In the present work we intend to investigate how to detect the behaviour of the immune system reaction to an external stimulus in terms of phase transitions. The immune model considered follows Jerne’s idiotypic network theory. We considered two graph complexity measures—the connectivity entropy and the approximate von Neumann entropy—and one entropy for topological spaces, the so-called persistent entropy. The simplicial complex is obtained enriching the graph structure of the weighted idiotypic network, and it is formally analyzed by persistent homology and persistent entropy. We obtained numerical evidences that approximate von Neumann entropy and persistent entropy detect the activation of the immune system. In addition, persistent entropy allows also to identify the antibodies involved in the immune memory

Complex networks equipped with topological data analysis are one of the promising tools in the study of biological systems (e.g. evolution dynamics, brain correlation, breast cancer diagnosis, etc…). In this paper, we propose jHoles, a new version of Holes, an algorithms based on persistent homology for studying the connectivity features of complex networks. jHoles fills the lack of an efficient implementation of the filtering process for clique weight rank homology. We will give a brief overview of Holes, a more detailed description of jHoles algorithm, its implementation and the problem of clique weight rank homology. We present a biological case study showing how the connectivity of epidermal cells changes in response to a tumor presence. The biological network has been derived from the proliferative, differentiated and stratum corneum compartments, and jHoles used for studying variation of the connectivity

In this paper, we propose a methodology for deriving a model of a complex system by exploiting the information extracted from topological data analysis. Central to our approach is the S[B] paradigm in which a complex system is represented by a two-level model. One level, the structural S one, is derived using the newly-introduced quantitative concept of persistent entropy, and it is described by a persistent entropy automaton. The other level, the behavioral B one, is characterized by a network of interacting computational agents. The presented methodology is applied to a real case study, the idiotypic network of the mammalian immune system.

ObjectiveAn innovative method based on topological data analysis is introduced for classifying EEG recordings of patients affected by epilepsy. We construct a topological space from a collection of EEGs signals using Persistent Homology; then, we analyse the space by Persistent entropy, a global topological feature, in order to classify healthy and epileptic signals.ResultsThe performance of the resulting one-feature-based linear topological classifier is tested by analysing the Physionet dataset. The quality of classification is evaluated in terms of the Area Under Curve (AUC) of the receiver operating characteristic curve. It is shown that the linear topological classifier has an AUC equal to while the performance of a classifier based on Sample Entropy has an AUC equal to 62.0%.

Objective: Laser ablation is a therapeutic modality used to reduce the volume of large benign thyroid nodules. Unsatisfactory reduction and regrowth are observed in some treated nodules. The aim of the study was to evaluate the long-term outcomes of laser treatment for solid nodules during a 5-year follow-up period, the regrowth rate, and the predictive risk factors of nodule regrowth. Materials and Methods: We retrospectively evaluated patients with benign, solid, cold thyroid nodules who underwent laser ablation and were followed-up for 5 years. According to the selection criteria, 104 patients were included (median baseline nodule volume, 12.5 mL [25.0-75.0%, 8-18 mL]; median energy delivered, 481.5 J/mL [25.0-75.0%, 370-620 J/mL]). Nodule volume, thyroid function test results, and ultrasound were evaluated at baseline and then annually after the procedure. Results: Of 104 patients, 31 patients (29.8%) had a 12-month volume reduction ratio (VRR) < 50.0% and 39 (37.5%) experienced nodule regrowth. Of these 39 patients, 17 (43.6%) underwent surgery and 14 (35.9%) underwent a second laser treatment. The rate of nodule regrowth was inversely related to the 12-month VRR, i.e., the lower the 12-month VRR, the higher the risk of regrowth (p < 0.001). The mean time for nodule regrowth was 33.5 ± 16.6 months. The 12-month VRR was directly related to time to regrowth, i.e., the lower the 12-month VRR, the shorter the time to regrowth (p < 0.001; R 2 = 0.3516). Non-spongiform composition increased the risk of regrowth with an odds ratio of 4.3 (95% confidence interval [CI] 1.8-10.2; p < 0.001); 12-month VRR < 50.0% increased the risk of regrowth with an odds ratio of 11.7 (95% CI 4.2-32.2; p < 0.001). Conclusion: The VRR of thyroid nodules subjected to similar amounts of laser energy varies widely and depends on the nodule composition; non-spongiform nodules are reduced to a lesser extent and regrow more frequently than spongiform nodules. A 12-month VRR < 50.0% is a predictive risk factor for regrowth and correlates with the time to regrowth.

In this paper we present a novel methodology based on a topological entropy, the so-called persistent entropy,\ud for addressing the comparison between discrete piecewise linear functions. The comparison is certified by the\ud stability theorem for persistent entropy that is presented here. The theorem is used in the implementation of a\ud new algorithm. The algorithm transforms a discrete piecewise linear function into a filtered simplicial complex\ud that is analyzed via persistent homology and persistent entropy. Persistent entropy is used as a discriminant\ud feature for solving the supervised classification problem of real long-length noisy signals of DC electrical motors.\ud The quality of classification is stated in terms of the area under receiver operating characteristic curve\ud (AUC=93.87%)

Persistent homology studies the evolution of k-dimensional holes along a nested sequence of simplicial complexes (called a filtration). The set of bars (i.e. intervals) representing birth and death times of k-dimensional holes along such sequence is called the persistence barcode. k-Dimensional holes with short lifetimes are informally considered to be "topological noise", and those with long lifetimes are considered to be "topological features" associated to the filtration. Persistent entropy is defined as the Shannon entropy of the persistence barcode of a given filtration. In this paper we present new important properties of persistent entropy ofČech and Vietoris-Rips filtrations. Among the properties, we put a focus on the stability theorem that allows to use persistent entropy for comparing persistence barcodes. Later, we derive a simple method for separating topological noise from features in Vietoris-Rips filtrations.

In this paper we propose a methodology based on Topogical\ud Data Analysis (TDA) for capturing when a complex system,\ud represented by a multivariate time series, changes its inter-\ud nal organization. The modication of the inner organization\ud among the entities belonging to a complex system can induce\ud a phase transition of the entire system. In order to identify\ud these reorganizations, we designed a new methodology that\ud is based on the representation of time series by simplicial\ud complexes. The topologization of multivariate time series\ud successfully pinpoints out when a complex system evolves.\ud Simplicial complexes are characterized by persistent homo-\ud logy techniques, such as the clique weight rank persistent\ud homology and the topological invariants are used for com-\ud puting a new entropy measure, the so-called weighted per-\ud sistent entropy. With respect to the global invariants, e.g.\ud the Betti numbers, the entropy takes into account also the\ud topological noise and then it captures when a phase transi-\ud tion happens in a system. In order to verify the reliability of\ud the methodology, we have analyzed the EEG signals of Phy-\ud sioNet database and we have found numerical evidences that\ud the methodology is able to detect the transition between the\ud pre-ictal and ictal states