On the Complexity of Finite Sequences

Lempel, A.; Ziv, J.

doi:10.1109/tit.1976.1055501

Cited by 2,321 publications

(1,687 citation statements)

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“…This is an alternative to the analysis based on firing rate methods. It was shown that Lempel-Ziv complexity as defined in Lempel and Ziv (1976) can be used successfully as such an estimator (Amigó et al 2004).…”

Section: The Entropy Rate and The Lempel-ziv Estimatormentioning

confidence: 99%

“…The entropy rate of a locally stationary neural signal at time t is then estimated in the time window [t, t + T ]. We will use for this purpose the normalized Lempel-Ziv 76 complexity (Lempel and Ziv 1976), a technique we applied and tested in Amigó et al (2004). Precisely, the explicit analysis of locally stationary neural signals (along with the introduction of RMI) is perhaps the most important aspect of this article.…”

Section: Introductionmentioning

confidence: 99%

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Mutual information and redundancy in spontaneous communication between cortical neurons

et al. 2011

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An important question in neural information processing is how neurons cooperate to transmit information. To study this question, we resort to the concept of redundancy in the information transmitted by a group of neurons and, at the same time, we introduce a novel concept for measuring cooperation between pairs of neurons called relative mutual information (RMI). Specifically, we studied these two parameters for spike trains generated by neighboring neurons from the primary visual cortex in the awake, freely moving rat. The spike trains studied here were spontaneously generated in the cortical network, in the absence of visual stimulation. Under these conditions, our analysis revealed that while the value of RMI oscillated slightly around an average value, the redundancy exhibited a behavior character- ized by a higher variability. We conjecture that this combination of approximately constant RMI and greater variable redundancy makes information transmission more resistant to noise disturbances. Furthermore, the redundancy values suggest that neurons can cooperate in a flexible way during information transmission. This mostly occurs via a leading neuron with higher transmission rate or, less frequently, through the information rate of the whole group being higher than the sum of the individual information rates-in other words in a synergetic manner. The proposed method applies not only to the stationary, but also to locally stationary neural signals.

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Section: The Entropy Rate and The Lempel-ziv Estimatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mutual information and redundancy in spontaneous communication between cortical neurons

et al. 2011

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“…In addition, time series need to be stationary, something that is usually not true with physiological signals. With these limitations in mind, Lempel-Ziv Complexity (LZC), a complexity estimator introduced by Lempel and Ziv (1976), has been proposed for EEG/MEG signals analysis. The LZC is a metric that, similar to the algorithmic complexity, reflects the number of distinct substrings and the rate of their recurrence along the given sequence (Radhakrishnan & Gangadhar, 1998).…”

Section: Introductionmentioning

confidence: 99%

“…If the length of the sequence is n and the number of different symbols is α, it has been proved (Lempel and Ziv, 1976) that the upper bound of c(n) is given by:…”

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confidence: 99%

Brain oscillatory complexity across the life span

Fernández

Zuluaga

Abásolo

et al. 2012

Clinical Neurophysiology

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Objective: Considering the increasing use of complexity estimates in neuropsychiatric populations, a normative study is critical to define the 'normal' behaviour of brain oscillatory complexity across the life span. Method:This study examines changes in resting-state magnetoencephalogram (MEG) complexity -quantified with the Lempel-Ziv complexity (LZC) algorithm -due to age and gender in a large sample of 222 (100 males/122 females) healthy participants with ages ranging from 7 to 84 years.Results: A significant quadratic (curvilinear) relationship (p < 0.05) between age and complexity was found, with LZC maxima being reached by the sixth decade of life.Once that peak was crossed, complexity values slowly decreased until late senescence.Females exhibited higher LZC values than males, with significant differences in the anterior, central and posterior regions (p < 0.05). Conclusions:These results suggest that the evolution of brain oscillatory complexity across the life span might be considered a new illustration of a 'normal' physiological rhythm.Significance: Previous and forthcoming clinical studies using complexity estimates might be interpreted from a more complete and dynamical perspective. Pathologies not 3 only cause an 'abnormal' increase or decrease of complexity values but they actually 'break' the 'normal' pattern of oscillatory complexity evolution as a function of age.Keywords: Life Span, Ageing, Complexity, Brain Development, White Matter Development Highlights 1. A significant quadratic (curvilinear) relationship between age and oscillatory complexity exists, with complexity maxima reached by the sixth decade of life.2. As in previous studies, females exhibit higher complexity values than males, at least in some brain regions.3. The evolution of oscillatory complexity across the life span is interpreted as a physiological rhythm which is altered by several brain pathologies.

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“…In particular, the complexity as defined by Lempel and Ziv (1976) counts the number of new patterns along a discrete sequence, time series or, in more physical terms, digital signal. A related quantity, the normalized complexity, provides a lower bound for the compression ratio of the signal by optimal coding (Ziv and Lempel, 1978), so that the higher the normalized complexity of a discrete signal, the more information it conveys.…”

Section: Introductionmentioning

confidence: 99%

On the number of states of the neuronal sources

et al. 2003

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In a previous paper (Proceedings of the World Congress on Neuroinformatics (2001)) the authors applied the socalled Lempel Á/Ziv complexity to study neural discharges (spike trains) from an information-theoretical point of view. Along with other results, it is shown there that this concept of complexity allows to characterize the responses of primary visual cortical neurons to both random and periodic stimuli. To this aim we modeled the neurons as information sources and the spike trains as messages generated by them. In this paper, we study further consequences of this mathematical approach, this time concerning the number of states of such neuronal information sources. In this context, the state of an information source means an internal degree of freedom (or parameter) which allows outputs with more general stochastic properties, since symbol generation probabilities at every time step may additionally depend on the value of the current state of the neuron. Furthermore, if the source is ergodic and Markovian, the number of states is directly related to the stochastic dependence lag of the source and provides a measure of the autocorrelation of its messages. Here, we find that the number of states of the neurons depends on the kind of stimulus and the type of preparation ( in vivo versus in vitro recordings), thus providing another way of differentiating neuronal responses. In particular, we observed that (for the encoding methods considered) in vitro sources have a higher lag than in vivo sources for periodic stimuli. This supports the conclusion put forward in the paper mentioned above that, for the same kind of stimulus, in vivo responses are more random (hence, more difficult to compress) than in vitro responses and, consequently, the former transmit more information than the latter. #

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On the Complexity of Finite Sequences

Cited by 2,321 publications

References 4 publications

Mutual information and redundancy in spontaneous communication between cortical neurons

Mutual information and redundancy in spontaneous communication between cortical neurons

Brain oscillatory complexity across the life span

On the number of states of the neuronal sources

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