Cross-sample entropy statistic as a measure of complexity

Transcription

659
Exp Physiol 92.4 pp 659–669
Experimental Physiology
Cross-sample entropy statistic as a measure of complexity
and regularity of renal sympathetic nerve activity in the rat
Tao Zhang1,2 , Zhuo Yang 1,3 and John H. Coote1
1
2
Division of Neuroscience, School of Medicine, University of Birmingham, Birmingham B15 2TT, UK
College of Life Sciences and 3 Medical College, University of Nankai, Tianjin 300071, PR China
In this study, we employed both power spectral analysis and cross-sample entropy measurement
to assess the relationship between two time series, arterial blood pressure (ABP) and renal
sympathetic nerve activity (RSNA), during a mild haemorrhage in anaesthetized Wistar rats.
Removal of 1 ml of venous blood decreased BP (by 7.1 ± 0.7 mmHg) and increased RSNA (by
25.9 ± 2.4%). During these changes, the power in the RSNA signal at heart rate frequency was
reduced but coherence between the spectra at heart rate frequency in RSNA and ABP remained
unchanged. Cross-sample entropy was significantly increased (by 10%) by haemorrhage,
revealing that there was greater asynchrony between ABP and the RSNA time series. Intrathecal
administration of the glutamate receptor antagonist kynurenic acid (2 mM) almost halved
(P < 0.01) the reflex increase in RSNA. Also during kynurenic acid block, haemorrhage failed
to change total power, power at heart rate frequency, coherence at heart rate frequency, or the
cross-sample entropy measurements. We conclude that the increase in asynchrony between ABP
and RSNA during the reflex increase in RSNA was a consequence of an increase in synaptic
input to the spinal renal neurones. The data show that the cross-sample entropy calculations
can characterize the non-linearities of neural mechanisms underlying cardiovascular control
and have a potential to reveal how some aspects of homeostatic regulation of kidney function is
achieved by the autonomic nervous system.
(Received 24 January 2007; accepted after revision 10 April 2007; first published online 13 April 2007)
Corresponding author J. H. Coote: Division of Neuroscience, School of Medicine, University of Birmingham,
Birmingham B15 2TT, UK. Email: [email protected]
It has become apparent that the impact of renal
sympathetic nerve activity (RSNA) on kidney function
may not be determined solely by the absolute level of
activity within the nerve signal, but more by how the
pattern of the energy is distributed across the signal and
how it changes dynamically (Stauss et al. 1997; Zhang et al.
1997; Malpas et al. 1998). Mathematical techniques are
helpful to address this issue. One of the most developed
approaches is that of power spectral analysis. Studies in
dogs and rats have shown that, in the renal sympathetic
nerve, major energy peaks occur at heart rate (HR)
frequency, respiratory frequency and also at very low
frequencies (Persson et al. 1992; Brown et al. 1994; Zhang
et al. 1997; Malpas et al. 1998; Burgess et al. 1999). The
energy distribution which is coherent with heart rate is
of special significance because it is strongly influenced by
feedback from arterial receptors (Gebber & Barman, 1980;
Davis & Johns, 1995; Zhang & Johns, 1996).
C 2007 The Authors. Journal compilation C 2007 The Physiological Society
Theoretically, however, coherence measurement in the
frequency domain fails to provide useful information
about whether signals are non-linearly correlated,
especially for detecting any non-linear changes of the
underlying network activated. A better understanding of
the underlying mechanisms regulating the cardiovascular
system may be obtained by measurement of the non-linear
relationship between arterial blood pressure (ABP) and
sympathetic vasomotor nerve activity (Zhang & Johns,
1998; Ringwood & Malpas, 2001). Zhang & Johns (1998)
assessed the pattern of sympathetic activity in a renal
nerve (RSNA) using chaos theory. It was shown that an
increase in RSNA, induced by stimulating a brachial nerve,
resulted in a decrease in the value of the largest Lyapunov
exponent and in the correlation dimension, implying there
was less chaos. The result is puzzling, because intuitively
an increase in nerve traffic might be expected to be more
chaotic, unless the activity becomes more synchronized
Downloaded from Exp Physiol (ep.physoc.org) by guest on October 6, 2014
DOI: 10.1113/expphysiol.2007.037150
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T. Zhang and others
by the stimulus. It is unlikely that brachial nerve afferents
would normally be activated simultaneously at high
frequency, under natural circumstances. Therefore, we
considered it important to study the non-linear dynamic
behaviour of a more naturally induced increase in RSNA.
Also, to overcome the problem of high dimensionality
of the raw nerve signal, there was a need to use another
mathematical approach, different from chaos theory. One
such method that appears to achieve this is conditional
entropy which, with a correction factor, allows estimation
of synchronicity over short data sequences and has the
advantage that it is not influenced by aberrant points
in a sequence of data and avoids an a priori selection
of the embedding dimension (Porta et al. 1998, 1999,
2000). When these authors applied this method to
measure the degree of coupling between sympathetic
nerve activity and ventilation in decerebrate cats during
manoeuvres that either increased or decreased sympathetic
discharge, the uncoupling function decreased, suggesting
that there was stronger synchrony and lower entropy
during both experimental situations (Porta et al. 1998,
1999). Whilst there may be a rational explanation for this,
we considered that a different non-linear approach was
required. For this, a method known as sample entropy
(SampEn), an event-counting statistic, based on earlier
studies of approximate entropy (Pincus, 1994; Pincus &
Goldberger, 1994), provides a useful approach. This was
extended to determine the relation between two time
series, by a correlation of their SampEn, to give a statistic
termed cross-SampEn, which provides an indication of the
degree of synchronizing between the signals (Richman &
Moorman, 2000). A significant drawback of cross-SampEn
is the estimation of the embedding dimension. However, a
method to optimize this was subsequently devised by Lake
et al. (2002).
Therefore, in the present study, we use cross-SampEn
to examine changes in the relationship between ABP and
RSNA during an excitation of sympathetic nerve activity,
reflexly induced by a mild haemorrhage (Yang & Coote,
2006). The main aim was to determine whether crossSampEn can provide a sensitive method for detecting
non-linear dynamic changes between sympathetic nerve
activity and blood pressure and to compare it with the more
traditional linear method, fast Fourier transform. Thus, we
hypothesized that an analysis of the non-linear features
of the relationship between RSNA and blood pressure
signals would reveal significant physiological changes that
coherence could not. Furthermore, we wished to observe
how the sympathetic network oscillators respond to a more
naturally induced increase in activity in view of the data
suggesting that an increase in sympathetic activity, elicited
by electrical stimulation of a somatic afferent nerve (Zhang
& Johns, 1998), or by occlusion of the inferior vena cava
(Porta et al. 1998, 1999), was associated with a decrease
in entropy. Would there be more irregularity or less in the
matching of the RSNA signals with blood pressure during
a brief episode of a mild haemorrhage that induces an
increase in RSNA and a small fall in blood pressure (Yang
& Coote, 2006)?
Methods
The experiments were approved by the local ethical
committee of the University of Birmingham and were
performed under a Home Office license of the UK.
Animal preparation
The experiments were performed on 10 male rats (Wistar)
weighing 305.8 ± 6.3 g, anaesthetized with a mixture
of urethane and α-chloralose (650 and 50 mg kg−1 ,
respectively (Sigma, Poole, Dorset, UK)) given i.v. after
initial induction with 4% enflurane (Abbot, Kent, UK) in
oxygen. An adequate depth of anaesthesia was monitored
by observing arterial blood pressure (ABP), heart rate
(HR) and the absence of corneal and paw-pinch reflexes,
and was maintained by regular 1 h administration of
additional anaesthetic (10% of original dose i.v.) or
sooner if needed. The femoral artery was cannulated
with a polyethylene catheter (PE-50 tubing), which was
connected to a pressure transducer (Capto SP 844, AD
Instruments, Chalgrove, UK), for continuous recording of
ABP. A femoral vein was also cannulated for withdrawing
blood and administration of drugs. Electrocardiogram
(ECG) signals were obtained from an ECG monitor via two
platinum electrodes inserted under the skin of the limbs.
The trachea was cannulated and spontaneous respiration
maintained throughout the experiment. The head of the
rat was mounted in a stereotaxic instrument (Narishige,
London, UK). Rectal temperature was continuously
monitored and maintained at 37◦ C by a heating blanket.
At the end of the experiment the animal was killed by an
overdose of anaesthetic (urethane and chloralose, i.v.).
Recordings of renal sympathetic nerve activity
The left kidney was exposed retroperitoneally and, with the
aid of an operating microscope, a branch of the nerve to the
kidney was dissected free and placed on a bipolar electrode
made from silver wire. After the condition for optimal
nerve recording had been established, both the nerve and
the electrode were covered with paraffin. The neural signals
were amplified (gain 5000, Neurolog, Digitimer, Welwyn
Garden City, UK) with bandpass filter set at 50 Hz low
frequency and 3000 Hz high frequency, digitized with
a sampling frequency of 1000 Hz and displayed on an
oscilloscope and stored for later analysis using a PowerLab
data acquisition system (AD Instruments).
Renal nerve activation and synchrony analysis
Spinal cord exposure and perfusion
The spinal subarachnoid space was cannulated for the
intrathecal (i.t.) administration of drugs as previously
described (Yang et al. 2002a) after the rat was placed
in the stereotaxic frame. The atlanto-occipital membrane
was exposed by removing the overlying muscle through a
mid-line dorsal incision. A laminectomy was performed at
C1 to provide clear access to the subarachnoid space for an
i.t. catheter. A 14 cm length of PP-10 tubing (total volume
15 μl) filled with artificial cerebrospinal fluid (ACSF:
NaCl, 7.42 g; KCl, 0.14 g; KH2 PO4 , 0.163 g; CaCl2 , 0.27 g;
MgSO4 , 0.319 g; NaHCO3 , 2.18 g; and d-glucose, 1.8 g;
dissolved in 1 l distilled water, pH 7.4) was introduced
under the dura and advanced caudally along the dorsal
surface so that the tip lay over the T10 segment. The
position was confirmed on necropsy at the termination
of the experiment. In some experiments, 10 μl of a 2 mm
solution of the excitatory amino acid antagonist kynurenic
acid (Sigma) was administered intrathecally by displacing
it from the i.t. cannula with ACSF. Each infusion took up
to 1 min.
Haemorrhage
After recording stable baseline levels of ABP, HR and
RSNA for 15 min, the effect on RSNA of i.t. application
of ACSF (10 μl) was tested. Next, a mild haemorrhage
was produced by withdrawing 1 ml of venous blood
(representing about 5% of total blood volume; Yang &
Coote, 2006, 2007), into a preheparinized syringe, over
1 min, from the femoral venous catheter and 5 min later
slowly re-infused. After a further 15 min, providing ABP
and RSNA had recovered to the control values, a second
haemorrhage test was performed.
Following these tests, the effect of i.t. application of the
glutamate antagonist kynurenic acid (10 μl, 2 mm) was
tested alone and then, after a period of recovery (15 min),
its effect on the RSNA, ABP and HR response when given
i.t. immediately before haemorrhage was examined.
Data analysis
The recorded signals from the amplifier outputs were taken
at high frequency (1000 Hz), digitized and stored on hard
disk for off-line processing (PowerLab data acquisition
system, AD Instruments). Data sets were taken up to 1 min
before haemorrhage (control), from 10 s after removal of
blood (haemorrhage) and then 10 min after re-infusion
of blood when ABP, HR and RSNA had recovered close
to control values (recovery) and were stable. The ABP,
HR and RSNA responses to each test procedure (repeated
twice) were expressed as a percentage change compared
with the basal value for the period immediately before
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each test procedure, as we have previously described (Yang
& Coote, 2006, 2007).
To measure changes in RSNA, for the time domain
analysis, the raw nerve signal was passed through a spike
discriminator (PowerLab) to remove background noise
and then integrated (time constant 20 ms, PowerLab) and
the total nerve activity in spikes per second, from when it
changed from basal value to when it returned to basal value,
was computed (PowerLab). The mean value obtained was
compared with the mean value during a similar period
before each test.
Frequency domain analysis. Rectified and digitized RSNA
signals and digitized ABP signals were subjected off-line
to a fast Fourier transformation to produce a power
spectrum. For this, approximately 33 s recording of 215
data points for each variable was divided into three 50%
overlapping segments. Next, 214 data points from each
of the three segments was passed through a Hanning
smoothing window, to minimize ‘end leakage’, which may
result from a finite length of data collection and a lack of
symmetry (Zhang et al. 1997). A spectrum was generated,
and the average value for the three overlapping segments
used to estimate the relative amount of energy in the signals
at frequencies from 0 to 10 Hz in 0.06 Hz increments. The
total power of RSNA in the spectrum was calculated as the
area under the curve, from 0 to 10 Hz. The power at heart
rate frequency was derived by determining the frequency
of the maximum peak in the blood pressure recordings
and the power in the area of the renal nerve spectra that
coincided with the heart rate frequency (±0.06 Hz). The
percentage power at heart rate frequency was measured as
a proportion of the total power in the frequency band 0–
10 Hz. The value probably reflects the degree of influence
on sympathetic activity of the arterial baroreceptor reflex.
Coherence analysis. Measurements between blood
pressure signals and renal sympathetic nerve activity
signals were performed in order to generate coherence
between the two signals in the frequency domain (Davis
& Johns, 1995; Zhang & Johns, 1996). The coherence
values for the peak at heart rate frequency, as well as
coherence for the power spectrum meaned between 0 and
10 Hz, were averaged for the three overlapping segments
analysed in each animal, providing the values were within
the 95% confidence limits. These coherence values were
then averaged for the 10 animals.
Cross-sample entropy. To measure asynchrony, we use
statistics known as sample entropy (Richman & Moorman,
2000), based on approximate entropy (Pincus, 1991).
Basically, this determines how many vectors (values with
magnitude and direction) in a time series of data occur
within a statistically significant range that can be defined
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T. Zhang and others
as similarity. An illustration of the method is shown
diagrammatically in Fig. 1.
A 33 s time series of a steady state recording of two
variables (in the present study these were ABP and RSNA)
was subjected to a statistical analysis as depicted in Fig. 1.
The computer program took measurements of voltage
levels (y) at adjacent points (set by sampling frequency) in
the signal, to give values y at times t (Fig. 1A). These were
then compared with voltage levels at the next interval, i.e.
t + 1. From this, a two-dimensional plot was obtained,
providing at any y(t) value, the number of vectors
lying within a determined tolerance range (Fig. 1B). The
Figure 1. Schematic diagram
illustrating the method of sample
entropy analysis of a time series of data
A, hypothetical record of voltage changes in
a multifibre nerve preparation, showing
voltage y(t) plotted against time t. y(42) is a
one-dimensional vector (in this case a
voltage) at t(42). B, two-dimensional plot
where the voltage level at the next interval,
y = t + 1, is plotted against y(t), e.g. y(42),
y(43). C, three-dimensional plot of the
vectors for three consecutive data points (t,
t + 1, t + 2), e.g. y(42), y(43), y(44).
statistics are further improved by measuring vectors for
three consecutive data points (t, t + 1, and t + 2) and
constructing a three-dimensional plot with axes y(t),
y(t + 1) and y(t + 2) as illustrated in Fig. 1C. The program
then can construct a series of spheres surrounding each
vector with radii set by the tolerance limits that we define
as similarity (see Fig. 1C). We can then identify the number
of vectors (values) that are within the tolerance limit for
each sphere constructed throughout the time series. From
this, we can calculate a conditional probability for sums
of patterns of voltage values that remain close in the next
interval comparison, i.e. next time points t + 1, t + 2. The
algorithm then calculates the average negative logarithm
of this conditional probability after removing self matches
to the conditional vectors to prevent bias (Richman &
Moorman, 2000). A low value indicates more similarity
and a high value more asynchrony.
To quantify asynchrony between two distinct but
interactive variables, such as ABP and RSNA, we use
cross-sample entropy as defined by Richman & Moorman
(2000). Here, a similar analysis of parallel sequences in
the two time series is cross-correlated. Mathematically, the
analysis we used is as follows.
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where CP is the conditional probability of length m + 1
given there is a match of length m, so that CP = matches
m + 1/matches m.
This criterion is equal to the condition:
95%CI = − log(CP) ± 1.96(σCP /CP) ⊂
− log(CP) ± 0.1 log(CP)
where 95% CI means the 95% confidence interval of the
sample entropies, assumed to be normally distributed.
This is a simple and practical method that we applied to
test whether the m and r are properly selected.
The analysis, in conjunction with the choice of
m = 2 and r = 0.05, was found acceptable, ensuring
good replicability for Cross-SampEn for the data lengths
studied. For example, suppose the number of data points
of each time series (ABP or RSNA) is 1000 and, with m = 2,
then we need (1000 − 2) × (1000 − 2) = 996 004 matches
(loops) to calculate Cross-SampEn.
This analysis was repeated for each condition tested and
the average values calculated for each of the 10 rats. All data
are expressed as the means ± s.e.m. Data were analysed
Let u = (u(1), u(2), . . . u(N )) and v = (v(1),v(2), . . . v(N )). Fix input parameters m and r .
Vector sequences : x(i) = (u(i), u(i + 1), . . . u(i + m − 1))
y( j) = (v( j),v( j + 1), . . . v( j + m − 1))
N is the number of data points of time series, i, j = N − m + 1.
For each i ≤ N − m, set
Bim (r )(v u) = (number of j ≤ N − m such that d [xm (i),ym ( j)] ≤ r )/(N − m),
where j ranges from 1 to N − m. And then
N −m
Bim (r )(v u)
i=1
B m (r )(vu) =
, which is the averaged value of Bim (v u).
N −m
Similarly, we define Aim and Am :
Aim (r )(v u) = (number of j ≤ N − m such that d [xm+1 (i),ym+1 ( j)] ≤ r )/(N − m)
N −m
Aim (r )(v u)
i=1
m
A (r )(vu) =
, which is averaged values of Aim (r )(v u), and then :
N −m
m
A (r )(v u)
Cross − SampEn(m,r ,N ) = −ln
B m (r )(v u)
We applied cross-sample entropy with m = 2 and r = 0.05
(where m is embedding dimension and r is tolerance limits
of similarity) to a standardized ABP and RSNA time series.
The selection of parameters for m and r is critical.
A good SampEn should meet the following criterion
defined by Lake et al. (2002):
max(σCP /CP, σCP / − log(CP)CP) ≤ 0.05
using repeated measures ANOVA. Statistical differences
were considered significant when P < 0.05.
Results
Renal sympathetic nerve activity and ABP
Sympathetic activity recorded in a renal nerve is a complex
signal comprised of action potentials in many nerve fibres.
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T. Zhang and others
Table 1. Effect of haemorrhage on cardiovascular variables and frequency domain measurement
Coh-Mean
Coh-HR
Baseline
0.492 ± 0.01 0.94 ± 0.03
Haemorrhage 0.491 ± 0.01 0.93 ± 0.02
Re-infusion
0.494 ± 0.01 0.96 ± 0.01
% Power-HR of RSNA
3.32∗
13.77 ±
6.35 ± 1.34
16.41 ± 2.80†
TP
HR
ABP
5∗
2.16 ± 0.41 377 ±
81.7 ± 2.1
2.24 ± 0.44 386 ± 6 74.6 ± 1.2
2.14 ± 0.43 367 ± 7† 78.5 ± 1.1
Values are means ± S.E.M. (n = 10 rats). Abbreviations: Coh-Mean, averaged coherence in the
range of 0–10 Hz between ABP and RSNA; Coh-HR, coherence between spectral power at
HR frequency in ABP and RSNA spectral plots; and % Power-HR of RSNA, RSNA peak at
HR frequency; with TP (total power) in volts squared, HR in beats per minute and ABP in
mmHg. Statistical comparisons are within groups, comparing the baseline immediately prior to
haemorrhage (∗ P < 0.01) with the values during 5 min after haemorrhage and after re-infusing
the blood (†P < 0.01). Signal sample frequency is 1 kHz and the length of each data file is about 33 s.
A typical recording during a control period, showing
baseline activity together with the ABP trace, is shown
in Fig. 2A. Withdrawal of 1 ml of blood from a femoral
vein induced a maintained decrease of ABP with a mean
of 7.1 ± 0.7 mmHg in 10 rats and simultaneously caused
a maintained increase of RSNA (Fig. 2B) of 25.9 ± 2.4%
for the 10 rats (P < 0.01 compared with baseline). There
was also a small but significant (P < 0.01) increase in
HR (Table 1). Re-infusion of the 1 ml of blood reversed
the change in RSNA in the first minute, RSNA falling to
3.5 ± 2.3% below the control value, and HR similarly was
Figure 2. Typical examples of recordings of arterial blood
pressure (ABP) and renal sympathetic nerve activity (RSNA) that
were subjected to power spectral analysis and cross-sample
entropy measurement in one rat
A, recordings from a control period immediately prior to haemorrhage.
B, recording during a period immediately following removal of 1 ml of
blood from a femoral vein (haemorrhage).
reversed to a level below the control value, whereas the ABP
decrease was reduced but still remained below the control
level (−3.2 ± 0.5 mmHg; Table 1). Thereafter, these values
slowly returned to control levels over the next 10 min.
Frequency domain analysis. Power spectral analysis of the
changes in RSNA (Fig. 3) induced by haemorrhage and
following re-infusion revealed that over the range 0–10 Hz
there was no significant change in total power of RSNA, it
being 2.24 ± 0.44 V2 for haemorrhage and 2.14 ± 0.43 V2
for re-infusion, compared with 2.16 ± 0.41 V2 for baseline
(Table 1). However, there was a significant decrease in the
power of RSNA at heart rate frequency (around 6 Hz) from
13.77 ± 3.32 to 6.35 ± 1.34 V2 (55%; P < 0.01, Table 1 and
Fig. 3B). There were no significant changes in the averaged
coherence over the range 0–10 Hz (from 0.492 ± 0.01 to
0.491 ± 0.01), or in the coherence at heart rate frequency
(from 0.94 ± 0.03 to 0.93 ± 0.02). After re-infusion of
the blood, the percentage power at heart rate frequency
returned to baseline levels (Fig. 3C).
Cross-sample
entropy
(cross-SampEn). Non-linear
dynamic analysis of the ABP and RSNA time series using
cross-SampEn measurements showed that the values
during baseline conditions were quite different in each
rat, varying from 0.52 in one rat to 0.865 in two other rats
(Fig. 4), with a mean for the 10 rats of 0.72 ± 0.05. During
haemorrhage there was a change in the relationship
between ABP signals and RSNA, the values increasing
from 0.72 ± 0.05 at baseline to 0.79 ± 0.06 during the
challenge (P < 0.05, Fig. 4), a mean change of 0.07 (range
0.04–0.17) or 10% (range 6–20%) which was significant
(P < 0.05). After re-infusion of the blood via the femoral
vein, this non-linear index returned to baseline levels
(0.71 ± 0.06).
Effect of blocking spinal glutamate receptors
Kynurenic acid given i.t. alone was without significant
effect on the baseline value of ABP or heart rate, or on the
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Figure 3. Examples of power spectra in
one anaesthetized Wistar rat (typical of
all rats), at baseline (A), after
haemorrhage (B) and following
re-infusion of blood (C)
Values are expressed as percentage (%)
power of RSNA at frequencies from 0 to
10 Hz.
level of RSNA, at the 2 mm dose used, as we have previously
reported (Yang et al. 2002a; Yang & Coote, 2006, 2007). The
following studies were done on eight of the rats.
During haemorrhage in the presence of kynurenic
acid (i.t. 10 μl, 2 mm), the changes in RSNA were
markedly reduced to 13.4 ± 2.9% (P < 0.01 compared
with haemorrhage alone), and RSNA returned to near
control values (0.2 ± 3.3%) on re-infusion. Similarly,
during haemorrhage, the change in ABP was smaller, APB
falling by 3.1 ± 0.8 mmHg. In contrast, heart rate change
was greater, being significantly increased from 383 ± 6 to
400 ± 9 beats min−1 (P < 0.05; Table 2).
Intrathecal application of ACSF had no significant effect
on baseline or haemorrhage measurements.
Frequency domain analysis. Following kynurenic acid,
averaged coherence in the range of 0–10 Hz was
0.531 ± 0.01 for baseline, 0.528 ± 0.01 for haemorrhage
and 0.529 ± 0.01 for re-infusion, whereas coherence
at heart rate frequency was 0.95 ± 0.02, 0.93 ± 0.02
and 0.96 ± 0.01, respectively, for each condition (not
significant). Total power over 0–10 Hz of RSNA was
reduced but not significantly (Table 2).
After kynurenic acid and haemorrhage, the percentage
power of the peak in RSNA at HR frequency displayed a
small but non-significant reduction from 15.06 ± 2.26 to
13.29 ± 1.42% (Table 2; Fig. 5B). This suggests that the
effect of haemorrhage to reduce the oscillation of RSNA at
HR frequency had been prevented by blocking glutamate
receptor activation with kynurenic acid (compare the
example in Fig. 3 with that in Fig. 5).
Cross-sample entropy. As shown by the group data
(n = 8) in Fig. 6, with kynurenic acid alone, there was an
increase in cross-SampEn. This was despite it appearing
that there was no change in the amount of RSNA measured
by integrating the raw nerve signals. Haemorrhage induced
after pretreatment with kynurenic acid (i.t.) failed to
significantly change cross-SampEn values in the eight rats.
This is shown by the group data in Fig. 6, where the values
are 0.814 ± 0.05 before haemorrhage and 0.812 ± 0.048
during haemorrhage.
Figure 4. Individual values for the cross-sample entropy (Y axis)
between RSNA and ABP obtained from each of 10 rats (X axis)
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T. Zhang and others
Table 2. Effect of haemorrhage following intrathecal application of kynurenic acid on
cardiovascular variables and frequency domain measurement
Coh-Mean
Coh-HR
% Power-HR of RSNA
Baseline
0.531 ± 0.01 0.95 ± 0.02
Haemorrhage 0.528 ± 0.01 0.93 ± 0.02
Re-infusion
0.529 ± 0.01 0.96 ± 0.01
15.06 ± 2.26
13.29 ± 1.42
17.15 ± 3.14
TP
HR
ABP
6∗
2.23 ± 0.53 383 ±
83.3 ± 2.5
2.12 ± 0.59 400 ± 9 80.2 ± 1.3
1.90 ± 0.57 382 ± 8† 85.2 ± 2.8
Values are means ± S.E.M. (n = 8 rats). Abbreviations and units as in Table 1. Statistical comparisons
are within groups, following spinal cord glutamate receptor blockade with kynurenic acid (2 mM,
10 μl), comparing the baseline immediately prior to haemorrhage with the values during 5 min
after haemorrhage (∗ P < 0.05) and after re-infusing the blood (†P < 0.05). Signal sample frequency
is 1 kHz and the length of each data file is about 33 s.
Discussion
In the control of arterial blood pressure, the kidney plays
a key role via its influence, both short- and long-term,
on plasma volume regulation (DiBona & Kopp, 1997).
The renal sympathetic nerves have an important part to
play via effects on renal vascular resistance, renal blood
flow and sodium excretion (Miki et al. 1993; DiBona &
Kopp, 1997; Malpas et al. 1998). Activity in the renal
nerves increases in response to haemorrhage and decreases
in response to volume expansion (Clement et al. 1972;
Miki et al. 1991, 1993; DiBona & Kopp, 1997; Malpas
et al. 1998; Pyner et al. 2002; Yang & Coote, 2006, 2007).
These changes were confirmed in the present experiments,
in which removal and subsequent re-infusion of 1 ml of
venous blood induced significant reciprocal changes in
RSNA. Such reflexly induced alterations in RSNA can
be considered as part of the action of a homeostatic
regulator in the brain, in response to error signals arising
from receptors in the circulation (Yang & Coote, 2006,
2007). However, the impact of RSNA on kidney function
is not solely determined by the number of nerve action
potentials arriving at the target site, but also by the
pattern of activity (Malpas et al. 1998). Examination of
multifibre renal nerve recordings, as illustrated in Fig. 2,
shows them to be complex signals, primarily pulsatile
in nature and continuously variable. Because of the
complexity, we subjected the RSNA and ABP responses
to a non-linear dynamic analysis, cross-SampEn, recently
developed for analysing short clinical data sets (Richman
& Moorman, 2000). Cross-sample entropy was increased
during haemorrhage, suggesting that there was more
asynchrony between RSNA and ABP when the system
responded to the challenge of a small reduction in blood
volume. Thus, not only is there an increase in sympathetic
nerve activity but also an increase in entropy; hence, there
is more power to restore blood pressure/blood volume to
some set point.
Several studies have previously attempted to determine
how the short-term regulation of blood volume/blood
pressure by the renal nerves is achieved (reviewed by
Dibona & Kopp, 1997). Some of these studies have focused
on the oscillatory pattern of action potential traffic in
Figure 5. Typical example showing the
effect of kynurenic acid on power
spectra in one anaesthetized Wistar rat
Values are shown for baseline (A),
haemorrhage challenge during spinal
glutamate receptor blockade with kynurenic
acid given intrathecally (B) and after
re-infusion of blood (C), expressed as
percentage (%) power of RSNA at
frequencies from 0 to 10 Hz.
a renal nerve and mathematically quantified, either in
the time domain or in the frequency domain, the linear
trends between the renal nerve signals and blood pressure
(reviewed by Malpas, 1998; Malpas et al. 1998; Malpas &
Burgess, 2000). In the present study, using power spectral
analysis, we concentrated on the spectral peak at heart rate
frequency, which is the point at which most energy in the
signal is contained, and we showed that the oscillations
in RSNA at this frequency (related to the pulsatile input
from arterial and cardiac baroreceptors) are reduced in
power during haemorrhage. However, coherence between
the same frequency components in RSNA and blood
pressure time series was not significantly altered. Thus,
coherence measurement at heart rate frequency provided
little further insight into the reflex adjustments, other
than to suggest that, even though sympathetic activity
was increased, arterial baroreceptor influence on basic
oscillatory rhythms was still present, after the small fall in
blood pressure. We are, however, aware that RSNA displays
a number of lower frequency oscillations (< 0.5 Hz) that
are of low power but are considered to play a role in
controlling some aspects of renal function (Malpas et al.
1998), but we did not study these.
The increasing realization that the neural control of the
cardiovascular system comprises non-linear dynamics led
Zhang & Johns (1998) to assess the pattern of RSNA using
chaos theory. So far, all chaos mathematical theory and
approaches have been based on low-dimension systems.
However, the multiunit RSNA signal is a non-linear
dynamic system with a high dimensionality, making it
difficult to estimate chaotic characteristics. To circumvent
this difficulty, Zhang & Johns (1998) reduced the high
dimension of the RSNA signal to a simpler form by
subjecting the filtered and rectified signal to cluster analysis
(Malpas & Ninomiya, 1992). From the resultant smoother
waves of activity, they measured peak-to-peak intervals
which, according to Tang et al. (2003), are likely to provided
a fair indication of frequency of action potential traffic.
Zhang & Johns (1998) showed that an increase in RSNA,
induced reflexly by stimulating a brachial nerve bundle,
resulted in a decrease in the value of the largest Lyapunov
exponent and in the correlation dimension, suggesting
that there was less chaos. The result from this important
first attempt to apply chaos theory to RSNA is puzzling,
however, since intuitively an increase in nerve traffic might
be expected to be more chaotic. A possible explanation
is that the brachial nerve induced a grouping of RSNA
that was more synchronized to the frequency at which the
afferents were stimulated (Coote & Perez-Gonzalez, 1970;
Davis & Johns, 1995).
This is in accord with the explanation provided by
Porta et al. (1998, 1999), who applied cross-conditional
entropy analysis to ventilation and sympathetic activity
in the T3 white ramus in decerebrate cats. They showed
that during increased sympathetic activity induced by
667
inferior vena cava occlusion, the uncoupling function
decreased, suggesting that there was stronger synchrony.
However, they obtained a similar result even when
sympathetic activity was decreased. These data may have
been influenced by the absence of forebrain regulation,
which recent studies indicate makes a critical contribution
to cardiovascular reflexes (Yang & Coote, 2006, 2007).
With respect to the study by Zhang & Johns (1998), it
is unlikely that brachial nerve afferents would normally be
activated simultaneously at high frequency under natural
circumstances.
Therefore, in the present study we have examined
how the system responds to more realistic activation
of receptors. Also, to overcome the problem of high
dimensionality, we have subjected the raw renal nerve
signal and the blood pressure to a non-linear dynamic
analysis using an event-counting statistic, cross-SampEn,
originally developed for short clinical data sets by
Richman & Moorman (2000). It was shown that crossSampEn increased during haemorrhage, suggesting that
asynchrony between the two time series, RSNA and ABP,
was increased by the stimulus. In applying this analysis, the
selection of parameters for the embedding dimension m
and tolerance limits r are critical in determining the extent
to which the data do not arise from a random process but
express the non-linear dynamic behaviour. To address this
issue, we used the method described by Lake et al. 2002; see
Methods) and showed that the confidence interval (CI) of
SampEn was within the 95% limits for the values we chose
of m = 2 and r = 0.05. Thus, we consider it unlikely that
the data reflect a random process but instead provide a
Figure 6. Group values of cross-sample entropy for
haemorrhage only (n = 10) and haemorrhage during spinal
glutamate receptor blockade with kynurenic acid (KYN) given
intrathecally
n = 8 rats. Open columns, baseline; filled columns, during
haemorrhage; and hatched columns, after re-infusion of blood.
∗ P < 0.05 comparison between baseline and haemorrhage. ξ P < 0.05
comparison between haemorrhage and re-infusion. In
Haemorrhage + KYN group, all values are following I.T. kynurenic acid.
668
T. Zhang and others
clear indication of the degree of order and complexity in
the pattern of RSNA and how it changed during a mild
haemorrhage.
A further observation in this study is that the asynchrony
is introduced because of an increase in excitatory
amino acid-dependent synaptic input to the spinal renal
sympathetic network of neurones. This conclusion was
reached because the effect was prevented by the glutamate
antagonist kynurenic acid, given intrathecally, close to
their locality. An interesting feature of the application
of kynurenic acid was that alone it increased crossSampEn, even though there was apparently no change
in the overall level of RSNA or ABP. The greater degree
of asynchrony may possibly have been caused by the
antagonist uncoupling some supraspinal rhythmic inputs
that synchronize RSNA. This is to some extent supported
by the small reduction in the peak synchronized at heart
rate frequency in the power spectrum of RSNA (Table 2
and Fig. 5). It might be argued that as a consequence
of the change in cross-SampEn induced by kynurenic
acid alone, the system was unable to exhibit a further
increase during haemorrhage. We consider this unlikely
because the magnitude of cross-SampEn was relatively low
(Richman & Moorman, 2000) and the lack of an increase
is consistent with the data showing a reduction of the
haemorrhage-induced excitatory drive to renal neurones
caused by kynurenic acid.
Logically, an increase in RSNA might be associated with
more entropy, although this is not always so, as discussed
earlier (Zhang & Johns, 1998; Porta et al. 1998, 1999).
Much depends on the degree of inhibitory feedback from
cardiovascular receptors that can powerfully induce an
ordered grouping of activity in sympathetic nerves. The
concept of entropy as it applies to signals in sympathetic
nerves is to quantify the repetition of patterns in the signals.
Larger values of entropy correspond to more variability
and complexity, suggesting greater coupling between
communicating pathways in a network of neurones, hence
responsiveness to feedback from afferent systems (Pincus,
1994). Therefore, such a system has greater capacity to
respond. Thus, although the functional importance of
the lack of synchrony remains to be established, the data
suggest that the central brain circuits become more primed
to defend the body against a decrease in blood volume.
The opposite is likely to accompany an increase in blood
volume, as we recently showed by applying the same nonlinear dynamic statistical method to a reduction in RSNA
induced by stimulating cardiac atrial receptors (Yang et al.
2002b).
A more speculative implication of the present results
relates to the interpretation of large interindividual
differences in levels of sympathetic vasomotor nerve
activity at rest, observed in microneurographic recordings
in normotensive humans (Charkoudian et al. 2005). So
far, emphasis has been placed on the rhythmic modulation
of sympathetic activity and the influence of ABP, cardiac
output and stroke volume. However, the present data
suggest that it is the degree of order in the system that
is likely to be of equal importance. Higher levels of
synchronized activity could indicate less irregularity, but
if the higher level of activity was less synchronized, it
might reflect more complexity and a system more sensitive
to disturbing inputs. This could predispose the system
to further increases in activity, which would have more
serious consequences, such as hypertension and heart
failure, in the long term.
In summary, this study is the first to show that
cross-SampEn analysis can be applied to analysis of
the relationship between increased activity in a renal
sympathetic nerve and blood pressure. Unlike a linear
dynamic measure of coherence, it was sensitive enough
to show a significant decrease in synchrony or more
entropy even for quite small increases in nerve activity
induced by a mild haemorrhage. This result contrasts with
a decrease in chaos and entropy accompanying somatic
afferent stimulation in the rat reported by Zhang & Johns
(1998). The present study suggested that the changes
in synchrony were mediated in part by glutamatergic
synapses on spinal sympathetic neurones. The analysis
indicates that during increases in sympathetic activity
associated with haemorrhage, the brain circuits become
less entrained and have greater potential to nullify the
disturbance in blood volume.
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Acknowledgements
This work was supported a grant from the Welcome Trust to Z.Y.
and J.H.C. Z.T. was supported by the NSFC (30370386) and the
TSTC (06yfjmjc09400).

Cross-sample entropy statistic as a measure of complexity

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