I. Aviram

Bacterial debris—an ecological mechanism for coexistence of bacteria and their viruses

A model of bacteria and phage survival is developed based on the idea of shielding by bacterial debris in the system. This model is mathematically formulated by a set of four nonlinear difference equations for susceptible bacteria, contaminated bacteria, bacterial debris and phages. Simulation results show the possibility of survival, and domains of existence of stable and unstable solutions

Bactria and Lytic Phage Coexistence in a Chemostat with Periodic Nutrient Supply

The dynamics of bacteria and bacteriophage coexistence was examined in a chemostat in which the externally driven supply of nutrient for bacteria, and washout rate oscillates periodically. The proposed mathematical model for three interacting variables, bacteria, phage, and nutrient, consists of 3 differential equations with time delay, due to the phage latent period of lysing. The study was carried out in an interval of physical parameters where an equivalent model with constant supply of nutrient and washout rate is mathematically unstable, running in limit cycle regimes, with known self-frequencies. It addresses mainly the asymptotically persistent dynamics of the system.Bifurcation maps in terms of two externally controlled parameters, the amplitude and frequency of the controlled nutrient supply were constructed for various latent lysis periods, in order to determine the frequency entrainment, i.e., the resulting main operating frequency of the system, relative to the known external and self-frequencies. Also presented are bifurcation maps for the rich variety of dynamical types observed in the study. Bifurcation diagrams in terms of the lysing time delay were also included for completion.A new type of entrainment, combining in a simple way the external and self-periods (reciprocal frequencies), is shown to exist for a range of parameters.

Singular value decomposition of optically-mapped cardiac rotors and fibrillatory activity

Our progress of understanding how cellular and structural factors contribute to the arrhythmia is hampered in part because of controversies whether a fibrillating heart is driven by a single, several, or multiple number of sources, and whether they are focal or reentrant, and how to localize them. Here we demonstrate how a novel usage of the neutral singular value decomposition (SVD) method enables the extraction of the governing spatial and temporal modes of excitation from a rotor and fibrillatory waves. Those modes highlight patterns and regions of organization in the midst of the otherwise seemingly-randomly propagating excitation waves. We apply the method to experimental models of cardiac fibrillation in rabbit hearts. We show that the SVD analysis is able to enhance the classification of the heart electrical patterns into regions harboring drivers in the form of fast reentrant activity and other regions of by-standing activity. This enhancement is accomplished without any prior assumptions regarding the spatial, temporal or spectral properties of those drivers. The analysis corroborates that the dominant mode has the highest activation rate and further reveals a new feature: A transfer of modes from the driving to the passive regions resulting in a partial reaction of the passive region to the driving region.

Dynamics of a spiral pair source and its interaction with plane waves

Spiral pair creation and dynamics is a widely occurring phenomenon in nature. It can appear in the heart tissue, causing severe arrhythmia, known as a figure-eight reentry. We consider the appearance of a spiral pair source, its minimal strength for survival, and the possible results of its interaction with a plane wave. In particular, its ability to outlast such an encounter is of interest. We also consider the question of exposing the source to a train of pulses, in terms of the frequency and angle of encounter. Results show different regimes of behavior, e.g. source annihilation, motion of the source away from, or towards the origin of the plane waves, its breaking and multiplication. Relevance of these results to heart arrhythmia and their possible cancellation by external pacing are briefly discussed.

The Weiss–Lapicque and the Lapicque–Blair strength—duration curves revisited

The Weiss–Lapicque and Lapicque–Blair relations connecting the strength of a stimulation pulse and its duration in order to attain a threshold are examined here in the case of the nonlinear FitzHugh–Nagumo (FHN) model system. The relation parameters can easily be derived and explained by analyzing the dynamical behavior at very short and very long pulse durations. This explanation should hold true for most nonlinear biological systems. It is seen that, for the FHN, both relations provide only approximations (albeit good ones) to the actual strength-duration curve.

New Generation Methods of Spiral-Pairs and 3D Patterns

Reaction-Diffusion systems of nonlinear partial differential equations are used to model quite a few scientific phenomena. We have used a specific system, the FitzHugh-Nagumo one, to study new methods of generating spiral pairs and spatiotemporal periodic structures in 2D, and 3D repeat ing scroll rings in excitable media. Results are important in different fields, the electrical information transfer and arr hythmias in the heart, chemical reactions such as the Belousov-Zhabotinsky ones and possibly abnormal brain waves.

Causality analysis of leading singular value decomposition modes identifies rotor as the dominant driving normal mode in fibrillation

Cardiac fibrillation is a major clinical and societal burden. Rotors may drive fibrillation in many cases, but their role and patterns are often masked by complex propagation. We used Singular Value Decomposition (SVD), which ranks patterns of activation hierarchically, together with Wiener-Granger causality analysis (WGCA), which analyses direction of information among observations, to investigate the role of rotors in cardiac fibrillation. We hypothesized that combining SVD analysis with WGCA should reveal whether rotor activity is the dominant driving force of fibrillation even in cases of high complexity. Optical mapping experiments were conducted in neonatal rat cardiomyocyte monolayers (diameter, 35 mm), which were genetically modified to overexpress the delayed rectifier K+ channel IKr only in one half of the monolayer. Such monolayers have been shown previously to sustain fast rotors confined to the IKr overexpressing half and driving fibrillatory-like activity in the other half. SVD analysis of the optical mapping movies revealed a hierarchical pattern in which the primary modes corresponded to rotor activity in the IKr overexpressing region and the secondary modes corresponded to fibrillatory activity elsewhere. We then applied WGCA to evaluate the directionality of influence between modes in the entire monolayer using clear and noisy movies of activity. We demonstrated that the rotor modes influence the secondary fibrillatory modes, but influence was detected also in the opposite direction. To more specifically delineate the role of the rotor in fibrillation, we decomposed separately the respective SVD modes of the rotor and fibrillatory domains. In this case, WGCA yielded more information from the rotor to the fibrillatory domains than in the opposite direction. In conclusion, SVD analysis reveals that rotors can be the dominant modes of an experimental model of fibrillation. Wiener-Granger causality on modes of the rotor domains confirms their preferential driving influence on fibrillatory modes.

Reentry as an Origin for Rotors

The aim of the study is to understand in depth the meaning of “reentry”, and to decipher if and how it can lead to malfunctions of the heart and possibly of the brain. A simple model is used to reveal the mechanism by which a single pulse of action potential rotating around a ring of excitable medium, the latter simulating a reentry circuit, can generate spirals (single and/or double) when the pulse can emerge from and develop outside the ring. Two mechanisms of spiral generation are demonstrated: (1) a mechanism in which a source of single spirals is created at the contact with the core soon after the pulse freeing action, their chirality being due to the sense of the preceding pulse rotation. Interestingly, these spirals, adhering to the core, become “double-spiral patterns” while leaving behind the seeds of the new single spirals. (2) A second possible mechanism, similar to the known “arms encountering methods”, in which a double spiral (a figure of eight) is repeatedly created on the other side of the core. Similar procedures are assumed to occur in the heart, leading to tachycardia and fibrillation and possibly in the brain leading to epilepsy. The exact processes of the hitherto assumed spiral generations by reentry were established. The novel deep understanding of the mechanisms involved in these processes can lead to new methods of treating heart fibrillation (e.g., by judicial ablation).

A simple simulation model for spiral induced epilepsy

regionwas intact, structuralMRI datawere obtained during the pretreatment scanning session. To establish whether tDCS supports re-recruitment of LH structures, lateralisation indices (LI) will be calculated as a measure for the relative contributionof the left and righthemisphere. LIswill be compared for patients receiving tDCS versus patients receiving sham, both pre and post treatment. We will unveil the significantly activated brain areas during both paradigms, in relation to behavioural performance on these language tasks. Results: At present, the analysis is ongoing. Results will be available in March 2017. Discussion: The results of this study will improve our understanding of tDCS-induced language reorganisation, and will contribute to the ongoing discussion among aphasiologists on the roles of the RH and LH in poststroke language reorganisation.

Maximizing yields of virulent phage: the T4/Escherichia coli system as a test case.

A hybrid mathematical model was devised to obtain optimal values for bacterial doubling time and initial phage/bacteria multiplicity of infection for the purpose of reaching the highest possible phage titers in steady-state exponentially growing cultures. The computational model consists of an initial probabilistic stage, followed by a second one processed by a system of delayed differential equations. The models approach can be used in any phage/bacteria system for which the relevant parameters have been measured. Results of a specific case, based on the detailed, known information about the interactions between virulent T4 phage and its host bacterium Escherichia coli, display a range of possible such values along a highlighted strip of parameter values in the relevant parameter plane. In addition, times to achieve these maxima and gains in phage concentrations are evaluated.