a second filter passing only frequencies above 0.12 Hz might be used to study respiratory sinus arrhythmia.
5. The twice modal duration rule should be used in designing the moving polynomial filter. This rule states that the duration of the polynomial should be twice the period of the modal frequency being evaluated. This rule has been developed as a guideline to deal with the shape of the transfer function of the filter and the fact that the frequency of periodic physiological processes is not constant but varies within a predictable a band of frequencies.
This chapter has briefly described in non-mathematical terms methods to quantify periodic physiological activity. For a number of years progress to this goal was slowed by the absence of appropriate statistical and hardware methods to deal with non-stationary trends and non-sinusoidal periodicities characteristic of physiological response systems. The method, described above as the moving polynomial filter, works well in many situations. No method is perfect, and we have identified problems in simulated situations when we apply this filter to trends or slow periodic processes which have rapid transitions (i.e., step functions or square waves). The moving polynomial can not fit rapid transitions because the slope of the transition mimics the frequency band of interest. Attempts to redesign the moving polynomial to fit rapid transitions, changes the transfer function and greatly attenuates the variances in the frequency band of interest. In spite of this limitation, we believe that the application of the moving polynomial method to detect and quantify periodic physiological activity is a significant advancement over other existing time series methods (1).
Acknowledgements
The preparation of this chapter was supported, in part, by grant HD22628 from the National Institute of Child Health and Human Development awarded to Stephen W. Porges. The authors are grateful to Evan A. Byrne for his diligent work in developing the examples and illustrations. A special thanks is offered to Carol Sue Carter for editorial comments.
Footnote
(1) The application of the moving polynomial procedure to evaluate the amplitude of periodic physiological processes is protected by a U.S. patent. Individuals interested in applying this methodology independent of the commercial products currently available (i.e., the Vagal Tone Monitor and the Vagal Tone Analysis Program manufactured and distributed by Delta- Biometrics, Inc. 9411 Locust Hill Road, Bethesda, MD 20814) should contact the patent holder (SWP) for guidelines.