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The classical Kuramoto model consists of finitely many pairwise coupled oscillators on the circle. In many applications a simple pairwise coupling is not sufficient to describe real-world phenomena as higher-order (or group) interactions take place. Hence, we replace the classical coupling law with a very general coupling function involving higher-order terms. Furthermore, we allow for multiple populations of oscillators interacting with each other through a very general law. In our analysis, we focus on the characteristic system and the mean-field limit of this generalized class of Kuramoto models. While there are several works studying particular aspects of our program, we propose a general framework to work with all three aspects (higher-order, multi-population, and mean-field) simultaneously. Assuming identical oscillators in each population, we derive equations for the evolution of oscillator populations in the mean-field limit. First, we clarify existence and uniqueness of our set of characteristic equations, which are formulated in the space of probability measures together with the bounded-Lipschitz metric. Then, we investigate dynamical properties within the framework of the characteristic system. We identify invariant subspaces and stability of the state, in which all oscillators are synchronized within each population. Even though it turns out that this so called all-synchronized state is never asymptotically stable, under some conditions and with a suitable definition of stability, the all-synchronized state can be proven to be at least locally stable. In summary, our work provides a rigorous mathematical framework upon which the further study of multi-population higher-order coupled particle systems can be based., 35 pages, 0 figures

We develop a general theory of random walks on hypergraphs which includes, as special cases, the different models that are found in literature. In particular, we introduce and analyze general random walk Laplacians for hypergraphs, and we compare them to hypergraph normalized Laplacians that are not necessarily related to random walks, but which are motivated by biological and chemical networks. We show that, although these two classes of Laplacians coincide in the case of graphs, they appear to have important conceptual differences in the general case. We study the spectral properties of both classes, as well as their applications to Coupled Hypergraph Maps: discrete-time dynamical systems that generalize the well-known Coupled Map Lattices on graphs. Our results also show why for some hypergraph Laplacian variants one expects more classical results from (weighted) graphs to generalize directly, while these results must fail for other hypergraph Laplacians.

Many components of the Earth system are thought to be prone to dangerous transitions, presenting a big challenge for human societies. Known as tipping elements, those form an intricate network of interacting subsystems, creating the possibility of cascading critical transitions. The presence of those interacting tipping events makes it hard to predict the outcome of climate change. In this research, we investigate those phenomena above the usual approach of linearly interacting bistable components.We propose to study generic nonlinear systems under generic nonlinear interaction. As a first step, we focus on unilaterally coupled components, where a leading and a following subsystem are naturally identified. Using singular perturbation methods, we show how the stability landscape can be approached semi-analytically when considering the weak and strong coupling limit. With only limited knowledge about the system's structure, this method applies to a wide class of interacting systems and allows for approaching steady states with a controlled error. This provides information on important structural features of the bifurcation diagram such as the presence of steady branches, their stability, and bifurcations. Finally, we illustrate our results using climate relevant conceptual models.

Bistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has taken on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. We generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

Objectives Total ischemic time is considered a limiting factor in lung transplantation. In this retrospective study we investigate effects of ischemic time and disease burden on outcomes after bilateral lung transplantation. Methods 1,298 patients undergoing bilateral lung transplantation between January 2010 and May 2022 (Follow-up 100%, median 54 months) were included. Pre-transplant diseases ‘severity (recipient body mass index, recipient age, previous lung transplantation, Tacrolimus immunosuppression, preoperative recipient extracorporeal membrane oxygenation support, lung volume reduction) for graft failure was individually calculated and- as ischemic time- categorised. Vice-versa adjusted Cox models were calculated. Considering competing risks, we assessed cumulative incidences of airway obstructive complications and chronic lung allograft dysfunction with death as competing risk factors for primary graft dysfunction were assessed by binary logistic regression. Results Higher disease burden significantly accelerated chronic lung allograft dysfunction and death occurrence (p Conclusion The eventual graft survival disadvantage that results from an ischemic time between 7 and at least 11 hours is negligible in contrast to frequent recipients’ disease-based risk levels.

Donor shortages have led transplant centers to extend their criteria for lung donors. Accepting lung donors ≥70 years of age has previously shown good short-term outcomes; however, no mid- and long-term outcome data on these extended criteria donors has been published to date. In this study, all patients who underwent lung transplantation between 06/2010 and 12/2019 were included in the analysis, and the outcomes were compared between patients receiving organs from donors

OBJECTIVE Paediatric lung transplantation poses unique management challenges. Experience regarding indications and outcome is scarce, especially in younger children. The primary aim of this study was to investigate outcome after first lung transplantation in children

Bistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has taken on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau, Φ4, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. In this paper we generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain, the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

We review a range of reduction methods that have been, or may be useful for connecting models of the Earth's climate system of differing complexity. We particularly focus on methods where rigorous reduction is possible. We aim to highlight the main mathematical ideas of each reduction method and also provide several benchmark examples from climate modelling.

In this paper, we discuss an original covariance boundary-value problem that captures the effects of white noise on the local behavior near stable periodic orbits of the corresponding deterministic dynamical systems. The methodology relies on suitably constructed adjoint variables to project the dynamics onto locally transversal hyperplanes and ensure the existence of a unique solution. For each such hyperplane, the computed co-variance matrix describes an approximately Gaussian, stationary distribution that highlights directions of particular sensitivity to noise. In contrast to previous formulations, the boundary-value problem analyzed in this paper makes predictions in the original state-space variables rather than in terms of a reduced set of local coordinates for hyperplanes perpendicular to the local vector field. The formulation is compatible with the general form of an augmented continuation problem for the software package COCO; a non-adaptive version has been implemented as a general-purpose constructor for the periodic-orbit toolbox included in the COCO release. We illustrate the efficacy of the proposed formulation and toolbox by analyzing noise-induced behavior near limit cycles in two examples of nonlinear dynamical systems with autonomous drift terms. The analysis presented in this study can be used to design safe operating regimes for systems working in stochastic environments, as well as to design optimum working conditions for systems utilizing noise, such as energy-harvesting applications.

Venous-arterial extracorporeal membrane oxygenation (ECMO) is an established technique for intraoperative cardiopulmonary support in patients undergoing lung transplantation. Patients with pulmonary fibrosis have a higher risk to require it. The aim of this study was to identify risk factors for the need of intraoperative ECMO use.Records of patients undergoing lung transplantation for pulmonary fibrosis at our institution between January 2010 and May 2018 were retrospectively reviewed. Univariate logistic regression analysis was used for statistical identification of risk factors.There were 105 patients (34%) who required intraoperative ECMO support (ECMO+ group), and 203 (66%) did not (ECMO- group). Preoperative proof of pulmonary hypertension was identified as a risk factor for intraoperative ECMO support (odds ratio [OR], 3.8; 95% confidence interval [CI], 2.2-6.5; P.01). Revealed mean pulmonary arterial pressure values exceeding 50 mm Hg and pulmonary vascular resistance values exceeding 9.4 Wood units were identified as risk factors for the need of intraoperative ECMO use with a prediction probability of 70%. Increased recipient body surface area (OR, 0.2; 95% CI, 0.1-0.5; P.01) emerged as a protective factor against intraoperative ECMO (Hosmer-Lemeshow statistic, P = .71) as well as higher cardiac output (OR, 0.7; 95% CI, 0.6-0.9; P.01). The postoperative course was more complicated in the ECMO+ group, whereas survival at 5 years did not differ among groups (70% vs 69%, P = .79).Pulmonary hypertension with elevated pulmonary vascular resistance values predicts the need of intraoperative ECMO in patients receiving lung transplantation for pulmonary fibrosis. Although the postoperative course was more complicated in the ECMO+ group, long-term survival did not differ significantly.

Since the breakthrough in rough paths theory for stochastic ordinary differential equations (SDEs), there has been a strong interest in investigating the rough differential equation (RDE) approach and its numerous applications. Rough path techniques can stay closer to deterministic analytical methods and have the potential to transfer many pathwise ordinary differential equation (ODE) techniques more directly to a stochastic setting. However, there are few works that analyze dynamical properties of RDEs and connect the rough path / regularity structures, ODE and random dynamical systems approaches. Here we contribute to this aspect and analyze invariant manifolds for RDEs. By means of a suitably discretized Lyapunov-Perron-type method we prove the existence and regularity of local center manifolds for such systems. Our method directly works with the RDE and we exploit rough paths estimates to obtain the relevant contraction properties of the Lyapunov-Perron map., To appear in SIAM Journal on Mathematical Analysis

We study dynamic networks under an undirected consensus communication protocol and with one state-dependent weighted edge. We assume that the aforementioned dynamic edge can take values over the whole real numbers, and that its behaviour depends on the nodes it connects and on an extrinsic slow variable. We show that, under mild conditions on the weight, there exists a reduction such that the dynamics of the network are organized by a transcritical singularity. As such, we detail a slow passage through a transcritical singularity for a simple network, and we observe that an exchange between consensus and clustering of the nodes is possible. In contrast to the classical planar fast–slow transcritical singularity, the network structure of the system under consideration induces the presence of a maximal canard. Our main tool of analysis is the blow-up method. Thus, we also focus on tracking the effects of the blow-up transformation on the network’s structure. We show that on each blow-up chart one recovers a particular dynamic network related to the original one. We further indicate a numerical issue produced by the slow passage through the transcritical singularity.

Over the past decade, extracorporeal membrane oxygenation (ECMO) has replaced cardiopulmonary bypass (CPB) for cardiopulmonary support during lung transplantation at our institution. In this study, we present our experience using intraoperative ECMO in isolated lung transplantation and evaluate its impact on long-term graft function and survival.All patients undergoing isolated lung transplantation with or without ECMO support between January 2010 and June 2019 were evaluated. Patients transplanted using CPB were excluded. Peri-operative and follow-up results from our database and patient charts were analyzed. Follow-up continued until September 1, 2019 (median, 3.34 years).In total, 311 of 1,161 lung transplant recipients (27%) received intraoperative ECMO, with 24 (2%) patients further requiring CPB. None of the remaining 826 (71%) patients required intraoperative cardiopulmonary support. ECMO patients exhibited higher pre-transplant surgical risk profiles and endured more complicated early post-operative courses than those without ECMO (in-hospital mortality, 10.9% vs 2.3%; p0.001). Inevitably, this resulted in poorer overall graft survival among ECMO recipients (p = 0.0025). However, correcting for patients surviving to hospital discharge, no difference in survival between groups was observed (5-year survival, 71% vs 72%; p = 0.56). Similarly, freedom from chronic lung allograft dysfunction, biopsy-confirmed cellular rejection, or need for pulsed-steroid therapy did not differ between the groups (p = 0.99, p = 0.78, and p = 0.93, respectively).Compared with patients not requiring cardiopulmonary support, ECMO recipients endured a more complicated peri-operative and early post-operative course. However, among those surviving to hospital discharge, no differences in long-term complications or outcomes were observed.

Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the classical theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time showing that solutions can only be local-in-time for general quasilinear SPDEs, while they might be global-in-time for special subclasses of problems., Comment: 42 pages, preprint

Heart retransplantation remains a controversial issue, due to the overall shortage of donor organs. Many patients put on the waiting list for retransplantation, decompensate rapidly, and do not survive. The use of veno-arterial extracorporeal life support remains an option in such emergency situations as bridge-to-recovery or bridge-to-transplantation therapy. In peripheral femoral configuration, veno-arterial extracorporeal life support improves the patient’s condition by relieving low-cardiac output but immobilizes him or her for an uncertain period of time. The upper-body cannulation is an alternative approach, which allows to maintain the patient awake and mobile. We present two cases of midterm circulatory support as a bridge to heart retransplantation, using upper-body cannulation veno-arterial extracorporeal life support. Two male patients, presenting with progressive cardiac decompensation due to severe graft rejection, were placed on upper-body veno-arterial extracorporeal life support. The stabilization of hemodynamics and improvement of end-organ perfusion could be achieved after extracorporeal life support initiation. After 48 and 40 days, respectively, on extracorporeal life support with active physical therapy and no major adverse events, both patients received a cardiac retransplantation and were eventually discharged home. The presented cases are the first reported where a successful cardiac retransplant was performed following prolonged upper-body extracorporeal life support.

Background: Infection of vascular grafts is a life-threatening complication in cardiovascular surgical procedures. This experimental study tested the efficacy and possible harmful effects of daptom...

In this retrospective study, we analyzed the presence of any association of three CD4+ CD25high regulatory T-cell subpopulations at 3 weeks after lung transplantation with the later incidence of chronic lung allograft dysfunction and graft survival. Among lung-transplanted patients between January 2009 and April 2018, only patients with sufficient T-cell measurements at 3 weeks after transplantation were included into the study. Putative regulatory T cells were defined as CD4+ CD25high T cells, detected in peripheral blood and further analyzed for CD127low , FoxP3+ , and CD152+ using fluorescence-activated cell sorting (FACS) analysis. Associations of regulatory T cells with chronic lung allograft dysfunction (CLAD) and graft survival were evaluated using Cox analysis. During the study period, 724 (71%) patients were included into the study. Freedom from chronic lung allograft dysfunction (CLAD) and graft survival amounted to 66% and 68% at 5 years. At the multivariable analysis, increasing frequencies of CD127low were associated with better freedom from CLAD (hazard ratio for each 1% increase of %CD127low , HR = 0.989, 95% CI = 0.981-0.996, P = 0.003) and better graft survival (HR = 0.991, 95% CI = 0.984-0.999, P = 0.026). A higher frequency of CD127low regulatory T cells in peripheral blood early after lung transplantation estimated a protective effect against chronic lung allograft dysfunction, mortality, and re-transplantation.

A fundamental example of reaction-diffusion system exhibiting Turing type pattern formation is the Gierer-Meinhardt system, which reduces to the shadow Gierer-Meinhardt problem in a suitable singular limit. Thanks to its applicability in a large range of biological applications, this singularly perturbed problem has been widely studied in the last few decades via rigorous, asymptotic, and numerical methods. However, standard matched asymptotics methods do not apply (Ni 1998, Wei 1998), and therefore analytical expressions for single spike solutions are generally lacking. By introducing an ansatz based on generalized hyperbolic functions, we determine exact radially symmetric solutions to the one-dimensional shadow Gierer-Meinhardt problem for any $1 < p < \infty$, representing both inner and boundary spike solutions depending on the location of the peak. Our approach not only confirms numerical results existing in literature, but also provides guidance for tackling extensions of the shadow Gierer-Meinhardt problem based on different boundary conditions (e.g. mixed) and/or $n$-dimensional domains.

Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator’s phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions, including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean-field theory (Vlasov–Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean-field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence–coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean-field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.

Dynamics of large-scale network processes underlies crucial phenomena ranging across all sciences. Forward simulation of large network models is often computationally prohibitive. Yet, most networks have intrinsic community structure. We exploit these communities and propose a fast simulation algorithm for network dynamics. In particular, aggregating the inputs a node receives constitutes the limiting factor in numerically simulating large-scale network dynamics. We develop community integration algorithms (CIAs) significantly reducing function-evaluations. We obtain a substantial reduction from polynomial to linear computational complexity. We illustrate our results in multiple applications including classical and higher-order Kuramoto-type systems for synchronisation and Cucker--Smale systems exhibiting flocking behaviour on synthetic as well as real-world networks. Numerical comparison and theoretical analysis confirm the robustness and efficiency of CIAs., Comment: 12 pages, 6 figures

We study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.

The paper deals with the study of rate-induced tipping in asymptotically autonomous scalar ordinary differential equations. We prove that, in such a tipping scenario, a solution which limits at a hyperbolic stable equilibrium of the past limit-problem loses uniform asymptotic stability and coincides with a solution which limits at a hyperbolic unstable equilibrium of the future limit-problem. We use asymptotic series to approximate such pairs of solutions and characterize the occurrence of a rate-induced tipping by using only solutions calculable on finite time intervals. Moreover, we show that a Melnikov-inspired method employing the asymptotic series allows to asymptotically approximate the tipping point., Comment: 24 pages, 4 figures

By generating transient encounters between individuals beyond their immediate social environment, transport can have a profound impact on the spreading of an epidemic. In this work, we consider epidemic dynamics in the presence of the transport process that gives rise to a multiplex network model. In addition to a static layer, the (multiplex) epidemic network consists of a second dynamic layer in which any two individuals are connected for the time they occupy the same site during a random walk they perform on a separate transport network. We develop a mean-field description of the stochastic network model and study the influence the transport process has on the epidemic threshold. We show that any transport process generally lowers the epidemic threshold because of the additional connections it generates. In contrast, considering also random walks of fractional order that in some sense are a more realistic model of human mobility, we find that these non-local transport dynamics raise the epidemic threshold in comparison to a classical local random walk. We also test our model on a realistic transport network (the Munich U-Bahn network), and carefully compare mean-field solutions with stochastic trajectories in a range of scenarios., Version as to appear in the Journal of Mathematical Biology with revised figures

We analyze the influence of multiplayer interactions and network adaptation on the stability of equilibrium points in evolutionary games. We consider the Snowdrift game on simplicial complexes. In particular, we consider as a starting point the extension from only two-player interactions to coexistence of two- and three-player interactions. The state of the system and the topology of the interactions are both adaptive through best-response strategies of nodes and rewiring strategies of edges, respectively. We derive a closed set of low-dimensional differential equations using pairwise moment closure, which yields an approximation of the lower moments of the system. We numerically confirm the validity of these moment equations. Moreover, we demonstrate that the stability of the fixed points remains unchanged for the considered adaption process. This stability result indicates that rational best-response strategies in games are very difficult to destabilize, even if higher-order multiplayer interactions are taken into account.

Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has received considerable attention recently are travelling wave patterns occurring in stochastic partial differential equations (SPDEs), i.e., how deterministic travelling waves behave under stochastic perturbations. In this paper, we start the mathematical study of related class of problems: stochastic rotating waves generated by SPDEs. We combine deterministic dynamics PDE techniques with methods from stochastic analysis. We establish two different approaches, the variational phase and the approximated variational phase, for defining stochastic phase variables along the rotating wave, which track the effect of noise on neutral spectral modes associated to the special Euclidean symmetry group of rotating waves. Furthermore, we prove transverse stability results for rotating waves showing that over certain time scales and for small noise, the stochastic rotating wave stays close to its deterministic counterpart., 40 pages, preprint; comments and suggestions welcome

OBJECTIVES The management of severe coronary artery disease at the time of a lung transplant remains a challenge. We analysed the short- and long-term outcomes of lung transplant recipients with severe coronary artery disease. METHODS Records of adult patients who received transplants at our institution between April 2010 and February 2021 were reviewed retrospectively. Severe coronary artery disease was defined as coronary stenosis ≥70% (main stem ≥50%) seen on the coronary angiographic scans performed before or at the time of listing. Patient characteristics, perioperative and long-term outcomes were compared between patients with and without severe coronary artery disease. RESULTS Among 896 patients who received lung transplants who had undergone coronary angiography before the transplant, 77 (8.5%) had severe coronary artery disease; the remaining 819 (91.5%) did not. Patients with severe coronary artery disease were older (p CONCLUSIONS Severe coronary artery disease was not associated with decreased survival after a lung transplant. Concomitant coronary artery bypass graft surgery and pretransplant percutaneous transluminal coronary angioplasty can be used for revascularization.

OBJECTIVESThe wearable cardioverter defibrillator (WCD) is an established, safe, effective solution, protecting patients at risk of sudden cardiac death. We specifically investigated WCD use in cardiac surgery patients since data for this patient group are rare.METHODSRetrospective data analysis in 10 German cardiac surgery centres was performed. Cardiac surgery patients with left ventricular ejection fraction (LVEF) ≤35% or after implantable cardioverter defibrillator (ICD) explantation who received WCD between 2010 and 2020 were assessed using LifeVest Network data.RESULTSA total of 1168 patients with a median age of 66 years [interquartile range (IQR) 57–73] were enrolled; 87% were male. Clinical indications included coronary artery bypass grafting (43%), valve surgery (16%), combined coronary artery bypass graft/valve surgery (15%), ICD explantation (24%) and miscellaneous (2%). The median wear time of WCD was 23.4 h/day (IQR 21.7–23.8). A total of 106 patients (9.1%) exhibited ventricular tachycardia. A total of 93.2% of episodes occurred within the first 3 months. Eighteen patients (1.5%) received 26 adequate shocks. The inadequate shock rate was low (8 patients, 0.7%). LVEF improved from a median of 28% (IQR 22–32%) before WCD prescription to 35% (IQR 28–42%) during follow-up. Excluding ICD explant patients, 37% of patients received an ICD.CONCLUSIONSThe risk of sudden cardiac death is substantial within the first 3 months after cardiac surgery. Patients were protected effectively by WCD. Due to significant LVEF improvement, the majority did not require ICD implantation after WCD use. Compliance was high despite sternotomy. This multicentre experience confirms existing data regarding effectiveness, safety and compliance. Therefore, WCD should be considered in cardiac surgery patients with severely reduced LVEF.

Many science phenomena are described as interacting particle systems (IPS). The mean field limit (MFL) of large all-to-all coupled deterministic IPS is given by the solution of a PDE, the Vlasov Equation (VE). Yet, many applications demand IPS coupled on networks/graphs. It is interesting to know, how the limit of a sequence of digraphs associated with the IPS influences the macroscopic MFL. This paper studies VEs on a generalized digraph, regarded as limit of a sequence of digraphs, which we refer to as a digraph measure (DGM) to emphasize that we work with its limit via measures. We provide well-posedness and discretization of the solution of the VE by empirical distributions supported on solutions of an IPS via ODEs coupled on a sequence of digraphs converging to the given DGM. Our result extends existing results on one-dimensional Kuramoto-type networks coupled on dense graphs. Here we allow the underlying digraphs to be not necessarily dense which include many interesting graphical structures such that stars, trees and rings, which have been frequently used in many sparse network models in finance, telecommunications, physics, genetics, neuroscience, and social sciences. A key contribution of this paper is a nontrivial generalization of Neunzert's in-cell-particle approach for all-to-all coupled indistinguishable IPS with global Lipschitz continuity in Euclidean spaces to distinguishable IPS on heterogeneous digraphs with local Lipschitz continuity, via a measure-theoretic viewpoint. The approach together with the metrics is different from the known techniques in $L^p$-functions using graphons and their generalization via harmonic analysis of locally compact Abelian groups. Finally, to demonstrate the wide applicability, we apply our results to various models in higher-dimensional Euclidean spaces in epidemiology, ecology, and social sciences.

Since Noble adapted in 1962 the model of Hodgkin and Huxley to fit Purkinje fibres, the refinement of models for cardiomyocytes has continued. Most of these models are high-dimensional systems of coupled equations so that the possible mathematical analysis is quite limited, even numerically. This has inspired the development of reduced, phenomenological models that preserve qualitatively the main feature of cardiomyocyte’s dynamics. In this paper, we present a systematic comparison of the dynamics between two notable low-dimensional models, the FitzHugh–Nagumo model (FitzHugh in Bull Math Biophys 17:257–269, 1955, J Gen Physiol 43:867–896, 1960, Biophys J 1:445–466, 1961) as a prototype of excitable behaviour and a polynomial version of the Karma model (Karma in Phys Rev Lett 71(7):16, 1993, Chaos 4:461, 1994) which is specifically developed to fit cardiomyocyte’s behaviour well. We start by introducing the models and considering their pure ODE versions. We analyse the ODEs employing the main ideas and steps used in the setting of geometric singular perturbation theory. Next, we turn to the spatially extended models, where we focus on travelling wave solutions in 1D. Finally, we perform numerical simulations of the 1D PDE Karma model varying model parameters in order to systematically investigate the impact on wave propagation velocity and shape. In summary, our study provides a reference regarding key similarities as well as key differences of the two models.

Motivated by the normal form of a fast-slow ordinary differential equation exhibiting a pitchfork singularity we consider the discrete-time dynamical system that is obtained by an application of the explicit Euler method. Tracking trajectories in the vicinity of the singularity we show, how the slow manifold extends beyond the singular point and give an estimate on the contraction rate of a transition mapping. The proof relies on the blow-up method suitably adapted to the discrete setting where a key technical contribution are precise estimates for a cubic map in the central rescaling chart., 29 pages, 7 figures

We extend slow manifolds near a transcritical singularity in a fast-slow system given by the explicit Euler discretization of the corresponding continuous-time normal form. The analysis uses the blow-up method and direct trajectory-based estimates. We prove that the qualitative behaviour is preserved by a time-discretization with sufficiently small step size. This step size is fully quantified relative to the time scale separation. Our proof also yields the continuous-time results as a special case and provides more detailed calculations in the classical (or scaling) chart.

Background: Venoarterial extracorporeal membrane oxygenation support is a well-established tool in the care of severe refractory cardiac and respiratory failure. The application of this support may serve as a bridge to transplant, recovery or to implantation of a ventricular assist device. Venoarterial extracorporeal membrane oxygenation support can be administered through an open surgical access via the common femoral or axillary artery or a percutaneous approach using Seldinger technique. Both techniques may obstruct the blood flow to the lower limb and may cause a significant ischemia with possible limb loss. Malperfusion of the distal limb can be avoided using an ipsilateral distal limb perfusion, which may be established by adding a single-lumen catheter during venoarterial extracorporeal membrane oxygenation treatment to overcome the obstruction. The aim of this study is to distinguish the presence or absence of a distal limb perfusion regarding the incidence of distal limb ischemia. Furthermore, expected risk factors of open and percutaneous femoral venoarterial extracorporeal membrane oxygenation installation were evaluated for the development of distal limb ischemia. Methods: Between January 2012 and September 2015, 489 patients received venoarterial extracorporeal membrane oxygenation support at our institution. In total, 307 patients (204 male, 103 female) with femoral cannulation were included in the analysis. The cohort was distinguished by the presence (group A; n = 237) or absence (group B; n = 70) of a distal limb perfusion during peripheral venoarterial extracorporeal membrane oxygenation treatment. Furthermore, a risk factor analysis for the development of distal limb ischemia was performed. Results: The main indications for venoarterial extracorporeal membrane oxygenation therapy were a low cardiac output syndrome (LCOS) (53%) and failed weaning of extracorporeal circulation (23%). A total of 23 patients (7.49%) under venoarterial extracorporeal membrane oxygenation support developed severe distal limb malperfusion (3.38% in group A vs 21.42% in group B). Preemptive installation of distal limb perfusion extended the intervention-free intervals to 7.8 ± 19.3 days in group A and 6.3 ± 12.5 in group B. A missing distal limb perfusion (p = 0.001) was identified as a main risk factor for critical limb ischemia. Other comorbidities such as arterial occlusion disease (p = 0.738) were not statistically significantly associated. Surgical intervention due to vascular complications after extracorporeal membrane oxygenation explantation was needed in 14 cases (4.22% in group A and 5.71% in group B). Conclusion: We were able to identify the absence of distal limb perfusion as an independent risk factor for the development of critical distal limb ischemia during femoral venoarterial extracorporeal membrane oxygenation treatment. The application of a distal limb perfusion should be considered as a mandatory approach in the context of femoral venoarterial extracorporeal membrane oxygenation treatment regardless of the implantation technique.

In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $$\varepsilon $$ ε such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit $$\varepsilon $$ ε to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering $$\varepsilon $$ ε to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales.

7,7′-Disubstituted 2,2′-methylenedioxy-1,1′-binaphthyls are highly efficient chirality inducers in nematic liquid crystals. The absolute configuration of these compounds is, however, hard to determine as they only crystallize as racemic mixtures. In this work a Vibrational Circular Dichroism (VCD) study is reported that provides an unambiguous determination of the absolute configuration of these compounds. An in-depth General Coupled Oscillator (GCO) analysis of the source of the VCD signal reveals that the unusual structure of these binaphthyl compounds inherently leads to strong and robust VCD bands. Combined with linear transit calculations, our VCD studies allow for the determination of key structural parameters.

The existence of global-in-time bounded martingale solutions to a general class of cross-diffusion systems with multiplicative Stratonovich noise is proved. The equations describe multicomponent systems from physics or biology with volume-filling effects and possess a formal gradient-flow or entropy structure. This structure allows for the derivation of almost surely positive lower and upper bounds for the stochastic processes. The existence result holds under some assumptions on the interplay between the entropy density and the multiplicative noise terms. The proof is based on a stochastic Galerkin method, a Wong--Zakai type approximation of the Wiener process, the boundedness-by-entropy method, and the tightness criterion of Brze\'{z}niak and coworkers. Three-species Maxwell--Stefan systems and $n$-species biofilm models are examples that satisfy the general assumptions., Comment: To appear in Annales de l'Institut Henri Poincar\'e (B) Probabilit\'e s et Statistiques