Effective phase dynamics of noise-induced oscillations in excitable systems

Increased habitat connectivity induces diversity via noise-induced symmetry breaking

Stochasticity or noise is omnipresent in ecosystems that mediates community dynamics. The beneficial role of stochasticity in enhancing species coexistence and, hence, in promoting biodiversity is well recognized. However, incorporating stochastic birth and death processes in excitable slow-fast ecological systems to study its response to biodiversity is largely unexplored. Considering an ecological network of excitable consumer-resource systems, we study the interplay of network structure and noise on species' collective dynamics. We find that noise drives the system out of the excitable regime, and high habitat patch connectance in the ordered as well as random networks promotes species' diversity by inducing new steady states via noise-induced symmetry breaking.

Interspike interval correlations in neuron models with adaptation and correlated noise

The generation of neural action potentials (spikes) is random but nevertheless may result in a rich statistical structure of the spike sequence. In particular, contrary to the popular renewal assumption of theoreticians, the intervals between adjacent spikes are often correlated. Experimentally, different patterns of interspike-interval correlations have been observed and computational studies have identified spike-frequency adaptation and correlated noise as the two main mechanisms that can lead to such correlations. Analytical studies have focused on the single cases of either correlated (colored) noise or adaptation currents in combination with uncorrelated (white) noise. For low-pass filtered noise or adaptation, the serial correlation coefficient can be approximated as a single geometric sequence of the lag between the intervals, providing an explanation for some of the experimentally observed patterns. Here we address the problem of interval correlations for a widely used class of models, multidimensional integrate-and-fire neurons subject to a combination of colored and white noise sources and a spike-triggered adaptation current. Assuming weak noise, we derive a simple formula for the serial correlation coefficient, a sum of two geometric sequences, which accounts for a large class of correlation patterns. The theory is confirmed by means of numerical simulations in a number of special cases including the leaky, quadratic, and generalized integrate-and-fire models with colored noise and spike-frequency adaptation. Furthermore we study the case in which the adaptation current and the colored noise share the same time scale, corresponding to a slow stochastic population of adaptation channels; we demonstrate that our theory can account for a nonmonotonic dependence of the correlation coefficient on the channel's time scale. Another application of the theory is a neuron driven by network-noise-like fluctuations (green noise). We also discuss the range of validity of our weak-noise theory and show that by changing the relative strength of white and colored noise sources, we can change the sign of the correlation coefficient. Finally, we apply our theory to a conductance-based model which demonstrates its broad applicability.

Resolving molecular contributions of ion channel noise to interspike interval variability through stochastic shielding

Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM J Appl Math 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in Phys Rev Lett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."

Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences

Dynamical systems are widespread, with examples in physics, chemistry, biology, population dynamics, communications, climatology and social science. They are rarely isolated but generally interact with each other. These interactions can be characterized by coupling functions-which contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how each interaction occurs. Coupling functions can be used, not only to understand, but also to control and predict the outcome of the interactions. This theme issue assembles ground-breaking work on coupling functions by leading scientists. After overviewing the field and describing recent advances in the theory, it discusses novel methods for the detection and reconstruction of coupling functions from measured data. It then presents applications in chemistry, neuroscience, cardio-respiratory physiology, climate, electrical engineering and social science. Taken together, the collection summarizes earlier work on coupling functions, reviews recent developments, presents the state of the art, and looks forward to guide the future evolution of the field. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes

Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime. We illustrate the utility of the PDMP on a non-trivial dynamical system by analysing the properties of a titration-based oscillator in the non-adiabatic limit. We first show how to transform the underlying chemical master equation into a PDMP where the slow transitions between promoter states are stochastic, but whose rates depend upon the faster deterministic dynamics of the transcription factors regulated by these promoters. We show that the PDMP accurately describes the observed periods of stochastic cycles in activator and repressor-based titration oscillators. We then generalize our PDMP analysis to more complicated versions of titration-based oscillators to explain how multiple binding sites lengthen the period and improve coherence. Last, we show how noise-induced oscillation previously observed in a titration-based oscillator arises from non-adiabatic and discrete binding events at the promoter site.

Alteration in Brain Functional and Effective Connectivity in Subjects With Hypertension

To reveal the physiological mechanism of the cognitive decline in subjects with hypertension, the functional connectivity (FC) was assessed by using the wavelet phase coherence (WPCO), and effective connectivity (EC) was assessed by using the coupling strength (CS) of near-infrared spectroscopy (NIRS) signals. NIRS signals were continuously recorded from the prefrontal cortex, sensorimotor cortex, and occipital lobes of 13 hypertensive patients (hypertension group, 70 ± 6.5 years old) and 16 elderly healthy subjects (control group, 71 ± 5.5 years old) in resting and standing periods. WPCO and CS were calculated in four frequency intervals: I, 0.6–2; II, 0.145–0.6; III, 0.052–0.145; and IV, 0.021–0.052 Hz. CS quantifies coupling amplitude. In comparison with the control group, the hypertension group showed significantly decreased (p < 0.05) WPCO and CS in intervals III and IV and in the resting and standing states. WPCO and CS were significantly decreased in the resting state compared with those in the standing state in the hypertension group (p < 0.05). Decreased WPCO and CS indicated a reduced network interaction, suggesting disturbed neurovascular coupling in subjects with hypertension. Compared with the control group, the hypertension group showed significantly lower Mini-Mental State Examination (MMSE) (p = 0.028) and Montreal Cognitive Assessment (MoCA) scores (p = 0.011). In the hypertension group, correlation analysis showed that WPCO and CS were significantly positively correlated with MMSE and MoCA scores, respectively. These findings may provide evidence of impaired cognitive function in hypertension and can enhance the understanding on neurovascular coupling.

Alterations in the coupling functions between cerebral oxyhaemoglobin and arterial blood pressure signals in post-stroke subjects

Cerebral autoregulation (CA) is the complex homeostatic regulatory relationship between blood pressure (BP) and cerebral blood flow (CBF). This study aimed to analyze the frequency-specific coupling function between cerebral oxyhemoglobin concentrations (delta [HbO₂]) and mean arterial pressure (MAP) signals based on a model of coupled phase oscillators and dynamical Bayesian inference. Delta [HbO₂] was measured by 24-channel near-infrared spectroscopy (NIRS) and arterial BP signals were obtained by simultaneous resting-state measurements in patients with stroke, that is, 9 with left hemiparesis (L–H group), 8 with right hemiparesis (R–H group), and 17 age-matched healthy individuals as control (healthy group). The coupling functions from MAP to delta [HbO₂] oscillators were identified and analyzed in four frequency intervals (I, 0.6–2 Hz; II, 0.145–0.6 Hz; III, 0.052–0.145 Hz; and IV, 0.021–0.052 Hz). In L–H group, the CS from MAP to delta [HbO₂] in interval III in channel 8 was significantly higher than that in healthy group (p = 0.003). Compared with the healthy controls, the coupling in MAP→delta [HbO₂] showed higher amplitude in interval I and IV in patients with stroke. The increased CS and coupling amplitude may be an evidence of impairment in CA, thereby confirming the presence of impaired CA in patients with stroke. In interval III, the CS in L–H group from MAP to delta [HbO₂] in channel 16 (p = 0.001) was significantly lower than that in healthy controls, which might indicate the compensatory mechanism in CA of the unaffected side in patients with stroke. No significant difference in region-wise CS between affected and unaffected sides was observed in stroke groups, indicating an evidence of globally impaired CA. These findings provide a method for the assessment of CA and will contribute to the development of therapeutic interventions in stroke patients.

Time-varying coupling functions: Dynamical inference and cause of synchronization transitions

Interactions in nature can be described by their coupling strength, direction of coupling, and coupling function. The coupling strength and directionality are relatively well understood and studied, at least for two interacting systems; however, there can be a complexity in the interactions uniquely dependent on the coupling functions. Such a special case is studied here: synchronization transition occurs only due to the time variability of the coupling functions, while the net coupling strength is constant throughout the observation time. To motivate the investigation, an example is used to present an analysis of cross-frequency coupling functions between delta and alpha brain waves extracted from the electroencephalography recording of a healthy human subject in a free-running resting state. The results indicate that time-varying coupling functions are a reality for biological interactions. A model of phase oscillators is used to demonstrate and detect the synchronization transition caused by the varying coupling functions during an invariant coupling strength. The ability to detect this phenomenon is discussed with the method of dynamical Bayesian inference, which was able to infer the time-varying coupling functions. The form of the coupling function acts as an additional dimension for the interactions, and it should be taken into account when detecting biological or other interactions from data.

Asymptotic phase for stochastic oscillators

Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arise from a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of the asymptotic phase function. In the presence of noise, the asymptotic phase is no longer well defined. We introduce a new definition of asymptotic phase in terms of the slowest decaying modes of the Kolmogorov backward operator. Our stochastic asymptotic phase is well defined for noisy oscillators, even when the oscillations are noise dependent. It reduces to the classical asymptotic phase in the limit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or by empirical observation of an oscillating density's approach to its steady state.

Quantification of causal couplings via dynamical effects: a unifying perspective

Quantitative characterization of causal couplings from time series is crucial in studies of complex systems of different origin. Various statistical tools for that exist and new ones are still being developed with a tendency to creating a single, universal, model-free quantifier of coupling strength. However, a clear and generally applicable way of interpreting such universal characteristics is lacking. This work suggests a general conceptual framework for causal coupling quantification, which is based on state space models and extends the concepts of virtual interventions and dynamical causal effects. Namely, two basic kinds of interventions (state space and parametric) and effects (orbital or transient and stationary or limit) are introduced, giving four families of coupling characteristics. The framework provides a unifying view of apparently different well-established measures and allows us to introduce new characteristics, always with a definite "intervention-effect" interpretation. It is shown that diverse characteristics cannot be reduced to any single coupling strength quantifier and their interpretation is inevitably model based. The proposed set of dynamical causal effect measures quantifies different aspects of "how the coupling manifests itself in the dynamics," reformulating the very question about the "causal coupling strength."

Persistence and failure of mean-field approximations adapted to a class of systems of delay-coupled excitable units

We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays. The formulation of the two approximations, referred to as the Gaussian and the quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system. It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system. As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms. We also discuss how the fulfillment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfillment of the quasi-independence approximation and certain forms of synchronization between the individual units.

Phase coherence and attractor geometry of chaotic electrochemical oscillators

Chaotic attractors are known to often exhibit not only complex dynamics but also a complex geometry in phase space. In this work, we provide a detailed characterization of chaotic electrochemical oscillations obtained experimentally as well as numerically from a corresponding mathematical model. Power spectral density and recurrence time distributions reveal a considerable increase of dynamic complexity with increasing temperature of the system, resulting in a larger relative spread of the attractor in phase space. By allowing for feasible coordinate transformations, we demonstrate that the system, however, remains phase-coherent over the whole considered parameter range. This finding motivates a critical review of existing definitions of phase coherence that are exclusively based on dynamical characteristics and are thus potentially sensitive to projection effects in phase space. In contrast, referring to the attractor geometry, the gradual changes in some fundamental properties of the system commonly related to its phase coherence can be alternatively studied from a purely structural point of view. As a prospective example for a corresponding framework, recurrence network analysis widely avoids undesired projection effects that otherwise can lead to ambiguous results of some existing approaches to studying phase coherence. Our corresponding results demonstrate that since temperature increase induces more complex chaotic chemical reactions, the recurrence network properties describing attractor geometry also change gradually: the bimodality of the distribution of local clustering coefficients due to the attractor's band structure disappears, and the corresponding asymmetry of the distribution as well as the average path length increase.

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.

Optimal phase description of chaotic oscillators

We introduce an optimal phase description of chaotic oscillations by generalizing the concept of isochrones. On chaotic attractors possessing a general phase description, we define the optimal isophases as Poincaré surfaces showing return times as constant as possible. The dynamics of the resultant optimal phase is maximally decoupled from the amplitude dynamics and provides a proper description of the phase response of chaotic oscillations. The method is illustrated with the Rössler and Lorenz systems.

Synchronization of weakly perturbed Markov chain oscillators

Rate processes are simple and analytically tractable models for many dynamical systems that switch stochastically between a discrete set of quasistationary states; however, they may also approximate continuous processes by coarse-grained, symbolic dynamics. In contrast to limit-cycle oscillators that are weakly perturbed by noise, in such systems, stochasticity may be strong, and topologies more complicated than a circle can be considered. Here we apply a second-order time-dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete-state oscillators coupled or driven through weakly time-dependent transition rates. We also describe a method of global control to optimize the response of the mean frequency in complex transition networks.

Assessing directed interactions from neurophysiological signals--an overview

The study of synchronization phenomena in coupled dynamical systems is an active field of research in many scientific disciplines including the neurosciences. Over the last decades, a number of time series analysis techniques have been proposed to capture both linear and nonlinear aspects of interactions. While most of these techniques allow one to quantify the strength of interactions, developments that resulted from advances in nonlinear dynamics and in information and synchronization theory aim at assessing directed interactions. Most of these techniques, however, assume the underlying systems to be at least approximately stationary and require a large number of data points to robustly assess directed interactions. Recent extensions allow assessing directed interactions from short and transient signals and are particularly suited for the analysis of evoked and event-related activity.

Collective phase description of globally coupled excitable elements

We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit-cycle solutions in infinite-dimensional dynamical systems, such as the time-periodic solutions to nonlinear Fokker-Planck equations representing macroscopic rhythms. We demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation, namely, it is type I for the saddle-node bifurcation and type II for the Hopf bifurcation.

Phase resetting of collective rhythm in ensembles of oscillators

Phase resetting curves characterize the way a system with a collective periodic behavior responds to perturbations. We consider globally coupled ensembles of Sakaguchi-Kuramoto oscillators, and use the Ott-Antonsen theory of ensemble evolution to derive the analytical phase resetting equations. We show the final phase reset value to be composed of two parts: an immediate phase reset directly caused by the perturbation and the dynamical phase reset resulting from the relaxation of the perturbed system back to its dynamical equilibrium. Analytical, semianalytical and numerical approximations of the final phase resetting curve are constructed. We support our findings with extensive numerical evidence involving identical and nonidentical oscillators. The validity of our theory is discussed in the context of large ensembles approximating the thermodynamic limit.