Spatially discordant alternans (SDA) of action potential duration (APD) has been widely observed in cardiac tissue and is linked to cardiac arrhythmogenesis. Theoretical studies have shown that conduction velocity restitution (CVR) is required for the formation of SDA. However, this theory is not completely supported by experiments, indicating that other mechanisms may exist. In this study, we carried out computer simulations using mathematical models of action potentials to investigate the mechanisms of SDA in cardiac tissue. We show that when CVR is present and engaged, such as fast pacing from one side of the tissue, the spatial pattern of APD in the tissue undergoes either spatially concordant alternans or SDA, independent of initial conditions or tissue heterogeneities. When CVR is not engaged, such as simultaneous pacing of the whole tissue or under normal/slow heart rates, the spatial pattern of APD in the tissue can have multiple solutions, including spatially concordant alternans and different SDA patterns, depending on heterogeneous initial conditions or pre-existing repolarization heterogeneities. In homogeneous tissue, curved nodal lines are not stable, which either evolve into straight lines or disappear. However, in heterogeneous itssue, curved nodal lines can be stable, depending on their initial locations and shapes relative to the structure of the heterogeneity. Therefore, CVR-induced SDA and non-CVR-induced SDA exhibit different dynamical properties, which may be responsible for the different SDA properties observed in experimental studies and arrhythmogenesis in different clinical settings.

Excitable cells, such as cardiac myocytes, exhibit short-term memory, i.e., the state of the cell depends on its history of excitation. Memory can originate from slow recovery of membrane ion channels or from accumulation of intracellular ion concentrations, such as calcium ion or sodium ion concentration accumulation. Here we examine the effects of memory on excitation dynamics in cardiac myocytes under two diseased conditions, early repolarization and reduced repolarization reserve, each with memory from two different sources: slow recovery of a potassium ion channel and slow accumulation of the intracellular calcium ion concentration. We first carry out computer simulations of action potential models described by differential equations to demonstrate complex excitation dynamics, such as chaos. We then develop iterated map models that incorporate memory, which accurately capture the complex excitation dynamics and bifurcations of the action potential models. Finally, we carry out theoretical analyses of the iterated map models to reveal the underlying mechanisms of memory-induced nonlinear dynamics. Our study demonstrates that the memory effect can be unmasked or greatly exacerbated under certain diseased conditions, which promotes complex excitation dynamics, such as chaos. The iterated map models reveal that memory converts a monotonic iterated map function into a nonmonotonic one to promote the bifurcations leading to high periodicity and chaos.

Excitable systems display memory, but how memory affects the excitation dynamics of such systems remains to be elucidated. Here we use computer simulation of cardiac action potential models to demonstrate that memory can cause dynamical instabilities that result in complex excitation dynamics and chaos. We develop an iterated map model that correctly describes these dynamics and show that memory converts a monotonic first return map of action potential duration into a non-monotonic one, resulting in a period-doubling bifurcation route to chaos.

Mathematical models of chronic myeloid leukemia (CML) cell population dynamics are being developed to improve CML understanding and treatment. We review such models in light of relevant findings from radiobiology, emphasizing 3 points. First, the CML models almost all assert that the latency time, from CML initiation to diagnosis, is at most ϳ 10 years. Meanwhile, current radiobiologic estimates, based on Japanese atomic bomb survivor data, indicate a substantially higher maximum, suggesting longer-term relapses and extra resistance mutations. Second, different CML models assume different numbers, between 400 and 10 6 , of normal HSCs. Radiobiologic estimates favor values > 10 6 for the number of normal cells (often assumed to be the HSCs) that are at risk for a CML-initiating BCR-ABL translocation. Moreover, there is some evidence for an HSC dead-band hypothesis, consistent with HSC numbers being very different across different healthy adults. Third, radiobiologists have found that sporadic (background, agedriven) chromosome translocation incidence increases with age during adulthood. BCR-ABL translocation incidence increasing with age would provide a hitherto underanalyzed contribution to observed background adult-onset CML incidence acceleration with age, and would cast some doubt on stage-number inferences from multistage carcinogenesis models in general. (Blood. 2012; 119(19):4363-4371) IntroductionChronic myeloid leukemia (CML) is characterized by Ph ϩ cells, that is, cells having a Philadelphia (BCR-ABL) chromosome translocation. 1,2 Treatment with the tyrosine kinase inhibitor (TKI) imatinib mesylate ("imatinib"), which suppresses bcr-abl oncoprotein action, 3 improves patient prognosis dramatically. 4 However, in some cases this treatment fails, a problem mitigated but not fully solved by the use of more recently developed TKI. 5,6 Moreover, many patients may need to continue TKI treatment indefinitely to avoid relapse. 7 CML is one of the best understood cancers; it has a simpler etiology than most cancers 8 and its time course is comparatively easy to monitor in the clinic. 9,10 Consequently, despite being much less prevalent than major solid tumors, CML has often been regarded as a kind of "model organism" for quantitative modeling of human carcinogenesis. 11,12 CML cell population dynamicsIn this review, we emphasize how radiobiologic studies impact CML models grounded in understanding underlying cell population dynamics. These models track CML time evolution by differential equations and/or stochastic formalisms. Such biologically based quantitative models are more ambitious, more comprehensive, and as yet less definitive than models often used in statistical analyses, which emphasize correlations analyzed by adjusting parameters in functions chosen mainly for mathematical convenience.After work by Rubinow and Lebowitz 13 on hematopoiesis, biologically based, mathematical CML models were pioneered by Clarkson and coworkers. 14 Many additional models have been suggested in the last decade. Recent a...

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Spiral wave reentry as a mechanism of lethal ventricular arrhythmias has been widely demonstrated in animal experiments and recordings from human hearts. It has been shown that in structurally normal hearts, spiral waves are unstable, breaking up into multiple wavelets via dynamical instabilities. However, many of the second-generation action potential models give rise only to stable spiral waves, raising issues regarding the underlying mechanisms of spiral wave breakup. In this study, we carried out computer simulations of two-dimensional homogeneous tissues using five ventricular action potential models. We show that the transient outward potassium current (Ito), although it is not required, plays a key role in promoting spiral wave breakup in all five models. As the maximum conductance of Ito increases, it first promotes spiral wave breakup and then stabilizes the spiral waves. In the absence of Ito, speeding up the L-type calcium kinetics can prevent spiral wave breakup, however, with the same speedup kinetics, spiral wave breakup can be promoted by increasing Ito. Increasing Ito promotes single-cell dynamical instabilities, including action potential duration alternans and chaos, and increasing Ito further suppresses these action potential dynamics. These cellular properties agree with the observation that increasing Ito first promotes spiral wave breakup and then stabilizes spiral waves in tissue. Implications of our observations to spiral wave dynamics in the real hearts and action potential model improvements are discussed.

We develop an iterated map model to describe the bifurcations and complex dynamics caused by the feedbacks between voltage and intracellular Ca 2+ and Na + concentrations in paced ventricular myocytes. Voltage and Ca 2+ can form either a positive or a negative feedback loop, while voltage and Na + form a negative feedback loop. Under certain diseased conditions, when the feedback between voltage and Ca 2+ is positive, Hopf bifurcations occur, leading to periodic oscillatory behaviors. When this feedback is negative, period-doubling bifurcation routes to alternans and chaos occur. In excitable cells [1], ion concentration gradients across the cell membrane are required for a negative (polarized) resting potential and excitability. The major ions are sodium ion (Na +), potassium ion (K +), and calcium ion (Ca 2+), with concentrations in the extracellular space being roughly 140 mM, 4 mM, and 1.5 mM, and in intracellular space being roughly 10 mM, 150 mM, and 100 nM, respectively. These ion gradients are primarily maintained by ion pumps, namely, the Na +-K + pump and the Na +-Ca 2+ exchange (NCX). During an action potential (AP), Na + and Ca 2+ enter the cell via voltage-gated Na + channels and Ca 2+ channels, respectively, and K + exits the cell via K + channels, which then are extruded out or brought into the cell by the pumps, maintaining ion homeostasis of the cell. Since the intracellular ion concentrations affect both ionic currents via ion channels and pumps, feedback loops form between voltage and the ion concentrations. Moreover, the ion channels and different intracellular ion concentration dynamics exhibit distinct time scales. The feedback loops and the multiple time scales can result in very interesting dynamics, such as bursting behaviors seen in many biological cells, including neurons [2-4], pancreatic β-cells [2], and cardiac cells [5-7]. Although some of the complex AP dynamics have been understood via bifurcation analyses, much work is still needed to reveal how the feedbacks and different time scales interact to give rise to these dynamics.

and electrical function of the heart. Although there is reduction of Ca 2þ transients during ischemia, the underlying mechanism of this Ca 2þ mishandling is not fully understood. Currently, no reports exist in which Ca 2þ currents and SR Ca 2þ release are measured simultaneously in situ during global ischemia. Recently, we found that global ischemia induces large increases in diastolic Ca 2þ leading to an increase in sarcoplasmic reticulum (SR) Ca 2þ content. Contrary to what we expected there was a smaller fractional SR Ca 2þ depletion during an action potential (AP). However, Ca 2þ sparks measured in intact hearts during ischemia are kinetically undistinguishable from preischemic recordings. Our idea is that depressed Ca 2þ transients during ischemia may result from a reduction of L-type Ca 2þ currents and/or coupling between plasma membrane Ca 2þ influx and SR Ca 2þ release. Recently, we found using loose patch photolysis that Ca 2þ dependent currents during an AP showed a fast early component driven by an L-type Ca 2þ influx and a slower late component mediated by a Na þ-Ca 2þ exchanger (NCX) forward current. To understand why Ca 2þ transient decreases, we performed experiments to address if ischemia induces changes in Ca 2þ dependent currents. Our results indicate that ischemia produced a decrease of the early Ca 2þ current and the late NCX current. However, the reduction of the NCX current occurs before the L-type current attenuation. Although both Ca 2þ mediated currents recover during reperfusion after ischemia, L-type Ca 2þ currents recuperate faster. We conclude that the impairment of NCX during ischemia is an early event in both the contractility reduction and diastolic Ca 2þ increase and that this impairment is maintained during postischemic diastolic dysfunction.