Quantifying the complexity of the delayed logistic map

Masoller, Cristina; Rosso, Osvaldo A.

doi:10.1098/rsta.2010.0281

Cited by 19 publications

(7 citation statements)

References 51 publications

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“…There are some methodologic limitations to the analysis of ordinal patterns. For example, the occurrence of all possible ordinal patterns (put differently, the nonoccurrence of forbidden ordinal patterns) does not prove that the underlying process is fully random, because as a recent study demonstrates, a deterministic map with delay may produce all possible ordinal patterns (Masoller & Rosso, 2010). Therefore, if there are no forbidden patterns, the iEEG signals may still be partly due to deterministic neuronal activity.…”

Section: Discussionmentioning

confidence: 99%

Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

Schindler¹,

Gast²,

Stieglitz

et al. 2011

Epilepsia

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SUMMARYPurpose: Epileptic seizures typically reveal a high degree of stereotypy, that is, for an individual patient they are characterized by an ordered and predictable sequence of symptoms and signs with typically little variability. Stereotypy implies that ictal neuronal dynamics might have deterministic characteristics, presumably most pronounced in the ictogenic parts of the brain, which may provide diagnostically and therapeutically important information. Therefore the goal of our study was to search for indications of determinism in periictal intracranial electroencephalography (EEG) studies recorded from patients with pharmacoresistent epilepsy. Methods: We assessed the number of forbidden ordinal patterns of 110 periictal multichannel intracranial EEG studies of 16 patients. Ordinal patterns are derived from the rank order of short sequences of consecutive EEG values. Ordinal patterns are well suited for analyzing realworld time series, for they have low sensitivity for many forms of noise and are applicable to nonstationary data. Although Gaussian random dynamics generate all possible ordinal patterns for a given sequence length, deterministic dynamics typically manifest with less random and more regular signals that miss a certain number of all the possible ordinal patterns. These missing ordinal patterns are referred to as ''forbidden ordinal patterns.'' In this study, the number of forbidden ordinal patterns n fp of an EEG signal was interpreted as an indication of determinism, when it was larger than the number of forbidden patterns occurring in amplitude adjusted Fourier transform surrogates. We computed n fp for each EEG signal in a time-resolved way by using a moving-window approach. Then we specifically investigated n mean fp denoting the average number of forbidden patterns across all EEG signals, and n max fp , which represents the number of forbidden patterns occurring in the EEG signal with the largest n fp during the seizure-onset period. Key Findings: The average number of forbidden patterns of all EEG signals, n mean fp , typically first increased and then decreased during the seizures. However, these changes were not statistically significant relative to the preseizure time period. In contrast, n max fp typically increased significantly during the first third of the seizure period and then gradually decreased toward and beyond seizure termination. In those patients who became seizure free following surgery, a larger percentage of the EEG signals containing the maximal number of forbidden patterns during the seizure-onset period tended to be recorded from within the visually identified seizure-onset zones. Significance: Our findings demonstrate a spatiotemporally limited shift of neuronal dynamics toward a more deterministic dynamic regimen, specifically pronounced during the seizure-onset period. Assessing the number of forbidden ordinal patterns of intracranial EEG provides quantitative and observer-independent information. We propose that it is at least partially complementary to class...

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Section: Discussionmentioning

confidence: 99%

Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

Schindler¹,

Gast²,

Stieglitz

et al. 2011

Epilepsia

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“…The Jensen-Shannon divergence, which quantifies the difference between two (or more) probability distributions, is especially useful to compare the symbol-composition of different sequences [37]. Furthermore, the complexity-entropy plane, H × C JS , which represents the evolution of the complexity of the system as a function of its entropy, has been used to study changes in the dynamics of a system originated by modifications of some characteristic parameters (see, for instance, [12,15,[38][39][40] and references therein). The complexity measure constructed in this way has the intensive property found in many thermodynamic quantities [32].…”

Section: The Statistical Complexity and The Complexity-entropy Planementioning

confidence: 99%

Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review

Zanin

Zunino

Rosso

et al. 2012

Entropy

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552

440

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Entropy is a powerful tool for the analysis of time series, as it allows describing the probability distributions of the possible state of a system, and therefore the information encoded in it. Nevertheless, important information may be codified also in the temporal dynamics, an aspect which is not usually taken into account. The idea of calculating entropy based on permutation patterns (that is, permutations defined by the order relations among values of a time series) has received a lot of attention in the last years, especially for the understanding of complex and chaotic systems. Permutation entropy directly accounts for the temporal information contained in the time series; furthermore, it has the quality of simplicity, robustness and very low computational cost. To celebrate the tenth anniversary of the original work, here we analyze the theoretical foundations of the permutation entropy, Entropy 2012, 14 1554 as well as the main recent applications to the analysis of economical markets and to the understanding of biomedical systems.

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“…For future work, to shed light onto this issue, it would be interesting to analyse the polarization switching dynamics using nonlinear methods of time-series analysis, such as ordinal analysis [19,20]. Varying the laser pump current (a parameter that can be easily controlled experimentally) would allow exploration of the parameter region close to the laser threshold, where a stronger influence of noise could be expected, when compared with the region that has been studied so far, which is above twice the threshold [16][17][18].…”

Section: Discussionmentioning

confidence: 99%

Two-parameter study of square-wave switching dynamics in orthogonally delay-coupled semiconductor lasers

Masoller

Sciamanna

Gavrielides³

2013

Phil. Trans. R. Soc. A.

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One contribution of 15 to a Theme Issue 'Dynamics, control and information in delay-coupled systems' . We perform a detailed numerical analysis of square-wave (SW) polarization switching in two semiconductor lasers with time-delayed, orthogonal mutual coupling. An in-depth mapping of the dynamics in the two-parameter plane coupling strength versus frequency detuning shows that stable SWs occur in narrow parameter regions that are localized close to the boundary of stability of the pure-mode solution. In this steady state, the two coupled lasers emit orthogonal polarizations. We also show that there are various types of SW forms and that stable switching does not need the inclusion of noise or nonlinear gain in the model. As these narrow regions of deterministic and stable SWs occur for quite different combinations of parameters, they could potentially explain the waveforms that have been observed experimentally. However, on the other hand, these regions are narrow enough to be in fact considered as experimentally unreachable. Therefore, our results indicate that further experimental statistical studies are needed in order to distinguish deterministic and stationary square waveforms from long transients because of noise.

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Quantifying the complexity of the delayed logistic map

Cited by 19 publications

References 51 publications

Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review

Two-parameter study of square-wave switching dynamics in orthogonally delay-coupled semiconductor lasers

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