Markov chain Monte Carlo method in Bayesian reconstruction of dynamical systems from noisy chaotic time series

Data-driven stochastic model for cross-interacting processes with different time scales

In this work, we propose a new data-driven method for modeling cross-interacting processes with different time scales represented by time series with different sampling steps. It is a generalization of a nonlinear stochastic model of an evolution operator based on neural networks and designed for the case of time series with a constant sampling step. The proposed model has a more complex structure. First, it describes each process by its own stochastic evolution operator with its own time step. Second, it takes into account possible nonlinear connections within each pair of processes in both directions. These connections are parameterized asymmetrically, depending on which process is faster and which process is slower. They make this model essentially different from the set of independent stochastic models constructed individually for each time scale. All evolution operators and connections are trained and optimized using the Bayesian framework, forming a multi-scale stochastic model. We demonstrate the performance of the model on two examples. The first example is a pair of coupled oscillators, with the couplings in both directions which can be turned on and off. Here, we show that inclusion of the connections into the model allows us to correctly reproduce observable effects related to coupling. The second example is a spatially distributed data generated by a global climate model running in the middle 19th century external conditions. In this case, the multi-scale model allows us to reproduce the coupling between the processes which exists in the observed data but is not captured by the model constructed individually for each process.

Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition

Currently, causes of the middle Pleistocene transition (MPT) – the onset of large-amplitude glacial variability with 100 kyr time scale instead of regular 41 kyr cycles before – are a challenging puzzle in Paleoclimatology. Here we show how a Bayesian data analysis based on machine learning approaches can help to reveal the main mechanisms underlying the Pleistocene variability, which most likely explain proxy records and can be used for testing existing theories. We construct a Bayesian data-driven model from benthic δ¹⁸O records (LR04 stack) accounting for the main factors which may potentially impact climate of the Pleistocene: internal climate dynamics, gradual trends, variations of insolation, and millennial variability. In contrast to some theories, we uncover that under long-term trends in climate, the strong glacial cycles have appeared due to internal nonlinear oscillations induced by millennial noise. We find that while the orbital Milankovitch forcing does not matter for the MPT onset, the obliquity oscillation phase-locks the climate cycles through the meridional gradient of insolation.

Circadian variability of body temperature responses to methamphetamine

Vital parameters of living organisms exhibit circadian rhythmicity. Although rats are nocturnal animals, most of the studies involving rats are performed during the day. The objective of this study was to examine the circadian variability of the body temperature responses to methamphetamine. Body temperature was recorded in male Sprague-Dawley rats that received intraperitoneal injections of methamphetamine (Meth, 1 or 5 mg/kg) or saline at 10 AM or at 10 PM. The baseline body temperature at night was 0.8°C higher than during the day. Both during the day and at night, 1 mg/kg of Meth induced monophasic hyperthermia. However, the maximal temperature increase at night was 50% smaller than during the daytime. Injection of 5 mg/kg of Meth during the daytime caused a delayed hyperthermic response. In contrast, the same dose at night produced responses with a tendency toward a decrease of body temperature. Using mathematical modeling, we previously showed that the complex dose dependence of the daytime temperature responses to Meth results from an interplay between inhibitory and excitatory drives. In this study, using our model, we explain the suppression of the hyperthermia in response to Meth at night. First, we found that the baseline activity of the excitatory drive is greater at night. It appears partially saturated and thus is additionally activated by Meth to a lesser extent. Therefore, the excitatory component causes less hyperthermia or becomes overpowered by the inhibitory drive in response to the higher dose. Second, at night the injection of Meth results in reduction of the equilibrium body temperature, leading to gradual cooling counteracting hyperthermia.

Amphetamine enhances endurance by increasing heat dissipation

Athletes use amphetamines to improve their performance through largely unknown mechanisms. Considering that body temperature is one of the major determinants of exhaustion during exercise, we investigated the influence of amphetamine on the thermoregulation. To explore this, we measured core body temperature and oxygen consumption of control and amphetamine‐trea ted rats running on a treadmill with an incrementally increasing load (both speed and incline). Experimental results showed that rats treated with amphetamine (2 mg/kg) were able to run significantly longer than control rats. Due to a progressively increasing workload, which was matched by oxygen consumption, the control group exhibited a steady increase in the body temperature. The administration of amphetamine slowed down the temperature rise (thus decreasing core body temperature) in the beginning of the run without affecting oxygen consumption. In contrast, a lower dose of amphetamine (1 mg/kg) had no effect on measured parameters. Using a mathematical model describing temperature dynamics in two compartments (the core and the muscles), we were able to infer what physiological parameters were affected by amphetamine. Modeling revealed that amphetamine administration increases heat dissipation in the core. Furthermore, the model predicted that the muscle temperature at the end of the run in the amphetamine‐treated group was significantly higher than in the control group. Therefore, we conclude that amphetamine may mask or delay fatigue by slowing down exercise‐induced core body temperature growth by increasing heat dissipation. However, this affects the integrity of thermoregulatory system and may result in potentially dangerous overheating of the muscles.

Relating Granger causality to long-term causal effects

In estimation of causal couplings between observed processes, it is important to characterize coupling roles at various time scales. The widely used Granger causality reflects short-term effects: it shows how strongly perturbations of a current state of one process affect near future states of another process, and it quantifies that via prediction improvement (PI) in autoregressive models. However, it is often more important to evaluate the effects of coupling on long-term statistics, e.g., to find out how strongly the presence of coupling changes the variance of a driven process as compared to an uncoupled case. No general relationships between Granger causality and such long-term effects are known. Here, we pose the problem of relating these two types of coupling characteristics, and we solve it for a class of stochastic systems. Namely, for overdamped linear oscillators, we rigorously derive that the above long-term effect is proportional to the short-term effects, with the proportionality coefficient depending on the prediction interval and relaxation times. We reveal that this coefficient is typically considerably greater than unity so that small normalized PI values may well correspond to quite large long-term effects of coupling. The applicability of the derived relationship to wider classes of systems, its limitations, and its value for further research are discussed. To give a real-world example, we analyze couplings between large-scale climatic processes related to sea surface temperature variations in equatorial Pacific and North Atlantic regions.

Orexinergic neurotransmission in temperature responses to methamphetamine and stress: mathematical modeling as a data assimilation approach

Experimental Data

Orexinergic neurotransmission is involved in mediating temperature responses to methamphetamine (Meth). In experiments in rats, SB-334867 (SB), an antagonist of orexin receptors (OX1R), at a dose of 10 mg/kg decreases late temperature responses (t>60 min) to an intermediate dose of Meth (5 mg/kg). A higher dose of SB (30 mg/kg) attenuates temperature responses to low dose (1 mg/kg) of Meth and to stress. In contrast, it significantly exaggerates early responses (t<60 min) to intermediate and high doses (5 and 10 mg/kg) of Meth. As pretreatment with SB also inhibits temperature response to the stress of injection, traditional statistical analysis of temperature responses is difficult.

Mathematical Modeling

We have developed a mathematical model that explains the complexity of temperature responses to Meth as the interplay between excitatory and inhibitory nodes. We have extended the developed model to include the stress of manipulations and the effects of SB. Stress is synergistic with Meth on the action on excitatory node. Orexin receptors mediate an activation of on both excitatory and inhibitory nodes by low doses of Meth, but not on the node activated by high doses (HD). Exaggeration of early responses to high doses of Meth involves disinhibition: low dose of SB decreases tonic inhibition of HD and lowers the activation threshold, while the higher dose suppresses the inhibitory component. Using a modeling approach to data assimilation appears efficient in separating individual components of complex response with statistical analysis unachievable by traditional data processing methods.

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."

Random dynamical models from time series

In this work we formulate a consistent Bayesian approach to modeling stochastic (random) dynamical systems by time series and implement it by means of artificial neural networks. The feasibility of this approach for both creating models adequately reproducing the observed stationary regime of system evolution, and predicting changes in qualitative behavior of a weakly nonautonomous stochastic system, is demonstrated on model examples. In particular, a successful prognosis of stochastic system behavior as compared to the observed one is illustrated on model examples, including discrete maps disturbed by non-Gaussian and nonuniform noise and a flow system with Langevin force.

Prognosis of qualitative system behavior by noisy, nonstationary, chaotic time series

An approach to prognosis of qualitative behavior of an unknown dynamical system (DS) from weakly nonstationary chaotic time series (TS) containing significant measurement noise is proposed. The approach is based on construction of a global time-dependent parametrized model of discrete evolution operator (EO) capable of reproducing nonstationary dynamics of a reconstructed DS. A universal model in the form of artificial neural network (ANN) with certain prior limitations is used for the approximation of the EO in the reconstructed phase space. Probabilistic prognosis of the system behavior is performed using Monte Carlo Markov chain (MCMC) analysis of the posterior Bayesian distribution of the model parameters. The classification of qualitatively different regimes is supposed to be dictated by the application, i.e., it is assumed that some classifier function is predefined that maps a point of a model parameter space to a finite set of different behavior types. The ability of the approach to provide prognosis for times comparable to the observation time interval is demonstrated. Some restrictions as well as possible advances of the proposed approach are discussed.