Classification of coupling patterns among ... - Semantic Scholar

Biol. Cybern. 75, 163—172 (1996)

Classification of coupling patterns among spontaneous rhythms and ventilation in the sympathetic discharge of decerebrate cats A. Porta 1, G. Baselli 1, N. Montano 2, T. Gnecchi-Ruscone 2, F. Lombardi 2, A. Malliani 2, S. Cerutti 3 1 Dipartimento di Elettronica per l’Automazione, Universita` di Brescia, via Branze 38, I-25123 Brescia, Italy 2 Medicina Interna II, Ospedale L. Sacco, Universita` di Milano, Milan, Italy 3 Dipartimento di Bioingegneria, Politecnico di Milano, Milan, Italy Received: 19 July 1995/Accepted in revised form: 20 May 1996

Abstract. The spontaneous low- and high-frequency rhythms in the sympathetic discharge of decerebrate artificially ventilated cats are affected by external ventilation. Two graphical methods (i.e. the space-time separation plot and the frequency tracking locus) are used to classify the non-linear interactions. The observed behaviours in the sympathetic discharge consist of phase-locked periodic dynamics (at various frequency ratios with ventilation), quasiperiodic and aperiodic patterns. They depend on the experimental condition. In control condition the sympathetic discharge appears more frequently locked to each ventilatory cycle (1 : 1 dynamics). However, some cases of quasiperiodic dynamics are found. A sympathetic activation stimulus, such as inferior vena cava occlusion, is able to synchronise slow rhythms in the sympathetic discharge to a subharmonic of ventilation. During a sympathetic inhibition stimulus, such as aortic constriction, 1 : 1 dynamics is detected but the amplitude of the sympathetic responses can be modulated by unlocked slow rhythms. Moreover, some cases of aperiodic dynamics are observed. Vagotomy reduces the 1 : 1 coupling between sympathetic outflow and ventilation. Vagotomy plus spinalisation disrupts periodic dynamics in the sympathetic discharge so that irregular and complex patterns are found.

1 Introduction Perturbations of the rhythmic activity of a physiological oscillator are commonly used to derive information about the properties of the underlying mechanisms. While a single stimulus can be applied to reset or annihilate a rhythm (Jalife and Antzelevitch 1979), a periodical stimulus can interact with the spontaneous physiological oscillations and determine non-linear behaviours. Phase

Correspondence to: A. Porta

locking (Guevara et al. 1988) and period doubling (Glass et al. 1983) have been observed in aggregates of embryonic heart cells undergoing self-sustained pacemaker activity driven by periodical electrical stimuli. Phaselocked patterns have also been found in the phrenic nerve activity perturbed by ventilation in anaesthetised paralysed cats (Petrillo et al. 1983). Efferent sympathetic discharge has long been known to be characterised by spontaneous rhythms at a low frequency (LF; around 0.1 Hz) corresponding to that of slow arterial pressure oscillations (Fernandez de Molina and Perl 1965; Preiss and Polosa 1974) and at a high frequency (HF; around 0.3 Hz) related to respiration (Adrian et al. 1932). More recently it was found that the power of LF and HF fluctuations detectable in efferent sympathetic discharge variability of decerebrate artificially ventilated cats could be modified by interventions associated with changes in mean firing rate (Montano et al. 1992) : the power of LF rhythm was increased during reflex sympathetic activation induced by inferior vena cava occlusion, while an increase in the power of HF was detectable during reflex sympathetic inhibition obtained by aortic constriction. This spontaneous rhythmical activity was found to be influenced by periodical ventilation (Porta et al. 1994a), resulting in a variety of non-linear patterns ranging from coupling behaviours to irregular dynamics (Porta et al. 1994b, 1995). In the present study several classical methods for non-linear interaction enhancement are first recalled; next attention is focused on the space-time separation plot (Provenzale et al. 1992) and the frequency tracking locus (Kitney and Bignall 1992), which appear most adequate for the classification of phase-locked patterns and quasiperiodic dynamics. These methods are specifically adapted to analyse and classify the non-linear interactions between the patterns of discharge of preganglionic sympathetic fibres directed to the heart and ventilation in decerebrate artificially ventilated cats. In the most frequently used experimental models, the amplitude and frequency of the periodical forcing stimulus are changed while the spontaneous activity of the

164

perturbed oscillator is not modified. Guevara et al. (1988) presented a simple classification scheme that was able to predict the complete sequence of phase-locking rhythms in periodically stimulated heart cell aggregates as the stimulation frequency was changed. In a similar way, in paralysed anaesthetised cats, changes in the coupling between spontaneous respiratory rhythm and a mechanical ventilator were found when varying the ventilatory frequency and tidal volume (Petrillo et al. 1983). Similarly, in the giant axon of the squid, disruption of the rhythmic activity and a rise in chaotic behaviour were obtained by means of a suitable variation in the amplitude and frequency of the periodical forcing current (Aihara et al. 1986). In the present study it is hypothesised that at least two kinds of spontaneous oscillators are present, responsible for the vasomotor (LF) and respiratory-related (HF) oscillations observed in the sympathetic discharge by several authors (Adrian et al. 1932; Fernandez de Molina and Perl 1965; Preiss and Polosa 1974; Montano et al. 1992) and their coupling with external ventilation. The interference patterns between the spontaneous oscillators and ventilation are described when the amplitude of LF and HF oscillations is modified by means of manoeuvres altering the sympathetic activity while the frequency of the periodical stimulus is not varied. The interactions between sympathetic discharge and ventilation were analysed in five experimental conditions: (1) control, (2) during inferior vena cava occlusion (reflex sympathetic activation), increasing the amplitude of the LF rhythm, (3) during aortic constriction (reflex sympathetic inhibition), enhancing the HF rhythm, (4) after vagotomy (excluding parasympathetic control) and (5) after spinal section (interrupting the connections between supraspinal and spinal circuits). The paper is organised as follows. In Sect. 2 we illustrate the several tools used in the study of synchronisation phenomena. The performance of these methods in detecting phase-locking patterns and quasiperiodic dynamics is verified on simulated signals obtained from a forced Van Der Pol oscillator. In Sect. 3 we describe our experimental preparation and the physiological manoeuvres performed. In Sect. 4 we describe the variety of coupling patterns between sympathetic discharge and ventilation and propose a classification of the observed non-linear behaviours. In Sect. 5 we discuss the possible sources of interference between spontaneous rhythms detected in sympathetic discharge and external ventilation. 2 Methods Classical graphical tools, such as simple superposition of the signals, Lissajous figures and recurrence plot (briefly recalled in the following), can be used to detect coupling phenomena. Unfortunately, these methods are very sensitive to noise. This makes the classification of non-linear phenomena difficult and prompts the search for other methods able to describe phase-locked patterns. The proposed tools, i.e. space-time separation plot and frequency tracking locus, are tested on the x signal obtained from the differential equations xR "y

(1)

yR "(1!x2)y!x#u

(2)

describing the behaviour of the Van Der Pol oscillator forced by a sinusoidal input u"A cos(2nt/¹ ) of period ¹ . In the absence of u u input the Van Der Pol oscillator exhibits a limit cycle in a state plane (x, y), i.e. a periodical oscillation in the time domain. Entrainment may occur depending on forcing amplitude and the ratio between forcing and natural frequency; in this case the oscillator is phase-locked to u and a fixed integer ratio M : N is established with M cycles of the forced rhythm every N cycles of the forcing input. Phase-locking regions are separated by regions in which no integer ratio is established. In this case quasiperiodic dynamics is found (Hayashi 1964).

2.1 Superposition The state variable x can be overlapped with the forcing input u to reveal coupling patterns between the two signals. This technique is applied to an example of 1 : 2 (Fig. 1a), 1 : 4 (Fig. 1b) and quasiperiodic (Fig. 1c) dynamics. Unfortunately this method is useful only when the coupling is clearly visible and detectable. Moreover, the degree of coupling cannot be quantified, the description of the underlying dynamics is qualitative and the classification of the non-linear behaviours is difficult and subjective. These difficulties increase when several interacting rhythms have to be considered.

2.2 Lissajous figures Coupling behaviour can be detected by plotting the forced signal versus the forcing input. When synchronised behaviours are present, only a limited number of patterns, called Lissajous figures, are detectable. These take the form of loops in the plane (u, x) and their number is related to the number of different responses to the input cycles. In contrast when no coupling pattern is present (e.g. in quasiperiodic dynamics) no clear figure appears. The major drawbacks of this tool are that time is not represented and the plot is very sensitive to noise.

2.3 Recurrence map The recurrence plot is a useful tool for finding out whether repetitive patterns are present in the plane (u, x) while maintaining the time direction. Two points, P "(u(i), x(i)) and P "(u(j), x(j)), (where i and i j j are the sample number) are defined as recurrent if they are sufficiently close to each other in the plane (u, x). The distance between the two points P and P is derived by calculating the normalised Euclidean i j norm as (u(i)!u(j))2 (x(i)!x(j))2 r2 " # i,j var[u] var[x]

iOj

(3)

If the distance r is below a reference value (in Fig. 1d—f 15% of the i,j maximum value of the distance r on the data sequence considered), i,j a dot is plotted at the point (i, j). Recurrent patterns are transformed in lines parallel to the main diagonal j"i. The equation of these lines is j"i#k¹, where ¹ is the distance between two consecutive lines and represents the periodicity of the recurrent pattern (k is an integer). Because the map is symmetrical the lines j"i!k¹ are also present. If these lines are complete the pattern is present for the whole temporal window. This is the case in Fig. 1d and e, where 1 : 2 and 1 : 4 dynamics are represented by complete diagonal lines with a periodicity of two and four input cycles respectively. In contrast the presence of interrupted lines forming segments, as in Fig. 1e, reveals that there are patterns with a period of one input cycle repeating themselves for only a brief time interval. Noise can make the recurrence plot difficult to be read since it corrupts the integrity of the diagonal lines and introduces further dots in the plot. Quasiperiodic dynamics results in diagonal lines which gradually fade while phase shifts and then become clear again (Fig. 1f ). The apparent periodicity in the structure of the recurrence plot is related to the detection of close returns in the plane (u, x). Therefore quasiperiodic dynamics cannot be clearly distinguished from periodic dynamics. This prompts the design of tools able to enhance a progressive phase shift between the signals.

165

Fig. 1a–n. Superposition of the time course (continuous line) of a forced Van Der Pol oscillator and the forcing sinusoidal input (dashed line). Examples are given of 1:2 dynamics (a), 1 : 4 dynamics (b) and quasiperiodic dynamics (c). These examples were obtained by varying the amplitude and frequency of the forcing input. d, e, f Recurrence maps were plotted from the data in panels a–c. g, h, i Space-time separation plots were constructed using the data in panels a–c. The three curves represent three different percentiles of points closer than a distance r (space separation) at a given time separation s (5%, 50% and 95%, respectively, from the bottom to the top of the each panel). l, m, n Frequency tracking loci were drawn for the data in panels a–c

2.4 Space-time separation plot To exploit stationarity and obtain a tool less sensitive to noise than the recurrence map it is possible to evaluate the probability of having a distance r smaller than a reference value of each delay q. Thus a space-time separation plot is built, in which the reference distance r (space separation) is indicated on the vertical axis versus the time separation q and in which the 5th, 50th and 95th percentiles of points separated by time q and closer than distance r are plotted. When recurrences or close returns occur at a specific q, all percentile lines are compressed at small values of space separation. Synchronisation of the natural frequency to a period ¹, which is multiple of the forcing period, results in small values of the distance r(q) for q"k¹ (k is an integer). In Fig. 1g and Fig. 1h, since zero values of the separation in space occur periodically at multiples of two and four input cycles respectively, 1 : 2 and 1 : 4 dynamics can be detected. In both Fig. 1g and Fig. 1h the slight decrease in space separation corresponding to each multiple of the input cycle emphasises that there are responses to every input cycle as well. In analysing quasiperiodic dynamics (Fig. 1i) the time separation plot suffers the same limitations as the recurrence map because small phase shifts are not able to disrupt close returns completely. It is worth

noting that, while a percentile line represents the fraction of points in the embedding space which are closer than distance r and separated by time q, the correlation integral (Grassberger and Procaccia 1983) represents the fraction of points closer than a distance r independently of its separation in time.

2.5 Frequency tracking locus The frequency tracking locus is designed to track, visualise and quantify in each cycle of the forcing input u the response of the forced oscillation x, and namely its phase and amplitude compared with that of u. The forced behaviour of the Van Der Pol oscillator due to an input cycle of the forcing signal u is represented by a vector, symbolised by an arrow in the plot v"oejr. The vector amplitude o is obtained as o"x(i

)/u(n¹ ) (n!1) ¹ )i (n¹ .!9 u u .!9 u

(4)

where (n!1) ¹ and n¹ represent the time of occurrence of the (n!1)th u u and nth peak defining the nth cycle of the forcing input and i is the .!9 instant at which the forced signal x assumes the maximum value in the

166 nth input cycle. In the nth input cycle the phase u, ranging from 360° ((n!1)th input peak) to 0° (nth input peak), can be calculated as u"2n(n¹ !i )/¹ (5) u .!9 u The presence of a response to an input cycle is considered only if the forced activity exceeds a defined threshold. If no response to the forcing cycle is found, an open circle is drawn and the number of input cycles that do not produce forced responses is written near the symbol in the plot. If more than one forced response is detected during the same forcing cycle, only the highest one is considered. The first vector starts at the origin of the plot and each subsequent vector is added to the tip of the preceding one in order to maintain the time direction and indicate the total phase shift. An M : N phase-locked pattern appears as a sequence of arrows and open circles regularly repeated after N forcing cycles. The number of arrows and open circles depends on the selected value of the threshold. Amplitude modulation results in a modulation of the vector magnitude. In Fig. 1l the repetitive sequence is an arrow followed by an open circle (1 : 2 dynamics). All the vectors have the same phase and magnitude. In Fig. 1m 1 : 4 dynamics is symbolised by three arrows with similar but not equal phase and magnitude followed by an open circle. Thus this graphical tools permits the acquisition of, in addition to the periodicity of the repetitive pattern, the phase of the forced responses. This feature permits a description and quantification of the quasiperiodic dynamics in a very direct way: the progressive phase shift in the forced responses is displayed by a regular rotation of the arrows in the plot (Fig. 1n).

2.6 Spectral analysis Although power spectral analysis is not able to distinguish coupled rhythms from independent oscillations, it allows detection of frequencies of the oscillations and verifies whether they are close to harmonic ratios. If the frequency of a forced oscillator has a rational relationship with respect to that of the periodic input, phase locking can be hypothesised. Moreover, spectral analysis is useful for distinguishing periodic and quasiperiodic dynamics from aperiodic dynamics, which is revealed by broad-band power spectra. Spectral analysis is performed via autoregressive (AR) techniques (Kay and Marple 1981) in order to calculate the central frequency of each spectral component (i.e. the phase of the relevant real pole or complex and conjugated pole pair in the AR model). The parameters of the AR process y(i) [a , k"1, . . . , p and the variance j2 of the k zero-mean white noise w(i)] are connected through the following relationship p y(i)" + a y(i!k)#w(i) (6) k k/1 and are estimated via Levinson-Durbin recursion. The optimum order p of the model is chosen according to the Akaike criterion. The AR power spectral density (PSD) is expressed as P( f )"Dtj2 [H(z) H(z~1)] (7) z/%91 (j2nfD t) where H(z) represents the transfer function of AR process and Dt the sampling interval.

3 Experimental preparation All the animals were decerebrated under anaesthesia by brain stem transection at the intercollicular level. All the animals were artificially ventilated by a Harvard Pump ventilator. Ventilation was kept at a fixed rate of 18 cycles/min while ventilatory volume was adapted to maintain constant pCO and pO levels. Neural activity 2 2 was obtained from the cut central ends of small filaments from the third (T3) left thoracic white ramus communicans. The neural signal was processed by means of an analog counter evaluating the number of spikes in 20 ms and thus producing a step-wise signal of the counted

sympathetic discharge. Ventilation flow, counted neural activity, electrocardiogram (ECG) and arterial pressure (AP) were recorded 2 h after decerebration (Montano et al. 1992). These signals were A/D converted (600 Hz sampling rate, 12-bit resolution, $10 V acquisition range) and processed to extract beat-to-beat variability series of the cardiac cycle duration, Mt(i)N, where i is the progressive number of cardiac beats, and of the sympathetic discharge, Ms(i)N. Cardiac cycle duration was obtained as the temporal distance between two consecutive R peaks. Parabolic interpolation permitted detection of the R apex with minimum jitters. The variability series of the sympathetic discharge was obtained by filtering the counted neural activity at 1 Hz (FIR filter, Weber-Cappellini window, 2400 coefficients) and by sampling the filtered signal in correspondence to the R peak on the ECG. The sympathetic discharge values were expressed in spikes/s. In addition the ventilation flow was sampled at the R peak on the ECG to extract the sampled ventilation, Mr (i)N. A first group of animals (n"17) was recorded in control conditions. On these animals two manoeuvres were performed: a reflex sympathetic activation was obtained by decreasing the AP by means of an inferior vena cava occlusion (IVCO, n"17), while reflex sympathetic inhibition was carried out by increasing AP by means of aortic constriction (AC, n"13). The success of the manoeuvre was verified by controlling the presence of stable tachycardia, increased sympathetic activity and decreased AP during IVCO and the opposite response during AC. The AC manoeuvre failed in four animals in which an unstable increase in AP was observed. A second group of animals (n"26) had a spinal section performed at C1 level and vagotomy. The spinalization (SP) separated supraspinal centres from spinal ones. In these animals a reduction of sympathetic discharge was generally observed when stable conditions were reached. In 11 of the 26 animals sympathetic discharge was recorded after vagotomy (VG) but prior to spinalization. All protocols used in these experiments were approved by a special Committee on Animal Use and Care of the University of Milan. 4 Results In the decerebrate artificially ventilated cat, LF and HF oscillations are clearly detected in the beat-to-beat variability of sympathetic outflow, as already described by Montano et al. (1992) by means of spectral analysis. In the LF band spectral peaks due to spontaneous oscillations are present, while in the HF band both the effects of ventilation and spontaneous HF waves can be detected. The variety of the resulting interference patterns is analysed and classified by the proposed tools. 4.1 Non-linear interactions between HF rhythms and ventilation In control condition 1 : 1 dynamics is frequently detected (Fig. 2a). The sympathetic discharge increases every

167

Fig. 2a–f. Superposition of sympathetic discharge (continuous line) and ventilation (dashed line) in 1 : 1 (a) and 2 : 2 dynamics (d). Space-time separation plots were constructed for the 1 : 1 (b) and 2 : 2 dynamics (e) depicted in panels a and d. Frequency tracking loci were drawn for the 1:1 (c) and 2:2 dynamics (f ) shown in panels a and d

Fig. 3a–f. Superposition of sympathetic discharge (continuous line) and ventilation (dashed line) in 1 : 2 (a) and 2:4 dynamics (d). Space-time separation plots were constructed for the 1:2 (b) and 2:4 dynamics (e) shown in panels a and d. Frequency tracking loci were drawn for the 1 : 2 (c) and 2 : 4 dynamics (f ) depicted in panels a and d

ventilatory cycle with the same phase. This coupling is clearly visible on the time-space separation plot and the frequency tracking locus. The time-space separation plot (Fig. 2b) points out that close returns can be found every ventilatory period. Phase relationships are quantified by the frequency tracking locus (Fig. 2c); the vectors, representing the sympathetic responses to ventilatory cycles, are characterised by the same phase and magnitude. Moreover no open circle is drawn because no missed response is present. Only in one animal, during IVCO, does sympathetic discharge increase every ventilatory period but with alternated phases (2 : 2 dynamics) (Fig. 2d). This dynamics, not clearly detectable by means of the superposition of the two signals, is enhanced by the space-time separation plot (Fig. 2e). The space separation is closer to zero after two ventilatory cycles than after one ventilatory cycle. In the frequency tracking locus (Fig. 2f ) the recurrent pattern is symbolised by two vectors with two different phases and provides further evidence of 2 : 2 dynamics.

with fixed phase (1 : 2 dynamics). In Fig. 3d, the sympathetic response alternates its phase (2 : 4 dynamics). The 1 : 2 dynamics is easily detected from the space-time separation plot because the space separation decreases markedly when the separation in time approaches the multiples of two ventilatory cycles (Fig. 3b). Sympathetic responses to every ventilatory period are still present but not well defined (i.e. the decrease in the space separation at multiples of one ventilatory cycle is limited). The 2 : 4 dynamics can be appreciated by observing that the decrease in the space separation is more marked every multiple of four ventilatory periods than every multiple of two ventilatory cycles (Fig. 3e). In the frequency tracking locus 1 : 2 dynamics is represented by a sequence constituted by a fixed-phase arrow alternating with an open circle (Fig. 3c). The 2 : 4 dynamics is illustrated as two consecutive vectors with different phase followed by an open circle labelled by the number 2 (Fig. 3f ) to indicate 2 cycles with no response.

4.2 Non-linear interactions between LF rhythms and ventilation

4.3 Quasiperiodic dynamics

LF oscillations can be synchronised at subharmonics of ventilation. Two examples are depicted in Fig. 3a, d derived from two cats during IVCO. In Fig. 3a, the sympathetic discharge grows every two ventilatory cycles

In a few cases, rhythms of sympathetic discharge not entrained by ventilation are found. These oscillations can be at a frequency either near to that of ventilation (at HF) or near to its subharmonics (at LF). A case of HF quasiperiodic dynamics is depicted in Fig. 4a: the regular

168

discharge and ventilation indicate that the HF rhythms on the two signals have close but not equal frequencies (Fig. 4b). HF quasiperiodic dynamics can best be appreciated by means of the frequency tracking locus (Fig. 4c). Regular shifts in the phase are able to produce a sequence of arrows which form an arc of a circle. LF oscillations in the sympathetic discharge independent of ventilation can be observed in Fig. 4d (LF quasiperiodic dynamics). The two PSDs point out that the LF oscillation in the sympathetic discharge is not a subharmonic of ventilation: in fact no rational M : N relationship by small M and N with ventilatory frequency is detectable (Fig. 4e). LF quasiperiodic dynamics is stressed by the spiral-like figure formed by the arrows of the frequency tracking locus (Fig. 4f ): the absence of a clear repetitive vector sequence points out that privileged phase relationships do not exist. 4.4 Locked LF rhythm plus HF waves

Fig. 4a–f. Superposition of sympathetic discharge (continuous line) and ventilation (dashed line) in HF (a) and LF (d) quasiperiodic dynamics. Superposition of the power spectral densities (PSDs) of sympathetic discharge (continuous line) and ventilation (dashed line) was drawn for the HF (b) and LF (e) quasiperiodic dynamics shown in panels a and d. Frequency tracking loci were drawn for the HF (c) and LF (f ) quasiperiodic dynamics depicted in panels a and d

phase shift of sympathetic discharge with respect to ventilation indicates the absence of synchronisation between the two signals. Overlapped PSDs of the sympathetic

In several cases the HF and LF rhythms are clearly simultaneously present in the sympathetic discharge. The LF rhythm appears well synchronised as a subharmonic of ventilation with a period of N ventilatory cycles (N"2, 4, 6, 8). In contrast the HF sympathetic responses to each ventilatory cycle appear to be synchronised only at the rising edge of a LF wave; decreasing responses follow with changing phase. Thus the dynamics of the sympathetic responses to each ventilatory cycle is not periodic. In Fig. 5a, d a clear example of this phenomenon is depicted. A LF oscillation is locked to ventilation and starts every eight ventilatory cycles (Fig. 5a). HF spontaneous oscillation results in clearly detectable sympathetic responses with no repetitive phase to every ventilatory period. This coupling is clearly indicated by the marked decrease in the separation in space every eight ventilatory cycles and by the slight decrease every ventilatory cycle (Fig. 5d).

Fig. 5a–f. Superposition of sympathetic discharge (continuous line) and ventilation (dashed line) at control (a), during inferior vena cava occlusion (IVCO) (b), and during aortic constriction (AC) (c), performed on the same animal. Space-time separation plots (d, e, f ) were constructed for the same data shown in panels a–c

169 Table 1. Classification of coupling patterns found at control, during inferior vena cava occlusion (IVCO) and aortic constriction (AC) Animal no.

Control (n"17)

IVCO (n"17)

AC (n"13)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 : 4! 1:1 1 : 8! Quasiperiodicity Quasiperiodicity Aperiodicity 1:1 1:1 1:1 1:1 1 : 1" 1 : 1" 1:3 1:1 1 : 2! Quasiperiodicity Quasiperiodicity

1:4 1:2 1 : 6! 1 : 2! 1:2 2:4 1:3 1:1 1 : 2! 1 : 2! 1:1 1 : 6! Quasiperiodicity 1 : 1" 2:2 1:2 Quasiperiodicity

1 : 1" 1 : 1" 1 : 1" Aperiodicity 1:2 Aperiodicity 1:1 — 1:1 Aperiodicity 1 : 1" 1 : 1" 1 : 1" — Aperiodicity — —

! Presence of a superimposed HF rhythm " Presence of an unlocked LF modulation

4.5 Response to IVCO and AC Figure 5 provides an example of how the coupling patterns described above may occur when the experimental condition is varied from control (left panels) to IVCO (central panels) and to AC (right panels). In this case a LF rhythm entrained to ventilation is already present in the control (Fig. 5a). During IVCO the coupling between the HF rhythm and ventilation is reduced (Fig. 5b). The decrease in the separation in space every forcing period is not as regular as in the control condition, while close returns every six ventilatory cycles are detected (Fig. 5e). During AC the LF synchronisation (Fig. 5f ) is disrupted and the HF ripple in the space-time diagram becomes almost uniform. Nonetheless a LF rhythm in the sympathetic discharge is still present (Fig. 5c), as confirmed by spectral analysis (not shown), but appears as an unlocked amplitude modulation of HF responses. The patterns identified at control, IVCO and AC are detailed for each animal in Table 1, where the trend described in the example of Fig. 5, although with different patterns, is confirmed. A 1 : 1 dynamics found in the control condition often becomes, during IVCO, a LFlocked pattern (animals 2, 7, 9, 10, 12) or sometimes is not affected (animals 8, 11, 14). A quasiperiodic dynamics

seen in the control is either synchronised to a ratio of the ventilatory frequency during IVCO (animals 4, 5, 16) or does not change (animal 17). The LF-locked patterns already present in the control are not disrupted by the sympathetic activation stimulus (animals 1, 3, 15) but the coupling ratio may be modified. Only in animal 13 does a 1 : 3 synchronisation becomes a quasiperiodic dynamics. Coupling patterns at different ratios of the ventilatory frequency in the control become either 1 : 1 dynamics during AC (animals 1, 2, 3, 7, 9, 11, 12, 13) or aperiodic patterns (animals 10, 15). However 1 : 1 dynamics during AC is very different from that in the control condition: in fact the sympathetic responses occurring at fixed phase to every ventilatory cycle often appear modulated in amplitude by unlocked LF rhythms. The strong effect of ventilation found during AC is able to disrupt quasiperiodic dynamics (animals 4, 5). The observed phase-locking phenomena can mainly be grouped into three classes as in Table 2: (I) the sympathetic discharge grows every ventilatory cycle (mainly 1 : 1 dynamics), i.e. tight locking of HF rhythm to ventilation; (II) subharmonic phase locking (mainly 1 : N dynamics, where N"2, 3, 4), i.e. a locked LF rhythm with no evidence of HF oscillations; (III) a subharmonic locking again (1 : N dynamics, where N"2, 4, 6, 8) in which sympathetic bursts in response to each ventilatory cycle are still present and are synchronised only at the rising edge of LF waves. In addition quasiperiodicity and aperiodicity (see Sect. 4.6) are classified. The 2 : 2 and 2 : 4 dynamics are included in the type (I) and (II), respectively. As shown in Table 2, in the control HF-locked rhythms (47%, type (I)) are more frequent than LF synchronisations (23%, type (II) and (III)). Moreover, a few cases of quasiperiodic dynamics (24%) are found. HFlocked rhythms (23%, type (I)) and quasiperiodic dynamics (12%) are less frequent during IVCO and LF rhythms appear to be synchronised to subharmonics of ventilation (65%, type (II) and (III)). During AC, HF synchronisation is most frequent (62%, type (I)). VG is able to reduce the 1 : 1 coupling with ventilation (27%, type (I)) but LF locked patterns (45%, type (II) and (III)) can still be found. 4.6 Complex and aperiodic patterns Aperiodic behaviours can be detected by considering the PSD of the sympathetic discharge. In fact it is not composed solely of isolated peaks as in periodic and

Table 2. Summary of phase locking patterns, quasiperiodicity and aperiodicity cases Condition

HF-locked type (I)

LF-locked type (II)

LF-locked #HF type (III)

Quasiperiodic dynamics

Aperiodic dynamics

Control (n"17) IVCO (n"17) AC (n"13) VG (n"11) SP (n"26)

8 4 8 3 0

1 6 1 3 1

3 5 0 2 0

4 2 0 1 0

1 0 4 2 25

IVCO, inferior vena cava occlusion; AC, aortic constriction; VG, vagotomy; SP, spinalisation

170

parently stable periodic case found after SP is a clear 1 : 3 dynamics (Fig. 6d, e) with a single burst of activity every three ventilatory cycles. Even in this case, the frequency tracking locus is able to detect a considerable phase variability (Fig. 6f ), thus suggesting that this behaviour lies close to a boundary with intermittency or aperiodicity. 5 Discussion

Fig. 6a–f. Superposition of sympathetic discharge (continuous line) and ventilation (dashed line) in an aperiodic (a) and 1:3 (d) dynamics. Superposition of the PSDs of sympathetic discharge (continuous line) and ventilation (dashed line) for the aperiodic (b) and 1 : 3 (e) dynamics shown in panels a and d. Poincare´ map (c) was constructed for the aperiodic dynamics depicted in panel a. Frequency tracking locus (f ) was drawn for the 1 : 3 dynamics depicted in panel d

quasiperiodic dynamics, but has a continuous, broadband nature. This spectral pattern depends on the presence of high sympathetic bursts which are likely to reflect the response to a ventilatory cycle but with irregular periodicities. LF oscillations can still be present but are weak and unlocked to ventilation. Several segments of data are characterised by the intermittent occurrence of odd periodicities (mainly period-three periodicities). This dynamics is found in experimental conditions characterised by a strong reduction in the sympathetic discharge values such as during AC (30%) and after SP (96%). It is less represented in the control (6%), during IVCO (0%) and after VG (18%). In Fig. 6 an example of irregular dynamics after SP is shown. Bursts rising from aperiodic activity in correspondence to ventilatory peaks can be observed (Fig. 6a). The PSD reveals a broad band; nonetheless, evident peaks do emerge (Fig. 6b). When aperiodic dynamics is found, a Poincare´ map is constructed to search for a deterministic structure. The Poincare´ map can be obtained by sampling the beat-to-beat series of the sympathetic discharge in correspondence to the peaks of the ventilatory signal (i.e. at fixed phases) and by plotting the sympathetic value at the next ventilatory cycle versus that at the preceding one. In some case (Fig. 6c) the points do not spread randomly or form clusters but a deterministic structure seems to appear. The only ap-

These data indicate the presence of two main rhythmicities in the sympathetic discharge: the first one around the ventilatory frequency (HF oscillation) and the second one at a lower frequency (LF oscillation). This is in accordance with the findings of several authors (Adrian et al. 1932; Fernandez de Molina and Perl 1965; Preiss and Polosa 1974; Montano et al. 1992), who have found in the sympathetic discharge rhythms related to the spontaneous activity of respiratory centres (HF oscillations) and slow rhythms likely to reflect the spontaneous activity of vasomotor centres (LF oscillations). In addition these data indicate non-linear phenomena among ventilation and both the rhythms. A general scheme of the supposed interactions is indicated in Fig. 7a. Ventilation is able to affect and sometimes synchronise the LF and HF oscillators, but also some interferences between the two spontaneous oscillators has to be taken into account. The sympathetic discharge mirrors the activity and the interactions among these oscillators. Often the LF oscillation appears to be centred around an integer fraction of the ventilatory frequency; however, the LF rhythm cannot simply be considered a subharmonic of the ventilation but rather a spontaneous rhythm locked to it. This could be supported by the cases of quasiperiodic dynamics where either the LF or the HF rhythms are not synchronised by ventilation. Further evidence for this hypothesis is provided by the presence of LF rhythms at various ratios of the ventilatory frequency, the change in coupling patterns with manoeuvres known to enhance or depress LF rhythms and, finally, the persistence of LFunlocked modulation during AC. This evidence cannot be considered sufficient to demonstrate the existence of LF and HF oscillations independent from ventilation, but all these findings are in accordance with this hypothesis. Petrillo and Glass (1984), after observing phrenic discharge, suggested an integrate-and-fire model in which the threshold starting a spontaneous breath was modulated by ventilation. As a consequence they were able to explain the entrainment of HF oscillation to ventilation. The more complex patterns found in the sympathetic discharge can possibly be described by a similar model in which an endogenous LF rhythm is supposed to induce some kind of facilitation in the fast response to ventilation. This facilitation can occur either through summation of the two rhythms (LF and HF) or by threshold modulation. Such a model is compatible with type (I) cases of phase locking in which a large locked HF oscillation comes above threshold each ventilatory cycle, as well as with type (II) patterns in which

171

Fig. 7. A possible general scheme for the non-linear interactions between spontaneous LF and HF oscillators and external ventilation (» ) and how this scheme changes when the experimental conditions are varied. The involved oscillators at various frequencies are represented by open circles. The coupling is represented by arrows while strong and weak interferences are depicted by bold arrows and dashed arrows, respectively. a The general scheme, representing all possible influences. b, c The results relevant to the two stimuli IVCO and AO, respectively. d A possible hypothesis for complex interactions among residual oscillators after spinal section. The sympathetic discharge (Sy) reflects the activity and the interactions of the underlying oscillators

threshold can be exceeded only one time every N cycles due to a large locked LF modulation. Even more interesting in this regard are the mixed type (III) patterns in which a large and locked LF oscillation permits a large and synchronised HF response only at its rising edge, followed by fading responses in the subsequent ventilatory cycles. The experimental conditions are capable of altering the coupling between the spontaneous LF and HF rhythms and the external periodical input. In the control condition ventilation synchronises not only the spontaneous HF rhythms in a 1 : 1 fashion but also a usually weak LF rhythm. However, a few cases of quasiperiodic dynamics indicate that in some cases ventilation is not able to entrain the spontaneous rhythms of the sympathetic discharge. During the sympathetic activation induced by IVCO the LF oscillation is dominant over the HF rhythm and appears to be exactly locked to a subharmonic of ventilation. In Fig. 7b this synchronising effect is represented by the arrow from V to LF. In contrast, synchronisations between HF rhythms and ventilation are reduced: in fact the sympathetic response to each ventilatory cycle does not occur at fixed phase. However, it appears that HF activity is strongly influenced by LF

activity, which permits fixed-phase sympathetic responses to the ventilatory cycle only every N ventilatory periods. In Fig. 7b this result is indicated by the bold arrow from LF to HF, which summarises synchronisations, modulating and gating effects observed in the various locked patterns. During the sympathetic inhibition induced by AC the HF oscillation is clearly observed although the LF rhythm is present as well. The LF coupling with ventilation is lost and HF-locked patterns appear as indicated by the arrow from V to HF in Fig. 7c. Nonetheless the sympathetic responses to ventilation appear frequently amplitude-modulated by the unlocked LF rhythm; thus a residual effect of the LF oscillator on the HF oscillator can be supposed and is indicated by the arrow from LF to HF in Fig. 7c. It is worth noting that the effects of the LF oscillator over the HF oscillator can be different. While during IVCO the LF oscillator is able to lock the phase of the HF rhythm, during AO it can only modulate the amplitude of the sympathetic response to ventilation. The study of both these effects can be important for the better understanding of phenomena such as the periodical respiration (i.e. the LF amplitude modulation of the respiratory pattern) which frequently appears in pathological conditions such as the heart failure (Guzzetti et al. 1995). The opposite influence (i.e. a possible effect of the HF rhythm on the LF rhythm) is less evident in our data and further studies are necessary to ascertain whether this may reflect some structural properties of neural control mechanisms. After VG the coupling between LF and HF oscillations and ventilation is still recognised even if HF-locked patterns are less present. These results not only demonstrate the existence of interference between central oscillators and the external periodical stimulus but also that sympathetic activation or inhibition may vary the degree of synchronisation between spontaneous LF and HF rhythms and ventilation. In spinal cats periodic patterns are absent. A possible interpretation of the above finding is that spinal section, by interrupting the connection of spinal sympathetic circuits with supraspinal centres, markedly limits the influence of ventilation on the sympathetic outflow (Fig. 7d). Nonetheless, some deterministic dynamics and interactions with ventilation seem still to be present as revealed by bursts of activity synchronous with ventilatory cycles both in 1 : 3 and aperiodic patterns and also by the Poincare´ maps. It is possible that, in spinal cats, spontaneous oscillations in sympathetic activity can be still present (Fernandez de Molina and Perl 1965), although weak and less stable. This could be a result of the exclusion of supraspinal centres which are no longer able to synchronise the spinal circuits and of the persistence of either oscillating spinal circuits or afferents sensing vasomotor activity. However, different kinds of complex interactions at spinal level cannot be excluded (Fig. 7d). 6 Conclusions Non-linear interference between sympathetic discharge and ventilation is classified into phase-locking behav-

172

iours (periodic dynamics), quasiperiodic patterns and aperiodic dynamics. The observed dynamics depends on the experimental conditions. The space-time separation plot permits the locked periodic patterns to be detected much more easily, while the frequency tracking locus is useful for describing both periodic and quasiperiodic dynamics. The study of interference between ventilation and spontaneous LF and HF rhythms appears to be necessary to extract useful physiological information from these and similar experimental preparations. It can also be hypothesised that non-linear mechanisms similar to the ones described here can appear in several pathophysiological conditions of clinical relevance and in different variability signals. Acknowledgements. The present work was partially supported by a grant from the Italian Ministry of University and Science Research, special project on Cardiovascular System.

References Adrian ED, Bronk DW, Phillips G (1932) Discharges in mammalian sympathetic nerves. J Physiol (Lond) 74:115—133 Aihara K, Numajiri T, Matsumoto G, Kotani M (1986) Structures of attractors in periodically forced neural oscillators. Phys Lett A 116:313—317 Fernandez de Molina A, Perl ER (1965) Sympathetic activity and the systemic circulation in the spinal cat. J Physiol (Lond) 181:82—102 Glass L, Guevara MR, Shrier A, Perez R (1983) Bifurcations and chaos in a periodically stimulated cardiac oscillator. Physica D 7:89—101 Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189—208 Guevara MR, Shrier A, Glass L (1988) Phase-locked rhythms in periodically stimulated heart cell aggregates. Am J Physiol 254:H1—H10

Guzzetti S, Cogliati G, Turiel M, Crema G, Lombardi F, Malliani A (1995) Sympathetic predominance followed by functional denervation in the progression of chronic heart failure. Eur Heart J 16:1100—1107 Hayashi C (1964) Nonlinear oscillation in physical systems. McGrawHill, New York Jalife J, Antzelevitch C (1979) Phase resetting and annihilation of pacemaker activity in cardiac tissue. Science 206:695—697 Kay SM, Marple SL (1981) Spectrum analysis: a modern perspective. Proc IEEE 69:1380—1418 Kitney RI, Bignall S (1992) Techniques for studying short-term changes in cardio-respiratory data. In: Di Rienzo M, et al. (eds) Blood pressure and heart rate variability. IOS Press, Amsterdam, pp 1—23 Montano N, Lombardi F, Gnecchi-Ruscone T, Contini M, Finocchiaro ML, Baselli G, Porta A, Cerutti S, Malliani A (1992) Spectral analysis of sympathetic discharge, R-R interval and systolic arterial pressure in decerebrate cats. J Auton Nerv Syst 40:21—32 Petrillo GA, Glass L (1984) A theory for phase locking respiration in cats to a mechanical ventilator. Am J Physiol 246:R311—R320 Petrillo GA, Glass L, Trippenbach T (1983) Phase locking of the respiratory rhythm in cats to a mechanical ventilator. Can J Physiol Pharmacol 61:599—607 Porta A, Baselli G, Montano N, Gnecchi-Ruscone T, Lombardi F, Malliani A, Cerutti S (1994a) Non-linear dynamics in the beat-tobeat variability of sympathetic activity in decerebrate cats. Methods Inform Med 33:89—93 Porta A, Baselli G, Montano N, Gnecchi-Ruscone T, Lombardi F, Malliani A, Cerutti S (1994b) Low and high frequency non-linear interactions in the sympathetic discharge: contribution of spinal centres. Proc Comp in Cardiol Congress, Bethesda, Md, Sept 25—28, pp 141—144 Porta A, Baselli G, Montano N, Gnecchi-Ruscone T, Lombardi F, Malliani A, Cerutti S (1995) Different interference patterns among spontaneous low and high frequency oscillations and ventilation in sympathetic outflow. Proc Comp in Cardiol Congress, Vienna, Austria, Sept 10—12, pp 473—476 Preiss G, Polosa C (1974) Patterns of sympathetic neuron activity associated with Mayer waves. Am J Physiol 226:724—730 Provenzale A, Smith LA, Vio R, Morante G (1992) Distinguishing between low-dimensional dynamics and randomness in measured time series. Physica D 58:31—49

.