Dynamics of human atrial cell models: restitution, memory, and intracellular calcium dynamics in single cells

Mathematical models of cardiac cells have become important tools for investigating the electrophysiological properties and behavior of the heart. As the number of published models increases, it becomes more difficult to choose a model appropriate for the conditions to be studied, especially when multiple models describing the species and region of the heart of interest are available. In this paper, we will review and compare two detailed ionic models of human atrial myocytes, the Nygren et al. model (NM) and the Courtemanche et al. model (CM). Although both models include the same transmembrane currents and are largely based on the same experimental data from human atrial cells, the two models exhibit vastly different properties, especially in their dynamical behavior, including restitution and memory effects. The CM produces pronounced rate adaptation of action potential duration (APD) with limited memory effects, while the NM exhibits strong rate dependence of resting membrane potential (RMP), limited APD restitution, and stronger memory, as well as delayed afterdepolarizations and auto-oscillatory behavior upon cessation of rapid pacing. Channel conductance modifications based on experimentally measured changes during atrial fibrillation modify rate adaptation and memory in both models, but do not change the primary rate-dependent properties of APD and RMP for the CM and NM, respectively. Two sets of proposed changes to the NM that yield a spike-and-dome action potential morphology qualitatively similar to the CM at slow pacing rates similarly do not change the underlying dynamics of the model. Moreover, interchanging the formulations of all transmembrane currents between the two models while leaving calcium handling and ionic concentrations intact indicates that the currents strongly influence memory and the rate adaptation of RMP, while intracellular calcium dynamics primarily determine APD rate adaptation. Our results suggest that differences in intracellular calcium handling between the two human atrial myocyte models are responsible for marked dynamical differences and may prevent reconciliation between the models by straightforward channel conductance modifications.     

Discrete-time models with mosquitoes carrying genetically-modified bacteria

We formulate a homogeneous model and a stage-structured model for the interactive wild mosquitoes and mosquitoes carrying genetically-modified bacteria. We establish conditions for the existence and stability of fixed points for both models. We show that a unique positive fixed point exists and is asymptotically stable if the two boundary fixed points are both unstable. The unique positive fixed point exists and is unstable if the two boundary fixed points are both locally asymptotically stable. Using numerical examples, we demonstrate the models undergoing a period-doubling bifurcation.     

Locus dependence in epigenetic chromatin silencing

Current biological models of epigenetic switches built on chromatin modifications lead to strong constraints on the repertoire of dynamic behaviors for the system. We use the structure of the bifurcation diagram of the underlying dynamical system to explain the existing single cell data in silencing by the SIR system in yeast.     

Parametric dependence in model epidemics. I: Contact-related parameters

One of the interesting properties of nonlinear dynamical systems is that arbitrarily small changes in parameter values can induce qualitative changes in behavior. The changes are called bifurcations, and they are typically visualized by plotting asymptotic dynamics against a parameter. In some cases, the resulting bifurcation diagram is unique: irrespective of initial conditions, the same dynamical sequence obtains. In other cases, initial conditions do matter, and there are coexisting sequences. Here we study an epidemiological model in which multiple bifurcation sequences yield to a single sequence in response to varying a second parameter. We call this simplification the emergence of unique parametric dependence (UPD) and discuss how it relates to the model’s overall response to parameters. In so doing, we tie together a number of threads that have been developing since the mid-1980s. These include period-doubling; subharmonic resonance, attractor merging and subduction and the evolution of strange invariant sets. The present paper focuses on contact related parameters. A follow-up paper, to be published in this journal, will consider the effects of non-contact related parameters.     

Properties of two human atrial cell models in tissue: restitution, memory, propagation, and reentry

To date, two detailed ionic models of human atrial cell electrophysiology have been developed, the Nygren et al. model (NM) and the Courtemanche et al. model (CM). Although both models draw from similar experimental data, they have vastly different properties. This paper provides the first systematic analysis and comparison of the dynamics of these models in spatially extended systems including one-dimensional cables and rings, two-dimensional sheets, and a realistic three-dimensional human atrial geometry. We observe that, as in single cells, the CM adapts to rate changes primarily by changes in action potential duration (APD) and morphology, while for the NM rate changes affect resting membrane potential (RMP) more than APD. The models also exhibit different memory properties as assessed through S1-S2 APD and conduction velocity (CV) restitution curves with different S1 cycle lengths. Reentrant wave dynamics also differ, with the NM exhibiting stable, non-breaking spirals and the CM exhibiting frequent transient wave breaks. The realistic atrial geometry modifies dynamics in some cases through drift, transient pinning, and breakup. Previously proposed modifications to represent atrial fibrillation-remodeled electrophysiology produce altered dynamics, including reduced rate adaptation and memory for both models and conversion to stable reentry for the CM. Furthermore, proposed variations to the NM to reproduce action potentials more closely resembling those of the CM do not substantially alter the underlying dynamics of the model, so that tissue simulations using these modifications still behave more like the unmodified NM. Finally, interchanging the transmembrane current formulations of the two models suggests that currents contribute more strongly to RMP and CV, intracellular calcium dynamics primarily determine reentrant wave dynamics, and both are important in APD restitution and memory in these models. This finding implies that the formulation of intracellular calcium processes is as important to producing realistic models as transmembrane currents.     

Global dynamics and bifurcation analysis of a host–parasitoid model with strong Allee effect

In this paper, we study the global dynamics and bifurcations of a two-dimensional discrete time host–parasitoid model with strong Allee effect. The existence of fixed points and their stability are analysed in all allowed parametric region. The bifurcation analysis shows that the model can undergo fold bifurcation and Neimark–Sacker bifurcation. As the parameters vary in a small neighbourhood of the Neimark–Sacker bifurcation condition, the unique positive fixed point changes its stability and an invariant closed circle bifurcates from the positive fixed point. From the viewpoint of biology, the invariant closed curve corresponds to the periodic or quasi-periodic oscillations between host and parasitoid populations. Furthermore, it is proved that all solutions of this model are bounded, and there exist some values of the parameters such that the model has a global attractor. These theoretical results reveal the complex dynamics of the present model.     

The effect of predator limitation on the dynamics of simple food chains

We investigate the influence of competition between predators on the dynamics of bitrophic predator–prey systems and of tritrophic food chains. Competition between predators is implemented either as interference competition, or as a density-dependent mortality rate. With interference competition, the paradox of enrichment is reduced or completely suppressed, but otherwise, the dynamical behavior of the systems is not fundamentally different from that of the Rosenzweig–MacArthur model, which contains no predator competition and shows only continuous transitions between fixed points or periodic oscillations. In contrast, with density-dependent predator mortality, the system shows a surprisingly rich dynamical behavior. In particular, decreasing the density regulation of the predator can induce catastrophic shifts from a stable fixed point to a large oscillation where the predator chases the prey through a cycle that brings both species close to the threshold of extinction. Other catastrophic bifurcations, such as subcritical Hopf bifurcations and saddle-node bifurcations of limit cycles, do also occur. In tritrophic food chains, we find again that fixed points in the model with predator interference become unstable only through Hopf bifurcations, which can also be subcritical, in contrast to the bitrophic situation. The model with a density limitation shows again catastrophic destabilization of fixed points and various nonlocal bifurcations. In addition, chaos occurs for both models in appropriate parameter ranges.     

A model of beta-cell mass, insulin, and glucose kinetics: pathways to diabetes

Diabetes is a disease of the glucose regulatory system that is associated with increased morbidity and early mortality. The primary variables of this system are beta-cell mass, plasma insulin concentrations, and plasma glucose concentrations. Existing mathematical models of glucose regulation incorporate only glucose and/or insulin dynamics. Here we develop a novel model of beta -cell mass, insulin, and glucose dynamics, which consists of a system of three nonlinear ordinary differential equations, where glucose and insulin dynamics are fast relative to beta-cell mass dynamics. For normal parameter values, the model has two stable fixed points (representing physiological and pathological steady states), separated on a slow manifold by a saddle point. Mild hyperglycemia leads to the growth of the beta -cell mass (negative feedback) while extreme hyperglycemia leads to the reduction of the beta-cell mass (positive feedback). The model predicts that there are three pathways in prolonged hyperglycemia: (1) the physiological fixed point can be shifted to a hyperglycemic level (regulated hyperglycemia), (2) the physiological and saddle points can be eliminated (bifurcation), and (3) progressive defects in glucose and/or insulin dynamics can drive glucose levels up at a rate faster than the adaptation of the beta -cell mass which can drive glucose levels down (dynamical hyperglycemia).     

心肌细胞团搏动整数倍节律的非线性动力学机制

利用心肌细胞耦合模型研究心肌整数倍节律的动力学机理。确定性模型仿真揭示了心肌细胞团同步搏动加周期分岔的节律变化规律;随机模型仿真发现在加周期分岔序列中分岔点附近会出现整数倍节律,其中,0-1整数倍节律产生于从静息到周期1的Hopf分岔点附近,1-2整数倍节律产生于周期1和周期2极限环间的加周期分岔点附近;对系统相空间轨道的分析进一步揭示出整数倍节律是由系统运动在相邻的两个轨道之间随机跃迁形成的。上述分析结果不仅阐明了心肌整数倍节律的机理,并且揭示了各种整数倍节律与加周期分岔序列中相邻节律的内在联系,为重新认识心律变化的规律开辟了新的途径。     

Soliton behaviour in a bistable reaction diffusion model

We analyze a generic reaction-diffusion model that contains the important features of Turing systems and that has been extensively used in the past to model biological interesting patterns. This model presents various fixed points. Analysis of this model has been made in the past only in the case when there is only a single fixed point, and a phase diagram of all the possible instabilities shows that there is a place where a Turing-Hopf bifurcation occurs producing oscillating Turing patterns. In here we focus on the interesting situation of having several fixed points, particularly when one unstable point is in between two equally stable points. We show that the solutions of this bistable system are traveling front waves, or solitons. The predictions and results are tested by performing extensive numerical calculations in one and two dimensions. The dynamics of these solitons is governed by a well defined spatial scale, and collisions and interactions between solitons depend on this scale. In certain regions of parameter space the wave fronts can be stationary, forming a pattern resembling spatial chaos. The patterns in two dimensions are particularly interesting because they can present a coherent dynamics with pseudo spiral rotations that simulate the myocardial beat quite closely. We show that our simple model can produce complicated spatial patterns with many different properties, and could be used in applications in many different fields.      

Parametric dependence in model epidemics. I: contact-related parameters

One of the interesting properties of nonlinear dynamical systems is that arbitrarily small changes in parameter values can induce qualitative changes in behavior. The changes are called bifurcations, and they are typically visualized by plotting asymptotic dynamics against a parameter. In some cases, the resulting bifurcation diagram is unique: irrespective of initial conditions, the same dynamical sequence obtains. In other cases, initial conditions do matter, and there are coexisting sequences. Here we study an epidemiological model in which multiple bifurcation sequences yield to a single sequence in response to varying a second parameter. We call this simplification the emergence of unique parametric dependence (UPD) and discuss how it relates to the model's overall response to parameters. In so doing, we tie together a number of threads that have been developing since the mid-1980s. These include period-doubling; subharmonic resonance, attractor merging and subduction and the evolution of strange invariant sets. The present paper focuses on contact related parameters. A follow-up paper, to be published in this journal, will consider the effects of non-contact related parameters.     

On the relationship between the protein structure and protein dynamics

Recently, we have developed a method (Shih et al., Proteins: Structure, Function, and Bioinformatics 2007;68: 34-38) to compute correlation of fluctuations of proteins. This method, referred to as the protein fixed-point (PFP) model, is based on the positional vectors of atoms issuing from the fixed point, which is the point of the least fluctuations in proteins. One corollary from this model is that atoms lying on the same shell centered at the fixed point will have the same thermal fluctuations. In practice, this model provides a convenient way to compute the average dynamical properties of proteins directly from the geometrical shapes of proteins without the need of any mechanical models, and hence no trajectory integration or sophisticated matrix operations are needed. As a result, it is more efficient than molecular dynamics simulation or normal mode analysis. Though in the previous study the PFP model has been successfully applied to a number of proteins of various folds, it is not clear to what extent this model will be applied. In this article, we have carried out the comprehensive analysis of the PFP model for a dataset comprising 972 high-resolution X-ray structures with pairwise sequence identity or=0.5. Our result shows that the fixed-point model is indeed quite general and will be a useful tool for high throughput analysis of dynamical properties of proteins.     

Transferable Measurements of Heredity in Models of the Origins of Life

We propose a metric which can be used to compute the amount of heritable variation enabled by a given dynamical system. A distribution of selection pressures is used such that each pressure selects a particular fixed point via competitive exclusion in order to determine the corresponding distribution of potential fixed points in the population dynamics. This metric accurately detects the number of species present in artificially prepared test systems, and furthermore can correctly determine the number of heritable sets in clustered transition matrix models in which there are no clearly defined genomes. Finally, we apply our metric to the GARD model and show that it accurately reproduces prior measurements of the model’s heritability.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

Estimating the Stochastic Bifurcation Structure of Cellular Networks

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.     

Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models

The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcation. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a Ricker recruitment function in an SIS model and obtained a three component discrete Hopf (Neimark-Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.     

Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations

Neurophysiological and anatomical observations are used to derive a non-linear delay-differential equation for the pupil light reflex with negative feedback. As the gain or the time delay in the reflex is increased, a supercritical Hopf bifurcation occurs from a stable fixed point to a stable limit cycle oscillation in pupil area. A Hopf bifurcation analysis is used to determine the conditions for instability and the period and amplitude of these oscillations. The more complex waveforms typical of the occurrence of higher order bifurcations were not seen in numerical simulations of the model. This model provides a general framework to study the different types of dynamical behaviors which can be produced by the pupil light reflex, e.g. edge-light pupil cycling.