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J Appl Physiol 101: 1425-1431, 2006. First published June 22, 2006; doi:10.1152/japplphysiol.01364.2004
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Wavelet analysis of scaling properties of gastric electrical activity

Bruce J. West,1 Artur Maciejewski,2 Miroslaw Latka,2 Tadeusz Sebzda,3 Zbigniew Swierczynski,4 Sylwia Cybulska-Okolow,5 and Eugeniusz Baran5

1Mathematical and Information Science Directorate, Army Research Office, Research Triangle Park, North Carolina; 2Institute of Physics, Wroclaw University of Technology; 3Department of Pathophysiology, Wroclaw Medical University; 4Department of Electronic and Photonic Metrology, Wroclaw University of Technology; and 5Department of Dermatology, Venerology and Allergology, Wroclaw University of Medicine, Wroclaw, Poland

Submitted 8 December 2004 ; accepted in final form 10 May 2006


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
We present a novel approach to the analysis of fluctuations in human myoelectrical gastric activity measured noninvasively from the surface of the abdomen. The time intervals between successive maxima of the wavelet transformed quasi-periodic electrogastrographic waveform define the gastric rate variability (GRV) time series. By using the method of average wavelet coefficients, the statistical fluctuations in the GRV signal in healthy individuals are determined to scale in time. Such scaling was previously found in a variety of physiological phenomena, all of which support the hypothesis that physiological dynamics utilize fractal time series. We determine the scaling index in a cohort of 17 healthy individuals to be 0.80 ± 0.14, which compared with a set of surrogate data is found to be significant at the level P < 0.01. We also determined that the dynamical pattern, so evident in the spectrum of average wavelet coefficients of the GRV time series of healthy individuals, is significantly reduced in a cohort of systemic sclerosis patients having a scaling index 0.64 ± 0.17. These results imply that the long-term memory in GRV time series is significantly reduced from healthy individuals to those with systemic sclerosis. Consequently, this disease degrades the complexity of the underlying gastrointestinal control system and this degradation is manifest in the loss of scaling in the GRV time series.

electrogastrography; gastric rate variability; systemic sclerosis; fractal physiology; average wavelet coefficients; fractal time series

ONE OF THE MORE SIGNIFICANT influences that the concept of fractals has had on science in the past 20 years has to do with its twofold impact on physiology. First of all it influenced the description of anatomical structure such as the bronchial tree in the mammalian lung (30), the His-Purkinje conduction system in the human heart (8), the urinary collecting tubules of the kidney, and the folds of the surface of the brain (10), to name a few. Each of these anatomical structures was shown to lack a characteristic scale over a significant distance domain and to be usefully described by geometrical fractals. A fractal object is one in which the parts resemble the whole in some well-defined way, often referred to as self-similarity, self-affinity, or scaling. Of course, a real physiological structure would have a largest and a smallest scale, thereby violating the mathematical definition of a fractal. Consequently, whether the concept of a fractal is useful in a particular physiological context is determined by the span of scales over which the object is scale free or self-affine (2, 3).

Complementing the change in anatomical perspective was a potent reexamination of time series generated by a number of physiological phenomena whose fluctuations lack a characteristic scale over a significant time domain and that are well described by statistical fractals. See the APPENDIX for a mathematical discussion of the defining properties of statistical fractals, as for fractal statistical physiological phenomena we have the cardiac rhythm of a beating heart (22), cerebral blood flow (35), the respiratory rhythm of breathing (1), and the motor control rhythm of walking (12, 32), to name a few.

Herein we add the stomach to the growing list of organs that utilize fractal time series for their successful operation. This should come as no surprise, but it is important to note that the nature of the stomach's fractal dynamics had not been previously documented. The main function of the stomach is associated with what appeared to be periodic contractions that help to mix, grind, and move food toward the small bowel. These contractions are intimately related to spontaneous, rapid intracellular electrical depolarization caused by the exchange of ions through the membrane of smooth muscle cells.

Although the mechanism of propagation of gastric electrical activity from one cell to another is still unknown, it is believed that periodic waves of depolarization and repolarization originate from a pacemaker area located in the corpus around the greater curvature of the stomach and spread aborally toward the pylorus. The contractions are initiated by the vagus nerve. The visceromotor part of the vagus nerve innervates ganglion neurons that go to the lung for bronchoconstriction, to the heart for slowing the heart rate and to the stomach for secretion and constriction of smooth muscle.

The question arises whether cutaneous recordings of gastric electrical activity called electrogastrography (EGG) (5, 21) contain patterns that can be used to characterize physiological gastric activity, as well as to diagnose gastrointestinal pathologies (see for example Ref. 5 and references therein). Alvarez made the first EGG measurement in 1921, and during the last several years this technique has experienced renewed interest. To a large extent, this resurgence of interest can be attributed to the noninvasive character of this experimental method; see the experimental setup of the apparatus depicted in the schematic at the top of Fig. 1 and the placement of the electrodes shown at the bottom of this figure. Although there have been several attempts to apply the measured EGG time series for clinical assessment, most approaches involved qualitative and not quantitative evaluation of the recorded tracings.


Figure 1
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  		Fig. 1. Block diagram of the experimental setup (top). Placement of the electrodes (bottom). A/D, analog-to-digital converter.

 
Systemic sclerosis (SSc or scleroderma) is a clinically heterogeneous generalized disorder that affects the connective tissue of the skin and internal organs such as gastrointestinal tract, lungs, heart, and kidneys (13). It is characterized by alterations of the microvasculature, disturbances of the immune system, and massive deposition of collagen. The pathogenesis of scleroderma involves genetic, vascular, collagen, and immunological abnormalities. Progressive SSc is often accompanied by esophageal and gaster motor abnormality. The esophagus reveals atrophy and sclerosis of the distal smooth muscle with fragmentation of the connective tissue. The objective of the present study was to investigate antral myoelectrical activity in SSc patients, even though the EGG manifestations of scleroderma are not fully understood (17, 18, 20, 24, 25).

Herein we hypothesize that the time series generated by myoelectric gastric activity, the EGG time series, or more accurately the gastric rate variability (GRV) time series consisting of the time intervals between successive EGG maxima, is a statistical fractal and consequently the underlying dynamics have a long-time memory that can be measured by means of a scaling parameter. Furthermore, certain pathologies affecting the gastrointestinal system significantly change the scaling behavior of the GRV of normal healthy individuals. The former hypothesis was tested using a cohort group of 17 healthy individuals, and the latter hypothesis was tested using a cohort group of 24 SSc patients.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
The experimental measurements were carried out on a cohort group of 24 SSc patients (15 women, 9 men) with median age of 58 yr (range 32–74 yr). The control group was made up of 17 healthy students of the Technical University of Wroclaw, all of whom were between the ages of 21 and 27 and in good health. All subjects gave their written, informed consent. Fasting EGG was recorded for 1 h. In the experimental setup, shown in Fig. 1, three bipolar electrodes spaced ~3.5 cm apart and placed along the projection of the main stomach axis on the abdomen were used. The common reference electrode was placed ~6 cm above them. The electrodes, positioned with the help of USG, were connected to a Grass Instrument amplifier with built-in 30-Hz low-pass filters. Upon amplification the signal was sent to a 12-bit Convert Lab data-acquisition board via an anti-aliasing filter. The signal was sampled at 20 Hz and stored on a personal computer for data analysis. To increase the signal-to-noise ratio and to eliminate the inevitable cardiac and respiratory components of the cutaneous recording, the analyzed signal was passed through a low-pass digital FIR filter with cutoff frequency 0.25 Hz, the filter length 601, and the Kaiser window (beta = 4). The resulting signal was then processed by wavelet transform techniques.

The wavelet transform is an integral transform for which the set of basis functions, known as wavelets, are well localized both in time and frequency (16). Moreover, the wavelet basis can be constructed from a single function {psi}(t) by means of translation and dilation commonly referred to as the mother function or analyzing wavelet. The continuous wavelet transform of the function characterizing the data set s(t) is defined as

Formula
where {psi}*(t) denotes the complex conjugate of {psi}(t). The numerical analysis of real data requires that the discrete wavelet transform of a discrete time series Formula of length N and equal spacing {delta}t is defined as

Formula
and replaces the continuum wavelet transform. By sing a standard fast Fourier transform routine it is possible to efficiently calculate the above sum (28); see the APPENDIX. The standard deviation of wavelets coefficients {sigma}wav(a)

Formula
is frequently used to determine which scales (pseudo-frequencies) significantly contribute to system's dynamics. The angular brackets denote an average over the data points. In the present application the discrete wavelet transform is first applied to the discretely sampled continuous EGG time series to obtain {sigma}wav(a).

One may ask whether, for a given wavelet and sampling period {delta}t, it is possible to associate a pseudo-frequency with the scale a. Obviously, the dual localization of wavelets enables us to answer this question only in a broad sense, where we define

Formula
with fa being the pseudo-frequency corresponding to the scale a, and fc the frequency that maximizes the modulus of the Fourier transform of the wavelet. From a wide variety of possibilities, we choose the Morlet wavelet (whose center frequency fc is equal to 0.8125 Hz) as the mother function for the EGG analysis

Formula


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 APPENDIX
 REFERENCES
 
The type of depolarization associated with contractions of the stomach is referred to as electrical control activity (ECA) and is quasi-periodic with an average frequency of 0.05 Hz, which is quite low compared with the characteristic frequencies of other physiological signals (left potential in Fig. 2) (14). ECA is a precursor to stomach wall muscle contractions. However, contractions actually take place only when electrical response activity is superimposed on ECA. This second component of gastric electrical activity is made of a well-pronounced plateau upon which spike bursts may be superposed (middle and right potentials in Fig. 2). In the absence of spike bursts the strength of the contractions is small and usually does not exceed 0.25 N.


Figure 2
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  		Fig. 2. Three types of intracellular gastric electrical activity. Left: electrical control activity (ECA) is basic periodic activity of smooth muscle cells. ECA superimposed with electrical response activity (ERA) may lead to weak muscle fiber contractions (middle) and in the presence of spike burst to strong contractions (right).

 
Figure 3 shows an example of the Morlet wavelet transform applied to a typical EGG recording of a healthy subject. The signals from three bipolar electrodes attached to the surface of the abdomen were averaged (solid line) and wavelet transformed to obtain the waveform given by the dashed curve in that graph. The standard deviation of the wavelet coefficients, as a function of both the scale a and pseudofrequency fa, is shown in Fig. 4. The function {sigma}wav(a) peaks in the vicinity of 0.05 Hz, corresponding to the 20-s time interval for the basic periodic activity of smooth muscle cells (ECA). The width of the {sigma}wav(a) distribution indicates the number of scales (frequencies) contributing to the EGG signal and {sigma}wav(a) does not scale with a but has a distinctive peak. The scale that maximizes the standard deviation of the wavelet coefficients is denoted as amax. We used peak-to-peak intervals of the wavelet transform for amax to define a new time series that characterizes the statistical properties of gastric activity, which for notational consistency with other types of physiological variability we refer to as GRV. The results shown are typical of both the healthy and SSc cohort groups.


Figure 3
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  		Fig. 3. First 200 s of the electrogastrography (EGG) recording of the healthy subject (solid line). The displayed waveform is the average of signals acquired from 3 bipolar electrodes. Dashed line, wavelet transform of the EGG waveform.

 

Figure 4
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  		Fig. 4. Standard deviation of wavelet coefficients for the average signal from 3 electrodes is shown as a function of the wavelet scale. The upper x-axis gives the corresponding values of pseudo-frequency. {sigma}wav(a), Standard deviation of wavelets coefficients.

 
Note that the determination of the statistics of the variability in interpeak intervals in the GRV time series is consistent with that used in the analysis of heart rate variability (HRV) (22), breathing rate variability (BRV) (1), and stride rate variability (SRV) (11, 12).

In Table 1 the average interpeak interval PP for the 17 healthy volunteers is shown to be 20.7 ± 1.86 s. We can also see that amax varies between 24 and 40 over the cohort group. A typical time series of the GRV interpeak intervals (determined from the Morlet wavelet waveform) in Fig. 3 is given in Fig. 5. The average length N of such time series is 151 ± 14 data points. The example of average wavelet coefficients (AWC) analysis for the data shown in Fig. 5 is provided in Fig. 6. The values of the AWC scaling exponents are recorded in Table 1 and the average exponent, averaged over the cohort group, is {alpha} = 0.80 ± 0.14, indicating that the gastric myoelectric time series has a long-time memory, just as found in a variety of other physiological phenomena, as we now discuss below. We emphasize that the statistical properties of other physiological fluctuations found in ECG, cerebral blood flow and arterial blood pressure dynamics have two scaling regimes, one characterized by a short-time scaling exponent (STSE) and an asymptotic regime characterized by the Hurst exponent H (15). The crossover between these two regimes usually takes place between 32 and 64 heartbeats. The length of the GRV interpeak time series does not allow us to determine whether such a transition region exists for GRV. Therefore, if we err, it is on the side of caution and we interpret {alpha} to be a STSE.


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  		Table 1. Summary of data analysis for the 17 healthy subjects

 

Figure 5
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  		Fig. 5. Time series of peak-to-peak (PP) intervals of the EGG waveform from Fig. 3 to provide the gastric rate variability time series.

 

Figure 6
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  		Fig. 6. Average wavelet coefficient analysis of peak-to-peak interval time series from Fig. 5. W(a), ???.

 
Scaling can arise from correlations in the data or from the statistics of the time series, both of which can give rise to a second moment that increases as a power law in time. Randomly reordering the data points in a time series certainly destroys any correlations and consequently erases the scaling resulting from such correlations. However, because no data are created or destroyed in such random reordering this shuffling procedure leaves unaffected any scaling produced by the statistics of the time series. To determine whether the power-law scaling of an EGG time series is a consequence of noise or is due to a more fundamental dynamical process, we implement the surrogate data technique of shuffling the time-series data points to random positions in the sequence for each of the subjects. Ten realizations of the surrogate data were then processed in the same way as the original data and averaged over the ensemble of realizations to obtain the values of the exponents recorded in Table 1. The average of the surrogate short-time scaling exponents over the cohort group yields {alpha}r = 0.53 ± 0.10 as the average STSE. It is well known that an uncorrelated random process would have a scaling exponent equal to 0.5 (3). Comparing the exponents of the surrogate data with the original data we find statistical significance with P < 7.0 x 10–3 for each of the 17 members of the control group.

The above wavelet coefficient analysis was also carried out using the GRV time series data from a cohort group of SSc patients. The values of the AWC scaling exponents are recorded in Table 2 and the average exponent, averaged over the cohort group, is {alpha} = 0.64 ± 0.17. The typical time series for GRV interpeak time series did not visually appear significantly different from the results shown in Figs. 3–6 and so those figures are not reproduced here.


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  		Table 2. Summary of data analysis for the 24 SSc patients

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
APPENDIX
 REFERENCES
 
Although there is no doubt that the Fourier spectrum of the EGG records contain meaningful clinical information, it should be emphasized that EGG recordings are intrinsically nonstationary and consequently traditional spectral methods cannot reliably capture all their properties. The goal of the present work is to apply the technique of wavelet analysis to establish the scaling behavior in the variability of EGG tracings, using the GRV time series, and thereby quantify its fractal character in both normal and diseased individuals.

We determined that the variability of the EGG time series scales in time and the GRV time series is a random fractal with an average scaling index of {alpha} = 0.80 ± 0.14 corresponding to an early time fractal dimension D = 1.20 ± 0.14. In terms of random walk models the phenomenon of stomach contractions would be called persistent. A persistent random walker is more likely to continue in the direction of the preceding step than to change directions. In the present context persistence implies that if the time interval between the last two contractions was long, that the subsequent interval will statistically favor even longer contractions over shorter ones. This means that the GRV time series are not completely random. They are the result of processes with long-term correlations. The power-law form of the wavelet transform with scale a reveals that stomach contractions at any given time are influenced by fluctuations that occurred a substantial number of contractions earlier. This behavior is manifest in the fractal nature of GRV time series, the data showing long-time correlations that were statistically significant above the P = 0.01 level for each of the 17 healthy subjects.

The fractal nature of the fluctuations in time intervals between events, whether the event is a heartbeat, a breath, or a stride, has been shown to have clinical implications and to lead to scaling of the central moments of the distribution. These and other such phenomena motivated the coining of the term fractal physiology (3), which explicitly acknowledges the anticipated ubiquitous nature of fractal scaling in physiological time series and anatomical structures.

The idea that the fractal dimension interrogates the physiological control mechanisms of the body is of fairly recent vintage. Fluctuations of physiological variables are documented for the cardiovascular system (22, 31), gait (11, 12), and balance (6), among others (7, 29). One of the most significant results to come out of the flood of applications of concepts from nonlinear systems theory to physiology in the past two decades is the importance of variability over that of average values in physiological variables (33). This importance is demonstrated by the systematic change in the measure of variability, the fractal dimension, in HRV and BRV time series, with increasing levels of exercise (34). Consequently, there exists a complex of multiple states that determine the behavior of healthy physiological systems that we view as a wide temporal spectrum of internal control mechanisms. In this paradigm, the single-scale steady state of homeostasis is replaced by a multiplicity of nonequilibrium states correlated over many scales (3, 11). It is possible to conclude that fractal scaling is the hallmark of integrative control systems (3, 23).

The present results for inverse power-law physiological phenomena support the observation made by a number of investigators (9) that disease is not the loss of regularity, as had been asserted historically, but rather disease is the loss of complexity (9). This interpretation of disease is further bolstered by more recent theoretical developments involving complex networks with "small-world" topology and noise (2). Therefore, it is of some interest to discuss other physiological phenomena, which when diseased produce a change in the scaling index.

The autonomic nervous system innervates every organ in the body. Because autonomic disturbances ultimately affect patient survival, thorough understanding of such abnormalities is indispensable. The studies of gastric myoelectric activity in patients with diabetic autonomic neuropathy (19) or traumatic brain injury (27) clearly demonstrated the impact of vegetative function on the gastrointestinal system. It is well established that scleroderma frequently leads to degeneration of smooth muscle of gastrointestinal tract (myopathy). McNearney et al. (20) argue that reduction of slow-wave Fourier amplitudes of EGG is a direct manifestation of such degeneration. In the same work these authors also emphasize SSc-related abnormalities in broadly defined rhythmicity of EGG slow waves. This phenomenon is often observed in patients with neuropathy. Thus the fundamental question arises as to whether the impairment of slow-wave rhythmicity is caused by myopathy or neuropathy. The results involving HRV and SRV should stimulate the assessment of clinical applicability of fractal time series analysis to GRV.

In the present work we have made a first step in the direction of assessing the clinical applicability of fractal time series analysis to GRV, establishing that the peak-to-peak GRV activity for healthy individuals is a random fractal process and that its scaling properties change in pathology such as SSc. It is worth pointing out that detection of impaired gastric myolectrical activity with the help of EGG spectral analysis enable us to predict delayed gastric emptying with an accuracy approaching 85% (4). Unfortunately, the intact physiological power spectrum does not guarantee a normal emptying of a stomach. The most prevalent methods for the assessment of gastric motility or emptying (barium X-ray, radioisotope gastric-emptying scan, invasive gastric manometry, or endoscopy) are either radioactive or invasive. A noninvasive technique such as EGG accompanied by a reliable assessment of GRV properties intimately related to gastric motility would be a most welcome addition to these procedures.

In conclusion, one must distinguish between descriptive analyses intended to categorize particular phenomena and the more penetrating analyses that are able to discriminate among various kinds of dynamical behavior. Linear systems are often distinguished by the quantitative differences in time series: for example, differences between the means, variances, and correlations. In nonlinear dynamical phenomena, however, it is not only the quantitative but the qualitative differences as well that enable one to distinguish among dynamical modes, such as the dynamical branches in a bifurcating nonlinear system. We use the method of AWC to determine that the GRV time series scale and therefore establish that the underlying control process is a statistical fractal. This finding suggests that the EGG physiological control system is a nonlinear dynamical feedback process, distinct from the linear feedback hypothesis usually made for homeostatic control systems in a physiological context. Similar observations have been made by a number of investigators using a variety of arguments (see, for example, Refs. 2, 11).

The analysis presented herein shows a difference in fractal dimension between cohorts of healthy individuals and those with scleroderma. The difference in scaling between the two groups is statistically significant and supports the hypothesis that the scaling index, or equivalently the fractal dimension, can be used as an indicator of health. This index changes dramatically when an individual is diseased in such a way that the pathophysiology directly influences the gastrointestinal system. The change indicates the loss of long-term memory, making the contractions of the stomach muscle more random in nature than in a healthy person. These random uncoordinated contractions are part of the pathology.


   APPENDIX
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
REFERENCES
 
A statistical fractal defines a process that is described by a probability distribution whose variate scales in time. For example, if X(t) is a stochastic function of time, then if the process scales, such that for a constant {lambda} we obtain X({lambda}t) = {lambda}–{alpha}X(t) where {alpha} is a constant, the process is said to be fractal. The probability density p(x,t) for such a scaling process satisfies the relation

Formula
where F is an analytic function of its argument. An example of such scaling is observed in fractional Brownian motion, where the statistics are Gaussian, the second moment is a power law <X(t)2> {propto} t2H, the spectrum is an inverse power law, S(f) {propto} 1/f2H–1 and {alpha} = 2H. There is no unique way to generate inverse power-law fluctuations; they can result from random walks with memory (22), they can be generated by fractional random walks (31), another way is using discrete nonlinear dynamical equations (28), and they can also arise from the interaction topology of a complex network (2). Consequently, herein we do not promote a particular model for the scaling properties observed in the data, but leave that for a subsequent publication.

A random fractal time series is characterized by trends and discontinuities called singularities of the signal, and their strength is measured by Hölder exponents (2). Given a function s(t) with a singularity at t0, the Hölder exponent h(t0) at such a point is defined as the supremum of all exponents h that fulfills the condition

Formula
where Pn(t – t0) is a polynomial of degree n < h. Struzik (26) shows that h = 2 – D, so we can determine the fractal dimension D, which is a local property of the time series from the local Hölder exponent. The wavelet transform can be used to determine the Hölder exponent of a singularity because the wavelet kernel {psi}(u) can be chosen in such a way as to be orthogonal to polynomials up to degree n. In fact, for a wavelet chosen in this way, it is easy to prove that if the function s(t) fulfils the condition for a Hölder scaling function, its wavelet transform at t = t0 is given by

Formula
where u = (t – t0)/a. Therefore, at least theoretically, the Hölder exponent of a singularity that is localized at t0 can be evaluated as the scaling exponent of the wavelet transform coefficient, Ws(a,t0), for a -> 0. Thus it is apparent that the scaling exponent may be obtained from the slope of the linear part of the plot of the wavelet transform coefficient vs. a on a log-log scale. This approach is known as the AWC and can provide reliable estimates of scaling exponents for even relatively short time series (26).

The convolution in the series representation of the Wavelet transform can be evaluated for any of N values of the time index n. However, the convolution theorem allows us to compute all N convolutions simultaneously in Fourier space using a discrete Fourier transform:

Formula
where k = 0, ... N – 1 is the frequency index. If one notes that the Fourier transform of a function {psi}(t/a) is |a| Formula (af), then by the convolution theorem we can write

Formula
with frequencies defined in the conventional way. The discrete wavelet coefficient can be efficiently calculated by the standard fast Fourier transform routine. It should be emphasized that formally these coefficients do not yield the discrete linear convolution corresponding to the equation in the text, but rather to a discrete circular convolution in which the shift n' – n is taken modulo N. The numerical implementation of continuous wavelet transforms may be found at http://paos.colorado.edu/research/wavelets/ or http://www-stat.stanford.edu/~wavelab/. The Mathworks' MATLAB Wavelet Toolbox provides a host of wavelet-related routines.


   FOOTNOTES
 

Address for reprint requests and other correspondence: Bruce J. West, Mathematical and Information Science Directorate, Army Research Office, P.O. Box 12211, Research Triangle Park, NC 27709-2211 (e-mail: Bruce.J.West{at}us.army.mil)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


   REFERENCES
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 

  1. Altemeier WA, McKinney S, and Glenny RW. Fractal nature of regional ventilation distribution. J Appl Physiol 88: 1551–1557, 2000.[Abstract/Free Full Text]
  2. Amaral LA, Diaz-Guilera A, Moreira AA, Goldberger AL, and Lipsitz LA. Emergence of complex dynamics in a simple model of signaling networks. Proc Natl Acad Sci USA 101: 15551–15555, 2004.[Abstract/Free Full Text]
  3. Bassingthwaighte J, Liebovitch L, and West B. Fractal Physiology. New York: Oxford University Press, 1995.
  4. Chen JD, Lin Z, Pan J, and McCallum RW. Abnormal gastric myoelectrical activity and delayed gastric emptying in patients with symptoms suggestive of gastroparesis. Dig Dis Sci 41: 1538–1545, 1996.[CrossRef][ISI][Medline]
  5. Chen JZ and McCallum RW. Electrogastrography: Principles and Applications. New York: Raven, 1994.
  6. Collins JJ and De Luca CJ. Random walking during quiet standing. Phys Rev Lett 73: 764–767, 1994.[CrossRef][ISI][Medline]
  7. Goldberger AL. Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. Lancet 347: 1312–1314, 1996.[CrossRef][ISI][Medline]
  8. Goldberger AL, Bhargava V, West BJ, and Mandell AJ. On a mechanism of cardiac electrical stability. The fractal hypothesis. Biophys J 48: 525–528, 1985.[Abstract/Free Full Text]
  9. Goldberger AL, Rigney DR, and West BJ. Chaos and fractals in human physiology. Sci Am 262: 42–49, 1990.[Medline]
  10. Goldberger AL and West BJ. Fractals in physiology and medicine. Yale J Biol Med 60: 421–435, 1987.[ISI][Medline]
  11. Griffin L, West DJ, and West BJ. Random stride intervals with memory. J Biol Phys 26: 185–202, 2000.[CrossRef]
  12. Hausdorff JM, Peng CK, Ladin Z, Wei JY, and Goldberger AL. Is walking a random walk? Evidence for long-range correlations in stride interval of human gait. J Appl Physiol 78: 349–358, 1995.[Abstract/Free Full Text]
  13. Haustein UF. Systemic sclerosis-scleroderma. Dermatol Online J 8: 3, 2002.[Medline]
  14. Koch KL and Stern RM. Handbook of Electrogastrography. Oxford, UK: Oxford University Press, 2003.
  15. Latka M, Glaubic-Latka M, Latka D, and West BJ. Fractal rigidity in migraine. Chaos Soliton Fractals 20: 165–170, 2003.
  16. Mallat SG. A Wavelet Tour of Signal Processing. San Diego, CA: Academic, 1999.
  17. Marie I, Levesque H, Ducrotte P, Denis P, Hellot MF, Benichou J, Cailleux N, and Courtois H. Gastric involvement in systemic sclerosis: a prospective study. Am J Gastroenterol 96: 77–83, 2001.[CrossRef][ISI][Medline]
  18. Marycz T, Muehldorfer SM, Gruschwitz MS, Katalinic A, Herold C, Ell C, and Hahn EG. Gastric involvement in progressive systemic sclerosis: electrogastrographic and sonographic findings. Eur J Gastroenterol Hepatol 11: 1151–1156, 1999.[ISI][Medline]
  19. Mayaudon H, Bauduceau B, Dupuy O, Cariou B, Ceccaldi B, Farret O, and Molinie C. Assessment of gastric neuropathy using electrogastrography in asymptomatic diabetic patients. Correlation with cardiac autonomic neuropathy. Diabetes Metab 25: 138–142, 1999.[ISI][Medline]
  20. McNearney T, Lin X, Shrestha J, Lisse J, and Chen JD. Characterization of gastric myoelectrical rhythms in patients with systemic sclerosis using multichannel surface electrogastrography. Dig Dis Sci 47: 690–698, 2002.[CrossRef][ISI][Medline]
  21. Mintchev MP, Kingma YJ, and Bowes KL. Accuracy of cutaneous recordings of gastric electrical activity. Gastroenterology 104: 1273–1280, 1993.[ISI][Medline]
  22. Peng CK, Mietus JE, Hausdorff JM, Havlin S, Stanley HE, and Goldberger AL. Long-range anticorrelations and non-Gaussian behavior of the heartbeat. Phys Rev Lett 70: 1343–1346, 1993.[CrossRef][ISI][Medline]
  23. Peng CK, Mietus JE, Liu Y, Lee C, Hausdorff JM, Stanley HE, Goldberger AL, and Lipsitz LA. Quantifying fractal dynamics of human respiration: age and gender effects. Ann Biomed Eng 30: 683–692, 2002.[CrossRef][ISI][Medline]
  24. Pfaffenbach B, Adamek RJ, Hagemann D, and Wegener M. Gastric myoelectrical activity and gastric emptying in patients with progressive systemic sclerosis. Am J Gastroenterol 91: 411–413, 1996.[ISI][Medline]
  25. Sallam H, McNearney TA, and Chen JD. Systematic review: pathophysiology and management of gastrointestinal dysmotility in systemic sclerosis (scleroderma). Aliment Pharmacol Ther 23: 691–712, 2006.[CrossRef][ISI][Medline]
  26. Struzik ZR. Revealing local variability properties of human heartbeat intervals with the local effective Hölder exponent. Fractals 8: 163–179, 2000.[CrossRef]
  27. Thor PJ, Goscinski I, Kolasinska-Kloch W, Madroszkiewicz D, Madroszkiewicz E, and Furgala A. Gastric myoelectric activity in patients with closed head brain injury. Med Sci Monit 9: CR392–CR395, 2003.[Medline]
  28. Torrence C and Compo GP. A practical guide to wavelet analysis. Bull Am Meteorol Soc 79: 61–78, 1998.[CrossRef]
  29. West BJ. Physiology, Promiscuity, and Prophecy at the Millennium: A Tale of Tails. River Edge, NJ: World Scientific, 1999.
  30. West BJ, Bhargava V, and Goldberger AL. Beyond the principle of similitude: renormalization in the bronchial tree. J Appl Physiol 60: 1089–1097, 1986.[Abstract/Free Full Text]
  31. West BJ and Deering W. Fractal physiology for physicists: Lévy statistics. Phys Rept 246: 1–100, 1994.[CrossRef]
  32. West BJ and Griffin L. Allometric control of human gait. Fractals 6: 101–108, 1998.[Medline]
  33. West BJ and Griffin L. Biodynamics: Why the Wirewalker Doesn't Fall. New York: Wiley, 2003.
  34. West BJ, Griffin LA, Frederick HJ, and Moon RE. The independently fractal nature of respiration and heart rate during exercise under normobaric and hyperbaric conditions. Respir Physiol Neurobiol 145: 219–233, 2005.[CrossRef][ISI][Medline]
  35. West BJ, Zhang R, Sanders AW, Miniyar S, Zuckerman JH, and Levine BD. Fractal fluctuations in transcranial Doppler signals. Physical Review E 59: 3492–3498, 1999.[CrossRef]




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