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.

State space reconstruction of spatially extended systems and of time delayed systems from the time series of a scalar variable.

The space-time representation of high-dimensional dynamical systems that have a well defined characteristic time scale has proven to be very useful to deepen the understanding of such systems and to uncover hidden features in their output signals. By using the space-time representation many analogies between one-dimensional spatially extended systems (1D SESs) and time delayed systems (TDSs) have been found, including similar pattern formation and propagation of localized structures. An open question is whether such analogies are limited to the space-time representation, or it is also possible to recover similar evolutions in a low-dimensional pseudo-space. To address this issue, we analyze a 1D SES (a bistable reaction-diffusion system), a scalar TDS (a bistable system with delayed feedback), and a non-scalar TDS (a model of two delay-coupled lasers). In these three examples, we show that we can reconstruct the dynamics in a three-dimensional phase space, where the evolution is governed by the same polynomial potential. We also discuss the limitations of the analogy between 1D SESs and TDSs.

Differentiating resting brain states using ordinal symbolic analysis.

Symbolic methods of analysis are valuable tools for investigating complex time-dependent signals. In particular, the ordinal method defines sequences of symbols according to the ordering in which values appear in a time series. This method has been shown to yield useful information, even when applied to signals with large noise contamination. Here, we use ordinal analysis to investigate the transition between eyes closed (EC) and eyes open (EO) resting states. We analyze two electroencephalography datasets (with 71 and 109 healthy subjects) with different recording conditions (sampling rates and the number of electrodes in the scalp). Using as diagnostic tools the permutation entropy, the entropy computed from symbolic transition probabilities, and an asymmetry coefficient (that measures the asymmetry of the likelihood of the transitions between symbols), we show that the ordinal analysis applied to the raw data distinguishes the two brain states. In both datasets, we find that, during the EC-EO transition, the EO state is characterized by higher entropies and lower asymmetry coefficient, as compared to the EC state. Our results thus show that these diagnostic tools have the potential for detecting and characterizing changes in time-evolving brain states.

Experimental characterization of the transition to coherence collapse in a semiconductor laser with optical feedback.

Semiconductor lasers with time-delayed optical feedback display a wide range of dynamical regimes, which have found various practical applications. They also provide excellent testbeds for data analysis tools for characterizing complex signals. Recently, several of us have analyzed experimental intensity time-traces and quantitatively identified the onset of different dynamical regimes, as the laser current increases. Specifically, we identified the onset of low-frequency fluctuations (LFFs), where the laser intensity displays abrupt dropouts, and the onset of coherence collapse (CC), where the intensity fluctuations are highly irregular. Here we map these regimes when both, the laser current and the feedback strength vary. We show that the shape of the distribution of intensity fluctuations (characterized by the standard deviation, the skewness, and the kurtosis) allows to distinguish among noise, LFFs and CC, and to quantitatively determine (in spite of the gradual nature of the transitions) the boundaries of the three regimes. Ordinal analysis of the inter-dropout time intervals consistently identifies the three regimes occurring in the same parameter regions as the analysis of the intensity distribution. Simulations of the well-known time-delayed Lang-Kobayashi model are in good qualitative agreement with the observations.

Unveiling Temporal Correlations Characteristic of a Phase Transition in the Output Intensity of a Fiber Laser.

We use advanced statistical tools of time-series analysis to characterize the dynamical complexity of the transition to optical wave turbulence in a fiber laser. Ordinal analysis and the horizontal visibility graph applied to the experimentally measured laser output intensity reveal the presence of temporal correlations during the transition from the laminar to the turbulent lasing regimes. Both methods unveil coherent structures with well-defined time scales and strong correlations both, in the timing of the laser pulses and in their peak intensities. Our approach is generic and may be used in other complex systems that undergo similar transitions involving the generation of extreme fluctuations.

Effects of periodic forcing on the temporally correlated spikes of a semiconductor laser with feedback.

Optical excitable devices that mimic neuronal behavior can be building-blocks of novel, brain-inspired information processing systems. A relevant issue is to understand how such systems represent, via correlated spikes, the information of a weak external input. Semiconductor lasers with optical feedback operating in the low frequency fluctuations regime have been shown to display optical spikes with intrinsic temporal correlations similar to those of biological neurons. Here we investigate how the spiking laser output represents a weak periodic input that is implemented via direct modulation of the laser pump current. We focus on understanding the influence of the modulation frequency. Experimental sequences of inter-spike-intervals (ISIs) are recorded and analyzed by using the ordinal symbolic methodology that identifies and characterizes serial correlations in datasets. The change in the statistics of the various symbols with the modulation frequency is empirically shown to be related to specific changes in the ISI distribution, which arise due to different phase-locking regimes. A good qualitative agreement is also found between simulations of the Lang and Kobayashi model and observations. This methodology is an efficient way to detect subtle changes in noisy correlated ISI sequences and may be applied to investigate other optical excitable devices.

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.

Multidimensional subwavelength position sensing using a semiconductor laser with optical feedback.

We demonstrate experimentally how to harness quasi-periodic dynamics in a semiconductor laser with dual optical feedback for measuring subwavelength changes in each arm of the cavity simultaneously. We exploit the multifrequency spectrum of quasi-periodic dynamics and show that independent frequency shifts are mapped uniquely to two-dimensional displacements of the arms in the external cavity. Considering a laser diode operating at telecommunication wavelength λ≈1550 nm, we achieve an average nanoscale resolution of approximately 9.8 nm (~λ/160).

Dual-lag synchronization between coupled chaotic lasers due to path-delay interference.

We study experimentally the synchronization dynamics of two semiconductor lasers coupled unidirectionally via two different delayed paths. The emitter laser operates in a chaotic regime characterized by low-frequency fluctuations due to optical feedback and induces a synchronized dynamical activity in the receiver laser, which operates in the continuous-wave regime when uncoupled. Different delays in the two coupling paths lead to the coexistence of two time lags in the synchronized dynamics of the oscillators. This dual-lag synchronization degrades the average synchronization quality of the system of coupled lasers and hinders the transmission of information between them. Numerical simulation results agree with the experimental observations, and allow us to explore this phenomenon in a wide parameter range, and quantify the degree of signal transmission degradation caused by this chaotic path-delay interference.

Quantifying the degree of locking in weakly forced stochastic systems.

Controlling an stochastic nonlinear system with a small amplitude signal is a fundamental problem with many practical applications. Quantifying locking is challenging, and current methods, such as spectral or correlation analysis, do not provide a precise measure of the degree of locking. Here we study locking in an experimental system, consisting of a semiconductor laser with optical feedback operated in the regime where it randomly emits abrupt spikes. To quantify the locking of the optical spikes to small electric perturbations, we use two measures, the success rate (SR) and the false positive rate (FPR). The SR counts the spikes that are emitted shortly after each perturbation, while the FPR counts the additional extra spikes. We show that the receiver operating characteristic (ROC) curve (SR versus FPR plot) uncovers parameter regions where the electric perturbations fully control the laser spikes, such that the laser emits, shortly after each perturbation, one and only one spike. To demonstrate the general applicability of the ROC analysis we also study a stochastic bistable system under square-wave forcing and show that the ROC curve allows identifying the parameters that produce best locking.

Interplay of subthreshold activity, time-delayed feedback, and noise on neuronal firing patterns.

Feedback connections and noise are ubiquitous features of neuronal networks and affect in a determinant way the patterns of neural activity. Here we study how the subthreshold dynamics of a neuron interacts with time-delayed feedback and noise. We use a Hodgkin-Huxley-type model of a thermoreceptor neuron and assume the feedback to be linear, corresponding effectively to a recurrent electrical connection via gap junctions. This type of feedback can model electrical autapses, which connect the terminal fibers of a neuron's axon with dendrites from the same neuron. Thus the delay in the feedback loop is due basically to the axonal propagation time. We chose model parameters for which the neuron displays, in the absence of feedback and noise, only subthreshold oscillations. These oscillations, however, take the neuron close to the firing threshold, such that small perturbations can drive it above the level for generation of action potentials. The resulting interplay between weak delayed feedback, noise, and the subthreshold intrinsic activity is nontrivial. For negative feedback, depending on the delay, the firing rate can be lower than in the noise-free situation. This is due to the fact that noise inhibits feedback-induced spikes by driving the neuronal oscillations away from the firing threshold. For positive feedback, there are regions of delay values where the noise-induced spikes are inhibited by the feedback; in this case, it is the feedback that drives the neuronal oscillations away from the threshold. Our study contributes to a better understanding of the role of electrical self-connections in the presence of noise and subthreshold activity.

Ghost stochastic resonance with distributed inputs in pulse-coupled electronic neurons.

We study experimentally the phenomenon of ghost stochastic resonance in pulse-coupled excitable systems, for input signals distributed among different elements. Specifically, two excitable electronic circuits are driven by different sinusoidal signals that produce periodic spikes at distinct frequencies. Their outputs are sent to a third circuit that processes these spiking signals and is additionally perturbed by noise. When the input signals are harmonics of a certain fundamental (that is not present in the inputs) the processing circuit exhibits, for an optimal amount of noise, a resonant response at the frequency of the missing fundamental (ghost frequency). In contrast with the standard case in which the signals being directly integrated are sinusoidal, this behavior relies here on a coincidence-detection mechanism. When the input signals are homogeneously shifted in frequency, the processing circuit responds with pulse packages composed of spikes at a frequency that depends linearly on the frequency shift. Expressions for the dependence of the package period and duration on the frequency shift and spike width, respectively, are obtained. These results provide an experimental verification of a recently proposed mechanism of binaural pitch perception.

Emergence of spike correlations in periodically forced excitable systems.

In sensory neurons the presence of noise can facilitate the detection of weak information-carrying signals, which are encoded and transmitted via correlated sequences of spikes. Here we investigate the relative temporal order in spike sequences induced by a subthreshold periodic input in the presence of white Gaussian noise. To simulate the spikes, we use the FitzHugh-Nagumo model and to investigate the output sequence of interspike intervals (ISIs), we use the symbolic method of ordinal analysis. We find different types of relative temporal order in the form of preferred ordinal patterns that depend on both the strength of the noise and the period of the input signal. We also demonstrate a resonancelike behavior, as certain periods and noise levels enhance temporal ordering in the ISI sequence, maximizing the probability of the preferred patterns. Our findings could be relevant for understanding the mechanisms underlying temporal coding, by which single sensory neurons represent in spike sequences the information about weak periodic stimuli.

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.

Delay-induced resonances in an optical system with feedback.

We study the influence of the delay time in the response of a delayed feedback system to external periodic driving. The nonlinear system we consider is a semiconductor laser with optical feedback operating in the low-frequency fluctuation regime. We numerically examine the consequences of varying the external cavity length of the system when a weak modulation is introduced through the laser's pump current. The harmonic modulation is seen to lead to a partial periodic entrainment of power dropouts, and the distribution of time intervals between the dropouts exhibits resonances with certain delay times. In other words, the response of the system to the external modulation is enhanced for particular values of the external cavity length. The same effect can be observed in the presence of noise, indicating that stochastic resonance can be enhanced or degraded depending on the feedback time.

Stochastic entrainment of optical power dropouts.

We show that the natural pulsed behavior, in the form of sudden power dropouts, exhibited by semiconductor lasers subject to optical feedback can be entrained by the joint action of external noise and weak periodic driving. These power dropouts, which in the absence of forcing do not occur periodically, acquire the periodicity of the harmonic driving for an optimal amount of external noise, in what constitutes a form of stochastic resonance. This phenomenon is analyzed by means of a generalized Lang-Kobayashi model with external nonwhite noise in the modulated pump current, in terms of both the temporal correlation and the amplitude of the noise.

Effect of external noise correlation in optical coherence resonance.

Coherence resonance occurring in semiconductor lasers with optical feedback is studied via the Lang-Kobayashi model with external nonwhite noise in the pumping current. The temporal correlation and the amplitude of the noise have a highly relevant influence in the system, leading to an optimal coherent response for suitable values of both the noise amplitude and correlation time. This phenomenon is quantitatively characterized by means of several statistical measures.

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.

Zero-lag synchronization and bubbling in delay-coupled lasers.

We show experimentally that two semiconductor lasers mutually coupled via a passive relay fiber loop exhibit chaos synchronization at zero lag, and study how this synchronized regime is lost as the lasers' pump currents are increased. We characterize the synchronization properties of the system with high temporal resolution in two different chaotic regimes, namely, low-frequency fluctuations and coherence collapse, identifying significant differences between them. In particular, a marked decrease in synchronization quality develops as the lasers enter the coherence collapse regime. Our high-resolution measurements allow us to establish that synchronization loss is associated with bubbling events, the frequency of which increases with increasing pump current.

Language organization and temporal correlations in the spiking activity of an excitable laser: experiments and model comparison.

We introduce a method, based on symbolic analysis, to characterize the temporal correlations of the spiking activity exhibited by excitable systems. The technique is applied to the experimentally observed dynamics of a semiconductor laser with optical feedback operating in the low-frequency fluctuations regime, where the laser intensity displays irregular trains of sudden dropouts that can be interpreted as excitable pulses. Symbolic analysis transforms the series of interdropout time intervals into sequences of words, which represent the local ordering of a certain (small) number of those intervals. We then focus on the transition probabilities between pairs of words, showing that certain transitions are overrepresented (resulting in others being underrepresented) with respect to the surrogate series, provided the laser injection current is above a critical value. These experimental observations are in very good agreement with numerical simulations of the delay-differential Lang-Kobayashi model that is commonly used to describe this laser system, which supports the fact that the language organization reported here is generic and not a particular feature of the specific laser employed or the experimental time series analyzed. We also present results of simulations of the phenomenological nondelayed Eguia-Mindlin-Giudici(EMG) model and find that in this model the agreement between the experiments and the simulations is good at a qualitative, but not at a quantitative, level.