Psychophysiology, 42 (2005), 246–252. Blackwell Publishing Inc. Printed in the USA.
Copyright r 2005 Society for Psychophysiological Research
DOI: 10.1111/j.1469-8986.2005.00277.x
and
Department of Psychology, The Ohio State University, Columbus, Ohio, USA
Abstract
The root mean square successive difference (RMSSD) in heart period series is a time domain measure of heart period variability. The RMSSD is sensitive to high-frequency heart period fluctuations in the respiratory frequency range and has been used as an index of vagal cardiac control. By transfer function simulations, the RMSSD statistic is shown to represent a high-pass filter that effectively captures respiratory sinus arrhythmia but also passes lower frequency fluctuations that can include sympathetic influences. These simulations, together with analysis of actual heart period series, reveal that the RMSSD is biased by basal heart period. Although between-subjects levels of RMSSD covary highly with spectral estimates of high-frequency variability, within-subject RMSSD change scores account for only 50–
60% of the variance in spectral estimates. The present findings raise caveats in the applications and interpretation of the
RMSSD statistic.
Descriptors: Electrocardiogram, Heart rate variability, Successive difference, Root mean square successive differences,
Respiratory sinus arrhythmia
The square root of the mean squared differences of successive heart periods (root mean square successive differences, or
RMSSD) is a time domain measure of heart period (HP) variability that is sensitive to short-term, high-frequency HP fluctuations (Ewing, Neilson, & Travis, 1984; Kleiger, Stein, Bosner,
& Rottman, 1992; Porges & Byrne, 1992; Task Force of the
European Society for Cardiology and the North American Society of Pacing and Electrophysiology [Task Force], 1996).
RMSSD is a commonly used index in clinical cardiology (Bigger et al., 1988; Task Force, 1996) as well as in psychophysiology
(e.g., Owen & Steptoe, 2003), and is often included in analytical systems for measuring heart rate variability (e.g., AMS, Vrije
University, Amsterdam; AccuPlus Holter Analysis System, Delmar Medical Systems, Irvine, CA). The RMSSD measure correlates well with frequency domain measures of high-frequency heart rate variability reflecting respiratory sinus arrhythmia
(RSA), at least under constrained conditions (Penttila¨ et al.,
2001; Task Force, 1996). Pharmacological blockade studies further indicate that the RMSSD statistic is sensitive to vagal cardiac control, and it has even been suggested to be superior to spectral methods as it may be less sensitive to variations in respiratory patterns (Penttila¨ et al., 2001).
The statistical properties of the RMSSD remain to be fully clarified, however, and an important issue arises as to whether it represents an index of cardiac vagal control uncontaminated by sympathetically mediated heart rate variability. Spectral estimates of vagal cardiac control, derived from measures of respiratory sinus arrhythmia (Berntson, Cacioppo, & Quigley, 1993;
Porges & Bohrer, 1990), typically employ a low-frequency cutoff of 0.12 to 0.15 Hz, as there is little sympathetic contribution to heart period variability beyond that frequency (for reviews, see
Berntson et al., 1993, 1997; Porges & Bohrer, 1990). The present article evaluates the frequency-filter characteristics of the
RMSSD statistic, compares the RMSSD to a standard spectral estimate of respiratory sinus arrhythmia (high-frequency heart rate variability), and considers the implications of the results for the application of the RMSSD as a measure of cardiac vagal control. The primary aim of the work is to identify characteristics of the RMSSD statistic, however, not to model the mapping of vagal control onto heart rate.
We thank Justin Tawil for contributions to analysis. This research was supported in part by a grant from the National Heart, Lung, and Blood
Institute (HL54428).
Address reprint requests to: Gary G. Berntson, Ohio State University, 1885 Neil Avenue, Columbus, OH 43210, USA. E-mail: berntson@osu.edu.
246
Methods
Transfer Ratio Modeling
To clarify the relationship between successive heart period differences and respiratory sinus arrhythmia, we modeled the transfer functions of RSA onto successive heart period differences. As input functions, sinusoids at a range of frequencies that span the typical respiratory frequency band (0.1–1.5 Hz) were used to simulate respiratory modulations of heart period (see Figure 1

RMSSD and the Appendix). The plus/minus peaks of these sinusoids represent the maxima and minima of the respiratory cycle modulation of heart period, normalized to 1. The resulting transfer functions reflect the relative proportion of the respiratory modulation of heart period (i.e., the magnitude of RSA) that is captured by the RMSSD statistic. This proportion would apply regardless of the absolute amplitude of the fluctuation, within the limits of measurement error at low levels of modulation. Different absolute amplitude values would be characterized by a coefficient that would be multiplied by the normalized values to restore the absolute values. As this coefficient is a constant for any given amplitude and is a multiplier of both the input and the output, it would not affect the relative transfer as derived below
(and shown in Figure 2).
As more fully described in the Appendix, output functions were derived by averaging the relative amplitude values of the input sinusoids over the duration of successive heart period sample intervals and the basal heart period values were modulated by this output.
1,2
Successive heart period differences were then derived from these time-varying heart period series. The transfer function was calculated as the ratio of the output (successive differences) to the input, for different respiratory frequencies. Two different basal heart period values were employed
(1000 and 500 ms, corresponding to basal heart rates of 60 and
120 bpm, respectively) to evaluate the effects of baseline heart rates on the transfer functions, and differences in the phase relation (0 to 90
1
) between the heart cycles and the respiratory cycles were also modeled. Heart period levels and RSA are not independent in nature, of course, and changes in basal heart period may be associated with alterations in RSA amplitude
(e.g., the amplitude of respiratory modulation of HP would likely be smaller at 500 compared to 1000 ms basal heart periods). This interdependence does not compromise the present simulations, as the transfer ratio represents the relative transfer of the modulation signal, irrespective of absolute magnitude of that modulation.
Respiratory fluctuations are not pure sinusoids and other sources of variation (including random fluctuations) in successive heart periods would introduce additional noise in actual cardiac functions. The goal of the present approach, however, is not to capture all sources of variation or to define an absolute transfer function, but rather to characterize the relative transfer of respiratory vagal modulations onto heart period variability.
This would not be corrupted by the simplified conditions of the modeling.
An additional consideration is that the cardiac impact of vagal modulations in vivo would have some finite phase lag, and a momentary level of vagal outflow could impact not only the ongoing beat but the next beat as well. This also would not corrupt the present approach, as it would be formally equivalent
1
Equivalent results are obtained by simply taking the midpoint, endpoint, or any other standard point within the heart period, rather than the average of the function over the heart period interval. Moreover, the phase dependency remains the same, although the phase angle may be shifted depending on the sample point.
2
A simplification of the modeling concerns the fact that the modulation of the basal heart period would change the heart period sampling time, and this in turn could alter the magnitude of modulation. This effect would be minimal, however, because the magnitude of RSA is generally a very small percentage of the basal heart period. Moreover, the effect would be a slight amplification of the transfer ratio on the rising modulation function and a slight negative amplification on the falling phase of the function, so the net effect over the modulation cycle (and hence the effect on RMSSD) would be trivial.
A
+1.0
0
−
1.0
B
ECG
B1 B2
HP
HP
B1-B2
= +0.64
HP
B2-B3
=
−
0.64
B3
1.28
b1 b2 b3 b4 b5
HP
HP b1/b2
= +0.86
HP b2/b3
= +0.32
+0.54
HP b3/b4
= − 0.32
HP b4/b5
=
−
0.86
+0.54
Effect of Phase
+1.0
0
Effect of Sample Interval (Heart Period)
247
−
1.0
EC G
B1 B2 B3 B1’ B2’ B3’
HP
HP
B1-B2
= +0.64
HP
B2-B3
= − 0.64
1.28
HP
HP
B1’-B2’
= +0.86
HP
B2’-B3’
= +0.00
+0.86
RMSSD = 1.28; RMSSD = 0.00; RMSSD = 0.64
Figure 1.
Illustration of the modeling of heart period differences. Identical basal heart periods, defined as interbeat intervals (B1 to B2, B2 to B3, etc.), are modulated by a sine function representing respiratory fluctuations. The degree of modulation is taken as the average of the sine wave function over the heart period epoch (dashed arrows). Resulting differences between successive heart periods are expressed as relative proportions of the maximal sine wave function that represents vagal modulation of heart period
(normalized to 1.0 for illustration). Thus, the transfer function units represent a proportionality from 1 1 (inspiratory peak) to 1 (expiratory trough), regardless of the absolute magnitude of modulation. A: Effects of different heart period sample intervals on successive heart period differences.
Left: Three beats (B1–B3) define two basal heart period intervals that are modulated by the modeled respiratory function, yielding the illustrated successive heart period difference. Right: Corresponding heart period intervals and successive differences for a basal heart period sample one-half of that on left (twice the frequency). The successive differences of the shorter heart periods are considerably smaller than for the longer heart periods on the left. B: Effects of phase relations between the modeled respiratory modulation and the heart beat. Beats B1–B3 (left) are identical to B1
0
–B3
0
(right) except for phase. B1–B3 have a 0
1 phase shift (alternate beats fall on zero crossings) whereas B1
0
–B3
0 are phase shifted and yield different successive beat differences. Large solid arrows on ECG timeline: Maximal
RMSSD is achieved when the interbeat interval is one-half of the modulatory period and in a 0
1 phase relation. In this case, one beat extends wholly across the peak half of the function and the next across the trough, yielding the largest possible difference between adjacent beats. Large open arrows on ECG timeline: A null difference occurs when the same heart periods are shifted 90
1
, in which case the means of the modulation functions for all heart periods are zero and hence beat differences are zero. RMSSD values: for beats indicated by solid arrows, for beats indicated by open arrows, and for a beat series comprised of all (solid and open) arrows.

248
1.00
Filter Characteristics of RMSSD
0.75
90
°
0
°
90 °
0
°
0.50
HP=1000 ms
0.25
HP=500 ms
0.00
0.0
0.1
0.2
0.3
0.4
Respiratory Frequency (Hz)
0.5
Figure 2.
Transfer of modeled respiratory modulations to successive beat differences as a function of respiratory frequency, basal heart period, and phase relation between the respiratory and cardiac cycles. Solid lines illustrate the overall functions collapsed across phase and the dashed lines illustrate two of the modeled phase shifts (0 and 90
1
). For a given basal heart period sample time, there is a progressively increasing transfer as respiratory frequency increases, until the aliasing frequency is approached (Nyquist frequency, where cardiac frequency is less than twice the respiratory frequency). At that point, phase-dependent aliasing yields widely divergent values. As illustrated by the two sets of functions
(500 and 1000 ms basal heart periods) the RMSSD statistic is biased by the sample intervals imposed by different heart periods, even for identical respiratory modulation functions.
G.G. Berntson, D.L. Lozano, and Y.-J. Chen ciated with real differences in vagal cardiac control or respiratory parameters. Consequently, to examine the role of heart period as an independent determinant of RMSSD, we compared RMSSD values from actual heart period series to those derived from the same data sets in which every other HP was deleted and time was added to the prior period, yielding a series with a mean heart period twice that of the original data. This approach permits evaluation of the impact of different heart period sample times, as it preserves the time line as well as the temporal pattern of respiratory modulation. For a given individual, such a lengthening of heart period would likely be achieved in part by increases in vagal control, which would in turn increase RSA. That consideration does not compromise the present analysis, however, as the results are expressed in relative terms. The question is not whether higher amplitude RSA would yield a difference in
RMSSD, as surely it would. Rather the question concerns the effect of a change in the heart period sample time on the relative transfer of a given RSA modulation function. Differences in
RSA amplitude would not change the relative transfer functions.
Two methods of deleting alternate heart periods were employed. The first was the simple deletion of alternate R waves, resulting in the summing of adjacent heart periods. This yields a simulated HP series with a mean twice that of the original series and a higher variance. The latter would be expected given the results of the transfer ratio analyses, that longer heart periods more effectively transfer systematic trends into higher successive beat differences (and hence variance). An additional method was also employed, in which alternate heart periods were deleted and the mean of the entire series was added to the remaining periods.
This again yields a simulated series with a heart period twice that of the original series, but with a variance that is equivalent to the original series (see Table 1).
to shifting the phase relations of the cardiac and respiratory cycles, which is included as a dimension in the present modeling.
Human Heart Period Data
Data processing.
Archival data were analyzed from 20 participants (13 females) tested in the laboratory under sitting baseline, sitting stress (mental arithmetic), and standing conditions (3 min in each condition). ECG was recorded (Biopac model
MP150, Biopac Systems, Inc., Goleta, CA) throughout the session and analyzed by the MindWare analysis system (MindWare,
Gahana, OH). The resulting heart period series was examined for artifacts visually and by an automated algorithm (Berntson,
Quigley, Jang, & Boysen, 1990), and corrected when necessary.
High-frequency (HF) heart rate variability (0.15–0.4 Hz), corresponding to respiratory sinus arrhythmia, was then determined on a minute by minute basis by standard methods (Berntson et al., 1997). Briefly, the heart period series was time sampled (at
250 ms) by an interpolation algorithm (see Berntson et al., 1995).
The time series was then de-meaned, linearly detrended, tapered, and submitted to a fast Fourier transform. The resulting spectral function was integrated within the respiratory frequency band to provide an index of RSA. RMSSD was also derived over the same epochs, as the square root of the mean of the squared differences between adjacent heart periods.
Simulated doubling of the heart period sample time.
Transfer ratio analyses indicated that the RMSSD statistic is biased by the prevailing basal heart period, with longer heart periods more effectively transferring the modulation function into higher values of RMSSD. This prediction was further evaluated by analysis of archival heart period data from human subjects. Comparisons of RMSSD across individuals or conditions with differing heart periods would be problematic, as these contrasts may be asso-
Correlations between RSA and RMSSD.
RMSSD values derived from the original heart period series were also compared to estimates of RSA derived from HF heart rate variability (correlation and regression analyses were by SPSS v11.0; SPSS, Chicago, IL). These analyses were pursued merely to provide a
Table 1.
RMSSD Is Increased with Longer Heart Period Sample
Times
Baseline Stress Stand
Original HP series
Heart period a
Heart period variance b
RMSSD
Simulation (summed beats)
Heart period a
Heart period variance b
RMSSD
RMSSD change
(simulated/original)
Simulation (alternate beats 1 mean)
Heart period a
Heart period variance b
RMSSD
RMSSD change
(simulated/original)
852 (35)
4978 (1300)
62.2 (11.3)
1703 (70) 1492 (62) 1288 (37)
12934 (2638) 21547 (8492) 5370 (986)
121.3 (18.3) 106.9 (17.2) 46.8 (4.6)
1.95
2.28
2.82
1702 (70)
4958 (1298)
80.8 (13.4)
1.30
746 (31)
6219 (2301)
46.8 (8.8)
1501 (62)
1.40
644 (18)
1414 (259)
16.6 (1.6)
1288 (37)
6370 (2460) 1432 (262)
65.5 (11.1) 25.9 (2.6)
1.56
Means and (standard errors); all values in milliseconds.
a b
Overall mean heart period.
Average within-subjects variance of successive heart periods.

RMSSD comparison among measures and not for the purpose of validation, as all indirect measures (including spectral estimates) have limits and caveats. Validation studies of RMSSD would require more direct measures, such as pharmacological blockades, and are beyond the scope of the present effort.
Results
Transfer Ratio Modeling
As illustrated in Figure 2, the transfer of modeled vagal fluctuations to heart period variation is not flat across the respiratory frequency band. For a given basal heart period time sample, slower respiratory fluctuations are generally less effectively transferred. This is because successive heart period intervals fall over proportionally shorter segments of longer respiratory cycles, and adjacent heart periods are thus closer together on the model sinusoidal (respiratory modulation) input functions. This constitutes a low-cut filter function that confers on successive heart period differences (RMSSD) its preferential sensitivity to higher frequency fluctuations, including those associated with RSA.
Nevertheless, the RMSSD statistic does pass some variance below the typical respiratory band (see Figure 2). The transfer functions are shown in relative units, as the important question concerns the effects of respiratory frequency on RMSSD for a given respiratory amplitude. This effect is sizable, as the relative transfer is greater than twice as large for the highest frequencies compared to the lowest frequencies, in the typical respiratory frequency band. That difference in proportionality would hold regardless of the absolute value of the respiratory modulation.
The obverse of the above is that, for a given respiratory frequency, the filter function associated with the RMSSD yields a more effective transfer of the modeled vagal fluctuations onto longer heart periods, as is illustrated by the generally higher transfer function values for the long compared to the short basal heart periods in Figure 2. This is attributable to the fact that longer heart periods fall over more disparate amplitudes of the modeled respiratory fluctuations, and hence have larger successive differences than do short heart periods (see Figure 1A).
There is a symmetrical trade-off between the transfer functions across respiratory frequencies and the functions for different heart periods. Transfer values of the shorter (500 ms) heart periods at a given frequency (e.g., 0.4 Hz) are equivalent to those for the longer heart periods (1000 ms) at half the respiratory frequency (e.g., 0.2 Hz; see Figure 2), because the relevant dimension is the proportion of the respiratory cycle sampled by the heart period. The effect of heart period is not trivial, as the relative transfer at 1000 ms can be about twice the transfer at 500 ms
(at the lowest respiratory frequencies). Again, the proportionality illustrated in Figure 2 would hold regardless of the absolute magnitude of RSA.
With increasing respiratory frequencies, there is a progressive increase in transfer reaching a maximum when the heart period is one-half the respiratory period, and at a 0
1 phase lag, so that each beat falls precisely at the zero crossings of the modeled respiratory function. This is the condition in which successive beats show the greatest differences, as one heart period extends wholly over the peak half of the sine function and the next over the trough (see Figure 1B, solid heavy arrows).
A notable contribution of phase becomes apparent when the heart period approaches one-half of the respiratory period (see
Figure 2). At this point the sample rate (heart rate) approaches
249 the Nyquist frequency (twice the respiratory frequency), which is the minimal sampling frequency required to avoid aliasing. As discussed above, the maximal beat difference is seen when the heart period sample time corresponds to one-half of the respiratory period, with a 0 1 phase shift. With a 90 1 phase shift, successive beats fall not at the zero crossings, but alternately at the peak and trough of the respiratory modulation (see Figure 1B, open heavy arrows). Each heart period would thus be subject to an identical modulation function (peak to trough or trough to peak, with each having a mean of zero), and heart periods would be equivalent except for noise or other nonrespiratory trends.
Similar functions are seen with the 500-ms heart periods, but because this represents a doubling of the sample rate, the phase dependency is not apparent until twice the respiratory frequency, which is beyond what is generally observed experimentally.
3
Human Heart Period Data
Simulated doubling of heart period.
Predictions from the transfer function analyses presented above were evaluated by heart period data from human subjects. Comparisons of
RMSSD across individuals or conditions with differing heart periods would be problematic, as either of these contrasts may entail differences in vagal cardiac control and/or respiratory parameters. Consequently, a simulation of the effect of doubling of heart period sample times on the RMSSD statistic, in the absence of changes in RSA or vagal control, was accomplished by deleting alternate R waves and summing the values of the two adjacent heart periods. As illustrated in Table 1, this doubled the heart period and increased the variance of the series. It also increased the RMSSD value of the simulated series by about twoto threefold, despite the fact that the underlying respiratory modulation was identical in the two cases as the analyses were based on the same data set.
Summing adjacent beats also increased the variance of the series, as is apparent in Table 1. Although this may reflect a bias similar to that on the RMSSD, a further simulation was pursued in which the heart period was doubled but the variance of the series was maintained (by deleting alternate heart periods and adding the mean of the original series to the remaining periods).
This resulted in a doubling of heart period, with a comparable variance to that of the original series. Although this procedure did reduce the increment in RMSSD with the (simulated) longer heart period sampling times, it did not eliminate the bias. As illustrated in Table 1, the RMSSD was still increased between 30 and 60% with the longer heart period sampling times.
For both simulations, the RMSSD ratio becomes progressively larger with lower RMSSD levels (from baseline to stress to stand). The origin of this feature is not clear, but it may reflect a signal-to-noise issue where longer heart periods more effectively pull the RSA variance out of noise.
Relation between RMSSD and RSA.
Despite the considerations outlined above, there was a high between-subjects correlation between absolute RMSSD and spectral estimates of high frequency (HF) variability (RSA) in the human data (Table 2).
The average correlation across conditions was .85, with the
RMSSD accounting for about 75% of the variance in the spectral estimate. The correlation was lowest under the standing
3
Higher frequency respiratory rates are not illustrated as they would exceed what is typically observed naturally. At higher rates, however, there is a progressively dampening and frequency-dependent ‘‘ringing’’ of the transfer function that ultimately becomes flat and null.

250
Table 2.
Correlations between RMSSD and HF Variability
Condition
Baseline
Stress
Stand
Mean
D bsl/stress
D bsl/stand
D stress/stand
Mean r
.75
.76
.74
.75
.87
.93
.74
.85
condition, where values were small due to the minimal vagal control under this condition. This likely reflects a restricted range limitation. As illustrated in Table 2, however, correlations of within-subjects changes in RMSSD and HF did not fare as well. For change scores, variations in the RMSSD statistic account for only 50–60% of the variance in HF variability, which is no better than the correlation between changes in
HF variability and changes in heart period ( r 5 .75 for RMSSD vs. .74 for HP). This may be related in part to the restricted range of the change scores, relative to between-subjects differences. It may also be due, at least in part, to the fact that withinsubjects changes in RSA are generally associated with changes in basal heart period. This could introduce biases into RMSSD measures and degrade the correlations with HF variability. Decreases in RSA, for example, may be associated with increases in heart period, which lead to a less effective transfer of RSA to
RMSSD.
Discussion
Transfer function modeling reveals that the RMSSD statistic acts as a low-cut, high-pass filter that captures respiratory sinus arrhythmia, but also passes lower frequency fluctuations ( o
0.12
Hz) that can reflect sympathetic influences. This suggests that the
RMSSD may not represent a pure index of vagal cardiac control.
Moreover, simulation studies together with analyses of human heart period data show that the RMSSD statistic is biased by basal heart period, independent of variations in RSA. These effects can be rather large as evidenced by the approximate doubling of transfer from lower to higher respiratory frequencies and from shorter to longer heart periods. For a given amplitude of respiratory modulation, regardless of its absolute magnitude, the
RMSSD statistic may differ by twofold over normal ranges of respiratory frequency and heart period levels. This may be more of a problem for between-subjects analyses, as respiratory frequencies and basal heart periods may differ more across than within subjects.
Despite these findings, the present results confirm a high correlation between the absolute values of the RMSSD statistic and
HF variability in the respiratory frequency range. This is consistent with the existing literature (see Penttila¨ et al., 2001; Task
Force, 1996). These findings have led to the suggestion that the
R
2
.56
.58
.54
.56
.76
.87
.55
.72
G.G. Berntson, D.L. Lozano, and Y.-J. Chen
RMSSD may be useful as a time-domain index of vagal control of the heart. Although absolute between-subjects levels of
RMSSD covary closely with estimates of HF variability, within-subject RMSSD change scores did not fare nearly as well, accounting for only 50–60% of the variance in the spectral estimates. This may be due to a restriction of range limitation. This was not a validation study, in any event, and it is important to note that spectral measures have their own caveats and limitations. The main point is that RMSSD and HF variability may intercorrelate well under some circumstances, but less so under others.
Absolute levels of RSA and/or RMSSD are influenced by a variety of factors independent of vagal cardiac control. These include age, physical conditioning, physiology of the sinoatrial node, and rate and depth of respiration, among other variables.
Individual differences in these variables may confound estimates of absolute levels of vagal control and limit interpretation of group differences. For this reason, within-subjects change or lability scores are generally considered more valid as indices of relative vagal control across conditions than are estimates of absolute levels (see SPR Committee report, Berntson et al.,
1997). Pharmacological blockade studies provide further support for this conclusion (Berntson et al., 1994; Cacioppo et al., 1994;
Grossman & Kollai, 1993).
Despite these considerations, present results reveal that between-subjects correlations of absolute levels of RMSSD and HF variability were higher than within-subjects changes in these measures. That is, under the very conditions where HF variability has been shown to have the greatest validity as an index of cardiac vagal control, its intercorrelation with the RMSSD statistic is diminished.
A change in vagal control across states or conditions is generally associated with a change in heart rate, which may arise from the alteration in vagal control itself and/or from a change in sympathetic control. As revealed by the present transfer functions (Figure 2) and by analysis of simulations based on human data, changes in heart rate can bias RMSSD estimates independent of variations in autonomic control. In addition, there can be an appreciable contribution of lower frequency ( o
0.15
Hz) sympathetic cardiac control to successive beat differences
(see Figure 2), and these contributions may vary across states.
Consequently, heart period contributions may confound RMSSD estimates of vagal control, and successive difference statistics such as the RMSSD conflate sympathetic and vagal influences on heart rate variability. This becomes particularly problematic when comparing across conditions with different heart rate levels, which is common in psychophysiological applications.
Although these considerations do not rule out the use of
RMSSD as an index of vagal control, they do raise important caveats that must be considered in interpreting this statistic.
Specifically, RMSSD comparisons may be problematic across wide differences or variations in heart period and/or respiratory frequencies. Although the studies outlined here suggest that these can be large, at least under the simulated conditions, their actual magnitude in typical experimental contexts has yet to be confirmed.
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RMSSD
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APPENDIX
Sine Wave Input Function
The simulation entailed the derivation of a transfer ratio of output, as reflected in RMSSD, to input (respiratory modulation of heart period). Initially, a sine wave array of amplitude 1.0 was generated to model respiratory modulation of heart period, as follows: f ð x Þ ¼ sin ð p = 180 x Þ ; ð 1 Þ for x 5 1 to 360. The amplitude ( 1.0) was selected for computational simplicity and does not impact the simulation, as the transfer ratio reflects a proportionality of the output to the input and is equivalent regardless of the absolute magnitude of the input modulation.
RMSSD Statistic and the Transfer Ratio
The next step was to determine the dependency of the transfer ratio on the respiratory period and the heart period sample interval. The heart period sample frequency can be expressed as a proportion of the respiratory period (samples/cycle):
S ¼ respiratory period = heart period
¼ heart rate = respiratory rate ð 2 Þ
For example, a 1-s heart period and a 10-s respiratory period yield a sample interval ( S ) of 1/10 of the respiratory period or 10 heart periods/respiratory cycle. This corresponds to a sampling of the above function f ( x ) every 36 sample points (360 points in the array divided by 10 samples). Consequently, the first heart period would extend along the function f ( x ) from x 5 1 to x 5 36; the second from x 5 37 to x 5 72, and so forth. The deviations of the successive heart periods from the mean were then taken as the average of the function f ( x ) between x 5 1 to 36, 37 to 72, and so forth. The deviation of each heart period from the mean was calculated as follows: b 5 ( a 1) 1 360/ S , ending x point in f ( x ); c 5 360/ S , number of points in interval a to b of f ( x ).
Only the deviation from the mean was modeled, as the
RMSSD statistic only reflects the differences between adjacent heart periods, irrespective of the basal heart period.
Heart period differences across the IBI series are then derived as follows:
D i
¼ j HP i
HP i þ 1 j ;
Finally, the transfer ratio ( TR
S
) was derived as
TR
S
¼ output ð RMSSD Þ = input ¼ S ð D Þ = 2
ð
ð
4 for i 5 1, 2, . . . , I 1, where I 5 number of heart periods in the series. The RMSSD statistic S ( D ), which approximates the standard deviation when I is large, was then calculated by the standard formula
S ð D Þ ¼ v t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
X
I 1
D 2 i i ¼ 1
ð 5 Þ
6
Þ
Þ
This ratio can range from 0, when there is no manifestation in
RMSSD, to a maximum of 0.707, which is the RMS amplitude value of a sign wave of unity amplitude, when the input is fully captured. The latter would occur, for example, when there are exactly two heart periods in a respiratory cycle, one falling wholly in the positive half of the cycle and the other falling wholly in the negative half. The means of these heart periods would be
0.707, the difference would be 1.414, and the transfer ratio would be 1.414/2 5 0.707.
HP i
¼ HP
B i
B i
þ 1
¼ x ¼ a f x = c ð 3 Þ where B i
5 heart beat at the onset of HP i
; B i 1 1
5 heart beat terminating HP i
; a 5 1 1 ( i –1) 360/ S , starting x point in f ( x );
Transfer Ratio Functions
The S index on TR
S reflects the fact that the output/input relation holds only for a given heart period sampling rate ( S ) over the respiratory cycle (10 in the present example). The transfer ratio would be identical for any respiratory/cardiac cycle combination that yields 10 heart periods for each respiratory cycle. Examples in Figure 2 include the points at 1 Hz cardiac rate (1000 ms heart period) at 0.1 Hz respiratory rate, as well

252 as a 2 Hz cardiac rate (500 ms heart period) and a 0.2 Hz respiratory rate. Values for these two combinations are equivalent, with the 500-ms point merely shifted to the right along the abscissa. To define the transfer ratio functions ( TR
S
) of
Figure 2, over the full range of respiratory frequencies, the computations of equations 2–6 were iterated for different values of S .
G.G. Berntson, D.L. Lozano, and Y.-J. Chen
Modeling of Phase Relations
The modeling described above was based on a null phase lag.
That is, the start of the first cardiac cycle was coincident with the start of the respiratory cycle. To characterize the effects of phase relations, additional simulations were also run with phase lags ranging from 0 to 90
1
. These were pursued as described above, but with different offsets of the starting cardiac cycle.