f Information domain analysis of

cardiovascular variability signals: evaluation of regularity,

synchronisation and co-ordination

A. Porta I S. Guzzet t i 2 N. M o n t a n o I M. Pagani I V. Somers 4 A. Mal l ian i 1"2 G. Baselli 3 S. Cerut t i 3

1Dipartimento di Scienze Precliniche, UniversitY' degli Studi di Milano, LITA di Vialba, Milan, Italy 2Centro Ricerche Cardiovascolari CNR, Medicina Interna II, Ospedale L. Sacco, Universita' degli Studi di

Milano, Milan, Italy 3Dipartimento di Bioingegneria, Politecnico di Milano, Milan, Italy

4Department of Internal Medicine, Mayo Clinic, Rochester, Minnesota, USA

Abst rac t - -A unifying genera/ approach to measure regularity, synchronisation and co-ordination is proposed. This approach is based on conditional entropy and is specifically designed to deal with a small amount of data (a few hundred samples). Quantitative and reliable indexes of regularity, synchronisation and co-ordination (ranging from 0 to 1) are derived in a domain (i.e. the information domain) different from time and frequency domains. The method is applied to evaluate regularity, synchronisation and co-ordination among cardiovascular beat-to-beat variability signals during sympathetic activation induced by head-up tilt (T), during the perturbing action produced by controlled respiration at 10, 15 and 20 breaths/min (CR10, CR15 and CR20), and after peripheral muscarinic blockade provoked by the administration of low and high doses of atropine (LD and HD). It is found that: (1) regularity of the RR interval series is around 0.209; (2) this increases during 7-, CRIO and HD; (3) the systolic arterial pressure (SAP) series is more regular (0.406) and its regularity is not affected by the specified experimental conditions; (4) the muscle sympathetic (MS) series is a complex signal (0.093) and its regularity is not influenced by HD and LD; (5) the RR interval and SAP series are significantly, though weakly, synchronised (0.093) and their coupling increases during 7-, CRIO and CR15; (6) the RR interval and respiration are coupled (0.152) and their coupling increases during CRIO; (7) SAP and respiration are significantly synchronised (0.108) and synchronisation increases during CRIO; (8) MS and respiration are uncoupled and become coupled (0.119) after HD; (9) the RR interval, SAP and respiration are significantly co-ordinated (0.118) and co-ordination increases during CRIO and CR 15; (10) during HD the co-ordination among SAP, MS and the respiratory signal is larger than that among the RR interval, SAP, MS and the respiratory signal, thus indicating that the RR interval contributes towards reducing co-ordination.

Keywords--Condit ional entropy, Corrected conditional entropy, Regularity, Synchro- nisation, Co-ordination, Complexity, Cardiovascular control, Cardiovascular variability

Med. Biol. Eng. Comput., 2000, 38, 180-188

1 In t roduct ion

REGULARITY, SYNCHRONISATION and co-ordination are concepts which play an important role in biology and medicine ( SINGER, 1993; GLASS and MACKEY, 1988; RAPP, 1987; KELSO, 1995). Regularity can be defined as the degree of recurrence of a pattern in a signal. Synchronisation is related to the repetition of a complex pattern involving two signals. Co-ordination can be

Correspondence should be addressed to Dr A. Porta; emaih alberto.porta@unimi.it

First received 20 September 1999 and in final form 8 November 1999

© IFMBE:2000

J

defined as the degree of recurrence of a complex scheme involving several signals. These concepts could be very impor- tant in describing the cardiovascular system because of the presence of several mechanisms and circuits co-operating and competing with each other for its regulation (KOEPCHEN, 1991 ). Oscillators located at the brain stem level (PREISS and PO£OSA, 1974) are able to modify respiratory and vasomotor activities, oscillators located at a peripheral level can change microvascular resistances and modify local flows (MEYER et at., 1987); feed- back control loops with afferent limbs projecting to supraspinal and spinal structures (e.g. baroreceptive, chemoreceptive and brain perfusion feedback loops) are able to regulate arterial pressure and respiratory activity (GUYTON and HARRIS, 1951; ANDERSON et at., 1950; MIYAKAWA, 1988). Changes in regu- larity may occur when one of these mechanisms takes priority

180 Medical & Biological Engineering & Computing 2000, Vol. 38

and entrains the others. Synchronisation leads two of these mechanisms to operate synchronously and allows one mechanism to pace the activity of the other. Co-ordination among control mechanisms is necessary to allow adequate functioning of the organism in any condition. However, the level of autonomy necessary to fulfil the local demands produces changes in co-ordination.

The aim of the study is to evaluate changes of regularity, synchronisation and co-ordination among cardiovascular varia- bility signals (PORTA, 1998) as a result of modifications in the regulation of the cardiovascular function obtained by means of two experimental protocols. The first set of data is obtained from healthy young subjects undergoing traditional physiological manoeuvres (controlled respiration and head-up tilt). This protocol allows one to evaluate the effect of sympathetic activation produced by tilt and a perturbing input (controlled respiration) on regularity, synchronisation and co-ordination of heart period, systolic arterial pressure and respiration. The second set of data is obtained from young healthy subjects undergoing a pharmacological intervention (the administration of different doses of atropine). The co-ordination among heart period, systolic arterial pressure, muscle sympathetic activity and respiration is expected to depend on the dose of atropine. Indeed, at high doses the heart period becomes independent of the other cardiovascular variables due to the peripheral muscarinic blockade.

Recently, tools have been provided to measure regularity (or its opposite, complexity). For example, correlation dimension (GRASSBERGER and PROCACCIA, 1983), because it is related to the number of independent variables necessary to describe the dynamics of the process, can be considered as a measure of complexity. Unfortunately, this index needs a large number of samples to be reliably calculated (GUEVARA, 1997), thus preventing its application on short biological series. To over- come this limitation PINCUS (1995) has defined a function referred to as approximate entropy to measure complexity over short data series (about 100 points). HOYER et al. (1998) has proposed measures of coupling strength based on reconstruction of the joint phase space of the two processes and on the calculation of the correlation integral over about 1000 points.

Two methodological issues are usually not addressed: (i) the embedding dimension should not be fixed because too small an embedding dimension leads to an overestimation of complexity due to the inability to unfold complex structures, while the selection of too large an embedding dimension leads to a loss of reliability of the estimate, thus resulting in an underestimation of complexity; (ii) measures of regularity, synchronisation and co- ordination should be derived from a common basic function because these concepts address the same entity (i.e. the degree of recurrence of a pattern) over a different number of signals.

in this study a new approach is followed: a function (the conditional entropy) evaluating regularity is generalised to measure synchronisation, and the function quantifying synchro- nisation is utilised to derive an index of co-ordination. The method exploits two functions previously defined and validated (the corrected conditional entropy (PORTA et al., 1998a) and the uncoupling ftmction (PORTA et al., 1999) to extract indexes of regularity and synchronisation over short sequences of data. The indexes are calculated without any a-priori selection of the embedding dimension.

2 Methods

2.1 Condit ional entropy

The function used to measure regularity is the conditional entropy (CE). Given the normalised process {x(i), i = 1 . . . . . N } with i temporal index, C E describes the amount of information

Medical & Biological Engineering & Computing 2000, Vol. 38

carried by the most recent sample x(i) of a length L pattern xc( i ) = (x(i), x( i - 1) . . . . . x ( i - L + 1)) when the previous L - 1 samples (i.e. the pattern x c 1 ( i - 1) = ( x ( i - 1) . . . . . x ( i - L ÷ 1))) are known. C E is defined (PAPOULIS, 1984) as

C E ( L ) = - ~ p ( x L 1) ~ P (x ( i ) / xL 1)log p ( x ( i ) / x L 1) L 1 i / (L 1)

(1)

where log performs the natural logarithm, p(x c 1) denotes the joint probability of the pattern x L 1(i) and p ( x ( i ) / x c 1) symbo- lises the conditional probability of the sample x(i) given the pattern XL 1. As a result of the use of log e, C E is expressed in nats (natural logarithm) instead of bits. CE(1) is equal to

E(1 ) = - ~ p(x( i ) ) log p(x( i ) ) (2)

representing the Shannon entropy of the process (i.e. the amount of information carried by the sample x(i) when no previous sample is given). CE(1) depends only on the probability distribution of the process and quantifies the reliability of the mean as a predictor of the furore behaviour (it is larger in an identically distributed process than in a Gaussian one).

C E furnishes a measure of complexity of the process x based on conditional probability. Indeed, if x is complex and com- pletely unpredictable (e.g. x is white noise), C E is constant and equal to E(1) (i.e. the knowledge of past samples is not able to reduce the uncertainty of the process x). In contrast, if x is regular and completely predictable (e.g. a rigorously periodic signal), C E reaches zero as soon as a new sample can be exactly predicted from the previous L - 1 ones.

2.2 Corrected condit ional entropy and regulari O, index

Unfortunately, C E cannot be used as a measure of complexity over short data series. Indeed, the estimate of C E (CE) can be unreliable due to the approximation of the conditional prob- ability with the conditional sample frequency. The frequency distribution of x( i ) given the pattern x L 1 ( i - 1) provides an erroneous certainty when the conditioning pattern x L 1 (i - 1 ) is found only once in the ser iesx (i.e. D ( x ( i ) / x L 1 ) = 1), thus causing an underestimate of C E and its decrease to zero with L (PORTA et al., 1998a). AAsa consequence, both periodic signals and white noise exhibit C E equal to zero for sufficiently large L. Therefore, to perform a reliable measure of complexity over short data series an estimator of C E referred to as the corrected conditional entropy ( C C E ) is defined (PORTA et al., 1998a)

C C E ( L ) = CAE(L) + p e r c ( L ) . E(1 ) (3)

C C E is the sum of two terms: C E and the corrective term, where perc(L)-E(1), where perc(L) is the percentage of length L patterns found only one time in the data set and E(1) is the estimate of Shannon entropy of the process x. Each time a pattern is found only once (it results in a misleading certainty as a side effect of the shortness of data), the null information associated to this pattern is substituted with the maximum information carried by the series considered (E(1): i.e. the information carried by white noise with the same probability distribution as the series considered), in other words, when no reliable statistic can be obtained due to the lack of data, the randomness is privileged over a misleading full predictability. CCE: (i) decreases to zero for a periodic signal; (ii) is flat for an unpredictable signal such as white noise; and (iii) exhibits a minimum smaller than E(1) for those processes in which the knowledge of past samples is useful to predict the furore dynamics. C C E m i n i m u m (m in (CCE(L) ) ) is taken as an index of complexity (PORTA et al., 1998a): the larger the index, the more complex the series. Therefore, for a periodic signal m i n ( C C E ( L ) ) = 0; for white noise m i n ( C C E ( L ) ) =

181

E(1 ); and lies between 0 and E( 1 ) for a process which is neither completely predictable nor fully unpredictable. Obviously, an index of regularity (the opposite of complexity) o f the process x can be oAbtained as E(1) - min(CCE(L)).

As CE( 1 ) = E(1 ) and depends on the shape of the probability distribution of the process, min(CCE(L)) depends not only on the process dynamics but also on the static properties o f the process (the probability distribution). A dynamic index of complexity can be obtained by minimising the normalised CCE (NCCE) (PORTA et al., 1998b)

CAE(L) + perc(L). E(1) NCCE(L) = /~(1 ) (4)

Therefore, independently of the distribution of the process, N C C E ranges from 0 to 1. Therefore, the minimum of N C C E appears more useful than that o f CCE when processes with different probability distributions are analysed, in this case, the index of regularity of the process x is

Px = 1 - min(NCCE(L)) (5)

2.3 Cross-conditional entropy

The function utilised to quantify the coupling between two series is the cross-conditional entropy (PORTA et al., 1999). Given a pair ofnormalised signals (x, y) with x = {x(i), i = 1 . . . . . N} and y = {y(i), i = 1 . . . . . N} the cross-conditional entropy o f x given a pattern of y (CEx/),) is defined as

CE~/),(L) = - ~ P(YL 1) ~ P(x(i)/yL 1)log p(x( i ) /y L 1) L 1 i / L 1

(6)

This represents the amount of information carried by the present sample of the signal x (i.e. x(i)) when a pattern o f L - 1 samples ofthe signaly (i.e. YL 1(i) = (y(i) . . . . . y(i -- L + 2)))is known. CE~/),(1 ) is equal to Ex(1 ) provided by eqn 2 (i.e. the amount of information carried by the sample x(i) when no information about y is supplied). CEx/), is a measure of the complexity o f x with respect to y. Indeed, it: (i) reaches zero when a sufficient number of samples o f ) ' allows one to completely predict x; (ii) remains high and constant if x and y are independent processes; and (iii) decreases toward a value between these extremes when the knowledge o f ) ' is useful to partially estimate x.

2.4 Uncoupling function and synchronisation index

As a measure of synchronisation the uncoupling function (UF) (PORTA et al., 1999)

UFx,),(L) = min(NCE),/x(L), NCE~/),(L)) (7)

is used, where min takes the minimum value between NCE),/x(L) and NCEx/),(L) and the functions NCE),/x and NCEx/), are obtained by normalising CE),/x and CEx/), by Ev(1) and Ex(1) respectively. Normalisation is necessary because the amount of information carried by x and y (quantified by Ex(1 ) and E),(1) respectively) is different. The rationale of this definition comes from the necessity to measure the amount of information carried by one signal that cannot be derived from the knowledge of past samples of the other signal (i.e. the cross- conditional entropy) independently from the causal relationships between them (i.e. which is the input and which the output). Therefore, both causal directions are tested by choosing the one which leads to the best prediction (i.e. to the least cross- conditional entropy). UF is 1 when x and y are independent, while it is 0 for synchronised periodic dynamics (e.g. 1N phase locking).

182

As UF is based on the CE definition, it suffers from the same limitation when calculated over short data segments (it decreases towards zero independently of the amount of interactions between x and y as an effect of the shortness of the data). Therefore, its estimate (PORTA et al., 1999)

UFx,),(L) = min(NCCEy/x(L), NCCE~/),(L)) (8)

is based on NCCE),/.~(L) and NCCE~/),(L). These f a c t i o n s can be obtained from eqn 4 simply by substituting CE with CAE),/~and CAEx/), and E(1 ) with E),(1 ) and Ex(1) respectively. The UFx,),: (i) decreases to zero for strongly coupled signals; (ii) is flat and equal to 1 in the case of independent processes; and (iii) exhibits a minimum smaller than 1 for those processes in which the knowledge of past samples is useful to predict the furore dynamics. Therefore, the minimum of UFx, ), (min(UF~,),(L))) is taken as an index of uncoupling between x and y: the l ~ r the index, the more independent the signals. Indeed, min(UFx,) , (L))=0 for fully synchronised signals; min(UFx,),(L))= 1 for independent processes; and lies between 0 and 1 for pairs of signals able to exchange a certain amount of information.

Obviously, an index of synchronisation (the opposite of uncoupling) (PORTA et al., 1999), quantifying the maximum amount of information exchange between the two signals, can be defined as

)/x,), = 1 - min(UFx,y(L)) (9)

2.5 Co-ordination index

We consider K normalised signals, x I ={xm(i)}, x 2 = {x2(i)} . . . . . x/~ = {xK(i)} with i = 1 . . . . . N. We group the signals in pairs, (x i, xj) with i ~ j, and calculate the uncoupling function UF~ ~ for all the K ( K - 1)/2 different pairs. The isolation inde~("(I I )

2 Ilxl"X: . . . . . . I" - K ( K - 1) ~i,j min(UFx''xi(L)) with i ~ . ]

(10)

represents the mean uncoupling in the set o f signals considered, i.e. the degree of isolation of the signals in terms of lack of influences among them. The hypothesis underlying this defini- tion is that the influences of several signals on the same variable can be analysed separately and reduced to the action of one signal at a time. When all the signals are independent, I I is 1 because all UFs are flat and equal to 1. in contrast, when the signals interact, UFs decrease with L, and I I ranges from 0 to 1.

A reliable estimate of I I over short segments of data can be obtained by substituting UFs with their estimates UFs. Obviously, an index of co-ordination quantifying the average amount of information exchanged inside the set o f signals can be defined as

)~X 1 . . . . . . K = 1 - I l x l . . . . . . K

2 (11) --K(K-1) ~ Z .... ~ w i t h i ~ j

i,j

and, therefore, it can be obtained simply by averaging all the synchronisation indexes.

3 Experimental protocol and data analysis

3.1 Experimental protocols

Two experimental protocols were considered, in the first experimental protocol (type-1 protocol) the ECG, the arterial

Medical & Biological Engineering & Computing 2000, Vol. 38

pressure (AP) (via a plethysmographic non-invasive device, Finapres) and the respiratory signal (via a nasal thermistor) were recorded and sampled at 300 Hz in nine healthy young subjects. After detecting the QRS complex, the RR interval was calculated. The ith systolic AP (SAP) was searched inside the ith RR intervals. The respiratory signal was sampled (R(i)) at the first R peak defining the ith RR interval. The RR interval was expressed in seconds, SAP in mmHg and the respiratory signal in arbitrary units (a.u.). The protocol included four recordings in the supine position: the first one breathing spontaneously (R) and the others with controlled metronome breathing at 10, 15 and 20 breaths/ min (CR10, CR15 and CR20 respectively). The last recording was performed during 80: passive head-up tilt (T). The mean RR period was reduced during T. No significant difference was found during CR even though a tendency towards bradycardia was observed. The mean SAP did not change during the whole protocol.

in the second protocol (type-2 protocol) the ECG, AP (from a catheter inserted into the radial artery), respiration (using a pneumatic chest belt) and efferent muscle sympa- thetic nerve activity (MS) were recorded and sampled at 500Hz in seven healthy young subjects (MONTANO et al., 1998). MS was obtained from a sympathetic nerve fascicle to muscle blood vessels in the peroneal nerve by means of tungsten microelectrodes (WAIIIN, 1983). The beat-to-beat variability series of the RR interval and SAP values were extracted with sampled respiration. The ith MS measure was obtained by integrating the MS signal over the two conse- cutive diastolic values delimiting the ith pressure wave and by dividing by the integration interval (PAGANI et al., 1997). The basal condition (B) was obtained after the administration of a saline bolus. Then two intravenous boluses (2 mg/kg and 15 mg/kg) of atropine, corresponding to a low and high dose (LD and HD), were administered in a stepwise fashion starting from the lower dose. After each bolus, recordings lasting 15min were carried out. LD caused bradycardia without any changes of the SAP. HD induced an important tachycardia, an increase of the SAP and a reduction of the burst rate in the MS signal.

3.3 Data analysis

To exploit stationarity, all the processes were normalised by subtracting the mean and by dividing by the standard deviation. The calculation of the conditional sample frequencies implied quantisation. As the CCE function depended on the quantisation levels ~ and on the data series length N (PORTA et al., 1998a), they were maintained fixed in all the analysis (4 = 6 and N ~ 300). All segments of data were subjected to a linear detrending procedure thus preventing the effect of trends on calculation of the indexes.

All results were given as mean (-4-standard deviation). Comparison between the indexes obtained at control (R or B) and in the other experimental conditions was performed via one- way repeated measures analysis of variance (Bonferroni test).

4 Results

4.1 Regulari O, synchronisation and co-ordination analysis: an example

Fig. 1 shows an example of regularity analysis performed on an RR series (Fig. la) at R. N C C E (Fig. lb, solid line) showed a deep minimum with a large regularity index (Prr = 0.24) clearly different from that extracted from N C C E calculated over a shuffled version of the original series (Fig. lb, broken line).

Fig. 2 shows an example of synchronisation analysis carried out on an RR series and a respiratory signal (Figs 2a, b respectively) at R. UF (Fig. 2c, solid line) showed an important minimum ()~rr,r = 0.15) largely different from that detected between a pair of relevant independent phase-randomised surrogate series (Fig. 2c, broken line).

Fig. 3 shows an example of co-ordination analysis performed on RR and SAP series and a respiratory signal (Figs 3a, b,c) at R. it was performed by calculating UFs on the pairs (rr, sap), (rr, r) and (sap,r) (Figs 3d, e,f, solid line). UF between RR series and a respiratory signal exhibited the deepest minimum (Fig. 3e). The coupling between the respiratory signal and SAP series was small ()~sap,r = 0.08 ) but significant (i. e. it was different from that calculated on a surrogate pair, broken line, Fig. 3f). The RR and SAP series were well synchronised ()~ . . . . p = 0.11, Fig. 3d). The co-ordination index )~ . . . . p,r was 0.113 and significantly different from that calculated from the surrogate data (0.033).

3.2 Surrogate data

Two types of surrogate data were used. The first type was used to test if the calculated regularity index was different from that obtained from a random sequence with the same frequency distribution as the original series. This random sequence was simply obtained by shuffling the time order of the original series (THEILER et al., 1992). The second type of surrogate was utilised to verify if the synchronisation index was different from that calculated over two independent processes with the same frequency distribution and power spectra as the original pair of signals. This type of surrogate was obtained by randomising the Fourier phases (OPPENHEIM and SCHAFEa, 1975) of the two original series with two independent realisations of white noise (PALUS, 1997) and by performing the inverse Fourier transform (OPPENHEIM and SCHAVEa, 1975). These surrogate data were also utilised to test the co-ordination index against that obtained from the same number of independent signals. The test proposed by THEILER et al. (1992) was used to verify the significant difference (p <0.05) between the original and surrogate data. Ten independent surrogate realisations for each data series were utilised.

Medical & Biological Engineering & Computing 2000, Vol. 38

4.2 Type-1 protocol: results

Table 1 summarises the results of the analysis of regularity, synchronisation and co-ordination obtained from the type-1 protocol. Surrogate data analysis confirmed that all series in all experimental conditions exhibited significant regularity (i.e. they were different from white noise). The RR series was a process neither completely predictable nor fully random (Prr = 0.209). Regularity increased during T and CR10, but was unchanged during CR15 and CR20. The SAP regularity was larger (0.406) than that of the RR series. Moreover, the SAP regularity was not modified by the experimental conditions. Surprisingly, the respiratory signal was not completely regular (0.309) and its regularity increased only during CR10.

Surrogate data analysis proved that any pairs of series were significantly coupled (i.e. they were not completely indepen- dent). Synchronisation between the RR and SAP series increased during T and CR10. Synchronisation between respira- tion and both RR and SAP series increased only during CR10.

RR and SAP series were significantly co-ordinated with respiration at R (Z . . . . p,r=0.118) and the co-ordination increased during CR10 and CR15. During T and CR20 no variation in co-ordination was observed.

183

1.05 1.2

Fig. 1

0.08 0.5 250 1 12

sample number k a b

I0rr

Example o f regulariO; analysis performed on an RR series at R. (a) RR series: (b) relevant NCCE (solid line). A cleat" minimum can be observed, thus permitting derivation o f a large index o f regulariO; Prr" The broken line is relevant to NCCE calculated on a surrogate series obtained by shuffling samples of the RR series. It is flat, around 1 and with a minimum value clearly diff8rent fi'om that relevant to NCCE calculated over the original series

4.3 Type-2 protocol: results

Table 2 summarises the results of the analysis of regularity, synchronisation and co-ordination obtained from the type-2 protocol. The analysis of regularity at B confirmed that the RR series was neither completely regular nor fully unpredictable (0.247) and the SAP series was more regular (0.317). The MS series was quite complex but some repetitive patterns were still present (Dins was low but significant). LD did not affect regularity but HD was able to increase the regularity of the RR series and the respiratory signal. The SAP and MS regularities remained unchanged.

The RR and SAP series were coupled and LD and HD did not affect their link. Synchronisation between the RR series and the respiratory signal was significant at B, increased, but not significantly, after LD and decreased below the level of signifi- cance after HD. In contrast, the coupling between SAP and the respiratory signal was not modified by atropine. The MS series was significantly linked with the RR series at B. This coupling did not change after LD and decreased below the level of significance after HD. The MS series and the respiratory signal were independent at B and after LD but the coupling increased and became significant after HD. Synchronisation between the MS and SAP series was not modified by atropine.

The RR, SAP and MS series and the respiratory signal were significantly co-ordinated at B (Z . . . . p ..... = 0.122) and co- ordination was not varied by atropine. The co-ordination indexes calculated over triplets of signals were always signifi- cant and no clear difference in co-ordination was found while varying the experimental conditions. During HD, co-ordination among SAP, MS and respiration was larger (Z~p,r, = 0.133) than that among RR, SAP, MS and respiration (Zrr,~p . . . . . =

0.113 ), thus suggesting that the loss o fco-ordination is a result of an independent RR series.

5 Discussion

5.1 Effects o f tilt and controlled respiration on regulari O, synchronisation and co-ordination o f cardiovascular variables

The RR series is regular but its regularity is largely smaller than 1. The complexity of the RR interval series can derive from the complex regulation of this cardiovascular parameter invol- ving several different mechanisms operating over different temporal scales: direct influences arising from central oscillators (respiratory and vasomotor oscillators), feedback loops (e.g. baroreceptive and chemoreceptive closed-loop circuits),

1.05

Fig. 2

0.80 250 250

1.2

0 . ~

1 sample number sample number L

a b c

I Zrr, r

12

Example o f synchronisation analysis pelformed oil an RR series and respiratoo; signal at R. (a) RR series: (b) respiratoo; signal: (c) relevant UF (solid line). A clear minimum can be observed, thus permitting derivation o f a large index of o,'nchronisation X~r,r" The broken line is relevant to UF calculated on two independent surrogate series obtained fi'om the RR series and respiratoo; signal by phase randomisation. It is flat, a twmd 1 and with a minimum value clearly diffErent f ivm that relevant to UF calculated over the original series

184 Medical & Biological Engineering & Computing 2000, Vol. 38

1.05

0.80

1.2

104

9 2 250 250 1 250

sample number sample number sample number a b c

I J_

Fig. 3

0.5

Zrr, sa~ Zrr, r IZsap, r

12 12 12 k L L

d e f

Example o f co-ordination analysis pelformed on RR and SAP series and respiratoly signal at R. (a) RR series: (b) SAP series: ((9 respiratoo; signal. UFs between the original signals (RR and SAP, RR and respiration and SAP and respiration) are plotted at (d), (e) and ( f) respectively (solid line) superimposed on UFs (broken line) obtained fi'om their surrogate data. The index o f co-ordination )~rr,sap,r = ()~r~,sap + )~rr,r + )~s~p,r)/3 is" 0.113 and is" significantly diffbrent fi'om that obtained bv analysis o f the surrogate data (0.033)

complex mechanisms adjusting the peripheral resistances and flow based on local demands and humoral regulation. The rise o f a dominant low frequency (LF, from 0.04 to 0.14 Hz) oscillation during T (PAGANI et al., 1986; MONTANO et al., 1994) is able to reduce the complexity o f the RR interval series by entraining all the mechanisms operating in this band and by reducing the high frequency (HF, at respiratory rate) oscillation. An exogenous slow respiratory oscillation at 10 acts /min (i.e. 0.167Hz) increases regularity on the RR series by pacing a small LF oscillation and producing a large and highly regular HF rhythm (BROWN et al., 1993). in contrast, exogenous fast respiratory rhythms (15 and 20 acts/min) cannot reduce the complexity of the RR series because the only partially reduced LF rhythm remains unlocked to respiration. The SAP series is more regular than the RR series. As a direct vagal control on SAP is absent, the

lack o f interaction between vagal and sympathetic control could explain the increased regularity of SAP. However, as several control mechanisms simultaneously act to regulate SAP, it can be hypothesised that they operate synchronously, thus keeping the SAP complexity low. As further evidence of the high degree of co-ordination among SAP regulatory mechanisms, neither the sympathetic activation produced by T nor the perturbing action of controlled respiration modify SAP regularity. The respiratory signal is not fully regular: it is less regular than the SAP series and only a little more regular than the RR series. The large amount of information carried by this signal can be related to changes in its morphology due to the different and independent regulation o f inspiratory and expiratory phases, to adjustments of the breathing rate around the control frequency and to the type of measurement.

Table 1 Summaly o f the lesults o f the regularity synchronisation and co-ordination analysis pelformed on O.'pe-1 protocol

R T CR10 CR15 CR20

Prr 0.209 ± 0.049 + 0.401 ± 0.097 +### 0.308 ± 0.082 +# 0.248 ± 0.092 + 0.205 ± 0.094 + /)sap 0.406 ± 0.057 + 0.354 ± 0.085 + 0.40 ± 0.071 + 0.366 ± 0.093 + 0.315 ± 0.098 + Pr 0.309 ± 0.072 + 0.318 ± 0.096 + 0.416 ± 0.084 +# 0.365 ± 0.091 + 0.281 ± 0.079 + )~rr, sap 0.093 ± 0.03 + 0.163 ± 0.052 +## 0.165 ± 0.064 +## 0.151 ± 0.071 +# 0.129 ± 0.051 + )~rr,r 0.152 ± 0.082 + 0.116 ± 0.063 + 0.28 ± 0.116 +## 0.228 ± 0.088 + 0.136 ± 0.077 + )~sap,r 0.108 ± 0.043 + 0.121 ± 0.059 + 0.186 ± 0.062 +# 0.17 ± 0.069 + 0.12 ± 0.053 + )~rr, sap,r 0.118 ± 0.041 + 0.133 ± 0.036 + 0.21 ± 0.067 +## 0.183 ± 0.062 +## 0.128 ± 0.052 +

R, rest; T, tilt; CR10, CR15, CR20, controlled respiration at 10, 15 and 20 breaths/min +p<0.05 original versus surrogate data

## ### #p<0.05, p<0.01, p<0.001 experimental condition versus R

Medical & Biological Engineering & Computing 2000, Vol. 38 185

Table 2 Summaly of the results of the regulariO.', synchronisation and co-ordination analysis pelformed on O.'pe-2 protocol

B LD HD

Prr 0"247±0"046+ 0"242±0"071+ 0"542±0"084+### /)sap 0,317±0.07+ 0.31±0,079+ 0,385±0,081+ Pr 0"25±0"067+ 0"277±0"057+ 0"427±0"067+### Pms 0"093±0"036+ 0"073±0"032+ 0"129±0"034+ )~rr, sap 0,138±0,062+ 0,125±0,029+ 0.12±0,045+ )~rr,r 0"144±0"046+ 0"198±0"074+ 0"085±0"057# )~rr,ms 0"129±0"06+ 0"127±0"051+ 0"076±0"032# )~sap,r 0"139±0"05+ 0"139±0"065+ 0"159±0"051+ )~sap,ms 0"103±0"04+ 0"099±0"028+ 0"122±0"067+ )~r,ms 0,078±0,034 0,065±0,032 0,119±0,028 +## )~rr, sap,r 0,141±0,026+ 0,154±0,034+ 0,121±0,032+ )~rr, sap,ms 0"123±0"028+ 0"117±0"024+ 0"106±0"025+ )~rr,r,ms 0"117±0"033+ 0"13±0"018+ 0"093±0"022+ )~sap,r,ms 0"107±0"021+ 0"101±0"026+ 0"133±0"037+ )~rr, sap,r,ms 0"122±0"018+ 0"125±0"013+ 0"113±0"018+

B, baseline; LD, low dose of atropine; HD, high dose of atropine p<0.05 original versus surrogate data

< 0.05, ##p < 0.01, ###p < 0.001 experimental condition versus B

At R, the RR and SAP series are coupled, although weakly. This relationship comes from the RR-SAP mechanical coupling and from the baroreceptor feedback. The two paths (from RR to SAP series and vice versa) cannot be disentangled by UF and, therefore, the synchronisation index accounts for the information exchange on both causal directions. The degree of synchronisation between the RR and SAP series increases during T, CR10 and CR15 as a result of increased coupling in the LF band during T and in the HF band during CR10 and CR15. During T, the baroreflex gain is known to decrease (COOKE et al., 1999) according to the sympathetic activation. Therefore, it appears that the information exchange between the RR interval and SAP is not related to the baroreflex gain. CR is not always able to increase synchronisa- tion between respiratory activity and cardiovascular variables (RR or SAP). Only slow breathing frequencies (e.g. 10 acts/min) increase synchronisation. As tidal volume decreases with breathing frequency, the coupling between RR and respiration may depend on the amount of stretch of the lung receptors. Also a reduced Bainbridge reflex with the breathing rate (due to the decrease of stimulation of the atrial stretch receptors) could produce decoupling at higher breathing frequencies. Both these two mechanisms contribute to synchronise the activity of the central oscillators responsible for the generation of the HF rhythm with the exogenous periodical input (i.e. CR). Mechanical influences on the arterial vessels may be responsible for the synchronisation between SAP and respiration. The increased mechanical influences at low breathing rates (e.g. 10 acts/min) due to an augmented tidal volume may produce the increased coupling between SAP and respiration.

The analysis of co-ordination proves that the co-ordination effect of free respiration on the cardiovascular variables is small. Indeed, )~ . . . . p,r is not significantly larger than )~ . . . . p. These results can be explained by considering that, although free respiration is a strong input perturbing cardiovascular variables, it is not able to entrain or phase-lock completely cardiovascular variabilities, in contrast, a large increase in ){rr, sap,r is found during CR10 and CR15. A more regular respiratory pattern cannot be the unique explanation (regularity of respiration increases only du6ng CR10). More likely, CR10 and CR15 evoke more powerful reflexes in response to the mechanical action, thus being more effective in driving cardiovascular variables.

186

5.2 Effects q f atropine on regulari O, synchronisation and co-ordination qfl cardiovascular variables

HD increases RR regularity. This significant change in regularity is indicated by the presence of a dramatic decrease of the variance, thus proving that there is no positive correlation between a time domain parameter like variance and the regu- larity index. This increase in regularity can be ascribed to the reduction of complexity of the regulation of RR interval related to peripheral muscarinic blockade (the HF oscillations disap- pear). The MS regularity is very low. The LF and HF rhythm observed in the MS series (MONTANO et al., 1998) are not well- defined repetitive oscillations (they continuously change in phase, frequency and amplitude). These rhythms, usually quite repetitive, contribute significantly to the complexity of this signal. The increase in the HF rhythms observed after LD and HD and related to the vagotonic effects of atropine (MONTANO et al., 1998) does not determine changes in regularity, thus proving that regularity cannot be completely inferred from the power spectrum.

The only effect of LD is an increase, though not significant, of the coupling between the RR interval and respiration, probably related to the vagotonic effects of LD enhancing rhythmicities synchronous with respiration (MONTANO et al., 1998). After HD, the RR interval is independent of respiration, while MS and respiration become strongly coupled. Therefore, it appears that crosstalk phenomena occurring between respiratory oscillators and sympathetic circuits permit the central vagotonic effects of atropine to be observed. These effects cannot be appreciated at the level of RR variability due to peripheral muscarinic blockade produced by HD.

The RR, SAP, MS and respiration are significantly co- ordinated. They appear slightly more co-ordinated after HD, when the RR series is excluded from the set of variables over which co-ordination is calculated ()~sap . . . . . = 0.133). This might suggest that the RR series contributes to the increase in the total amount of information. Therefore, the RR series appears to be independent of the others (i.e. it is not able to exchange information with the others) due to peripheral muscarinic blockade.

5.3 Unified general approach to the measure q f regulari O, synchronisation and co-ordination over short data series

The proposed unified general approach to the measure of regularity, synchronisation and co-ordination is based on the quantification of the degree of recurrence of a pattern assessed by CE. When the regularity of the signal y is evaluated, the pattern is a sequence of L delayed samples of the signal y (i.e. YL(i)) and CE measures the amount of information carried by the most recent sample of the sequence (i.e. y(i)) when the previous L - 1 samples (i.e. YL 1 ( i - 1 ) ) are known. When evaluating the synchronisation between a pair of signals (e.g. x and y), the scheme consists of relating one sample from signal x and L - 1 past samples of signal y (i.e. (x(i)),3Z 1(i)). With this definition of the pattern, CE evaluates the amount of information carried by signal x which cannot be derived from a knowledge of signal y. In this study UF (PORTA et al., 1999) is used because no causal relationship is privileged in the interactions between x and y (it is unknown i fx affects y or vice versa).

Theoretically, CE and U F have to be evaluated from an infinite amount of data. Over a short data set they do not hold due to bias in the CE estimate: the conditional sample frequency used to approximate the conditional probability is one when the pattern is found only once in the series, thus interpreting the presence of a single pattern as an index of full regularity (PORTA et al., 1998a). The estimators of these functions over short data series are based on the definition o f C C E (PORTA et al., 1998a).

Medical & Biological Engineering & Computing 2000, Vol. 38

This function is designed to counteract the bias: every time a pattern is detected only once, the certainty associated with this pattern is substituted with the maximum uncertainty detectable in the signal. Based on the definition o f C C E and on a procedure o f minimum extraction, indexes o f regularity and synchronisa- tion can be reliably obtained from short segments of data.

When the problem of measuring co-ordination is addressed, the pattern to be considered is a joint pattern formed by the current sample of the signal xk(i) and a certain amount o f past samples extracted from all the others (e.g. (xk(i)), xl,L 1(i), x2, L 1(i) . . . . . xK, L 1(i))), thus permitting one to model the contemporaneous effects o f K - 1 signals on xl-. Obviously, this approach requires a large amount o f data because the pattern length is multiplied for K - 1. In contrast, the proposed approach gives up the construction o f joint patterns. The relationships among the signals are modelled as interactions between all possible pairs o f signals according to the U F definition. Therefore, an index of co-ordination is simply obtained as the average synchronisation measured inside the set o f signals, i f contemporaneous influences are present, the co- ordination quantified by the proposed index is smaller than that extracted by considering the whole joint process. Moreover, this index cannot determine if a weak correlation arises from a set o f signals weakly linked or from a subset o f signals strongly interacting while the remaining ones are independent. These different types o f co-ordination can be detected by comparing co-ordination indexes calculated on subsets with increasing numbers o f signals.

The proposed indexes are defined in the information domain as they come from functions measuring information inside the data set and provide an indication different from traditional indexes obtained in the time (e.g. the mean and variance) and frequency (e.g. the power o f the oscillations inside well-defined frequency bands) domains. Indeed, C E is independent o f the mean and the variance o f the process, o f the definition of the frequency bands and of the gain o f the relationships between the signals.

6 Conclusions

A general approach to the measure o f regularity, synchronisa- tion and co-ordination has been proposed, it is fully quantitative and can be applied over short segments of data (about 300 samples). This approach could provide new insights into the analysis of complex systems (i.e. systems characterised by several interacting subsystems co-operating and/or competing in accomplishing a specific task) in several fields including the biological one. For the analysis o f cardiovascular control, the proposed method has been shown to be useful in interpreting the differences between the series considered and the changes induced by the experimental conditions. Therefore, information domain indexes should be added to more traditional parameters derived from time and frequency domains. However, more specific studies are needed to compare their performance with the performance of conventional linear methods such as the power spectrum or coherence function in assessing cardiovas- cular variability.

References

ANDERSON, B., KENNEY, R. A., and NEIL, E. (1950): 'The role of the chemoreceptors of the carotid and aortic regions in the production of Mayer waves', Acta Physiol. Stand., 20, pp. 203-220

BROWN, T. E., BEIGHTOL, L. A., KOB, J., and ECKBERG, D. L. (1993): 'Important influence of respiration on human RR interval power spectra is largely ignored', J Appl. Physiol., 75, pp. 2310-2317

Medical & Biological Engineering & Computing 2000, Vol. 38

COOKE, W. H., HOAG, J. B., CROSSMAN, A. A., KUUSELA, T. A., TAHVANAINEN, K. U. O., and ECKBERG, D. L. (1999): 'Human responses to upright tilt: a window on central autonomic integra- tion', J Plo.'siol., 517, pp. 617-628

GLASS, L., and MACKEY, M. C. (1988): 'From clock to chaos. The rhythms of life' (University Press, Princeton, NJ)

GRASSBERGER, R, and PROCACCIA, I. (1983): 'Measuring the strange- ness of strange attractors', Phvsica D, 9, pp. 189-208

GUEVARA, M. R. (1997): 'Chaos in electrophysiology', in BILLETTE, J., and LEBLANC, A. R. (Eds), 'Concepts and techniques in bio- electric measurements: is the medium caxrying the message?' (Editions de l't~cole Polytechnique, Montr6al), pp. 67-87

GUYTON, A. C., and HARRIS, J. H. (1951 ): 'Pressoreceptor-autonomic oscillation: a probable cause of vasomotor waves', Am. J Pl¢vsiol., 165, pp. 158-166

HOYER, D., BAUER, R., WALTER, B., and ZWIENER, U. (1998): 'Estimation of nonlinear couplings on the basis of complexity and predictability: a new method applied to cardiorespiratory co-ordina- tion', IEEE Trans. Biomed. Eng., 45, pp. 545-552

I~LSO, S. (1995): 'Dynamic pattern' (MIT Press, Cambridge, Mas- sachusetts).

KOEPCHEN, H. E (1991): 'Physiology of rhythms and control systems: an integrative approach' in HAKEN, H., and KOEPCHEN, H. E (Eds), 'Rhythms in physiological systems' (Springer-Verlag, Berlin), pp. 3-20

MEYER, J. U., LIMDBOM, L., and INTIGLIETTA, M. (1987): 'Coordi- nated diameter oscillations at axteriolar bifurcation in skeletal muscle', Am. J Physiol., 253, pp. H568-H573

MIYAKAWA, K. (1988): 'Mechanisms of blood pressure oscillation caused by central nervous system ischemic response', Jpn. J Physiol., 38, pp. 399-425

MONTANO, N., GNECCHI-RUSCONE, T., PORTA, A., LOMBARDI, E, PAGAN1, M., and MALLIANI, A. (1994): 'Power spectrum analysis of heart rate variability to assess changes in sympatho-vagal balance during graded orthostatic tilt', Circulation, 90, pp. 1826-1831

MONTANO, N., PAGANI, M., PORTA, A., COGLIATI, C., MALLIANI, A., NARKIEWICZ, K., ABBOUD, E M., BIRKETT, C., and SOMERS, V K. (1998): 'Vagotonic effects of atropine modulate spectral oscillations of sympathetic nerve activity', Circulation, 98, pp. 1394-1399

OPPENHEIM, A. V, and SCHAFER, R.W. (1975): 'Digital signal processing' (Prentice Hall, Englewood Cliffs, New Jersey)

PAGANI, M., LOMBARDI, F., GUZZETTI, S., RIMOLDI, O., FURLAN, R., PIZZINELLI, P., SANDRONE, G., MALFATTO, G., DELL'ORTO, S., PICCALUGA, E., TURIEL, M., BASELLI, G., CERUTTI, S., and MAL- LIANI, A. (1986): 'Power spectral analysis of heart rate and arterial pressure variabilities as a marker of sympatho-vagal interaction in man and conscious dog', Circ. Res., 59, pp. 178-193

PAGANI, M., MONTANO, N., PORTA, A., MALLIANI, A., ABBOUD, E M., BIRKETT, C., and SOMERS, V K. (1997): 'Relationship between spectral components of cardiovascular variabilities and direct mea- sures of muscle sympathetic nerve activity in humans', Circulation, 95, pp. 1441-1448

PALUS, M. (1997): 'Detecting phase synchronisation in noisy sys- tems', Phys. Lett. A, 235, pp. 341-351

PAPOULIS, A. (1984): 'Probability, random variables and stochastic processes' (McGraw-Hill, New York)

PINCUS, S. M. (1995): 'Approximated entropy (ApEn) as a complexity measure', Chaos, 5, pp. 110-117

PORTA, A. (1998): 'Multivariate method based on conditional entropy estimate for measuring regularity, synchronisation and co-ordina- tion in cardiovascular variability signals'. PhD Thesis, Dipaxtimento di Bioingegneria, Politecnico di Milano

PORTA, A., BASELLI, G., LIBERATI, D., MONTANO, N., COGLIATI, C., GNECCHI-RUSCONE, T., MALLIANI, A., and CERUTTI, S. (1998a): 'Measuring regularity by means of a corrected conditional entropy in sympathetic outflow', Biol. Qvbern., 78, pp. 71-78

PORTA, A., BASELLI, G., LOMBARDI, E, and CERUTTI, S. (1998b): 'Quantifying regularity and synchronisation by using entropy rates in the cardiovascular variabiliy data' in 'Information processing and management of uncertainty' (EDK, Paris), pp. 555-561

PORTA, A., BASELLI, G., LOMBARDI, E, MONTANO, N., MALLIANI, A., and CERUTTI, S. (1999): 'Conditional entropy approach for the evaluation of the coupling strength', Biol. Qvbern., 81, pp. 119-129

187

PREISS, G., and POLOSA, C. (1974): 'Patterns of sympathetic neuron activity associated with Mayer waves', Am. J. Physiol., 226, pp. 724-730

RAPR P. E. (1987): 'Why are so many biological systems periodic?', Prog. Neurobiol., 29, pp. 261-273

SINGER, W. (1993): 'Synchronisation of cortical activity and its putative role in information processing and learning', Annu. Rm: PhysioL, 55, pp. 349-374

THEILER, J., EUBANK, S., LONGTIN, A., GALDRIKIAN, J., and FARMER, D. (1992): 'Testing for nonlinearity in time series: the method of surrogate data', Phvsica D, 58, pp. 77-94

WALLIN, B. G., (1983): 'Intrax~eural recording and autonomic function in man' in BANNISTER, R. (Ed.): 'Autonomic failure' (Oxford University Press, London), pp. 36-51

Author's biography

ALBERTO PORTA was born in 1964. He graduated in Electronic Engineering at the Politecnico di Milaxlo, Milan, Italy in 1989. He was a Research Fellow on Automatic Control and System Theory at the Dipaxtimento di Elettronica per l'Automa- zione, Universita' di Brescia, Brescia, Italy, until

J~ 1994. He received his PhD degree in Biomedical Engineering from the Politecnico di Milano, Milaxlo, in 1998. Since 1999 he has been with

the Dipaxtimento di Scienze Precliniche, Universita' degli Studi di Milaxlo, Milano, Italy. His primary interests include time series analysis and system identification applied to cardiovascular control mechanisms.

188 Medical & Biological Engineering & Computing 2000, Vol. 38