ABSTRACT
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.
ABSTRACT
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.
