Valentina Ticcinelli

Coupling functions in networks of oscillators

Networks of interacting oscillators abound in nature, and one of the prevailing challenges in science is how to characterize and reconstruct them from measured data. We present a method of reconstruction based on dynamical Bayesian inference that is capable of detecting the effective phase connectivity within networks of time-evolving coupled phase oscillators subject to noise. It not only reconstructs pairwise, but also encompasses couplings of higher degree, including triplets and quadruplets of interacting oscillators. Thus inference of a multivariate network enables one to reconstruct the coupling functions that specify possible causal interactions, together with the functional mechanisms that underlie them. The characteristic features of the method are illustrated by the analysis of a numerically generated example: a network of noisy phase oscillators with time-dependent coupling parameters. To demonstrate its potential, the method is also applied to neuronal coupling functions from single- and multi-channel electroencephalograph recordings. The cross-frequency δ, α to α coupling function, and the θ, α, γ to γ triplet are computed, and their coupling strengths, forms of coupling function, and predominant coupling components, are analysed. The results demonstrate the applicability of the method to multivariate networks of oscillators, quite generally.

Neural Cross-Frequency Coupling Functions

Although neural interactions are usually characterized only by their coupling strength and directionality, there is often a need to go beyond this by establishing the functional mechanisms of the interaction. We introduce the use of dynamical Bayesian inference for estimation of the coupling functions of neural oscillations in the presence of noise. By grouping the partial functional contributions, the coupling is decomposed into its functional components and its most important characteristics—strength and form—are quantified. The method is applied to characterize the δ-to-α phase-to-phase neural coupling functions from electroencephalographic (EEG) data of the human resting state, and the differences that arise when the eyes are either open (EO) or closed (EC) are evaluated. The δ-to-α phase-to-phase coupling functions were reconstructed, quantified, compared, and followed as they evolved in time. Using phase-shuffled surrogates to test for significance, we show how the strength of the direct coupling, and the similarity and variability of the coupling functions, characterize the EO and EC states for different regions of the brain. We confirm an earlier observation that the direct coupling is stronger during EC, and we show for the first time that the coupling function is significantly less variable. Given the current understanding of the effects of e.g., aging and dementia on δ-waves, as well as the effect of cognitive and emotional tasks on α-waves, one may expect that new insights into the neural mechanisms underlying certain diseases will be obtained from studies of coupling functions. In principle, any pair of coupled oscillations could be studied in the same way as those shown here.

Preliminary study of laser doppler perfusion signal by wavelet transform in patients with critical limb ischemia before and after revascularization

The haemodynamics of skin microcirculation can be quantitatively evaluated by Laser Doppler Fluxmetry (LDF). LDF signal in human skin shows periodic oscillations. Spectral analysis by wavelet transform displays six characteristic frequency intervals (FI) from 0.005 to 2 Hz, related to distinct vascular structures activities: heart (0.6-2 Hz), sympathetic respiratory (0.145-0.6 Hz), myogenic (0.052-0.145 Hz), local sympathetic nerve (0.021-0.052 Hz) and endothelial cells NO dependent (0.0095-0.021 Hz) and NO independent (0.005-0.0095 Hz). The most advanced stage of peripheral arterial obstructive disease is the critical limb ischemia (CLI), which causes the reduction of blood perfusion threatening limb viability. Besides macrocirculatory alterations, many studies have shown microvascular misdistribution of skin blood flow as the main factor that leads patients to CLI. Revascularization can save limb and patients life, too. In the present study, LDF signals have been recorded on the skin of the foot dorsum in 15 patients suffering from CLI. LDF signals have been analyzed before and after limb revascularization by means of the wavelet analysis. Significant changes in frequency distribution before and after limb revascularization have been detected: the median normalized values of spectral power increases for 49.8% (p = 0.0341) in the frequency range 0.050328-0.053707 Hz, whereas spectral power decreases for 77.1% (p = 0.0179) in the frequency range 0.018988-0.029284 Hz. We can conclude that changes in the frequency intervals occur after revascularization, shifting from a prevailing endothelial activity toward a prevailing sympathetic activity.

Changes of the cutaneous flowmotion pattern after limb revascularization in patients with critical ischemia

The skin flowmotion of 13 patients suffering from critical limb ischemia (CLI) was studied with wavelet analysis (WA) of the laser Doppler signals (LDS). The WA selects six different frequency components (FCs), each relating to a specific cardiovascular system structures activities; FC I 1-2 Hz heart, FC II 0.2 Hz respiratory, FC III 0.1 Hz myogenic, FC IV 0.04 Hz, sympathetic, FC V 0.01 Hz, and FC VI 0.007 Hz endothelial. The aim of the study was to observe which FC changed after the limb revascularization. The LDS was measured at the dorsum of the foot, one week before and no later than 30 days after revascularisation. The absolute and relative amplitude and energy of the flowmotion WA FCs, the ankle brachial pressure index (ABI) and the transcutaneous pressure of oxygen (TcpO2) were assessed before and after revascularization. The results showed that after successful revascularization ABI and TcpO2 increased from 0.34 ± 0.10 to 0.54 ± 0.09 (p 0.0003) and from 20.3 ± 13.4 to 43.8 ± 18.7 mmHg (p 0.0002) whereas only the absolute amplitude and energy of the cardiac FC I increased from 0.57 ± 0.44 to 1.07 ± 0.69 (P 0.002) AU and 1.14 ± 1.78 AU2 to 3.54 ± 3.78 AU2 (p 0.004). In conclusion after limb revascularization the cardiac component of the flowmotion increased maybe because the cardiac stroke volume had more influence over the skin arterioles.

Coherence and Coupling Functions Reveal Microvascular Impairment in Treated Hypertension

The complex interactions that give rise to heart rate variability (HRV) involve coupled physiological oscillators operating over a wide range of different frequencies and length-scales. Based on the premise that interactions are key to the functioning of complex systems, the time-dependent deterministic coupling parameters underlying cardiac, respiratory and vascular regulation have been investigated at both the central and microvascular levels. Hypertension was considered as an example of a globally altered state of the complex dynamics of the cardiovascular system. Its effects were established through analysis of simultaneous recordings of the electrocardiogram (ECG), respiratory effort, and microvascular blood flow [by laser Doppler flowmetry (LDF)]. The signals were analyzed by methods developed to capture time-dependent dynamics, including the wavelet transform, wavelet-based phase coherence, non-linear mode decomposition, and dynamical Bayesian inference, all of which can encompass the inherent frequency and coupling variability of living systems. Phases of oscillatory modes corresponding to the cardiac (around 1.0 Hz), respiratory (around 0.25 Hz), and vascular myogenic activities (around 0.1 Hz) were extracted and combined into two coupled networks describing the central and peripheral systems, respectively. The corresponding spectral powers and coupling functions were computed. The same measurements and analyses were performed for three groups of subjects: healthy young (Y group, 24.4 ± 3.4 y), healthy aged (A group, 71.1 ± 6.6 y), and aged treated hypertensive patients (ATH group, 70.3 ± 6.7 y). It was established that the degree of coherence between low-frequency oscillations near 0.1 Hz in blood flow and in HRV time series differs markedly between the groups, declining with age and nearly disappearing in treated hypertension. Comparing the two healthy groups it was found that the couplings to the cardiac rhythm from both respiration and vascular myogenic activity decrease significantly in aging. Comparing the data from A and ATH groups it was found that the coupling from the vascular myogenic activity is significantly weaker in treated hypertension subjects, implying that the mechanisms of microcirculation are not completely restored by current anti-hypertension medications.

Ageing of the couplings between cardiac, respiratory and myogenic activity in humans

The balance and functionality of the cardiovascular system are maintained by a network of couplings between the different oscillations involved. We study the effect of ageing on these interactions through the application of wavelet analysis, and by the use of dynamical Bayesian inference to compute coupling functions. The method, applied to phases extracted from microvascular flow recorded by laser Doppler flowmetry (LDF), reveals the coupling functions between oscillations propagated to the smallest vessels. Consistent with earlier work based on analysis of cardiac and respiratory phases obtained from direct measurements, our analysis demonstrates an impairment of the propagated cardio-respiratory coupling with ageing. The coupling weakens despite the increased cardiac component in the LDF with ageing. Our results bring new insight to the effect of ageing on cardiovascular regulation that might help improve the diagnostic potential of LDF monitors.