Basin entropy as an indicator of a bifurcation in a time-delayed system.

The basin entropy is a measure that quantifies, in a system that has two or more attractors, the predictability of a final state, as a function of the initial conditions. While the basin entropy has been demonstrated on a variety of multistable dynamical systems, to the best of our knowledge, it has not yet been tested in systems with a time delay, whose phase space is infinite dimensional because the initial conditions are functions defined in a time interval [-τ,0], where τ is the delay time. Here, we consider a simple time-delayed system consisting of a bistable system with a linear delayed feedback term. We show that the basin entropy captures relevant properties of the basins of attraction of the two coexisting attractors. Moreover, we show that the basin entropy can give an indication of the proximity of a Hopf bifurcation, but fails to capture the proximity of a pitchfork bifurcation. The Hopf bifurcation is detected because before the fixed points become unstable, a oscillatory, limit-cycle behavior appears that coexists with the fixed points. The new limit cycle modifies the structure of the basins of attraction, and this change is captured by basin entropy that reaches a maximum before the Hopf bifurcation. In contrast, the pitchfork bifurcation is not detected because the basins of attraction do not change as the bifurcation is approached. Our results suggest that the basin entropy can yield useful insights into the long-term predictability of time-delayed systems, which often have coexisting attractors.

Characterizing the spike timing of a chaotic laser by using ordinal analysis and machine learning.

Semiconductor lasers with optical feedback are well-known nonlinear dynamical systems. Under appropriate feedback conditions, these lasers emit optical pulses that resemble neural spikes. Influenced by feedback delay and various noise sources, including quantum spontaneous emission noise, the dynamics are highly stochastic. A good understanding of the spike timing statistics is needed to develop photonic systems capable of using the fast-spiking laser output for novel applications, such as information processing or random number generation. Here we analyze experimental sequences of inter-spike intervals (ISIs) recorded when a sinusoidal signal was applied to the laser current. Different combinations of the DC value and frequency of the signal applied to the laser lead to ISI sequences with distinct statistical properties. This variability prompts an investigation into the relationship between experimental parameters and ISI sequence statistics, aiming to uncover potential encoding methods for optical spikes, since this can open a new way of encoding and decoding information in sequences of optical spikes. By using ordinal analysis and machine learning, we show that the ISI sequences have statistical ordinal properties that are similar to Flicker noise signals, characterized by a parameter α that varies with the signal that was applied to the laser current when the ISIs were recorded. We also show that for this dataset, the (α, permutation entropy) plane is more informative than the (complexity, permutation entropy) plane because it allows better differentiation of ISI sequences recorded under different experimental conditions, as well as better differentiation of original and surrogate ISI sequences.

Permutation entropy analysis of EEG signals for distinguishing eyes-open and eyes-closed brain states: Comparison of different approaches.

Developing reliable methodologies to decode brain state information from electroencephalogram (EEG) signals is an open challenge, crucial to implementing EEG-based brain-computer interfaces (BCIs). For example, signal processing methods that identify brain states could allow motor-impaired patients to communicate via non-invasive, EEG-based BCIs. In this work, we focus on the problem of distinguishing between the states of eyes closed (EC) and eyes open (EO), employing quantities based on permutation entropy (PE). An advantage of PE analysis is that it uses symbols (ordinal patterns) defined by the ordering of the data points (disregarding the actual values), hence providing robustness to noise and outliers due to motion artifacts. However, we show that for the analysis of multichannel EEG recordings, the performance of PE in discriminating the EO and EC states depends on the symbols' definition and how their probabilities are estimated. Here, we study the performance of PE-based features for EC/EO state classification in a dataset of N=107 subjects with one-minute 64-channel EEG recordings in each state. We analyze features obtained from patterns encoding temporal or spatial information, and we compare different approaches to estimate their probabilities (by averaging over time, over channels, or by "pooling"). We find that some PE-based features provide about 75% classification accuracy, comparable to the performance of features extracted with other statistical analysis techniques. Our work highlights the limitations of PE methods in distinguishing the eyes' state, but, at the same time, it points to the possibility that subject-specific training could overcome these limitations.

Timing jitter reduction in semiconductor lasers induced by optical injection.

We present an experimental study of the effect of continuous-wave optical injection (OI) from a vertical-cavity surface-emitting laser (VCSEL) on the timing jitter of a gain-switched discrete-mode semiconductor laser (DML). Timing jitter was analyzed over a wide range of temperatures of the DML, which allowed tuning the detuning between the lasers emissions, and it was compared with the inter-pulse timing jitter. We have found that there is a range of detunings in which OI diminishes the jitter by 70% with respect to the jitter of the solitary DML. However, within this region, there are some detunings for which OI significantly increases the jitter.

Quantifying the synchronization of the spikes emitted by coupled lasers.

Synchronization phenomena is ubiquitous in nature, and in spite of having been studied for decades, it still attracts a lot of attention as is still challenging to detect and quantify, directly from the analysis of noisy signals. Semiconductor lasers are ideal for performing experiments because they are stochastic, nonlinear, and inexpensive and display different synchronization regimes that can be controlled by tuning the lasers' parameters. Here, we analyze experiments done with two mutually optically coupled lasers. Due to the delay in the coupling (due to the finite time the light takes to travel between the lasers), the lasers synchronize with a lag: the intensity time traces show well-defined spikes, and a spike in the intensity of one laser may occur shortly before (or shortly after) a spike in the intensity of the other laser. Measures that quantify the degree of synchronization of the lasers from the analysis of the intensity signals do not fully quantify the synchronicity of the spikes because they also take into account the synchronization of fast irregular fluctuations that occur between spikes. By analyzing only the coincidence of the spike times, we show that event synchronization measures quantify spike synchronization remarkably well. We show that these measures allow us to quantify the degree of synchronization and, also, to identify the leading laser and the lagging one.

Experimental study of spatial and temporal coherence in a laser diode with optical feedback.

Optical feedback can reduce the linewidth of a semiconductor laser by several orders of magnitude, but it can also cause line broadening. Although these effects on the temporal coherence of the laser are well known, a good understanding of the effects of feedback on the spatial coherence is still lacking. Here we present an experimental technique that allows discriminating the effects of feedback on temporal and spatial coherence of the laser beam. We analyze the output of a commercial edge-emitting laser diode, comparing the contrast of speckle images recorded using a multimode (MM) or single mode (SM) fiber and an optical diffuser, and also, comparing the optical spectra at the end of the MM or SM fiber. Optical spectra reveal feedback-induced line broadening, while speckle analyses reveal reduced spatial coherence due to feedback-excited spatial modes. These modes reduce the speckle contrast (SC) up to 50% when speckle images are recorded using the MM fiber, but do not affect the SC when the images are recorded using the SM fiber and diffuser, because the spatial modes that are excited by the feedback are filtered out by the SM fiber. This technique is generic and can be used to discriminate spatial and temporal coherence of other types of lasers and under other operating conditions that can induce a chaotic output.

Diffusion capacity of single and interconnected networks.

Understanding diffusive processes in networks is a significant challenge in complexity science. Networks possess a diffusive potential that depends on their topological configuration, but diffusion also relies on the process and initial conditions. This article presents Diffusion Capacity, a concept that measures a node's potential to diffuse information based on a distance distribution that considers both geodesic and weighted shortest paths and dynamical features of the diffusion process. Diffusion Capacity thoroughly describes the role of individual nodes during a diffusion process and can identify structural modifications that may improve diffusion mechanisms. The article defines Diffusion Capacity for interconnected networks and introduces Relative Gain, which compares the performance of a node in a single structure versus an interconnected one. The method applies to a global climate network constructed from surface air temperature data, revealing a significant change in diffusion capacity around the year 2000, suggesting a loss of the planet's diffusion capacity that could contribute to the emergence of more frequent climatic events.

Abrupt transition from low-coherence to high-coherence radiation in a semiconductor laser with optical feedback.

Semiconductor lasers are very sensitive to optical feedback. Although it is well known that coherent feedback lowers the threshold of the laser, the characteristics of the transition from low-coherence radiation-dominated by spontaneous emission-below threshold to high-coherence radiation-dominated by stimulated emission-above threshold have not yet been investigated. Here we show experimentally that, in contrast to the transition that occurs in the solitary laser, in the laser with feedback the transition to high-coherence emission can occur abruptly. We use the speckle technique to show that the transition varies from smooth to abrupt as the amount of light fed back to the laser increases.

Entropy-based early detection of critical transitions in spatial vegetation fields.

In semiarid regions, vegetated ecosystems can display abrupt and unexpected changes, i.e., transitions to different states, due to drifting or time-varying parameters, with severe consequences for the ecosystem and the communities depending on it. Despite intensive research, the early identification of an approaching critical point from observations is still an open challenge. Many data analysis techniques have been proposed, but their performance depends on the system and on the characteristics of the observed data (the resolution, the level of noise, the existence of unobserved variables, etc.). Here, we propose an entropy-based approach to identify an upcoming transition in spatiotemporal data. We apply this approach to observational vegetation data and simulations from two models of vegetation dynamics to infer the arrival of an abrupt shift to an arid state. We show that the permutation entropy (PE) computed from the probabilities of two-dimensional ordinal patterns may provide an early warning indicator of an approaching tipping point, as it may display a maximum (or minimum) before decreasing (or increasing) as the transition approaches. Like other spatial early warning indicators, the spatial permutation entropy does not need a time series of the system dynamics, and it is suited for spatially extended systems evolving on long time scales, like vegetation plots. We quantify its performance and show that, depending on the system and data, the performance can be better, similar or worse than the spatial correlation. Hence, we propose the spatial PE as an additional indicator to try to anticipate regime shifts in vegetated ecosystems.

Perspectives on the importance of complex systems in understanding our climate and climate change-The Nobel Prize in Physics 2021.

The Nobel Prize in Physics 2021 was awarded to Syukuro Manabe, Klaus Hasselmann, and Giorgio Parisi for their "groundbreaking contributions to our understanding of complex systems," including major advances in the understanding of our climate and climate change. In this Perspective article, we review their key contributions and discuss their relevance in relation to the present understanding of our climate. We conclude by outlining some promising research directions and open questions in climate science.

Time crystal dynamics in a weakly modulated stochastic time delayed system.

Time crystal oscillations in interacting, periodically driven many-particle systems are highly regular oscillations that persist for long periods of time, are robust to perturbations, and whose frequency differs from the frequency of the driving signal. Making use of underlying similarities of spatially-extended systems and time-delayed systems (TDSs), we present an experimental demonstration of time-crystal-like behavior in a stochastic, weakly modulated TDS. We consider a semiconductor laser near threshold with delayed feedback, whose output intensity shows abrupt spikes at irregular times. When the laser current is driven with a small-amplitude periodic signal we show that the interaction of delayed feedback and modulation can generate long-range regularity in the timing of the spikes, which lock to the modulation and, despite the presence of noise, remain in phase over thousands of modulation cycles. With pulsed modulation we find harmonic and subharmonic locking, while with sinusoidal modulation, we find only subharmonic locking, which is a characteristic feature of time-crystal behavior.

Dynamics of a semiconductor laser with feedback and modulation: experiments and model comparison.

We study experimentally and numerically the dynamics of a semiconductor laser near threshold, subject to optical feedback and sinusoidal current modulation. The laser operates in the low frequency fluctuation (LFF) regime where, without modulation, the intensity shows sudden spikes at irregular times. Under particular modulation conditions the spikes lock to the modulation and their timing becomes highly regular. While the modulated LFF dynamics has received a lot of attention, an in-depth comparison with the predictions of the Lang-Kobayashi (LK) model has not yet been performed. Here we use the LK model to simulate the laser dynamics and use the Fano factor to quantify the regularity of the timing of the spikes. The Fano factor is calculated by counting the number of spikes in successive segments of the intensity time-series and keeps information about temporal order in the spike sequence that is lost when the analysis is based on the distribution of inter-spike intervals. Here we compare the spike timing regularity in experimental and in simulated spike sequences as a function of the modulation amplitude and frequency and find a good qualitative agreement. We find that in both experiments and simulation for appropriate conditions the spike timing can be highly regular, as revealed by very small values of the Fano factor.

What Models and Tools can Contribute to a Better Understanding of Brain Activity?

Despite impressive scientific advances in understanding the structure and function of the human brain, big challenges remain. A deep understanding of healthy and aberrant brain activity at a wide range of temporal and spatial scales is needed. Here we discuss, from an interdisciplinary network perspective, the advancements in physical and mathematical modeling as well as in data analysis techniques that, in our opinion, have potential to further advance our understanding of brain structure and function.

Permutation entropy analysis of the output of a laser diode under stimulated Brillouin scattering optical feedback.

The chaotic output emitted by a diode laser with optical feedback has fascinated the community for decades. The external cavity delay time imparts a weak level of periodicity to the laser output (the so-called "time delay signature", TDS) that is a drawback for applications that require random optical signals. A lot of efforts have focused in suppressing the TDS either by post-processing the signal or by using alternative ways to generate random optical signals. Here, we compare the signals generated by two optical feedback setups: in the first one, the stimulated Brillouin backscattered light from a standard optical fibre is re-injected into the laser (stimulated Brillouin scattering optical feedback, SBSOF); in the second one, the light transmitted through the fibre is re-injected into the laser (conventional optical feedback, COF). We analyse the permutation entropy, a well-known measure of complexity that captures order relations between values of a time series. We find that, on average, the signal generated by the SBSOF setup has slightly lower PE than the one generated by the COF setup, except when the sampling time of the intensity signal is an exact multiple of the delay; in that case, due to TDS, the entropy of the COF signal is lower than that of the SBSOF signal. We interpret the lower entropy value of the SBSOF signal as due to oscillations at the Brillouin frequency shift. Taken together, our results show that TDS suppression can have an undesirable side effect: a decrease of the entropy of the signal.

Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning.

Time series analysis comprises a wide repertoire of methods for extracting information from data sets. Despite great advances in time series analysis, identifying and quantifying the strength of nonlinear temporal correlations remain a challenge. We have recently proposed a new method based on training a machine learning algorithm to predict the temporal correlation parameter, α, of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, xαFN(t), generated with different values of α. Then, the ordinal probabilities computed from the time series of interest, x(t), are used as input features to the trained algorithm and that returns a value, αe, that contains meaningful information about the temporal correlations present in x(t). We have also shown that the difference, Ω, of the permutation entropy (PE) of the time series of interest, x(t), and the PE of a FN time series generated with α=αe, xαeFN(t), allows the identification of the underlying determinism in x(t). Here, we apply our methodology to different datasets and analyze how αe and Ω correlate with well-known quantifiers of chaos and complexity. We also discuss the limitations for identifying determinism in highly chaotic time series and in periodic time series contaminated by noise. The open source algorithm is available on Github.

Fast and effective pseudo transfer entropy for bivariate data-driven causal inference.

Identifying, from time series analysis, reliable indicators of causal relationships is essential for many disciplines. Main challenges are distinguishing correlation from causality and discriminating between direct and indirect interactions. Over the years many methods for data-driven causal inference have been proposed; however, their success largely depends on the characteristics of the system under investigation. Often, their data requirements, computational cost or number of parameters limit their applicability. Here we propose a computationally efficient measure for causality testing, which we refer to as pseudo transfer entropy (pTE), that we derive from the standard definition of transfer entropy (TE) by using a Gaussian approximation. We demonstrate the power of the pTE measure on simulated and on real-world data. In all cases we find that pTE returns results that are very similar to those returned by Granger causality (GC). Importantly, for short time series, pTE combined with time-shifted (T-S) surrogates for significance testing strongly reduces the computational cost with respect to the widely used iterative amplitude adjusted Fourier transform (IAAFT) surrogate testing. For example, for time series of 100 data points, pTE and T-S reduce the computational time by [Formula: see text] with respect to GC and IAAFT. We also show that pTE is robust against observational noise. Therefore, we argue that the causal inference approach proposed here will be extremely valuable when causality networks need to be inferred from the analysis of a large number of short time series.

Neuronal Transmission of Subthreshold Periodic Stimuli Via Symbolic Spike Patterns.

We study how sensory neurons detect and transmit a weak external stimulus. We use the FitzHugh-Nagumo model to simulate the neuronal activity. We consider a sub-threshold stimulus, i.e., the stimulus is below the threshold needed for triggering action potentials (spikes). However, in the presence of noise the neuron that perceives the stimulus fires a sequence of action potentials (a spike train) that carries the stimulus' information. To yield light on how the stimulus' information can be encoded and transmitted, we consider the simplest case of two coupled neurons, such that one neuron (referred to as neuron 1) perceives a subthreshold periodic signal but the second neuron (neuron 2) does not perceive the signal. We show that, for appropriate coupling and noise strengths, both neurons fire spike trains that have symbolic patterns (defined by the temporal structure of the inter-spike intervals), whose frequencies of occurrence depend on the signal's amplitude and period, and are similar for both neurons. In this way, the signal information encoded in the spike train of neuron 1 propagates to the spike train of neuron 2. Our results suggest that sensory neurons can exploit the presence of neural noise to fire spike trains where the information of a subthreshold stimulus is encoded in over expressed and/or in less expressed symbolic patterns.

Success rate analysis of the response of an excitable laser to periodic perturbations.

We use statistical tools to characterize the response of an excitable system to periodic perturbations. The system is an optically injected semiconductor laser under pulsed perturbations of the phase of the injected field. We characterize the laser response by counting the number of pulses emitted by the laser, within a time interval, ΔT, that starts when a perturbation is applied. The success rate, SR(ΔT), is then defined as the number of pulses emitted in the interval ΔT, relative to the number of perturbations. The analysis of the variation of SR with ΔT allows separating a constant lag of technical origin and a frequency-dependent lag of physical and dynamical origin. Once the lag is accounted for, the success rate clearly captures locked and unlocked regimes and the transitions between them. We anticipate that the success rate will be a practical tool for analyzing the output of periodically forced systems, particularly when very regular oscillations need to be generated via small periodic perturbations.

Remote recovery of audio signals from videos of optical speckle patterns: a comparative study of signal recovery algorithms.

Optical remote sensors are nowadays ubiquitously used, thanks to unprecedented advances in the last decade in photonics, machine learning and signal processing tools. In this work we study experimentally the remote recovery of audio signals from the silent videos of the movement of optical speckle patterns. This technique can be used even when in between the source and the receiver there is a medium that does not allow for the propagation of sound waves. We use a diode laser to generate a speckle pattern on the membrane of a loudspeaker and a low-cost CCD camera to record the video of the movement of the speckle pattern when the loudspeaker plays an audio signal. We perform a comparative analysis of six signal recovery algorithms. In spite of having different complexity and computational requirements, we find that the algorithms have (except for the simplest one) good performance in terms of the quality of the recovered signal. The best trade-off, in terms of computational costs and performance, is obtained with a new method that we propose, which recovers the signal from the weighted sum of the intensities of all the pixels, where the signs of the weights are determined by selecting a reference pixel and calculating the signs of the cross-correlations of the intensity of the reference pixel and the intensities of the other pixels.