Distinguishing signatures of determinism and stochasticity in spiking complex systems

Analysis of the spiking dynamics of a diode laser with dual optical feedback

In a complex dynamical system, noise, feedback, and external forces shape behavior that can range from regularity to high-dimensional chaos. Multiple feedback sources can significantly alter its dynamics, potentially even suppressing the system's output. This study investigates the impact of competing feedback sources on a stochastic complex dynamical system using a photonic neuron-a diode laser with external optical feedback. By varying the feedback intensities from two external reflectors, we explore how dual feedback influences the system's behavior. Using ordinal analysis and advanced measures of complexity, we quantify the system's dynamics and uncover underlying symmetries. Our findings reveal that the interaction between the two feedback sources induces a more intense deterministic behavior, distinct from the dynamics produced by each feedback source individually. Additionally, clear temporal symmetries emerge across all dynamical regimes. By employing a novel entropy-vector representation, we are able to identify a unique signature that characterizes the system's dynamics.

Numerical Demonstration of the Transmission of Low Frequency Fluctuation Dynamics Generated by a Semiconductor Laser with Optical Feedback

In this paper, the transmission mechanism of the spike information embedded in the low frequency fluctuation (LFF) dynamic in a cascaded laser system is numerically demonstrated. In the cascaded laser system, the LFF waveform is first generated by a drive laser with optical feedback and is then injected into a response laser. The range of crucial system parameters that can make the response laser generate the LFF dynamic is studied, and the effect of parameter mismatch on the transmission of LFF dynamics is explored through a method of symbolic time-series analysis and the index, such as the spike rate and the cross-correlation coefficient. The results show that the mismatch of the pump current has a more significant influence on the transmission of LFF waveforms than that of the internal physical parameter of the laser, such as the linewidth enhancement factor. Moreover, increasing the injection strength can enhance the robustness of LFF transmission. As spikes of the LFF dynamic generated by lasers with optical feedback is similar to the spike of neurons, the results of this paper can help understanding the information transporting and processing inside the photonic neurons.

Correlations Preceding High-Intensity Events in the Chaotic Dynamics of a Raman Fiber Laser

We study the time series of the output intensity of a Raman fiber laser with an ordinal patterns analysis in the laminar-turbulent transition. We look for signatures among consecutive events that indicate when the system changes from triggering low-intensity to high-intensity events. We set two thresholds, a low one and a high one, to distinguish between low intensity versus high-intensity events. We find that when the time series is performing low-intensity events (below the low threshold), it shows some preferred temporal patterns before triggering high-intensity events (above a high threshold). The preferred temporal patterns remain the same all through the pump current range studied, even though two clearly different dynamical regimes are covered (laminar regime for low pump currents and turbulent regime for high pump currents). We also find that the turbulent regime shows clearer signatures of determinism than the laminar regime.

Forecasting Events in the Complex Dynamics of a Semiconductor Laser with Optical Feedback

Complex systems performing spiking dynamics are widespread in Nature. They cover from earthquakes, to neurons, variable stars, social networks, or stock markets. Understanding and characterizing their dynamics is relevant in order to detect transitions, or to predict unwanted extreme events. Here we study, under an ordinal patterns analysis, the output intensity of a semiconductor laser with feedback in a regime where it develops a complex spiking behavior. We unveil that, in the transitions towards and from the spiking regime, the complex dynamics presents two competing behaviors that can be distinguished with a thresholding method. Then we use time and intensity correlations to forecast different types of events, and transitions in the dynamics of the system.

Experimental study of modulation waveforms for entraining the spikes emitted by a semiconductor laser with optical feedback

The entrainment phenomenon, by which an oscillator adjusts its natural rhythm to an external periodic signal, has been observed in many natural systems. Recently, attention has focused on which are the optimal conditions for achieving entrainment. Here we use a semiconductor laser with optical feedback, operating in the low-frequency fluctuations (LFFs) regime, as a testbed for a controlled entrainment experiment. In the LFF regime the laser intensity displays abrupt spikes, which can be entrained to a weak periodic signal that directly modulates the laser pump current. We compare the performance of three modulation waveforms for producing 1:1 locking (one spike is emitted in each modulation cycle), as well as higher order locking regimes. We characterize the parameter regions where high-quality locking occurs, and those where the laser emits spikes which are not entrained to the external signal. The role of the modulation amplitude and frequency, and the role of the dc value of the laser pump current (that controls the natural spike frequency) in the entrainment quality are discussed.

Characterizing time series via complexity-entropy curves

The search for patterns in time series is a very common task when dealing with complex systems. This is usually accomplished by employing a complexity measure such as entropies and fractal dimensions. However, such measures usually only capture a single aspect of the system dynamics. Here, we propose a family of complexity measures for time series based on a generalization of the complexity-entropy causality plane. By replacing the Shannon entropy by a monoparametric entropy (Tsallis q entropy) and after considering the proper generalization of the statistical complexity (q complexity), we build up a parametric curve (the q-complexity-entropy curve) that is used for characterizing and classifying time series. Based on simple exact results and numerical simulations of stochastic processes, we show that these curves can distinguish among different long-range, short-range, and oscillating correlated behaviors. Also, we verify that simulated chaotic and stochastic time series can be distinguished based on whether these curves are open or closed. We further test this technique in experimental scenarios related to chaotic laser intensity, stock price, sunspot, and geomagnetic dynamics, confirming its usefulness. Finally, we prove that these curves enhance the automatic classification of time series with long-range correlations and interbeat intervals of healthy subjects and patients with heart disease.

Quantitative identification of dynamical transitions in a semiconductor laser with optical feedback

Identifying transitions to complex dynamical regimes is a fundamental open problem with many practical applications. Semi- conductor lasers with optical feedback are excellent testbeds for studying such transitions, as they can generate a rich variety of output signals. Here we apply three analysis tools to quantify various aspects of the dynamical transitions that occur as the laser pump current increases. These tools allow to quantitatively detect the onset of two different regimes, low-frequency fluctuations and coherence collapse, and can be used for identifying the operating conditions that result in specific dynamical properties of the laser output. These tools can also be valuable for analyzing regime transitions in other complex systems.

Heavy-Tailed Fluctuations in the Spiking Output Intensity of Semiconductor Lasers with Optical Feedback

Although heavy-tailed fluctuations are ubiquitous in complex systems, a good understanding of the mechanisms that generate them is still lacking. Optical complex systems are ideal candidates for investigating heavy-tailed fluctuations, as they allow recording large datasets under controllable experimental conditions. A dynamical regime that has attracted a lot of attention over the years is the so-called low-frequency fluctuations (LFFs) of semiconductor lasers with optical feedback. In this regime, the laser output intensity is characterized by abrupt and apparently random dropouts. The statistical analysis of the inter-dropout-intervals (IDIs) has provided many useful insights into the underlying dynamics. However, the presence of large temporal fluctuations in the IDI sequence has not yet been investigated. Here, by applying fluctuation analysis we show that the experimental distribution of IDI fluctuations is heavy-tailed, and specifically, is well-modeled by a non-Gaussian stable distribution. We find a good qualitative agreement with simulations of the Lang-Kobayashi model. Moreover, we uncover a transition from a less-heavy-tailed state at low pump current to a more-heavy-tailed state at higher pump current. Our results indicate that fluctuation analysis can be a useful tool for investigating the output signals of complex optical systems; it can be used for detecting underlying regime shifts, for model validation and parameter estimation.

Complexity in pulsed nonlinear laser systems interrogated by permutation entropy

Permutation entropy (PE) has a growing significance as a relative measure of complexity in nonlinear systems. It has been applied successfully to measuring complexity in nonlinear laser systems. Here, PE and weighted permutation entropy (WPE) are discovered to show an unexpected inversion to higher values, when characterizing the complexity at the characteristic frequencies of nonlinear drivers in laser systems, for output power sequences which are pulsed. The cause of this inversion is explained and its presence can be used to identify when irregular dynamics transform into a fairly regular pulsed signal (with amplitude and timing jitter). When WPE is calculated from experimental output power time series from various nonlinear laser systems as a function of delay time, both the minimum value of WPE, and the width of the peak in the WPE plot are shown to be indicative of the level of amplitude variation and timing jitter present in the pulsed output. Links are made with analysis using simulated time series data with systematic variation in timing jitter and/or amplitude variations.

Dynamical inference: where phase synchronization and generalized synchronization meet

Synchronization is a widespread phenomenon that occurs among interacting oscillatory systems. It facilitates their temporal coordination and can lead to the emergence of spontaneous order. The detection of synchronization from the time series of such systems is of great importance for the understanding and prediction of their dynamics, and several methods for doing so have been introduced. However, the common case where the interacting systems have time-variable characteristic frequencies and coupling parameters, and may also be subject to continuous external perturbation and noise, still presents a major challenge. Here we apply recent developments in dynamical Bayesian inference to tackle these problems. In particular, we discuss how to detect phase slips and the existence of deterministic coupling from measured data, and we unify the concepts of phase synchronization and general synchronization. Starting from phase or state observables, we present methods for the detection of both phase and generalized synchronization. The consistency and equivalence of phase and generalized synchronization are further demonstrated, by the analysis of time series from analog electronic simulations of coupled nonautonomous van der Pol oscillators. We demonstrate that the detection methods work equally well on numerically simulated chaotic systems. In all the cases considered, we show that dynamical Bayesian inference can clearly identify noise-induced phase slips and distinguish coherence from intrinsic coupling-induced synchronization.

Unveiling the complex organization of recurrent patterns in spiking dynamical systems

Complex systems displaying recurrent spike patterns are ubiquitous in nature. Understanding the organization of these patterns is a challenging task. Here we study experimentally the spiking output of a semiconductor laser with feedback. By using symbolic analysis we unveil a nontrivial organization of patterns, revealing serial spike correlations. The probabilities of the patterns display a well-defined, hierarchical and clustered structure that can be understood in terms of a delayed model. Most importantly, we identify a minimal model, a modified circle map, which displays the same symbolic organization. The validity of this minimal model is confirmed by analyzing the output of the forced laser. Since the circle map describes many dynamical systems, including neurons and cardiac cells, our results suggest that similar correlations and hierarchies of patterns can be found in other systems. Our findings also pave the way for optical neurons that could provide a controllable set up to mimic neuronal activity.

Experimental and numerical study of the symbolic dynamics of a modulated external-cavity semiconductor laser

We study the symbolic dynamics of a stochastic excitable optical system with periodic forcing. Specifically, we consider a directly modulated semiconductor laser with optical feedback in the low frequency fluctuations (LFF) regime. We use a method of symbolic time-series analysis that allows us to uncover serial correlations in the sequence of intensity dropouts. By transforming the sequence of inter-dropout intervals into a sequence of symbolic patterns and analyzing the statistics of the patterns, we unveil correlations among several consecutive dropouts and we identify clear changes in the dynamics as the modulation amplitude increases. To confirm the robustness of the observations, the experiments were performed using two lasers under different feedback conditions. Simulations of the Lang-Kobayashi (LK) model, including spontaneous emission noise, are found to be in good agreement with the observations, providing an interpretation of the correlations present in the dropout sequence as due to the interplay of the underlying attractor topology, the external forcing, and the noise that sustains the dropout events.

Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback

We implement a method to identify the deterministic nature of specific events in the dynamics of a semiconductor laser subject to time-delayed optical feedback. Specifically, we study the power dropouts in the low-frequency fluctuations regime on an individual event basis and identify whether the underlying dominant mechanism is deterministic. Our approach is based on sychronization with a twin system in a symmetric relay configuration. We investigate the dependence of the fraction of deterministically driven (i.e., synchronized) dropouts on the laser's pump current as a key parameter. Our experimental results are corroborated by numerical modeling based on rate equations. Our numerical findings also provide insights into the influence of spontaneous emission noise.