Synchronicity from Synchronized Chaos

A Note on the Reproducibility of Chaos Simulation

An evergreen scientific feature is the ability for scientific works to be reproduced. Since chaotic systems are so hard to understand analytically, numerical simulations assume a key role in their investigation. Such simulations have been considered as reproducible in many works. However, few studies have focused on the effects of the finite precision of computers on the simulation reproducibility of chaotic systems; moreover, code sharing and details on how to reproduce simulation results are not present in many investigations. In this work, a case study of reproducibility is presented in the simulation of a chaotic jerk circuit, using the software LTspice. We also employ the OSF platform to share the project associated with this paper. Tests performed with LTspice XVII on four different computers show the difficulties of simulation reproducibility by this software. We compare these results with experimental data using a normalised root mean square error in order to identify the computer with the highest prediction horizon. We also calculate the entropy of the signals to check differences among computer simulations and the practical experiment. The methodology developed is efficient in identifying the computer with better performance, which allows applying it to other cases in the literature. This investigation is fully described and available on the OSF platform.

Simple capacitor-switch model of excitatory and inhibitory neuron with all parts biologically explained allows input fire pattern dependent chaotic oscillations

Due to known information processing capabilities of the brain, neurons are modeled at many different levels. Circuit theory is also often used to describe the function of neurons, especially in complex multi-compartment models, but when used for simple models, there is no subsequent biological justification of used parts. We propose a new single-compartment model of excitatory and inhibitory neuron, the capacitor-switch model of excitatory and inhibitory neuron, as an extension of the existing integrate-and-fire model, preserving the signal properties of more complex multi-compartment models. The correspondence to existing structures in the neuronal cell is then discussed for each part of the model. We demonstrate that a few such inter-connected model units are capable of acting as a chaotic oscillator dependent on fire patterns of the input signal providing a complex deterministic and specific response through the output signal. The well-known necessary conditions for constructing a chaotic oscillator are met for our presented model. The capacitor-switch model provides a biologically-plausible concept of chaotic oscillator based on neuronal cells.

Supermodeling: The Next Level of Abstraction in the Use of Data Assimilation

Data assimilation (DA) is a key procedure that synchronizes a computer model with real observations. However, in the case of overparametrized complex systems modeling, the task of parameter-estimation through data assimilation can expand exponentially. It leads to unacceptable computational overhead, substantial inaccuracies in parameter matching, and wrong predictions. Here we define a Supermodel as a kind of ensembling scheme, which consists of a few sub-models representing various instances of the baseline model. The sub-models differ in parameter sets and are synchronized through couplings between the most sensitive dynamical variables. We demonstrate that after a short pretraining of the fully parametrized small sub-model ensemble, and then training a few latent parameters of the low-parameterized Supermodel, we can outperform in efficiency and accuracy the baseline model matched to data by a classical DA procedure.

The Complex Behaviour of a Simple Neural Oscillator Model in the Human Cortex

The brain is a complex organ responsible for memory storage and reasoning; however, the mechanisms underlying these processes remain unknown. This paper forms a contribution to a lot of theoretical studies devoted to regular or chaotic oscillations of interconnected neurons assuming that the smallest information unit in the brain is not a neuron but, instead, a coupling of inhibitory and excitatory neurons forming a simple oscillator. Several coefficients of variation for peak intervals and correlation coefficients for peak interval histograms are evaluated and the sensitivity of such oscillator units is tested to changes in initial membrane potentials, interconnection signal delays, and changes in synaptic weights based on known histologically verified neuron couplings. Results present only a low dependence of oscillation patterns to changes in initial membrane potentials or interconnection signal delays in comparison to a strong sensitivity to changes in synaptic weights showing the stability and robustness of encoded oscillating patterns to signal outages or remoteness of interconnected neurons. Presented simulations prove that the selected neuronal couplings are able to produce a variety of different behavioural patterns, with periodicity ranging from milliseconds to thousands of milliseconds between the spikes. Many detected different intrinsic frequencies then support the idea of possibly large informational capacity of such memory units.

Phase synchronization of baroclinic waves in a differentially heated rotating annulus experiment subject to periodic forcing with a variable duty cycle

A series of laboratory experiments in a thermally driven, rotating fluid annulus are presented that investigate the onset and characteristics of phase synchronization and frequency entrainment between the intrinsic, chaotic, oscillatory amplitude modulation of travelling baroclinic waves and a periodic modulation of the (axisymmetric) thermal boundary conditions, subject to time-dependent coupling. The time-dependence is in the form of a prescribed duty cycle in which the periodic forcing of the boundary conditions is applied for only a fraction δ of each oscillation. For the rest of the oscillation, the boundary conditions are held fixed. Two profiles of forcing were investigated that capture different parts of the sinusoidal variation and δ was varied over the range 0.1≤δ≤1. Reducing δ was found to act in a similar way to a reduction in a constant coupling coefficient in reducing the width of the interval in forcing frequency or period over which complete synchronization was observed (the "Arnol'd tongue") with respect to the detuning, although for the strongest pulse-like forcing profile some degree of synchronization was discernible even at δ=0.1. Complete phase synchronization was obtained within the Arnol'd tongue itself, although the strength of the amplitude modulation of the baroclinic wave was not significantly affected. These experiments demonstrate a possible mechanism for intraseasonal and/or interannual "teleconnections" within the climate system of the Earth and other planets that does not rely on Rossby wave propagation across the planet along great circles.

Attractor learning in synchronized chaotic systems in the presence of unresolved scales

Recently, supermodels consisting of an ensemble of interacting models, synchronizing on a common solution, have been proposed as an alternative to the common non-interactive multi-model ensembles in order to improve climate predictions. The connection terms in the interacting ensemble are to be optimized based on the data. The supermodel approach has been successfully demonstrated in a number of simulation experiments with an assumed ground truth and a set of good, but imperfect models. The supermodels were optimized with respect to their short-term prediction error. Nevertheless, they produced long-term climatological behavior that was close to the long-term behavior of the assumed ground truth, even in cases where the long-term behavior of the imperfect models was very different. In these supermodel experiments, however, a perfect model class scenario was assumed, in which the ground truth and imperfect models belong to the same model class and only differ in parameter setting. In this paper, we consider the imperfect model class scenario, in which the ground truth model class is more complex than the model class of imperfect models due to unresolved scales. We perform two supermodel experiments in two toy problems. The first one consists of a chaotically driven Lorenz 63 oscillator ground truth and two Lorenz 63 oscillators with constant forcings as imperfect models. The second one is more realistic and consists of a global atmosphere model as ground truth and imperfect models that have perturbed parameters and reduced spatial resolution. In both problems, we find that supermodel optimization with respect to short-term prediction error can lead to a long-term climatological behavior that is worse than that of the imperfect models. However, we also show that attractor learning can remedy this problem, leading to supermodels with long-term behavior superior to the imperfect models.

Simulating climate with a synchronization-based supermodel

The SPEEDO global climate model (an atmosphere model coupled to a land and an ocean/sea-ice model with about 250.000 degrees of freedom) is used to investigate the merits of a new multi-model ensemble approach to the climate prediction problem in a perfect model setting. Two imperfect models are generated by perturbing parameters. Connection terms are introduced that synchronize the two models on a common solution, referred to as the supermodel solution. A synchronization-based learning algorithm is applied to the supermodel through the introduction of an update rule for the connection coefficients. Connection coefficients cease updating when synchronization errors between the supermodel and solutions of the "true" equations vanish. These final connection coefficients define the supermodel. Different supermodel solutions, but with equivalent performance, are found depending on the initial values of the connection coefficients during learning. The supermodels have a climatology and a climate response to a CO₂ increase in the atmosphere that is closer to the truth as compared to the imperfect models and the standard multi-model ensemble average, showing the potential of the supermodel approach to improve climate predictions.

"FORCE" learning in recurrent neural networks as data assimilation

It is shown that the "FORCE" algorithm for learning in arbitrarily connected networks of simple neuronal units can be cast as a Kalman Filter, with a particular state-dependent form for the background error covariances. The resulting interpretation has implications for initialization of the learning algorithm, leads to an extension to include interactions between the weight updates for different neurons, and can represent relationships within groups of multiple target output signals.

Introduction to focus issue: Synchronization in large networks and continuous media-data, models, and supermodels

The synchronization of loosely coupled chaotic systems has increasingly found applications to large networks of differential equations and to models of continuous media. These applications are at the core of the present Focus Issue. Synchronization between a system and its model, based on limited observations, gives a new perspective on data assimilation. Synchronization among different models of the same system defines a supermodel that can achieve partial consensus among models that otherwise disagree in several respects. Finally, novel methods of time series analysis permit a better description of synchronization in a system that is only observed partially and for a relatively short time. This Focus Issue discusses synchronization in extended systems or in components thereof, with particular attention to data assimilation, supermodeling, and their applications to various areas, from climate modeling to macroeconomics.