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Error hypersurfaces are very valuable to study because of their unique status in multilayer perceptron research. Given the architecture of a multilayer perceptron, if the pattern sets are different, so are the respective error hypersurfaces in the multilayer perceptron. Using the theory of groups and Polya Theorem, this paper constructs classes of congruent pattern sets and classes of congruent error hypersurfaces, and proves that the number of classes of congruent pattern sets is equal to the number of congruent error hypersurfaces. Calculation results lead to much fewer classes of congruent error hypersurfaces than the total error hypersurfaces, and show that as the input dimension N increases, the former number increases at a much lower rate than the latter number, thus simplifying the understanding of the complexity of classes of error hypersurfaces.

Organized brain activity is the result of dynamical, segregated neuronal signals that may be used to investigate synchronization effects using sophisticated neuroengineering techniques. Phase synchrony analysis, in particular, has emerged as a promising methodology to study transient and frequency-specific coupling effects across multi-site signals. In this study, we investigated phase synchronization in intracellular recordings of interictal and ictal epileptiform events recorded from pairs of cells in the whole (intact) mouse hippocampus. In particular, we focused our analysis on the background noise-like activity (NLA), previously reported to exhibit complex neurodynamical properties. Our results show evidence for increased linear and nonlinear phase coupling in NLA across three frequency bands [theta (4–10 Hz), beta (12–30 Hz) and gamma (30–80 Hz)] in the ictal compared to interictal state dynamics. We also present qualitative and statistical evidence for increased phase synchronization in the theta, beta and gamma frequency bands from paired recordings of ictal NLA. Overall, our results validate the use of background NLA in the neurodynamical study of epileptiform transitions and suggest that what is considered "neuronal noise" is amenable to synchronization effects in the spatiotemporal domain.

A complex network approach is combined with time dynamics in order to conduct a space–time analysis applicable to longitudinal studies aimed to characterize the progression of Alzheimer's disease (AD) in individual patients. The network analysis reveals how patient-specific patterns are associated with disease progression, also capturing the widespread effect of local disruptions. This longitudinal study is carried out on resting electroence phalography (EEGs) of seven AD patients. The test is repeated after a three months' period. The proposed methodology allows to extract some averaged information and regularities on the patients' cohort and to quantify concisely the disease evolution. From the functional viewpoint, the progression of AD is shown to be characterized by a loss of connected areas here measured in terms of network parameters (characteristic path length, clustering coefficient, global efficiency, degree of connectivity and connectivity density). The differences found between baseline and at follow-up are statistically significant. Finally, an original topographic multiscale approach is proposed that yields additional results.

Background: Transcranial magnetic stimulation combined with electroencephalogram (TMS-EEG) can be used to explore the dynamical state of neuronal networks. In patients with epilepsy, TMS can induce epileptiform discharges (EDs) with a stochastic occurrence despite constant stimulation parameters. This observation raises the possibility that the pre-stimulation period contains multiple covert states of brain excitability some of which are associated with the generation of EDs. Objective: To investigate whether the interictal period contains "high excitability" states that upon brain stimulation produce EDs and can be differentiated from "low excitability" states producing normal appearing TMS-EEG responses. Methods: In a cohort of 25 patients with Genetic Generalized Epilepsies (GGE) we identified two subjects characterized by the intermittent development of TMS-induced EDs. The high-excitability in the pre-stimulation period was assessed using multiple measures of univariate time series analysis. Measures providing optimal discrimination were identified by feature selection techniques. The "high excitability" states emerged in multiple loci (indicating diffuse cortical hyperexcitability) and were clearly differentiated on the basis of 14 measures from "low excitability" states (accuracy = 0.7). Conclusion: In GGE, the interictal period contains multiple, quasi-stable covert states of excitability a class of which is associated with the generation of TMS-induced EDs. The relevance of these findings to theoretical models of ictogenesis is discussed.

A multivariate sample entropy metric of signal complexity is applied to EEG data recorded when subjects were viewing four prior-labeled emotion-inducing video clips from a publically available, validated database. Besides emotion category labels, the video clips also came with arousal scores. Our subjects were also asked to provide their own emotion labels. In total 30 subjects with age range 19–70 years participated in our study. Rather than relying on predefined frequency bands, we estimate multivariate sample entropy over multiple data-driven scales using the multivariate empirical mode decomposition (MEMD) technique and show that in this way we can discriminate between five self-reported emotions ($p<0.05$). These results could not be obtained by analyzing the relation between arousal scores and video clips, signal complexity and arousal scores, and self-reported emotions and traditional power spectral densities and their hemispheric asymmetries in the theta, alpha, beta, and gamma frequency bands. This shows that multivariate, multiscale sample entropy is a promising technique to discriminate multiple emotional states from EEG recordings.

Purpose: The main aim of this paper is to study the influence of temperature on multiscale entropy (MSE) and multiscale time irreversibility (MTI) through the use of short-term measurements. Methods: A total of 12 physically active, healthy, and nonsmoker individuals ($25.6\pm 3.9$ years old; $174.2\pm 7.5$cm of height; and $68.6\pm 11.1$kg of body mass) voluntarily participated in this study. Two beat-to-beat recordings of 15min length were performed on every participant, one under hot conditions ($35^{\circ }$C) and the other assessment under cool conditions ($19^{\circ }$C). The order of these two assessments was randomly assigned. Multiscale sample entropy and MTI were assessed in every measurement through 10 scales. Results: Entropy was significantly higher under hot conditions ($p<0.05$) from the fifth scale compared to cool conditions. On the contrary, MTI values were significantly lower under hotter conditions ($p<0.05$). Conclusions: The study of MSE and time irreversibility of short RR measurements presents consistent and reliable data. Moreover, exposures to hot conditions provoke an increment of interbeat complexity throughout larger scales and a decrease in the MTI in a healthy population.

This work aims to analyze the complexity of surface electromyography (sEMG) signals under muscle fatigue conditions using Hjorth parameters and bubble entropy (BE). Signals are recorded from the biceps brachii muscle of 25 healthy males during dynamic and isometric contraction exercises. These signals are filtered and segmented into 10 equal parts. The first and tenth segments are considered as nonfatigue and fatigue conditions, respectively. Activity, mobility, complexity, and BE features are extracted from both segments and classified using support vector machine (SVM), Naïve bayes (NB), k-nearest neighbor (kNN), and random forest (RF). The results indicate a reduction in signal complexity during fatigue. The parameter activity is found to increase under fatigue for both dynamic and isometric contractions with mean values of 0.35 and 0.22, respectively. It is observed that mobility, complexity, and BE are lowest during fatigue for both contractions. Maximum accuracy of 95.00% is achieved with the kNN and Hjorth parameters for dynamic signals. It is also found that the reduction of signal complexity during fatigue is more significant in dynamic contractions. This study confirms that the extracted features are suitable for analyzing the complex nature of sEMG signals. Hence, the proposed approach can be used for analyzing the complex characteristics of sEMG signals under various myoneural conditions.

Using the laser speckle contrast imaging and wavelet-based analyses, we investigate a latent (a "hidden") stage of the development of intracranial hemorrhages (ICHs) in newborn rats. We apply two measures based on the continuous wavelet-transform of blood flow velocity in the sagittal sinus, namely, the spectral energy in distinct frequency ranges and a multiscality degree characterizing complexity of experimental data. We show that the wavelet-based multifractal formalism reveals changes in the cerebrovascular blood flow at the development of ICH.

Based on the laser speckle contrast imaging (LSCI) and the multiscale entropy (MSE), we study in this work the blood flow dynamics at the levels of cerebral veins and the surrounding network of microcerebral vessels. We discuss how the phenylephrine-related acute peripheral hypertension is reflected in the cerebral circulation and show that the observed changes are scale-dependent, and they are significantly more pronounced in microcerebral vessels, while the macrocerebral dynamics does not demonstrate authentic inter-group distinctions. We also consider the permeability of blood–brain barrier (BBB) and study its opening caused by sound exposure. We show that alterations associated with the BBB opening can be revealed by the analysis of blood flow at the level of macrocerebral vessels.

All complex life on Earth is composed of ‘eukaryotic’ cells. Eukaryotes arose just once in 4 billion years, via an endosymbiosis — bacteria entered a simple host cell, evolving into mitochondria, the ‘powerhouses’ of complex cells. Mitochondria lost most of their genes, retaining only those needed for respiration, giving eukaryotes ‘multi-bacterial’ power without the costs of maintaining thousands of complete bacterial genomes. These energy savings supported a substantial expansion in nuclear genome size, and far more protein synthesis from each gene.

The efficiency of complex industrialized farming systems are compared to that of natural environmental systems while taking into account economic and environmental benefit as well as the needs of farmers and cattle.

We have recently started to understand that fundamental aspects of complex systems such as emergence, the measurement problem, inherent uncertainty, complex causality in connection with unpredictable determinism, time-irreversibility and non-locality all highlight the observer’s participatory role in determining their workings. In addition, the principle of ‘limited universality’ in complex systems, which prompts us to ‘search for the appropriate level of description in which unification and universality can be expected’, looks like a version of Bohr’s ‘complementarity principle’. It is more or less certain that the different levels of description possible of a complex whole — actually partial objectifications — are projected on to and even redefine its constituent parts. Thus it is interesting that these fundamental complexity issues don’t just bear a formal resemblance to, but reveal a profound connection with, quantum mechanics. Indeed, they point to a common origin on a deeper level of description.

Some critical issues about the relative portability of models and solutions across disciplinary barriers are discussed. The risks linked to the use of models and theories coming from different disciplines are evidentiated with a particular emphasis on biology. A metaphorical use of conceptual tools coming from other fields is suggested, together with the unescapable need to judge about the relative merits of a model on the basis of the amount of facts relative to the particular domain of application it explains.

Some examples of metaphorical modeling coming from biochemistry and psychobiology are briefly discussed in order to clarify the above positions.

Logical difference is an important notion to distinct different versions of knowledge-base systems. To capture such difference in terms of logic consequence, clause consequence and prime clause consequence respectively, this paper proposes three notions of difference over relevant signatures — logical difference, clausal difference and prime difference. They are closely related to forgetting. It is generally intractable to compute such differences, even for Horn theories. Preliminary experimental results on clausal difference and prime difference illustrate an interesting phase transition phenomenon over random 3-CNF theories.

All complex life on Earth is composed of ‘eukaryotic’ cells. Eukaryotes arose just once in 4 billion years, via an endosymbiosis — bacteria entered a simple host cell, evolving into mitochondria, the ‘powerhouses’ of complex cells. Mitochondria lost most of their genes, retaining only those needed for respiration, giving eukaryotes ‘multi-bacterial’ power without the costs of maintaining thousands of complete bacterial genomes. These energy savings supported a substantial expansion in nuclear genome size, and far more protein synthesis from each gene.

The unique dynamical features of the critical state can endow the brain with properties which are fundamental for adaptive behavior. This proposal, put forward with Per Bak several years ago, is now supported by a wide body of empirical evidence at different scales demonstrating that the spatiotemporal brain dynamics exhibits key signatures of critical dynamics previously recognized in other complex systems. The rationale behind this program is discussed in these notes, followed by an account of the most recent results, together with a discussion of the physiological significance of these ideas.

We describe herein a multidisciplinary research, as it developes and applies concepts of the theory of complexity, in turn stemming from recent advancements of statistical physics, onto cognitive neuroscience. We discuss (define) complexity, and how the human brain is a paradigm of it. We discuss how the hypothesis of brain activity dynamically behaving as a critical system is taking momentum in literature, then we focus on a feature of critical systems (hence of the brain), which is the intermittent passage between metastable states, marked by events, locally resetting the memory, but giving rise to correlation functions with infinite correlation times. The events, extracted from multi-channel ElectroEncephaloGrams, mark (are interpreted as) a birth/death process of cooperation, namely of system elements being recruited into collective states. Finally we discuss a recently discovered form of control (in the form of a new Linear Response Theory), that allows an optimized information transmission between complex systems, named Complexity Matching.