Phase coherence-A time-localized approach to studying interactions.

Coherence measures the similarity of progression of phases between oscillations or waves. When applied to multi-scale, nonstationary dynamics with time-varying amplitudes and frequencies, high values of coherence provide a useful indication of interactions, which might otherwise go unnoticed. However, the choice of analyzing coherence based on phases and amplitudes (amplitude-weighted phase coherence) vs only phases (phase coherence) has long been seen as arbitrary. Here, we review the concept of coherence and focus on time-localized methods of analysis, considering both phase coherence and amplitude-weighted phase coherence. We discuss the importance of using time-localized analysis and illustrate the methods and their practicalities on both numerically modeled and real time-series. The results show that phase coherence is more robust than amplitude-weighted phase coherence to both noise perturbations and movement artifacts. The results also have wider implications for the analysis of real data and the interpretation of physical systems.

Stabilization of cyclic processes by slowly varying forcing.

We introduce a new mathematical framework for the qualitative analysis of dynamical stability, designed particularly for finite-time processes subject to slow-timescale external influences. In particular, our approach is to treat finite-time dynamical systems in terms of a slow-fast formalism in which the slow time only exists in a bounded interval, and consider stability in the singular limit. Applying this to one-dimensional phase dynamics, we provide stability definitions somewhat analogous to the classical infinite-time definitions associated with Aleksandr Lyapunov. With this, we mathematically formalize and generalize a phase-stabilization phenomenon previously described in the physics literature for which the classical stability definitions are inapplicable and instead our new framework is required.

The discriminatory value of cardiorespiratory interactions in distinguishing awake from anaesthetised states: a randomised observational study.

Depth of anaesthesia monitors usually analyse cerebral function with or without other physiological signals; non-invasive monitoring of the measured cardiorespiratory signals alone would offer a simple, practical alternative. We aimed to investigate whether such signals, analysed with novel, non-linear dynamic methods, would distinguish between the awake and anaesthetised states. We recorded ECG, respiration, skin temperature, pulse and skin conductivity before and during general anaesthesia in 27 subjects in good cardiovascular health, randomly allocated to receive propofol or sevoflurane. Mean values, variability and dynamic interactions were determined. Respiratory rate (p = 0.0002), skin conductivity (p = 0.03) and skin temperature (p = 0.00006) changed with sevoflurane, and skin temperature (p = 0.0005) with propofol. Pulse transit time increased by 17% with sevoflurane (p = 0.02) and 11% with propofol (p = 0.007). Sevoflurane reduced the wavelet energy of heart (p = 0.0004) and respiratory (p = 0.02) rate variability at all frequencies, whereas propofol decreased only the heart rate variability below 0.021 Hz (p < 0.05). The phase coherence was reduced by both agents at frequencies below 0.145 Hz (p < 0.05), whereas the cardiorespiratory synchronisation time was increased (p < 0.05). A classification analysis based on an optimal set of discriminatory parameters distinguished with 95% success between the awake and anaesthetised states. We suggest that these results can contribute to the design of new monitors of anaesthetic depth based on cardiovascular signals alone.

Stationary and traveling wave states of the Kuramoto model with an arbitrary distribution of frequencies and coupling strengths.

We consider the Kuramoto model of an ensemble of interacting oscillators allowing for an arbitrary distribution of frequencies and coupling strengths. We define a family of traveling wave states as stationary in a rotating frame, and derive general equations for their parameters. We suggest empirical stability conditions which, for the case of incoherence, become exact. In addition to making new theoretical predictions, we show that many earlier results follow naturally from our general framework. The results are applicable in scientific contexts ranging from physics to biology.

Basal sympathetic activity to the microcirculation in tetraplegic man revealed by wavelet transform of laser Doppler flowmetry.

BACKGROUND: The 1984/86 published neurogram results showing only rare sympathetic nerve activity (SNA) to the muscles and skin in tetraplegia are still accepted. The present study by a different method attempted to confirm or deny those findings. METHODS AND RESULTS: The effect of basal SNA to the microcirculation of the feet and calf in 10 complete (AIS A) traumatic tetraplegic and 10 healthy age matched subjects were evaluated by wavelet transform of laser Doppler flowmetry (LDF) recordings. The results clearly indicated there is significant basal SNA from the decentralized spinal cord in tetraplegia. In addition, wavelet analysis allowed a study of other influences on the microcirculation besides SNA. Collectively, in tetraplegia compared with controls, the powers of the low frequency oscillations in blood flow were reduced; in that the endothelium caused less vasodilatation while the SNA and intrinsic vascular smooth muscles induced smaller degrees of vasoconstriction. However, the high frequency and especially the cardiac powers were greater. The latter presenting an obvious important factor for the preservation of blood flow in the microcirculation. CONCLUSIONS: It is suggested that basal SNA to the cutaneous microcirculation occurs in complete tetraplegia, and the significant levels of circulating noradrenaline reported by others indicate this is also true in other parts of the body. This may explain the usual absence of severe, incapacitating, autonomic deficiency in this condition.

Nonlinear dynamics of cardiovascular ageing.

The application of methods drawn from nonlinear and stochastic dynamics to the analysis of cardiovascular time series is reviewed, with particular reference to the identification of changes associated with ageing. The natural variability of the heart rate (HRV) is considered in detail, including the respiratory sinus arrhythmia (RSA) corresponding to modulation of the instantaneous cardiac frequency by the rhythm of respiration. HRV has been intensively studied using traditional spectral analyses, e.g. by Fourier transform or autoregressive methods, and, because of its complexity, has been used as a paradigm for testing several proposed new methods of complexity analysis. These methods are reviewed. The application of time-frequency methods to HRV is considered, including in particular the wavelet transform which can resolve the time-dependent spectral content of HRV. Attention is focused on the cardio-respiratory interaction by introduction of the respiratory frequency variability signal (RFV), which can be acquired simultaneously with HRV by use of a respiratory effort transducer. Current methods for the analysis of interacting oscillators are reviewed and applied to cardio-respiratory data, including those for the quantification of synchronization and direction of coupling. These reveal the effect of ageing on the cardio-respiratory interaction through changes in the mutual modulation of the instantaneous cardiac and respiratory frequencies. Analyses of blood flow signals recorded with laser Doppler flowmetry are reviewed and related to the current understanding of how endothelial-dependent oscillations evolve with age: the inner lining of the vessels (the endothelium) is shown to be of crucial importance to the emerging picture. It is concluded that analyses of the complex and nonlinear dynamics of the cardiovascular system can illuminate the mechanisms of blood circulation, and that the heart, the lungs and the vascular system function as a single entity in dynamical terms. Clear evidence is found for dynamical ageing.

Physics of brain dynamics: Fokker-Planck analysis reveals changes in EEG delta and theta activity during anaesthesia.

We use drift and diffusion coefficients to reveal interactions between different oscillatory processes underlying a complex signal and apply the method to EEG delta and theta frequencies in the brain. By analysis of data recorded from rats during anaesthesia we consider the stability and basins of attraction of fixed points in the phase portrait of the deterministic part of the retrieved stochastic process. We show that different classes of dynamics are associated with deep and light anaesthesia, and we demonstrate that the predominant directionality of the interaction is such that theta drives delta.

Intrinsic dynamics of heart regulatory systems on short time-scales: from experiment to modelling.

We discuss open problems related to the stochastic modeling of cardiac function. The work is based on an experimental investigation of the dynamics of heart rate variability (HRV) in the absence of respiratory perturbations. We consider first the cardiac control system on short time scales via an analysis of HRV within the framework of a random walk approach. Our experiments show that HRV on timescales of less than a minute takes the form of free diffusion, close to Brownian motion, which can be described as a non-stationary process with stationary increments. Secondly, we consider the inverse problem of modeling the state of the control system so as to reproduce the experimentally observed HRV statistics of. We discuss some simple toy models and identify open problems for the modelling of heart dynamics.

Low-frequency blood flow oscillations in congestive heart failure and after beta1-blockade treatment.

Laser Doppler flowmetry (LDF) of forearm skin blood flow, combined with iontophoretically-administered acetylcholine and sodium nitroprusside and wavelet spectral analysis, was used for noninvasive evaluation of endothelial function in 17 patients newly diagnosed with New York Heart Association class II-III congestive heart failure (CHF). After 20+/-10 weeks' treatment with a beta(1)-blocker (Bisoprolol), the measurements were repeated. Measurements were also made on an age- and sex-matched group of healthy controls (HC). In each case data were recorded for 30 min. In HC, the difference in absolute spectral amplitude of LDF oscillations between the two vasodilators manifests in the frequency interval 0.005-0.0095 Hz (p<0.01); this difference is initially absent in patients with CHF, but appears following the beta(1)-blocker treatment (p<0.01). For HC, the difference between the two vasodilators also manifests in normalised spectral amplitude in 0.0095-0.021 Hz (p<0.05). This latter difference is absent in CHF patients and is unchanged by treatment with beta(1)-blockers. It is concluded that there are two oscillatory skin blood flow components associated with endothelial function. Both are reduced in CHF. Activity in the lower frequency interval is restored by beta(1)-blocker treatment, confirming the association between CHF and endothelial dysfunction but suggesting the involvement of two distinct mechanisms.

Nonlinear statistical modeling and model discovery for cardiorespiratory data.

We present a Bayesian dynamical inference method for characterizing cardiorespiratory (CR) dynamics in humans by inverse modeling from blood pressure time-series data. The technique is applicable to a broad range of stochastic dynamical models and can be implemented without severe computational demands. A simple nonlinear dynamical model is found that describes a measured blood pressure time series in the primary frequency band of the CR dynamics. The accuracy of the method is investigated using model-generated data with parameters close to the parameters inferred in the experiment. The connection of the inferred model to a well-known beat-to-beat model of the baroreflex is discussed.

Maximum amplitude of limit cycles in Liénard systems.

We establish sufficient criteria for the existence of a limit cycle in the Liénard system x[over ̇]=y-ɛF(x),y[over ̇]=-x, where F(x) is odd. In their simplest form the criteria lead to the result that, for all finite nonzero ɛ, the amplitude of the limit cycle is less than ρ and 0≤a≤ρ≤u, where F(a)=0 and ∫(0)(u)F(x)dx=0. We take the van der Pol oscillator as a specific example and establish that for all finite, nonzero ɛ, the amplitude of its limit cycle is less than 2.0672, a value whose precision is limited by the capacity of our symbolic computation software package. We show how the criterion for the upper bound can be extended to establish a bound on the amplitude of a limit cycle in systems where F(x) contains both odd and even components. We also show how the criteria can be used to establish bounds for bifurcation sets.

The contribution of lumbar sympathetic neurones activity to rat's skin blood flow oscillations.

Skin blood flow on the rat's paws using laser Doppler flowmeter, electrical activity of the heart (ECG) and respiration were measured simultaneously. The signals were recorded for 20 minutes, both before and after denervation, at core temperature 37°C and 38.5°C, that was maintained constant during the recordings. Spinal nerve fibres, at the level L3-L4, were transected. Experiments were performed on 15 adult Wistar rats under general anaesthesia. The oscillations in the measured signals were analysed in the time-frequency domain using wavelet transform. On the frequency region from 0.7Hz to 5Hz two characteristic peaks were observed in the skin blood flow spectrum. They correspond to the main peaks in the spectra of the ECG (around 3.3Hz) and respiration (around 1.3Hz). Several additional peaks were observed in the low frequency region, from 0.01 to 0.7 Hz, in all measured signals. In this frequency region the relative energy contribution of the blood flow oscillations decreased after denervation only in the denervated left hind paw. This difference was not statistically significant at 37°C (p=0.098, Kruskal-Wallis test) but became statistically significant at 38.5°C (p=0.017). Relative energy contribution of the low frequency region, from 0.01 to 0.7Hz, decreased 2.5-fold in the blood flow of the denervated paw. Within this region the relative energy contribution decreased significantly in two intervals, from 0.01 to 0.08Hz and from 0.08 to 0.2Hz (p=0.023). In the higher frequency region, from 0.7 to 5Hz, o statistically significant differences were obtained in any paws when compared before and after denervation at the same core temperature. We conclude that the activity of lumbar sympathetic neurones contributes to low frequency skin blood flow oscillations.

Continuous low amplitude direct current stimulation of the crushed peripheral nerve accelerates the early recovery of choline acetyltransferase but not of acetylcholinesterase activity in fast and slow muscles.

We investigated if continuous 1 µA direct current stimulation of the injured nerve, with the cathode electrode at the distal end of the nerve crush injury (cathode stimulation), accelerated the recovery of choline acetyltransferase (ChAT) and acetylcholinesterase (AChE) activity in transiently denervated extensor digitorum longus (EDL) and soleus (SOL) rat muscles. ChAT is a specific marker of cholinergic nerve terminals and may reflect axon ingrowth, and AChE reflects the re-establishment of neuromuscular junctions and recovery of muscle activity. Compared to sham operated animals, the cathode (CA) stimulated rats had a statistically significant larger ChAT activity in the EDL and SOL muscles on days 12 and 14 after nerve crush (P < 0.01, n = 6). The difference in ChAT activity between the groups decreased thereafter. Regarding recovery of muscle AChE, CA stimulation of the crushed sciatic nerve did not detectably accelerate the normalization of activity and pattern of AChE molecular forms in the EDL and SOL muscles. This means that the early rise in ChAT muscle activity in CA stimulated rats was not followed by an accelerated normalization of the neuromuscular transmission in the same group. It is more likely that the higher ChAT activity observed after cathode stimulation indicates a higher ChAT content in regenerating motor nerve endings, rather than a greater number of motor axons entering the muscles. It seems possible that cathode stimulation increased ChAT axonal transport, causing the early increase of ChAT content in the nerve endings. This raises the possibility that the axon transport and subsequent secretion of a trophic factor(s) from the nerve to the reinnervated muscle are enhanced as well, thus shortening the overall time of muscle force recovery in the absence of an appreciable acceleration of recovery of the neuromuscular transmission.

Muscle force recovery after continuous direct current stimulation of a crushed nerve.

The purpose of this work was to examine whether, after sciatic axonotmesis, continuous low-amplitude direct current stimulation across the nerve crush lesion could affect the overall regeneration rate and shorten the time necessary to restore muscle force. Rats were randomly divided into cathode-stimulated (7 animals with a cathode stimulating electrode at the distal end of the nerve crush), anode-stimulated (6 animals with an anode stimulating electrode at the distal end of the nerve crush) and sham-treated (6 animals) groups. The recovery of muscle force was assessed by measuring the isometric tetanic contraction of the plantar flexor muscles once weekly, for five weeks in all groups. There was no statistically significant difference (P > 0.05) among the three groups from the first to the third week after the nerve crush. The cathode-stimulated animals had a statistically significantly enhanced muscle force recovery in the fourth (P = 0.023) and fifth week (P = 0.003) after the nerve crush, when compared to the anode-stimulated and sham-treated groups. From the first to the fifth week after the nerve crush there was no statistically significant difference (P > 0.05) in muscle force between the anode-stimulated and sham groups. After axonotmesis, the average ratio of normalized muscle force to normalized body weight in the cathode-stimulated group reached the pre-crush control value in the fourth week following the nerve injury. This ratio was significantly lower all five weeks compared to the initial one before axonotmesis in anode-stimulated and sham-treated groups. Placing the cathode-stimulating electrode at the distal end of the nerve crush seems to have shortened the overall time of muscle force recovery. A possible mechanism for the enhancement of muscle force recovery in the cathode-stimulated group is proposed.

Wavelet analysis of oscillations in the peripheral blood circulation measured by laser Doppler technique.

The wavelet transform technique, a time-frequency method with logarithmic frequency resolution, was used to analyze oscillations in human peripheral blood flow measured by laser Doppler flowmetry. The oscillations extended over a wide frequency scale and their periods varied in time. Within the frequency range studied, 0.0095-1.6 Hz, five characteristic oscillations were revealed, arising from both local and central regulatory mechanisms. After the insertion of endothelium-dependent and endothelium-independent vasodilators the spectra of blood flow markedly differed in the frequency interval 0.0095-0.02 Hz. In this way it was demonstrated that endothelial activity is a rhythmic process that contributes to oscillations in blood flow with a characteristic frequency of around 0.01 Hz. The study illustrates the potential of laser Doppler flowmetry combined with dynamical systems analysis for studies of both the micro- and macroscopic mechanisms of blood flow regulation in vivo.

FES and spasticity.

A model of hemiplegic spasticity based on electromyographical and biomechanical parameters measured during passive muscle stretching is presented. Two components of spasticity can be distinguished--phasic and tonic. This classification depends on the pattern of stretch reflex activity which can be either phasic or tonic as well as on the muscle stretch/tension characteristic. Stretch reflex, as a control loop, is in phasic spasticity characterized by increased sensitivity to velocity of stretching. In tonic spasticity, sensitivity to length of stretching is increased. After the injury, phasic spasticity appears first and invokes monosynaptic reflex pathways. The intensity of tonic spasticity increases with the duration of disability and hence causes changes in muscle fiber biomechanical properties. The model mentioned above has been used to evaluate the effects of FES on spasticity. Hemiplegic patients with implanted peroneal nerve stimulator for gait correction were followed up for one year starting a week before implantation. Long-term use of FES resulted in decrease of tonic spasticity in both ankle joint antagonistic muscle groups. In stimulated tibialis anterior muscle, the phasic type of spasticity increased. To obtain the correlation between changes in spasticity and functional abilities of patients, the maximal voluntary isometric contraction of both muscle groups was also measured. An improvement in voluntary strength was also observed. This can be taken as additional evidence that tonic spasticity is of greater physiological and clinical significance than phasic spasticity. It may be concluded that use of FES can decrease tonic spasticity and, if applied early after the injury, can prevent the appearance of tonic spasticity.

Modelling couplings among the oscillators of the cardiovascular system.

A mathematical model of the cardiovascular system is simulated numerically. The basic unit in the model is an oscillator that possesses a structural stability and robustness motivated by physiological understanding and by the analysis of measured time series. Oscillators with linear couplings are found to reproduce the main characteristic features of the experimentally obtained spectra. To explain the variability of cardiac and respiratory frequencies, however, it is essential to take into account the rest of the system, i.e. to consider the effect of noise. It is found that the addition of noise also results in epochs of synchronization, as observed experimentally. Preliminary analysis suggests that there is a mixture of linear and parametric couplings, but that the linear coupling seems to dominate.

Spectral components of heart rate variability determined by wavelet analysis.

Spectral components of heart rate variability (HRV) are determined in the time-frequency domain using a wavelet transform. Based on the finer estimation of low-frequency content enabled by the logarithmic resolution of the wavelet transform, corrections of spectral intervals, already defined by Fourier and model based methods, are proposed. The characteristic peaks between 0.0095 and 0.6 Hz are traced in time and four spectral intervals are defined, I (0.0095-0.021 Hz), II (0.021-0.052 Hz), III (0.052-0.145 Hz) and IV (0.145-0.6 Hz), within which peaks are located for all subjects included. These intervals are shown to be invariant regardless of the age and the state of the system. We also show that the frequency and power of the spectral components are related to age, AMI and particularly to type II diabetes mellitus.

The cardiovascular system as coupled oscillators?

Based on physiological knowledge, and on an analysis of signals related to its dynamics, we propose a model of the cardiovascular system. It consists of coupled oscillators. Each of them describes one of the subsystems involved in the regulation of one passage of blood through the circulatory system. The flow of blood through the system of closed tubes-the blood vessels-is described by wave equations.