Critical Cell Number Research Articles

Spatiotemporal summation of perimetric stimuli in healthy observers.

Spatial summation of perimetric stimuli has been used to derive conclusions about the spatial extent of retinal-cortical convergence, mostly from the size of the critical area of summation (Ricco's area, RA) and critical number of retinal ganglion cells (RGCs). However, spatial summation is known to change dynamically with stimulus duration. Conversely, temporal summation and critical duration also vary with stimulus size. Such an important and often neglected spatiotemporal interaction has important implications for modeling perimetric sensitivity in healthy observers and for formulating hypotheses for changes measured in disease. In this work, we performed experiments on visually heathy observers confirming the interaction of stimulus size and duration in determining summation responses in photopic conditions. We then propose a simplified computational model that captures these aspects of perimetric sensitivity by modelling the total retinal input, the combined effect of stimulus size, duration, and retinal cones-to-RGC ratio. We additionally show that, in the macula, the enlargement of RA with eccentricity might not correspond to a constant critical number of RGCs, as often reported, but to a constant critical total retinal input. We finally compare our results with previous literature and show possible implications for modeling disease, especially glaucoma.

The quasi‐static crush response of electron‐beam‐melt Ti6Al4V body‐centred‐cubic lattices: The effect of the number of cells, strut diameter and face sheet

AbstractThe effect of the number of cells, strut diameter and face sheet on the compression of electron‐beam‐melt (EBM) Ti6Al4V (Ti64) body‐centred‐cubic (BCC) lattices was investigated experimentally and numerically. The lattices with the same relative density (~0.182) were fabricated with and without 2‐mm‐thick face sheets in 10 and 5 mm cell size, 8–125 unit cell (two to five cells/edge) and 2 and 1 mm strut diameter. The experimental compression tests were further numerically simulated in the LS‐DYNA. Experimentally two bending‐dominated crushing modes, namely, lateral and diagonal layer crushing, were determined. The numerical models however exhibited merely a bending‐dominated lateral layer crushing mode when the erosion strain was 0.4 and without face‐sheet models showed a diagonal layer crushing mode when the erosion strain was 0.3. Lower erosion strains promoted a diagonal layer crushing mode by introducing geometrical inhomogeneity to the lattice, leading to strain localisation as similar to the face sheets which introduced extensive strut bending in the layers adjacent to the face sheets. The face‐sheet model showed a higher but decreasing collapse strength at an increasing number of cells, just as opposite to the without face‐sheet model, and the collapse strength of both models converged when the number of cells was higher than five‐cell/edge. The decrease/increase of the collapse strengths of lattices before the critical number of cells was claimed mainly due to the size‐imposed lattice boundary condition, rather than the specimen volume. The difference in the experimental collapse strengths between the 5‐ and the 10‐mm cell‐size lattices was ascribed to the variations in the microstructures—hence the material model parameters between the small‐diameter and the large‐diameter EBM‐Ti64 strut lattices.

Microcrater-Arrayed Chemiluminescence Cell Chip to Boost Anti-Cancer Drug Administration in Zebrafish Tumor Xenograft Model.

Simple SummaryPersonalized drug cocktails have been widely implemented in cancer treatment, due to their merits of using a drug synergistic combination rather than a single effector. Through the incorporation of a cell microarray chip, the usage of both cell amount and reagent can be optimized and effectively reduced in cost. Herein, we present a laser direct writing strategy to fabricate a microcrater-arrayed (µCA) chip incorporated with an automatic liquid handling platform. Each cell droplet with a critical volume of 200 nL containing 100 SK-N-DZ or MCF-7 cells was utilized. The drug synergy screening took less than 30 s for a total of 81 tests. The results show that the potent drug prediction of the µCA chip is more accurate than the conventional 96-well plate assay, which are all compared using zebrafish DiI-labelled tumor xenograft models. Taken together, these findings may impact high-throughput drug screening and personalized drug medicine.Purpose: The aim of this study was to develop a rapid and automatic drug screening platform using microcrater-arrayed (µCA) cell chips. Methods: The µCA chip was fabricated using a laser direct writing technique. The fabrication time required for one 9 × 9 microarray wax chip was as quick as 1 min. On a nanodroplet handling platform, the chip was pre-coated with anti-cancer drugs, including cyclophosphamide, cisplatin, doxorubicin, oncovin, etoposide, and 5-fluorouracil, and their associated mixtures. Cell droplets containing 100 SK-N-DZ or MCF-7 cells were then loaded onto the chip. Cell viability was examined directly through a chemiluminescence assay on the chip using the CellTiter-Glo assay. Results: The time needed for the drug screening assay was demonstrated to be less than 30 s for a total of 81 tests. The prediction of optimal drug synergy from the µCA chip was found by matching it to that of the zebrafish MCF-7 tumor xenograft model, instead of the conventional 96-well plate assay. In addition, the critical reagent volume and cell number for each µCA chip test were 200 nL and 100 cells, respectively, which were significantly lower than 100 µL and 4000 cells, which were achieved using the 96-well assay. Conclusion: Our study for the µCA chip platform could improve the high-throughput drug synergy screening targeting the applications of tumor cell biology.

A numerical study on thermal management of a lithium-ion battery module via forced-convective air cooling

• Air-cooled thermal management of battery module has been explored numerically. • Thermal performance for inline and staggered cell arrangement was examined. • Negligible effect of cell arrangement occurs beyond a critical number of cells. • For small number of cells, the staggered arrangement slightly improves thermal uniformity. Global environmental concerns over increased emissions from the transportation vehicles have driven research instincts towards development of clean energy vehicles. Pure and hybrid electric vehicles coupled with lithium-ion batteries for energy storage are environmental friendly options to realize the future dream of low carbon emissions. However, an efficient battery thermal management system (BTMS) is a prerequisite for safe operation of such high voltage batteries to avoid thermal runaway and degradation, thereby maintaining the temperature uniformity in the battery pack. Thus, the present work numerically investigates the performance of an air cooled battery module for efficient removal of heat generated during the discharge process at different values of air-flow velocity, cell configuration and the number of cells in the battery module. A coupled electrochemical-thermal model is employed to model the discharge behaviour of the battery in conjunction with the two-dimensional flow and heat transfer model to monitor the temperature in and out of each battery cell. A correlation is developed for the maximum temperature in the module as a function of the number of cells in the system. The flow configuration (inline or staggered arrangement) of cells in the stack is observed to have no influence over the velocity and thermal field beyond a critical value of number of cells in the module.

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

A MATHEMATICAL MODEL FOR TREATMENT OF PAPILLARY THYROID CANCER USING THE ALLEE EFFECT

The incidence of thyroid cancer is rising all over the world, and the papillary subtype (PTC) is the primary factor for this increase. The presence of thyroid tumors is commonly associated with increased levels of cytokines, such as interleukin 6 (IL-6). Considering PTC patients treated with thyroidectomy and radioactive iodine [Formula: see text]I (RAI), we propose an ordinary differential system using four variables: the RAI activity, the number of cancer cells and the serum concentrations of IL-6 and thyroglobulin (Tg). Our objective is to study the efficacy of different therapeutic doses of RAI in the treatment of thyroid cancer. The Allee effect is taken into account when modeling cancer cells growth under the influence of IL-6. From the results obtained, the main factors and conditions correlated with successful treatment, such as the RAI activity used and the tumor response are addressed. The detection of minimal doses of RAI that can cause tumor extinction is performed, though this has also meant longer periods for tumor cell elimination. The critical number of tumor cells due to the Allee effect is analyzed and linked to the immune system or biological factors that can slow the progression of the tumor but are insufficient after thyroid resection surgery.

Design and simulation of a high efficiency InGaP/GaAs multi junction solar cell with AlGaAs tunnel junction

In this work we designed and simulated an InGaP/GaAs multi junction solar cell with AlGaAs tunneling junction. We optimized the efficiency of solar cell by changing Window layer and adding back-surface field layers. The simulations are done by TCAD Silvaco software and the results are obtained by considering AM1.5 spectrum. By considering critical number of cells and also by optimizing the layer parameters, very high operation factors for multi junction solar cells are obtained. Dividing the tunneling zone to six layers, and using InAlGaP for back-surface field and window layer, the efficiency 35.5% was achieved. Finally, by increasing thickness of back-surface field layer the efficiency increased up to 36.24%. The open circuit voltage, short circuit current, and fill factor of proposed cell in the final design are 2.45 V, 17.41 mA/cm2, and 88.70% respectively. All of the layers are lattice matched and the device can be fabricated by nowadays technology.

Optimization of culture conditions for short-term maintenance, proliferation, and colony formation of porcine gonocytes

BackgroundGonocytes give rise to spermatogonial stem cells, and thereby play an essential role in establishing spermatogenesis. Optimized culture conditions for gonocytes provide an opportunity for their study and in vitro manipulation for potential application in reproductive technologies. Using six experiments in a step-wise design, we examined the effects of several culture conditions on the maintenance, proliferation, and colony formation of porcine gonocytes. Testis cells from neonatal piglets were cultured for 7 d in DMEM supplemented with 10% fetal bovine serum. The examined culture conditions included using different cell seeding densities, gonocyte proportions, incubation temperatures, sampling strategies, and medium changing regimens.ResultsConfluency of cells was optimal (>90% by ~6 d) when 3.0 × 104 testis cells/cm2 containing ~40% gonocytes were used. Incubating the cells at 35 °C or 37 °C resulted in similar cell number and viability at confluency, but incubation at 35 °C resulted in a delayed confluency. In the first 2 d of culture, gonocytes remained mostly floating in the medium and gradually settled over the next 5 d. Consequently, not changing the medium for 7 d (as opposed to changing it every 2 d) led to a significant increase in the number of gonocyte colonies by reducing the loss of “floating gonocytes”.ConclusionWe found that gonocytes require the presence of a critical minimum number of somatic cells for settlement, and can proliferate and form growing colonies even in a basic medium. Large numbers of viable gonocytes remain floating in the medium for several days. The optimized culture conditions in the present study included seeding with 3.0 × 104 testis cells/cm2 containing ~40% gonocytes, incubating at 37 °C, and without changing the medium in the first week, which can result in improved colony formation of porcine gonocytes.

Marangoni convection instability in a sessile droplet with low volatility on heated substrate

In order to investigate Marangoni convection in a sessile droplet on heated substrate, a series of three-dimensional numerical simulations are carried out in a wide range of Marangoni number. A low volatile silicone oil is adopted and the effect of evaporation is neglected. Bénard-Marangoni convection and thermocapillary convection may coexist in droplet because of its curve surface. When Marangoni number is low the flow is steady axisymmetric thermocapillary convection whereas the flow becomes to be asymmetric and loses its stability and Hopf bifurcation occurs when Marangoni number high. The oscillatory transition convection, steady Bénard-Marangoni convection and irregular oscillatory Bénard-Marangoni convection appear in sequence with increasing Marangoni number. The characteristics of each kind of convection modes are described and the critical Marangoni numbers for the incipience of them are determined in a wide range of contact angles. One of the most interesting phenomenon is that the shape of the Bénard-Marangoni convection cell in sessile droplet is circular, which is different from that of classical hexagonal cell in flat layer. With increasing contact angle, the critical Marangoni number for the onset of convection instabilities first decreases and then increases. Critical cell number of the steady Bénard-Marangoni convection keeps on only one for contact angle less than 15° whereas it increases sharply after angle lager than 15°. The numerical results are qualitatively verified by experiment.

Extremal Examples of Collapsible Complexes and Random Discrete Morse Theory

We present extremal constructions connected with the property of simplicial collapsibility. (1) For each \(d \ge 2\), there are collapsible (and shellable) simplicial d-complexes with only one free face. Also, there are non-evasive d-complexes with only two free faces (both results are optimal in all dimensions). (2) Optimal discrete Morse vectors need not be unique. We explicitly construct a contractible, but non-collapsible 3-dimensional simplicial complex with face vector \(f=(106,596,1064,573)\) that admits two distinct optimal discrete Morse vectors, (1, 1, 1, 0) and (1, 0, 1, 1). Indeed, we show that in every dimension \(d\ge 3\) there are contractible, non-collapsible simplicial d-complexes that have \((1,0,\dots ,0,1,1,0)\) and \((1,0,\dots ,0,0,1,1)\) as distinct optimal discrete Morse vectors. (3) We give a first explicit example of a (non-PL) 5-manifold, with face vector \(f=(5013,72300,290944,\) 495912, 383136, 110880), that is collapsible but not homeomorphic to a ball. Furthermore, we discuss possible improvements and drawbacks of random approaches to collapsibility and discrete Morse theory. We will introduce randomized versions random-lex-first and random-lex-last of the lex-first and lex-last discrete Morse strategies of Benedetti and Lutz (Exp Math 23(1):66–94, 2014), respectively—and we will see that in many instances the random-lex-last strategy works significantly better than Benedetti–Lutz’s (uniform) random strategy. On the theoretical side, we prove that after repeated barycentric subdivisions, the discrete Morse vectors found by randomized algorithms have, on average, an exponential (in the number of barycentric subdivisions) number of critical cells asymptotically almost surely.

Cytokine-induced killer cells hunt individual cancer cells in droves in a mouse model.

Cytotoxicity of cytokine-induced killer (CIK) cells depends mainly on their encounters with target cells, but how many CIK cells are required to kill an individual cancer cell is unknown. Here we used time-lapse imaging to quantify the critical effector cell number required to kill an individual target cell. CIK cells killed MHC-I-negative and MHC-I-positive cancer cells, but natural killer (NK) cells destroyed MHC-I-negative cells only. The average threshold number of CIK cells required to kill an individual cancer cell was 6.7 for MHC-I-negative cells and 6.9 for MHC-I-positive cells. That of NK cells was 2.4 for MHC-I-negative cells. Likely due to the higher threshold numbers, killing by CIK cells was delayed in comparison with NK cells: 40% of MHC-negative target cells were killed after 5h when co-cultured with CIK cells and after 2h with NK cells. Our data have implications for the rational design of CIK cell-based immunotherapy of cancer patients.

Reduced Efficacy of the KCC2 Cotransporter Promotes Epileptic Oscillations in a Subiculum Network Model.

Ion regulation in the brain is a major determinant of neural excitability. Intracellular chloride in neurons, a partial determinant of the resting potential and the inhibitory reversal potentials, is regulated together with extracellular potassium via kation chloride cotransporters. During temporal lobe epilepsy, the homeostatic regulation of intracellular chloride is impaired in pyramidal cells, yet how this dysregulation may lead to seizures has not been explored. Using a realistic neural network model describing ion mechanisms, we show that chloride homeostasis pathology provokes seizure activity analogous to recordings from epileptogenic brain tissue. We show that there is a critical percentage of pathological cells required for seizure initiation. Our model predicts that restoration of the chloride homeostasis in pyramidal cells could be a viable antiepileptic strategy.

Decreases in Gap Junction Coupling Recovers Ca2+ and Insulin Secretion in Neonatal Diabetes Mellitus, Dependent on Beta Cell Heterogeneity and Noise.

Diabetes is caused by dysfunction to β-cells in the islets of Langerhans, disrupting insulin secretion and glucose homeostasis. Gap junction-mediated electrical coupling between β-cells in the islet plays a major role in coordinating a pulsatile secretory response at elevated glucose and suppressing insulin secretion at basal glucose. Previously, we demonstrated that a critical number of inexcitable cells can rapidly suppress the overall islet response, as a result of gap junction coupling. This was demonstrated in a murine model of Neonatal Diabetes Mellitus (NDM) involving expression of ATP-insensitive KATP channels, and by a multi-cellular computational model of islet electrical activity. Here we examined the mechanisms by which gap junction coupling contributes to islet dysfunction in NDM. We first verified the computational model against [Ca2+] and insulin secretion measurements in islets expressing ATP-insensitive KATP channels under different levels of gap junction coupling. We then applied this model to predict how different KATP channel mutations found in NDM suppress [Ca2+], and the role of gap junction coupling in this suppression. We further extended the model to account for stochastic noise and insulin secretion dynamics. We found experimentally and in the islet model that reductions in gap junction coupling allow progressively greater glucose-stimulated [Ca2+] and insulin secretion following expression of ATP-insensitive KATP channels. The model demonstrated good correspondence between suppression of [Ca2+] and clinical presentation of different NDM mutations. Significant recoveries in [Ca2+] and insulin secretion were predicted for many mutations upon reductions in gap junction coupling, where stochastic noise played a significant role in the recoveries. These findings provide new understanding how the islet functions as a multicellular system and for the role of gap junction channels in exacerbating the effects of decreased cellular excitability. They further suggest novel therapeutic options for NDM and other monogenic forms of diabetes.

Molecular biology of endometriosis: from aromatase to genomic abnormalities.

Endometriosis has been initially described as the presence of ectopic endometrial tissue on pelvic organs or in extrapelvic sites; and this has been used as its key pathologic feature ever since. Endometriosis responds to fluctuations in estrogen and progesterone by growth and inflammation, leading to pain aggravated by menses. It was proposed that pelvic endometriosis primarily originate from retrograde menstruation of a critical number of eutopic endometrial cells with stem characteristics. This postulate is supported by the molecular defects found in ectopic endometriotic tissue. Genome-wide differences in CpG methylation between eutopic endometrial and endometriotic stromal cells are present. Defective CpG methylation affecting several genes that encode key transcription factors such as GATA6, steroidogenic factor-1, and estrogen receptor-β in endometriosis gives rise to overproduction of local estrogen and prostaglandins and suppression of progesterone receptor. Progesterone receptor deficiency leads to progesterone resistance, resulting in decreased retinol uptake and retinoic acid production and altered retinoic acid action. These molecular defects collectively give rise to poor cellular differentiation, enhanced survival, and increased inflammation, which are the biological hallmarks of endometriotic tissue.

Searching combinatorial optimality using graph-based homology information

This paper analyses the topological information of a digital object \(O\) under a combined combinatorial-algebraic point of view. Working with a topology-preserving cellularization \(K(O)\) of the object, algebraic and combinatorial tools are jointly used. The combinatorial entities used here are vector fields, \(V\)-paths and directed graphs. In the algebraic side, chain complexes with extra \(2\)-nilpotent operators are considered. By mixing these two perspectives we are able to explore the problems of combinatorial and homological optimality. Combinatorial optimality is understood here as the problem for constructing a discrete gradient vector field (DGVF) in the sense of Discrete Morse Theory, such that it has the least possible number of critical cells. Fixing \({\mathbb {Z}}/2{\mathbb {Z}}\) as field of coefficients, by homological ‘optimality’ we mean the problem of constructing a \(2\)-nilpotent codifferential map \(\phi :C_*(K(O)) \rightarrow C_{*+1}(K(O))\) for finite linear combinations of cells in \(K(O)\), called homology integral operator. The homology groups associated to the chain complex \((C(K(O)), \phi )\) are isomorphic to those of \((C(K(O)), \partial )\), being \(\partial \) the canonical boundary or differential operator of the cell complex \(K(O)\). Relations between these two problems are tackled here by using a type of discrete graphs associated to a homology integral operator, called Homological Spanning Forests (HSF for short). Informally, an HSF for a cell complex can be seen as a kind of combinatorial compressed representation of a homology integral operator. As main result, we refine the heuristic for computing DGVFs based on the iterative Morse complex reduction technique of [1], reducing the search space for an optimal DGVF to an HSF associated to a homology integral operator.

A novel method for isolation of epithelial cells from ovine esophagus for tissue engineering

The yield of a critical number of basal epithelial cells with high mitotic rates from native tissue is a challenge in the field of tissue engineering. There are many protocols that use enzymatic methods for isolation of epithelial cells with unsatisfactory results for tissue engineering. This study aimed to develop a protocol for isolating a sufficient number of epithelial cells with a high Proliferating Index from ovine esophagus for tissue engineering applications. Esophageal mucosa was pretreated with dispase-collagenase solution and plated on collagen-coated culture dishes. Distinction of the various types of epithelial cells and developmental stages was done with specific primary antibodies to Cytokeratins and to Proliferating Cell Nuclear Antigen (PCNA). Up to approximately 8100 epithelial cells/mm2 of mucosa tissue were found after one week of migration. Cytokeratin 14 (CK 14) was positive identified in cells even after 83 days. At the same time the Proliferating Index was 71%. Our protocol for isolation of basal epithelial cells was successful to yield sufficient numbers of cells predominantly with proliferative character and without noteworthy negative enzymatic affection. The results at this study offer the possibility of generation critical cell numbers for tissue engineering applications.

A Perspective on the Role of the Extracellular Matrix in Progressive Retinal Degenerative Disorders

Progressive inherited retinal degenerative disorders (PIRDDs) are the leading cause of blindness in developed countries, with AMD and RP constituting the majority of PIRDDs. Currently, over 8 million Americans have PIRDDs, and that number is estimated to drastically increase by the end of this decade. Although a mutant protein is expressed starting early during retinal development in patients with PIRDDs, symptoms of retinal degeneration do not manifest until much later. Historically, research has focused on understanding the role a mutation has in the function of a protein and what role the mutant protein has in the disease process. However, it remains unknown why the disease, irrespective of the mutation, manifests clinically much later in life, while cellular indicators of disease (e.g., accumulation of toxic protein products and cell death) occur throughout early and middle life. Herein, we propose that there exists a time point at which the degenerative process is accelerated, leading to the appearance of clinical symptoms. This point is defined by structural disruptions of the extracellular matrix (ECM). Death of a critical number of ECM-maintaining mutant protein-expressing retinal cells contributes to that break point in the degenerative process. Therefore, it is important to understand the changes occurring at the ECM during PIRDDs and to take that into account when therapeutic approaches are designed.

The Effect of Cell Cluster Size on Intracellular Nanoparticle-Mediated Hyperthermia: Is It Possible to Treat Microscopic Tumors?

To compare the measured surface temperature of variable size ensembles of cells heated by intracellular magnetic fluid hyperthermia with heat diffusion model predictions. Starch-coated Bionized NanoFerrite (Micromod Partikeltechnologie GmbH, Rostock, Germany) iron oxide magnetic nanoparticles were loaded into cultured DU145 prostate cancer cells. Cell pellets of variable size were treated with alternating magnetic fields. The surface temperature of the pellets was measured in situ and the associated cytotoxicity was determined by clonogenic survival assay. For a given intracellular nanoparticle concentration, a critical minimum number of cells was required for cytotoxic hyperthermia. Above this threshold, cytotoxicity increased with increasing cell number. The measured surface temperatures were consistent with those predicted by a heat diffusion model that ignores intercellular thermal barriers. These results suggest a minimum tumor volume threshold of approximately 1 mm(3), below which nanoparticle-mediated heating is unlikely to be effective as the sole cytotoxic agent.

Synchronization of Early Afterdepolarizations and Arrhythmogenesis in Heterogeneous Cardiac Tissue Models

Early afterdepolarizations (EADs) are linked to both triggered arrhythmias and reentrant arrhythmias by causing premature ventricular complexes (PVCs), focal excitations, or heterogeneous tissue substrates for reentry formation. However, a critical number of cells that synchronously exhibit EADs are needed to result in arrhythmia triggers and substrates in tissue. In this study, we use mathematical modeling and computer simulations to investigate EAD synchronization and arrhythmia induction in tissue models with random cell-to-cell variations. Our major observations are as follows. Random cell-to-cell variations in action potential duration without EAD presence do not cause large dispersion of refractoriness in well-coupled tissue. In the presence of phase-2 EADs, the cells may synchronously exhibit the same number of EADs or no EADs with a very small dispersion of refractoriness, or synchronize regionally to result in large dispersion of refractoriness. In the presence of phase-3 EADs, regional synchronization leads to propagating EADs, forming PVCs in tissue. Interestingly, even though the uncoupled cells exhibit either no EAD or only a single EAD, when these cells are coupled to form a tissue, more than one PVC can occur. When the PVCs occur at different locations and time, multifocal arrhythmias are triggered, with the foci shifting in space and time in an irregular manner. The focal arrhythmias either spontaneously terminate or degenerate into reentrant arrhythmias due to heterogeneities and spatiotemporal chaotic dynamics of the foci.

Discrete Morse theory for manifolds with boundary

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain Relative Morse Inequalities relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman's Sphere Theorem. The main corollaries of our work are: (1) For each d ≥ 3 and for each k ≥ 0, there is a PL d-sphere on which any discrete Morse function has more than k critical (d -1)-cells. (This solves a problem by Chari.) (2) For fixed d and k, there are exponentially many combinatorial types of simplicial d-manifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most k critical interior (d -1)-cells. (This connects discrete Morse theory to enumerative combinatorics/ discrete quantum gravity.) (3) The barycentric subdivision of any simplicial constructible d-ball is collapsible. (This almost solves a problem by Hachimori.) (4) Every constructible ball collapses onto its boundary minus a facet. (This improves a result by the author and Ziegler.) (5) Any 3-ball with a knotted spanning edge cannot collapse onto its boundary minus a facet. (This strengthens a classical result by Bing and a recent result by the author and Ziegler.)