AMERICAN JOURNAL OF PHYSIOLOGY Vol. 229, No. 3, September 1975. Printed in U.S.A.
Phase lock of electrical bursts
slow waves and spike
in cat duodenum ARTURO G. SANCHOLUZ, THOMAS E. CROLEY II, JAMES CHRISTENSEN, EN20 0. MACAGNO, AND JOHN R. GLOVER Gastroenterology Research Laboratory, Department of Internal Medicine, and Iowa Institute of Hydraulic Uniuersity of Iowa C&eges of Medicine and Engineering, Iowa City, Iowa 52242
SANCHOLUZ, ARTURO G., THOMAS E. CROLEY II, JAMES CHRISTENSEN, ENZO 0. MACAGNO, AND JOHN R. GLOVER. Phase lock of electrical slow waves and spike bursts in cat duodenum. Am. J. Physiol. 229(3): 608-612. 1975.-The phase, relationship between slow waves and spike bursts was studied in vitro in the cat duodenum. Electrodes were arranged radially about the duodenum. Records were read for Tr, the time between the slow-wave “valley” and the first spike, and T 2, the time between successive valleys. The distributions of the ratio Tr/Tz showed very small differences, not uniquely attributable to radial electrode position. The distribution of Tr/T2 indicated that spike bursts began 58.8 y0 of the way through a slow-wave cycle, measured from the valley. Also, electrodes were arranged longitudinally and records were read to determine ASW, the slow-wave delay time between adjacent electrode positions, and AS, the spike burst delay time between adjacent electrode positions. The correlation of AS and ASW suggested a linear relation, and a regression resulted in a good linear description. These results are consistent with an assumed phase lock of spike bursts to slow waves. electrophysiology;
peristalsis;
gastrointestinal
glas tube (4 mm OD) and fastened at each end so that it was held at the approximate length it had in situ. Thi: tube was mounted inside a large Plexiglas cylinder which was filled with Krebs solution. Thus, the duodenal segment was vertically held in the core of a cylindrical bath (Fig. 1). The Krebs solution was recirculated constantly through the bath, flowing in through the lumen of the perforated tube holding the duodenum and out through a hole at the bottom of the cylindrical tank. Flow outside the tank, accomplished by a peristaltic finger pump acting on rubber tubing, included passage through a glass heat exchanger whose jacket was continuously perfused with water from a circulating pump and heater. The Krebs solution in the bath was maintained at 36.5-37.5”C. It was continuously aerated with 95 70 02-5 % Con. The electrodes were silver-silver chloride glass pore electrodes, described elsewhere (5). While glass pore electrodes may not absolutely reflect the unconstrained electrical behavior of the duodenum, they are standard and more practical. (Microelectrodes would be impossible to handle in a preparation of this sort.) The walls of the bath contained holes, aligned both in the long axis and in the circumference of the cylinder, through which these electrodes could be inserted to touch the duodenum. These holes contained “0”-ring gaskets to prevent leakage around electrodes. Records of the electromyogram were made separately from each electrode against a common grounded reference electrode, which was a coil of chlorided silver wire immersed in the bath. Records were made on an 8-channel ink-writing curvilinear polygraph provided with R-C amplifiers and a bank of filters. A relatively high recording paper speed of 5 mm/s was used throughout the data collection in order to minimize interpretation errors. The frequency response was limited to a band pass of 0.16-150 Hz. Polarity was set so that an upward pen deflection represented depolarization in the slow-wave cycle. In six preparations, the eight electrodes were arranged at equal circumferential intervals at a cross section located about 5 cm from the proximal end of the duodenal segment. The line of the mesenteric insertion served as a reference point and always lay between electrodes 1 and 2. The relative positions of the eight electrodes were thus made constant throughout the six experiments. However, neither the same electrode nor the same channel was always used to record from the same position from experi-
motility
THE MUSCULAR WALL of- the duodenum generates two kinds of electrical signals: slow waves and spike bursts (4). These signals are closely related‘ to duodenal contractions. Slow waves are considered to represent migrating zones of enhanced muscle excitability which, when further intensified by other factors, lead to contractions. A spike burst precedes and signals the beginning of a contraction. At any single point in the duodenum, spike bursts are believed to have a constant range of location in time within the slowwave cycle (1, 2, 4, 6, 7); but a rigorous investigation of the phase relationship between these two electrical signals has not been made. The aim of this work was to investigate statistically the variability of the phase relationship between spike bursts and slow waves. MATERIALS
AND
Research,
METHODS
Eleven healthy cats, weighing 3.0-5.0 kg, were anesthetized by intrapleural injection of sodium pentobarbital, 40 mg/kg. In each, the abdomen was opened and a lo-cm segment of duodenum, extending from about 1 cm below the pylorus to about the ligament of Treitz, was quickly excised and emptied. It was slipped over a perforated Plexi608
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
CAT DUODENUM
DUODENUY -
FIG. 1. A diagram of bath. Duodenum was mounted over a perforated Plexiglas tube mounted vertically in core of a cylindrical Plexiglas bath containing Krebs solution. Bathing solution was circulated externally through a heat exchanger to maintain a constant bath temperature, and bath was continuously bubbled with 950/, O&y0 CO2. Electrodes were inserted through sealed holes in wall of bath to touch serosal surface of duodenal segment.
ment to experiment. Records were made for approximately 2 h. The records were analyzed visually. In these records, two time intervals were measured. Interval Tr was defined as the time between the valley of a slow wave (the nadir between two successive slow waves) and the first spike in the spike burst which followed it. Interval Tz was defined as the time between two successive valleys; it represented the period of the slow wave. The ration Tl/Tz is a variable independent of the time scale of any given slow-wave cycle and satisfies the inequality,
The ratio Tl/Tz, rather than Tl, was chosen to represent the initiation of the spike burst in the slow-wave cycle; it locates the spike bursts for any duration of the slow wave. The stochastic process represented by Tl/T% can . be considered as a simple random variable. This implies that the process is stationary and that successive observations are independent. Intervals T1 and Tz were measured for all slow-wave cycles that contained spike bursts on each of the eight channels separately, and the ratios TI/Ts were calculated. The visual measurement errors were estimated by comparing results, obtained by different people reading the same record, and obtained by the same person on different days. In the first case, the relative error was about 4.1 %, and in the second case, it was about 3.7 %. This was judged small enough to accept this method of measuring TI/T~Relative error is defined herein as the square root of the
609 mean square error divided by, the average mean of Tl/Tz. The mean values, variances, and standard deviations of Tr/Tz were estimated separately for each electrode position in each experiment; see equations 2 and 3 in APPENDIX. A small variability in the mean value estimates among the eight electrode positions existed, so the hypothesis of equality of the means of Tl/Tz among the eight electrode positions was examined. That is, it was considered necessary to investigate the possibility of electrode positiondependent distributions of Tl/Tz, in addition to investigating the general behavior represented by the combined sample from all eight electrodes. The tests used are detailed in APPENDIX. In another five preparations, a different approach was used. Sixteen glass pore electrodes were aligned in the long axis of the duodenum. They were set 5 mm apart with the most proximal electrode 1 cm below the pylorus. Simulas taneous monopolar electromyograms were recorded described above on a 16-channel ink-writing curvilinear polygraph. The delay time between slow waves in adjacent channels in seconds was defined as ASW. Similarly, AS was defined for spike bursts as the time in seconds between the appearances of a spike burst on two adjacent channels. If spike bursts are rigidly phase locked to slow the slow wave and waves (i.e., the phase angle between the spike burst is constant), then AS is equal to ASW. Thus, a rigid phase lock would imply linear - that in the following relation, .(2) AS =a+bASW a is equal to 0 and b is equal to 1. The existence of a linear relationship was acertained initially by inspection of the correlation coefficient (where high values imply a linear relationship) (8). A large number of slow-wave cycles in which spike bursts occurred were selected by taking one slow wave-spike burst complex every Z-rnin period throughWhen several such complexes in out the experiment. each period were available for choice, the one that was the most free of electrical noise and in which the slow waves had the most distinct point of initiation was selected. This procedure reduced the number of data points, thereby reducing computation, yet still gave acceptable confidence in the correlation analysis. The correlation was estimated by the following equation: R = 2
(ASWi
-
m)(ASi
i-l
l/2
-
AS> -A
5 (ASWi i=1
-
ASW)”
2
(ASj - aS)2
j=l
1
(3)
R is the sample correlation coeflicient In this equation, used to estimate the population correlation; AS i and ASW i represent the corresponding ith sample observations, and 5 and m are the corresponding sample means (defined similarly to equation 2 of APPENDIX). By least-squares regression, a straight line satisfying equation 2 was subsequently fitted by estimating the coeflcients a and b as follows: J=
eASiASWi--ndSZ C i=1
1
-1
n FW2
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
1
(4)
610
SANCHOLUZ
A-
ci= as -
bASW
(5)
The estimates defined by equations 3, 4, and for the five experimental runs of this second
5 were made analysis.
RESULTS
Distributions of Tl/T, at eight points in cross section. No relationship was generally apparent between electrode position and the distribution of T1/T2. Mean values from one of the six experiments appear in Table 1. The slight variation among electrode positions (Table 1) prompted examination of the hypothesis that Tl/Tz is dependent on electrode position. One-hundred sixty-eight tests (28 sample pairs for each experiment) were run to compare all electrode position pairs (see APPENDIX), and the null hypothesis for equality of means was rejected (at the 1 % significance level) 85 times, or about 50 %. Particular combinations of electrode position pairs were also investigated to test for possible symmetries. For example, tests for equality of the means for adjacent electrode positions as well as for opposite electrode positions were run. Percentages of rejection of the null hypothesis varied from 45 to 55 %. Thus, the distribution of Tl/Tz could not generally be shown either. . to be dependent on or to be independent of electrode posrtion relative to the mesenterrc insertion at any arbitrary locations in the duodenum. The tests suggest, however, that the distributions of Tl/Tz might be electrode position dependent for some specific locations in the duodenum, for some of the experiments. The normal distribution was considered as a possibility for TI/T~. Although TJTz is bounded and the normal is not, the T1/T2 histograms were very narrow. Therefore, a fitted normal would contain almost no weight in the tails, thereby resulting in little error. The hypothesis of normality was tested for TJT2 for each electrode from each experiment, and the hypothesis was rejected 18 out of 48 times at the 5 % significance level (see APPENDIX). This is more than expected if the normality assumption is true. The mean value estimate for each electrode generally ranged from 0.48 to 0.64 and the standard deviation (“spread”) for each electrode was generally about 0.1. Data were then grouped into six global samples corresponding to the six experiments by combining data from all electrodes in each experiment. The global histograms for Tl/Tz were roughly symmetrical in all six cases (Fig. Z), and the normal distribution was again considered. The goodness-of-fit test was applied and the hypothesis of normality was rejected at the 5 % significance level, 5 out of l
.
.
.
.
.
*
.
TABLE 1. Estimates of mean, variance, and standard deviation for Tl/Tg in one experiment Channel
P
a2
0.610 0.582 0.585 0.623 0.619 0.583 0.558 0.611
0.011 G.007 0.011 0.008 0.008 0.006 0.004 0.007
8
0.107 0.084 0.103 0.092 0.088 0.081 0.063 0.088
ET AL.
the 6 times. Although this implied that the normality assumption was most likely invalid, inspection of the global histograms indicated that the data appeared sufficiently near normal for calculation of confidence limits as outlined in APPENDIX. The 95 % confidence intervals for means and variances (see APPENDIX) were then determined (Table 2). By compiling all 48 sample means, the best estimate of the general mean of T1/Tz was set at 0.588 =t 0.036 (+ 1 SD> . Correlation of AS and AS W in longitudinal section. The correlation between ASW and AS was generally high as shown in Table 3, and the least-squares analysis yielded estimates for the x intercept, a, and for the slope, b, which were close to those values expected for a phase lock, which are 0 and 1, respectively (Table 3). A plot of the regression line for one experiment appears in Fig. 3, which illustrates that there were often negative values for ASW. These represent orad propagation of slow waves. Others have observed this phenomenon in the past (3, 6). DISCUSSION
The histograms of T1/T2 indicate not be rigidly fixed to slow waves, distributed over the whole slow-wave
that spike bursts may but neither are whey cycle. A rigid phase
40 -
r 5z 30L co ; L *O2
to -
I
I 1 II 0.4
02
0.6 T,/Tz
2. Global histogram 1 experiment. Class intervals FIG.
of TJT2 are 0.01.
for all 8 electrode
positions
2. Estimates of means, variances, and conjdence intervals for the six global distributions
TABLE
n
127 74 62 46 77 102 102 62
Expt
2 2 3 4 5 6
;
0.594 0.610 0.567 0.575 0.578 0.590
62
0.008 0.004 0.011 0.008 0.005 0.005
n
652 425 413 644 797 603
95% Confidence Intervals for ~2
(0.0075, (0.0038, (OcOO96, (0.0078, (0.0047, (0.0042,
0.0093) 0.0051) 0.0120) 0.0090) 0.0057) 0.0052)
957& Confidence Intervals for p
(0.587, (0.604, (0.557, (0.568, (0.573, (0.585,
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
0.601) 0.616) 0.577) 0.584) 0.583) 0.595)
in
CAT
611
DUODENUM
TABLE 3. Estimates of correlation coejicient, and slope (b) between AS W and AS
x intercept (a),
Expt
R
n
ii
7 8 9 10 11
0.81 0.89 0.88 0.85 0.84
53 56 45 51 50
0.19 0.01 0.09 -0.07 1.05
8
0.70 0.98 0.64 0.73 0.85
existence of a rigid phase phase lock might exist.
lock
and
suggest
that
a rigid
APPENDIX The following simplified tests for the equality of means were used. They are based on the central limit theorem of statistics (10). From this theorem, the following random variable, 2, can be shown to be approximately distributed with the standard normal distribution, regardless of the actual distribution of Ti/Tz (assuming equality of means) :
for the null hypothesis true. The degree for sample sizes (nl, nz) greater than 30 sample mean and 0 i2 is the variance for and sample variance, Q is, are defined as
pi =
l/Q
2 j-1
of approximation is excellent (10). In equation I, fli is the electrode i. The sample mean follows:
ITl/T21i,
(2)
j
and
ni a2 ui
0
0 c -8
FIG.
3. Regression
of As upon
ASW
in 1 experiment.
lock would produce a “spiked” distribution, (i.e., one in which Tr/Tz is constant), whereas a distribution over the whole cycle would result from a purely random phase relationship. The observed distribution approaches a spiked distribution; the range of the cycle over which spike bursts can be expected was restricted. The marked uniformity of mean value estimates of Tr/T2 and the very small estimates of the standard deviation indicate that the beginning of spike bursts are confined to a very narrow range on the slow-wave cycle. The differences in the distributions, though small, could be related to electrode position. The evidence presented here prevents drawing a conclusion of position dependence valid for all locations. Nevertheless, it does suggest that this dependence probably exists for some specific location relative to the mesenteric insertion. The variation in Tr/Tz could be related to data interpretation or other variables. The hypothesis of a normal distribution for Tr/Tz was rejected 34 % of the time for tests on individual electrode positions and most of the time for tests on global samples of each experiment. The tests on individual electrodes generally revealed small skewness toward smaller values for Tr/T,. In fact, Tr/T2 is actually a ratio of two separate random variables, which is seldom normally distributed. Initially, the correlation analysis for time lags of the slow wave and spike burst at successive locations indicated that a linear relationship between the lags was likely. Subsequently, the regression analysis further indicated that a phase lock between the slow wave and spike bursts was also likely. If a rigid phase lock really exists, the preceding analyses values. The interpretation errors, would not yield “perfect” inherent in the visual measurement method used, would result in random errors (which are generally normally distributed). Considering this, these data do not refute the
=
1 /ni
C j=l
([T1/T21i,
j -
I&j2
(3)
represents the jth observation of the In equations 2 and 3, [T,/Tz]i,j random variable T1/T2 for electrode i, By replacing ui2, and ~2~ in equation 2 by their estimators, the distribution of 2 is no longer a normal. By using, however, standard normal distribution quantiles and values of 2 (with 0~2 = &i2), approximate tests can be made (10). This is done in lieu of other standard tests for equality of means because the approximation is good for large sample sizes and because other tests have larger computation requirements. The tests are performed by calculating the value of Z in equation 1 (with ui2 = 6 i2) and comparing with an appropriate y quantile of the standard normal distribution. If the value of 2 is greater, then the hypothesis of equal means is rejected at the (1 - 7) significance level. This test was applied to every pair of, electrodes (28 combinations in each experiment) and to combinations selected to test for various symmetries. To test the hypothesis of a normal distribution for TJT2, a goodness-of-fit test was done with the experimental frequency distributions (histograms) obtained. It can be shown (9) that the following random variable Qk
=
C
(Nj
-
YlP
(4)
j)2/(YlPj)
j-1
is approximately distributed with a chi-square distribution with k - 3 degrees of freedom when the hypothesized distribution with unknown mean and variance is really the actual. In equation 4, k is the number of intervals (for values of Tl/T2) into which the range is nbroken; iVi is the number of observations which lie within interval j; Pi is the probability that an observation would lie within interval j, which is calculated from the hypothesized distribution with mean equal to fi and variance equal to a2. From experience guidelines, the number of intervals, k, is set at 10. Smaller numbers for k result in poor test significance and larger numbers accentuate extreme observations. Goodness-of-fit tests are performed by calculating fl and 62 from the sample and Pj from the hypothesized distribution with mean fl and variance a2, and then calculating the chi-square statistic, Qk from equation 4. The value of Qk is compared with an appropriate 7 quantile point of the chi-square distribution with k - 3 degrees of freedom. If Qk is greater, the hypothesis that the data came from the suggested distribution is rejected at the (1 - 7) significance level. This test was applied to the individual electrode samples and to the global samples (all electrodes) for each experiment, to test for normality. It can be shown (9) that if Tl/Tz is normally distributed (or approximately so), then the following approximate 100y~O confidence intervals for the mean and variance, respectively, of the global distribution of Tl/Tz can be written for large sample sizes: (P - y %f&
P +y
Ufi)
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
(5)
612
SANCHOLUZ
In equations 5 and 6, @ is the sample mean of Tl/Ts for the combined eight electrodes, k2 is the sample variance, n is the combined sample size for the eight electrodes, y is the 1 - [( 1 - #2] quantile point of the t distribution with n - 1 degrees of freedom (or the standard normal distribution for large sample sizes), and yl and y2 are the and 1 - [(l - r)/2] quantile points, respectively, of the (1 - r)/2 chi-square distribution with n - 1 degrees of freed0.m. The sample estimates, 4 and a2, are defined similarly as in equations 2 and 3.
ET AL.
This work was supported by National Institutes of Health Research Grant AM 08901 and Research Career Development Award AM 20547. Requests for reprints should be sent to: J. Christensen, Division of Gastroenterology, Dept. of Internal Medicine, University Hospitals, Iowa City, Iowa 52242. Received
for publication
19 July
1974.
REFERENCES 1. BASS, P., C. F. CODE, AND E. H. LAMBERT. Motor and electric activity of the duodenum. Am. J. Physiol. 201: 287-291, 1961. 2. BASS, P., AND J. N. WILEY. Electrical and extraluminal contractileforce activity of the duodenum of the dog. Am. J. Digest. Diseases 10: 183-199, 1965. 3. BORTOFF, A. Slow potentials variations of small intestine. Am. J. Physiol. 201: 203-208, 196 1. 4. CHRISTENSEN, J. The controls of gastrointestinal movements, some old and new views. New Engl. J. Med. 285 : 85-98, 1971. 5. CHRISTENSEN, J., AND R. L. HAUSER. Longitudinal axial coupling of slow waves in proximal cat colon. Am. J. Physiol. 221: 246-250, 1971.
6. DANIEL, E. Electrical activity of the alimentary tract. Am. J. Digest. Diseases 13 : 297-319, 1968. 7 DANIEL, E. The electrical and contractile activity of the pyloric ’ region. Gastroenterology 49 : 403-4 18, 1965. 8. JENKINS, G., AND D. WATTS. Spectral Analysis and Its Application. San Francisco : Holden-Day, 1968, p. 73-75. g . MOOD, A. M., F. A. GRAYBILL, AND D. C. BOES. Introduction to the Theory of Statistics. New York : McGraw, 1974, p. 372-481. R. E., AND R. H. MYERS. Probability and Statistics for 10. WALPOLE, Engineers and Scientists. New York : Macmillan, 1972, p. 135-227.
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
Phase lock of electrical bursts
slow waves and spike
in cat duodenum ARTURO G. SANCHOLUZ, THOMAS E. CROLEY II, JAMES CHRISTENSEN, EN20 0. MACAGNO, AND JOHN R. GLOVER Gastroenterology Research Laboratory, Department of Internal Medicine, and Iowa Institute of Hydraulic Uniuersity of Iowa C&eges of Medicine and Engineering, Iowa City, Iowa 52242
SANCHOLUZ, ARTURO G., THOMAS E. CROLEY II, JAMES CHRISTENSEN, ENZO 0. MACAGNO, AND JOHN R. GLOVER. Phase lock of electrical slow waves and spike bursts in cat duodenum. Am. J. Physiol. 229(3): 608-612. 1975.-The phase, relationship between slow waves and spike bursts was studied in vitro in the cat duodenum. Electrodes were arranged radially about the duodenum. Records were read for Tr, the time between the slow-wave “valley” and the first spike, and T 2, the time between successive valleys. The distributions of the ratio Tr/Tz showed very small differences, not uniquely attributable to radial electrode position. The distribution of Tr/T2 indicated that spike bursts began 58.8 y0 of the way through a slow-wave cycle, measured from the valley. Also, electrodes were arranged longitudinally and records were read to determine ASW, the slow-wave delay time between adjacent electrode positions, and AS, the spike burst delay time between adjacent electrode positions. The correlation of AS and ASW suggested a linear relation, and a regression resulted in a good linear description. These results are consistent with an assumed phase lock of spike bursts to slow waves. electrophysiology;
peristalsis;
gastrointestinal
glas tube (4 mm OD) and fastened at each end so that it was held at the approximate length it had in situ. Thi: tube was mounted inside a large Plexiglas cylinder which was filled with Krebs solution. Thus, the duodenal segment was vertically held in the core of a cylindrical bath (Fig. 1). The Krebs solution was recirculated constantly through the bath, flowing in through the lumen of the perforated tube holding the duodenum and out through a hole at the bottom of the cylindrical tank. Flow outside the tank, accomplished by a peristaltic finger pump acting on rubber tubing, included passage through a glass heat exchanger whose jacket was continuously perfused with water from a circulating pump and heater. The Krebs solution in the bath was maintained at 36.5-37.5”C. It was continuously aerated with 95 70 02-5 % Con. The electrodes were silver-silver chloride glass pore electrodes, described elsewhere (5). While glass pore electrodes may not absolutely reflect the unconstrained electrical behavior of the duodenum, they are standard and more practical. (Microelectrodes would be impossible to handle in a preparation of this sort.) The walls of the bath contained holes, aligned both in the long axis and in the circumference of the cylinder, through which these electrodes could be inserted to touch the duodenum. These holes contained “0”-ring gaskets to prevent leakage around electrodes. Records of the electromyogram were made separately from each electrode against a common grounded reference electrode, which was a coil of chlorided silver wire immersed in the bath. Records were made on an 8-channel ink-writing curvilinear polygraph provided with R-C amplifiers and a bank of filters. A relatively high recording paper speed of 5 mm/s was used throughout the data collection in order to minimize interpretation errors. The frequency response was limited to a band pass of 0.16-150 Hz. Polarity was set so that an upward pen deflection represented depolarization in the slow-wave cycle. In six preparations, the eight electrodes were arranged at equal circumferential intervals at a cross section located about 5 cm from the proximal end of the duodenal segment. The line of the mesenteric insertion served as a reference point and always lay between electrodes 1 and 2. The relative positions of the eight electrodes were thus made constant throughout the six experiments. However, neither the same electrode nor the same channel was always used to record from the same position from experi-
motility
THE MUSCULAR WALL of- the duodenum generates two kinds of electrical signals: slow waves and spike bursts (4). These signals are closely related‘ to duodenal contractions. Slow waves are considered to represent migrating zones of enhanced muscle excitability which, when further intensified by other factors, lead to contractions. A spike burst precedes and signals the beginning of a contraction. At any single point in the duodenum, spike bursts are believed to have a constant range of location in time within the slowwave cycle (1, 2, 4, 6, 7); but a rigorous investigation of the phase relationship between these two electrical signals has not been made. The aim of this work was to investigate statistically the variability of the phase relationship between spike bursts and slow waves. MATERIALS
AND
Research,
METHODS
Eleven healthy cats, weighing 3.0-5.0 kg, were anesthetized by intrapleural injection of sodium pentobarbital, 40 mg/kg. In each, the abdomen was opened and a lo-cm segment of duodenum, extending from about 1 cm below the pylorus to about the ligament of Treitz, was quickly excised and emptied. It was slipped over a perforated Plexi608
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
CAT DUODENUM
DUODENUY -
FIG. 1. A diagram of bath. Duodenum was mounted over a perforated Plexiglas tube mounted vertically in core of a cylindrical Plexiglas bath containing Krebs solution. Bathing solution was circulated externally through a heat exchanger to maintain a constant bath temperature, and bath was continuously bubbled with 950/, O&y0 CO2. Electrodes were inserted through sealed holes in wall of bath to touch serosal surface of duodenal segment.
ment to experiment. Records were made for approximately 2 h. The records were analyzed visually. In these records, two time intervals were measured. Interval Tr was defined as the time between the valley of a slow wave (the nadir between two successive slow waves) and the first spike in the spike burst which followed it. Interval Tz was defined as the time between two successive valleys; it represented the period of the slow wave. The ration Tl/Tz is a variable independent of the time scale of any given slow-wave cycle and satisfies the inequality,
The ratio Tl/Tz, rather than Tl, was chosen to represent the initiation of the spike burst in the slow-wave cycle; it locates the spike bursts for any duration of the slow wave. The stochastic process represented by Tl/T% can . be considered as a simple random variable. This implies that the process is stationary and that successive observations are independent. Intervals T1 and Tz were measured for all slow-wave cycles that contained spike bursts on each of the eight channels separately, and the ratios TI/Ts were calculated. The visual measurement errors were estimated by comparing results, obtained by different people reading the same record, and obtained by the same person on different days. In the first case, the relative error was about 4.1 %, and in the second case, it was about 3.7 %. This was judged small enough to accept this method of measuring TI/T~Relative error is defined herein as the square root of the
609 mean square error divided by, the average mean of Tl/Tz. The mean values, variances, and standard deviations of Tr/Tz were estimated separately for each electrode position in each experiment; see equations 2 and 3 in APPENDIX. A small variability in the mean value estimates among the eight electrode positions existed, so the hypothesis of equality of the means of Tl/Tz among the eight electrode positions was examined. That is, it was considered necessary to investigate the possibility of electrode positiondependent distributions of Tl/Tz, in addition to investigating the general behavior represented by the combined sample from all eight electrodes. The tests used are detailed in APPENDIX. In another five preparations, a different approach was used. Sixteen glass pore electrodes were aligned in the long axis of the duodenum. They were set 5 mm apart with the most proximal electrode 1 cm below the pylorus. Simulas taneous monopolar electromyograms were recorded described above on a 16-channel ink-writing curvilinear polygraph. The delay time between slow waves in adjacent channels in seconds was defined as ASW. Similarly, AS was defined for spike bursts as the time in seconds between the appearances of a spike burst on two adjacent channels. If spike bursts are rigidly phase locked to slow the slow wave and waves (i.e., the phase angle between the spike burst is constant), then AS is equal to ASW. Thus, a rigid phase lock would imply linear - that in the following relation, .(2) AS =a+bASW a is equal to 0 and b is equal to 1. The existence of a linear relationship was acertained initially by inspection of the correlation coefficient (where high values imply a linear relationship) (8). A large number of slow-wave cycles in which spike bursts occurred were selected by taking one slow wave-spike burst complex every Z-rnin period throughWhen several such complexes in out the experiment. each period were available for choice, the one that was the most free of electrical noise and in which the slow waves had the most distinct point of initiation was selected. This procedure reduced the number of data points, thereby reducing computation, yet still gave acceptable confidence in the correlation analysis. The correlation was estimated by the following equation: R = 2
(ASWi
-
m)(ASi
i-l
l/2
-
AS> -A
5 (ASWi i=1
-
ASW)”
2
(ASj - aS)2
j=l
1
(3)
R is the sample correlation coeflicient In this equation, used to estimate the population correlation; AS i and ASW i represent the corresponding ith sample observations, and 5 and m are the corresponding sample means (defined similarly to equation 2 of APPENDIX). By least-squares regression, a straight line satisfying equation 2 was subsequently fitted by estimating the coeflcients a and b as follows: J=
eASiASWi--ndSZ C i=1
1
-1
n FW2
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
1
(4)
610
SANCHOLUZ
A-
ci= as -
bASW
(5)
The estimates defined by equations 3, 4, and for the five experimental runs of this second
5 were made analysis.
RESULTS
Distributions of Tl/T, at eight points in cross section. No relationship was generally apparent between electrode position and the distribution of T1/T2. Mean values from one of the six experiments appear in Table 1. The slight variation among electrode positions (Table 1) prompted examination of the hypothesis that Tl/Tz is dependent on electrode position. One-hundred sixty-eight tests (28 sample pairs for each experiment) were run to compare all electrode position pairs (see APPENDIX), and the null hypothesis for equality of means was rejected (at the 1 % significance level) 85 times, or about 50 %. Particular combinations of electrode position pairs were also investigated to test for possible symmetries. For example, tests for equality of the means for adjacent electrode positions as well as for opposite electrode positions were run. Percentages of rejection of the null hypothesis varied from 45 to 55 %. Thus, the distribution of Tl/Tz could not generally be shown either. . to be dependent on or to be independent of electrode posrtion relative to the mesenterrc insertion at any arbitrary locations in the duodenum. The tests suggest, however, that the distributions of Tl/Tz might be electrode position dependent for some specific locations in the duodenum, for some of the experiments. The normal distribution was considered as a possibility for TI/T~. Although TJTz is bounded and the normal is not, the T1/T2 histograms were very narrow. Therefore, a fitted normal would contain almost no weight in the tails, thereby resulting in little error. The hypothesis of normality was tested for TJT2 for each electrode from each experiment, and the hypothesis was rejected 18 out of 48 times at the 5 % significance level (see APPENDIX). This is more than expected if the normality assumption is true. The mean value estimate for each electrode generally ranged from 0.48 to 0.64 and the standard deviation (“spread”) for each electrode was generally about 0.1. Data were then grouped into six global samples corresponding to the six experiments by combining data from all electrodes in each experiment. The global histograms for Tl/Tz were roughly symmetrical in all six cases (Fig. Z), and the normal distribution was again considered. The goodness-of-fit test was applied and the hypothesis of normality was rejected at the 5 % significance level, 5 out of l
.
.
.
.
.
*
.
TABLE 1. Estimates of mean, variance, and standard deviation for Tl/Tg in one experiment Channel
P
a2
0.610 0.582 0.585 0.623 0.619 0.583 0.558 0.611
0.011 G.007 0.011 0.008 0.008 0.006 0.004 0.007
8
0.107 0.084 0.103 0.092 0.088 0.081 0.063 0.088
ET AL.
the 6 times. Although this implied that the normality assumption was most likely invalid, inspection of the global histograms indicated that the data appeared sufficiently near normal for calculation of confidence limits as outlined in APPENDIX. The 95 % confidence intervals for means and variances (see APPENDIX) were then determined (Table 2). By compiling all 48 sample means, the best estimate of the general mean of T1/Tz was set at 0.588 =t 0.036 (+ 1 SD> . Correlation of AS and AS W in longitudinal section. The correlation between ASW and AS was generally high as shown in Table 3, and the least-squares analysis yielded estimates for the x intercept, a, and for the slope, b, which were close to those values expected for a phase lock, which are 0 and 1, respectively (Table 3). A plot of the regression line for one experiment appears in Fig. 3, which illustrates that there were often negative values for ASW. These represent orad propagation of slow waves. Others have observed this phenomenon in the past (3, 6). DISCUSSION
The histograms of T1/T2 indicate not be rigidly fixed to slow waves, distributed over the whole slow-wave
that spike bursts may but neither are whey cycle. A rigid phase
40 -
r 5z 30L co ; L *O2
to -
I
I 1 II 0.4
02
0.6 T,/Tz
2. Global histogram 1 experiment. Class intervals FIG.
of TJT2 are 0.01.
for all 8 electrode
positions
2. Estimates of means, variances, and conjdence intervals for the six global distributions
TABLE
n
127 74 62 46 77 102 102 62
Expt
2 2 3 4 5 6
;
0.594 0.610 0.567 0.575 0.578 0.590
62
0.008 0.004 0.011 0.008 0.005 0.005
n
652 425 413 644 797 603
95% Confidence Intervals for ~2
(0.0075, (0.0038, (OcOO96, (0.0078, (0.0047, (0.0042,
0.0093) 0.0051) 0.0120) 0.0090) 0.0057) 0.0052)
957& Confidence Intervals for p
(0.587, (0.604, (0.557, (0.568, (0.573, (0.585,
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
0.601) 0.616) 0.577) 0.584) 0.583) 0.595)
in
CAT
611
DUODENUM
TABLE 3. Estimates of correlation coejicient, and slope (b) between AS W and AS
x intercept (a),
Expt
R
n
ii
7 8 9 10 11
0.81 0.89 0.88 0.85 0.84
53 56 45 51 50
0.19 0.01 0.09 -0.07 1.05
8
0.70 0.98 0.64 0.73 0.85
existence of a rigid phase phase lock might exist.
lock
and
suggest
that
a rigid
APPENDIX The following simplified tests for the equality of means were used. They are based on the central limit theorem of statistics (10). From this theorem, the following random variable, 2, can be shown to be approximately distributed with the standard normal distribution, regardless of the actual distribution of Ti/Tz (assuming equality of means) :
for the null hypothesis true. The degree for sample sizes (nl, nz) greater than 30 sample mean and 0 i2 is the variance for and sample variance, Q is, are defined as
pi =
l/Q
2 j-1
of approximation is excellent (10). In equation I, fli is the electrode i. The sample mean follows:
ITl/T21i,
(2)
j
and
ni a2 ui
0
0 c -8
FIG.
3. Regression
of As upon
ASW
in 1 experiment.
lock would produce a “spiked” distribution, (i.e., one in which Tr/Tz is constant), whereas a distribution over the whole cycle would result from a purely random phase relationship. The observed distribution approaches a spiked distribution; the range of the cycle over which spike bursts can be expected was restricted. The marked uniformity of mean value estimates of Tr/T2 and the very small estimates of the standard deviation indicate that the beginning of spike bursts are confined to a very narrow range on the slow-wave cycle. The differences in the distributions, though small, could be related to electrode position. The evidence presented here prevents drawing a conclusion of position dependence valid for all locations. Nevertheless, it does suggest that this dependence probably exists for some specific location relative to the mesenteric insertion. The variation in Tr/Tz could be related to data interpretation or other variables. The hypothesis of a normal distribution for Tr/Tz was rejected 34 % of the time for tests on individual electrode positions and most of the time for tests on global samples of each experiment. The tests on individual electrodes generally revealed small skewness toward smaller values for Tr/T,. In fact, Tr/T2 is actually a ratio of two separate random variables, which is seldom normally distributed. Initially, the correlation analysis for time lags of the slow wave and spike burst at successive locations indicated that a linear relationship between the lags was likely. Subsequently, the regression analysis further indicated that a phase lock between the slow wave and spike bursts was also likely. If a rigid phase lock really exists, the preceding analyses values. The interpretation errors, would not yield “perfect” inherent in the visual measurement method used, would result in random errors (which are generally normally distributed). Considering this, these data do not refute the
=
1 /ni
C j=l
([T1/T21i,
j -
I&j2
(3)
represents the jth observation of the In equations 2 and 3, [T,/Tz]i,j random variable T1/T2 for electrode i, By replacing ui2, and ~2~ in equation 2 by their estimators, the distribution of 2 is no longer a normal. By using, however, standard normal distribution quantiles and values of 2 (with 0~2 = &i2), approximate tests can be made (10). This is done in lieu of other standard tests for equality of means because the approximation is good for large sample sizes and because other tests have larger computation requirements. The tests are performed by calculating the value of Z in equation 1 (with ui2 = 6 i2) and comparing with an appropriate y quantile of the standard normal distribution. If the value of 2 is greater, then the hypothesis of equal means is rejected at the (1 - 7) significance level. This test was applied to every pair of, electrodes (28 combinations in each experiment) and to combinations selected to test for various symmetries. To test the hypothesis of a normal distribution for TJT2, a goodness-of-fit test was done with the experimental frequency distributions (histograms) obtained. It can be shown (9) that the following random variable Qk
=
C
(Nj
-
YlP
(4)
j)2/(YlPj)
j-1
is approximately distributed with a chi-square distribution with k - 3 degrees of freedom when the hypothesized distribution with unknown mean and variance is really the actual. In equation 4, k is the number of intervals (for values of Tl/T2) into which the range is nbroken; iVi is the number of observations which lie within interval j; Pi is the probability that an observation would lie within interval j, which is calculated from the hypothesized distribution with mean equal to fi and variance equal to a2. From experience guidelines, the number of intervals, k, is set at 10. Smaller numbers for k result in poor test significance and larger numbers accentuate extreme observations. Goodness-of-fit tests are performed by calculating fl and 62 from the sample and Pj from the hypothesized distribution with mean fl and variance a2, and then calculating the chi-square statistic, Qk from equation 4. The value of Qk is compared with an appropriate 7 quantile point of the chi-square distribution with k - 3 degrees of freedom. If Qk is greater, the hypothesis that the data came from the suggested distribution is rejected at the (1 - 7) significance level. This test was applied to the individual electrode samples and to the global samples (all electrodes) for each experiment, to test for normality. It can be shown (9) that if Tl/Tz is normally distributed (or approximately so), then the following approximate 100y~O confidence intervals for the mean and variance, respectively, of the global distribution of Tl/Tz can be written for large sample sizes: (P - y %f&
P +y
Ufi)
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
(5)
612
SANCHOLUZ
In equations 5 and 6, @ is the sample mean of Tl/Ts for the combined eight electrodes, k2 is the sample variance, n is the combined sample size for the eight electrodes, y is the 1 - [( 1 - #2] quantile point of the t distribution with n - 1 degrees of freedom (or the standard normal distribution for large sample sizes), and yl and y2 are the and 1 - [(l - r)/2] quantile points, respectively, of the (1 - r)/2 chi-square distribution with n - 1 degrees of freed0.m. The sample estimates, 4 and a2, are defined similarly as in equations 2 and 3.
ET AL.
This work was supported by National Institutes of Health Research Grant AM 08901 and Research Career Development Award AM 20547. Requests for reprints should be sent to: J. Christensen, Division of Gastroenterology, Dept. of Internal Medicine, University Hospitals, Iowa City, Iowa 52242. Received
for publication
19 July
1974.
REFERENCES 1. BASS, P., C. F. CODE, AND E. H. LAMBERT. Motor and electric activity of the duodenum. Am. J. Physiol. 201: 287-291, 1961. 2. BASS, P., AND J. N. WILEY. Electrical and extraluminal contractileforce activity of the duodenum of the dog. Am. J. Digest. Diseases 10: 183-199, 1965. 3. BORTOFF, A. Slow potentials variations of small intestine. Am. J. Physiol. 201: 203-208, 196 1. 4. CHRISTENSEN, J. The controls of gastrointestinal movements, some old and new views. New Engl. J. Med. 285 : 85-98, 1971. 5. CHRISTENSEN, J., AND R. L. HAUSER. Longitudinal axial coupling of slow waves in proximal cat colon. Am. J. Physiol. 221: 246-250, 1971.
6. DANIEL, E. Electrical activity of the alimentary tract. Am. J. Digest. Diseases 13 : 297-319, 1968. 7 DANIEL, E. The electrical and contractile activity of the pyloric ’ region. Gastroenterology 49 : 403-4 18, 1965. 8. JENKINS, G., AND D. WATTS. Spectral Analysis and Its Application. San Francisco : Holden-Day, 1968, p. 73-75. g . MOOD, A. M., F. A. GRAYBILL, AND D. C. BOES. Introduction to the Theory of Statistics. New York : McGraw, 1974, p. 372-481. R. E., AND R. H. MYERS. Probability and Statistics for 10. WALPOLE, Engineers and Scientists. New York : Macmillan, 1972, p. 135-227.
Downloaded from www.physiology.org/journal/ajplegacy at Lunds Univ Medicinska Fak Biblio (130.235.066.010) on February 14, 2019.
