Parasympathetic modulation of sinoatrial node pacemaker activity in rabbit heart: a unifying model

A novel conceptual model of heart rate autonomic modulation based on a small-world modular structure of the sinoatrial node

The present view on heartbeat initiation is that a primary pacemaker cell or a group of cells in the sinoatrial node (SAN) center paces the rest of the SAN and the atria. However, recent high-resolution imaging studies show a more complex paradigm of SAN function that emerges from heterogeneous signaling, mimicking brain cytoarchitecture and function. Here, we developed and tested a new conceptual numerical model of SAN organized similarly to brain networks featuring a modular structure with small-world topology. In our model, a lower rate module leads action potential (AP) firing in the basal state and during parasympathetic stimulation, whereas a higher rate module leads during β-adrenergic stimulation. Such a system reproduces the respective shift of the leading pacemaker site observed experimentally and a wide range of rate modulation and robust function while conserving energy. Since experimental studies found functional modules at different scales, from a few cells up to the highest scale of the superior and inferior SAN, the SAN appears to feature hierarchical modularity, i.e., within each module, there is a set of sub-modules, like in the brain, exhibiting greater robustness, adaptivity, and evolvability of network function. In this perspective, our model offers a new mainframe for interpreting new data on heterogeneous signaling in the SAN at different scales, providing new insights into cardiac pacemaker function and SAN-related cardiac arrhythmias in aging and disease.

Synergy between Membrane Currents Prevents Severe Bradycardia in Mouse Sinoatrial Node Tissue

Bradycardia is initiated by the sinoatrial node (SAN), which is regulated by a coupled-clock system. Due to the clock coupling, reduction in the 'funny' current (I_f), which affects SAN automaticity, can be compensated, thus preventing severe bradycardia. We hypothesize that this fail-safe system is an inherent feature of SAN pacemaker cells and is driven by synergy between I_f and other ion channels. This work aimed to characterize the connection between membrane currents and their underlying mechanisms in SAN cells. SAN tissues were isolated from C57BL mice and Ca²⁺ signaling was measured in pacemaker cells within them. A computational model of SAN cells was used to understand the interactions between cell components. Beat interval (BI) was prolonged by 54 ± 18% (N = 16) and 30 ± 9% (N = 21) in response to I_f blockade, by ivabradine, or sodium current (I_Na) blockade, by tetrodotoxin, respectively. Combined drug application had a synergistic effect, manifested by a BI prolonged by 143 ± 25% (N = 18). A prolongation in the local Ca²⁺ release period, which reports on the level of crosstalk within the coupled-clock system, was measured and correlated with the prolongation in BI. The computational model predicted that I_Na increases in response to I_f blockade and that this connection is mediated by changes in T and L-type Ca²⁺ channels.

Models of the cardiac L-type calcium current: A quantitative review

The L-type calcium current (I CaL ) plays a critical role in cardiac electrophysiology, and models ofI CaL are vital tools to predict arrhythmogenicity of drugs and mutations. Five decades of measuring and modelingI CaL have resulted in several competing theories (encoded in mathematical equations). However, the introduction of new models has not typically been accompanied by a data-driven critical comparison with previous work, so that it is unclear which model is best suited for any particular application. In this review, we describe and compare 73 published mammalianI CaL models and use simulated experiments to show that there is a large variability in their predictions, which is not substantially diminished when grouping by species or other categories. We provide model code for 60 models, list major data sources, and discuss experimental and modeling work that will be required to reduce this huge list of competing theories and ultimately develop a community consensus model ofI CaL . This article is categorized under: Cardiovascular Diseases > Computational Models Cardiovascular Diseases > Molecular and Cellular Physiology.

The virtual sinoatrial node: What did computational models tell us about cardiac pacemaking?

Since its discovery, the sinoatrial node (SAN) has represented a fascinating and complex matter of research. Despite over a century of discoveries, a full comprehension of pacemaking has still to be achieved. Experiments often produced conflicting evidence that was used either in support or against alternative theories, originating intense debates. In this context, mathematical descriptions of the phenomena underlying the heartbeat have grown in importance in the last decades since they helped in gaining insights where experimental evaluation could not reach. This review presents the most updated SAN computational models and discusses their contribution to our understanding of cardiac pacemaking. Electrophysiological, structural and pathological aspects - as well as the autonomic control over the SAN - are taken into consideration to reach a holistic view of SAN activity.

Adenosine reduces sinoatrial node cell action potential firing rate by uncoupling its membrane and calcium clocks

The spontaneous action potential (AP) firing rate of sinoatrial nodal cells (SANC) is regulated by a system of intracellular Ca²⁺ and membrane ion current clocks driven by Ca²⁺-calmodulin-activated adenylyl cyclase-protein kinase-A signaling. The mean AP-cycle length (APCL) and APCL variability inform on the effectiveness of clock coupling. Endogenous ATP metabolite adenosine binds to adenosine receptors (A₁, A₃) that couple to G_i protein-coupled receptors, reducing spontaneous AP firing rate via G_βγ signaling that activates I_KAch,Ado. Adenosine also inhibits adenylyl cyclase activity via G_αi signaling, impacting cAMP-mediated protein kinase-A-dependent protein phosphorylation. We hypothesize that in addition to I_KAch,Ado activation, adenosine impacts also Ca²⁺ via G_αi signaling and that both effects reduce AP firing rate by reducing the effectiveness of the Ca²⁺ and membrane clock coupling. To this end, we measured Ca²⁺ and membrane potential characteristics in enzymatically isolated single rabbit SANC. 10 µM adenosine substantially increased both the mean APCL (on average by 43%, n = 10) and AP beat-to-beat variability from 5.1 ± 1.7% to 7.2 ± 2.0% (n = 10) measured via membrane potential and 5.0 ± 2.2% to 10.6 ± 5.9% (n = 40) measured via Ca²⁺ (assessed as the coefficient of variability = SD/mean). These effects were mediated by hyperpolarization of the maximum diastolic membrane potential (membrane clock effect) and suppression of diastolic local Ca²⁺releases (LCRs) (Ca²⁺-clock effect): as LCR size distributions shifted to smaller values, the time of LCR occurrence during diastolic depolarization (LCR period) became prolonged, and the ensemble LCR signal became reduced. The tight linear relationship of coupling between LCR period to the APCL in the presence of adenosine "drifted" upward and leftward, i.e. for a given LCR period, APCL was prolonged, becoming non-linear indicating clock uncoupling. An extreme case of uncoupling occurred at higher adenosine concentrations (>100 µM): small stochastic LCRs failed to self-organize and synchronize to the membrane clock, thus creating a failed attempt to generate an AP resulting in arrhythmia and cessation of AP firing. Thus, the effects of adenosine to activate G_βγ and I_KACh,Ado and to activate G_αi, suppressing adenylyl cyclase activity, both contribute to the adenosine-induced increase in the mean APCL and APCL variability by reducing the fidelity of clock coupling and AP firing rate.

A Numerical Model of the Acute Cardiac Effects Provoked by Cervical Vagus Nerve Stimulation

OBJECTIVE

Today, the diverse acute cardiac effects of vagus nerve stimulation (VNS) are still not fully understood. Therefore, we propose a numerical model that can predict the acute cardiac responses to VNS and explain the underlying mechanisms on different levels.

METHODS

We integrated a model of vagal nerve fiber recruitment and acetylcholine (ACh) kinetics at vagal nerve terminals into a cardiovascular system model. A sensitivity analysis was performed to identify the most important parameters of vagal cardiac pathways. These parameters were tuned, and the model was validated based on published data of experiments in anesthetized sheep.

RESULTS

The four most important parameters are related to vagus nerve anatomy (electrode-fiber distances, fiber diameters) and ACh kinetics in the vagal neuroeffector junction (rate of ACh release and -hydrolysis) which together explain >53% of the observed variability in acute cardiac responses to VNS. The mean electrode-fiber distance and nerve fiber diameters obtained from tuning are 1.3 ± 0.09 mm, and 4.9 ± 0.25 μm; the ACh release and -hydrolysis rate constants are 0.023 s-1 and 0.77 s-1, respectively. With this parameterization, the model could accurately predict published data on the acute cardiac effects of VNS.

CONCLUSIONS

The model can explain the cardiac responses to VNS on multiple levels. The results highlight the importance of four parameters tied to ACh dynamics and vagus nerve anatomy for predicting the cardiac effects of VNS.

SIGNIFICANCE

The model represents a substantial improvement in terms of comprehensibility of the underlying mechanisms of the acute cardiac responses to VNS.

Ivabradine increases the high frequency gain ratio in the vagal heart rate transfer function via an interaction with muscarinic potassium channels

Muscarinic potassium channels (IK,ACh ) are thought to contribute to the high frequency (HF) dynamic heart rate (HR) response to vagal nerve stimulation (VNS) because they act faster than the pathway mediated by hyperpolarization-activated cyclic nucleotide-gated (HCN) channels. However, the interactions between the two pathways have not yet been fully elucidated. We previously demonstrated that HCN channel blockade by ivabradine (IVA) increased the HF gain ratio of the transfer function from VNS to HR. To test the hypothesis that IVA increases the HF gain ratio via an interaction with IK,ACh , we examined the dynamic HR response to VNS under conditions of control (CNT), IK,ACh blockade by tertiapin-Q (TQ, 50 nM/kg), and TQ plus IVA (2 mg/kg) (TQ + IVA) in anesthetized rats (n = 8). In each condition, the right vagal nerve was stimulated for 10 min with binary white noise signals between 0-10, 0-20, and 0-40 Hz. On multiple regression analysis, the HF gain ratio positively correlated with the VNS rate with a coefficient of 1.691 ± 0.151 (×0.01) (p < 0.001). TQ had a negative effect on the HF gain ratio with a coefficient of -1.170 ± 0.214 (×0.01) (p < 0.001). IVA did not significantly increase the HF gain ratio in the presence of TQ. The HF gain ratio remained low under the TQ + IVA condition compared to controls. These results affirm that the IVA-induced increase in the HF gain ratio is dependent on the untethering of the hyperpolarizing effect of IK,ACh .

Electrical properties and synaptic transmission in mouse intracardiac ganglion neurons in situ

The intrinsic cardiac nervous system represents the final site of signal integration for neurotransmission to the myocardium to enable local control of cardiac performance. The electrophysiological characteristics and ganglionic transmission of adult mouse intrinsic cardiac ganglion (ICG) neurons were investigated using a whole-mount ganglion preparation of the excised right atrial ganglion plexus and intracellular microelectrode recording techniques. The passive and active electrical properties of ICG neurons and synaptic transmission including synaptic response strength and efficacy as a function of stimulation frequency were examined. The resting membrane potential and input resistance of ICG neurons were -47.9 ± 4.0 mV and 197.2 ± 81.5 MΩ, respectively. All neurons had somatic action potentials with overshoots of >+15 mV and after-hyperpolarizations having an average of 10 mV amplitude and ~45 ms half duration. Phasic discharge activities were recorded from the majority of neurons studied and several types of excitatory synaptic responses were recorded following inputs from the vagus or interganglionic nerve trunk(s). Most postganglionic neurons (>75%) received a strong, suprathreshold synaptic input and reliably followed high-frequency repetitive nerve stimulation up to at least 50 Hz. Nerve-evoked synaptic transmission was blocked by extracellular Cd2+ , ω-conotoxin CVIE, or α-conotoxin RegIIA, a selective α3-containing nicotinic acetylcholine receptor antagonist. Synaptic transmission and the electrical properties of murine ICG neurons contribute to the pattern of discharge which regulates chronotropic, dromotropic, and inotropic elements of cardiac function.

β-Adrenergic Stimulation Synchronizes a Broad Spectrum of Action Potential Firing Rates of Cardiac Pacemaker Cells toward a Higher Population Average

The heartbeat is initiated by pacemaker cells residing in the sinoatrial node (SAN). SAN cells generate spontaneous action potentials (APs), i.e., normal automaticity. The sympathetic nervous system increases the heart rate commensurate with the cardiac output demand via stimulation of SAN β-adrenergic receptors (βAR). While SAN cells reportedly represent a highly heterogeneous cell population, the current dogma is that, in response to βAR stimulation, all cells increase their spontaneous AP firing rate in a similar fashion. The aim of the present study was to investigate the cell-to-cell variability in the responses of a large population of SAN cells. We measured the βAR responses among 166 single SAN cells isolated from 33 guinea pig hearts. In contrast to the current dogma, the SAN cell responses to βAR stimulation substantially varied. In each cell, changes in the AP cycle length were highly correlated (R2 = 0.97) with the AP cycle length before βAR stimulation. While, as expected, on average, the cells increased their pacemaker rate, greater responses were observed in cells with slower basal rates, and vice versa: cells with higher basal rates showed smaller responses, no responses, or even decreased their rate. Thus, βAR stimulation synchronized the operation of the SAN cell population toward a higher average rate, rather than uniformly shifting the rate in each cell, creating a new paradigm of βAR-driven fight-or-flight responses among individual pacemaker cells.

Ivabradine augments high-frequency dynamic gain of the heart rate response to low- and moderate-intensity vagal nerve stimulation under β-blockade

Our previous study indicated that intravenously administered ivabradine (IVA) augmented the dynamic heart rate (HR) response to moderate-intensity vagal nerve stimulation (VNS). Considering an accentuated antagonism, the results were somewhat paradoxical; i.e., the accentuated antagonism indicates that an activation of hyperpolarization-activated cyclic nucleotide-gated (HCN) channels via the accumulation of intracellular cyclic adenosine monophosphate (cAMP) augments the HR response to VNS, whereas the inhibition of HCN channels by IVA also augmented the HR response to VNS. To remove the possible influence from the accentuated antagonism, we examined the effects of IVA on the dynamic vagal control of HR under β-blockade. In anesthetized rats (n = 7), the right vagal nerve was stimulated for 10 min according to binary white noise signals between 0 and 10 Hz (V_0-10), between 0 and 20 Hz (V_0-20), and between 0 and 40 Hz (V_0-40). The transfer function from VNS to HR was estimated. Under β-blockade (propranolol, 2 mg/kg iv), IVA (2 mg/kg iv) did not augment the asymptotic low-frequency gain but increased the asymptotic high-frequency gain in V_0-10 (0.53 ± 0.10 vs. 1.74 ± 0.40 beats/min/Hz, P < 0.01) and V_0-20 (0.79 ± 0.14 vs. 2.06 ± 0.47 beats/min/Hz, P < 0.001). These changes, which were observed under a minimal influence from sympathetic background tone, may reflect an increased contribution of the acetylcholine-sensitive potassium channel (I_K,ACh) pathway after IVA, because the HR control via the I_K,ACh pathway is faster and acts in the frequency range higher than the cAMP-mediated pathway.NEW & NOTEWORTHY Since ivabradine (IVA) inhibits hyperpolarization-activated cyclic nucleotide-gated channels, interactions among the sympathetic effect, vagal effect, and IVA can occur in the control of heart rate (HR). To remove the sympathetic effect, we estimated the transfer function from vagal nerve stimulation to HR under β-blockade in anesthetized rats. IVA augmented the high-frequency dynamic gain during low- and moderate-intensity vagal nerve stimulation. Untethering the hyperpolarizing effect of acetylcholine-sensitive potassium channels after IVA may be a possible underlying mechanism.

A Novel Computational Model of the Rabbit Atrial Cardiomyocyte With Spatial Calcium Dynamics

Models of cardiac electrophysiology are widely used to supplement experimental results and to provide insight into mechanisms of cardiac function and pathology. The rabbit has been a particularly important animal model for studying mechanisms of atrial pathophysiology and atrial fibrillation, which has motivated the development of models for the rabbit atrial cardiomyocyte electrophysiology. Previously developed models include detailed representations of membrane currents and intracellular ionic concentrations, but these so-called “common-pool” models lack a spatially distributed description of the calcium handling system, which reflects the detailed ultrastructure likely found in cells in vivo. Because of the less well-developed T-tubular system in atrial compared to ventricular cardiomyocytes, spatial gradients in intracellular calcium concentrations may play a more significant role in atrial cardiomyocyte pathophysiology, rendering common-pool models less suitable for investigating underlying electrophysiological mechanisms. In this study, we developed a novel computational model of the rabbit atrial cardiomyocyte incorporating detailed compartmentalization of intracellular calcium dynamics, in addition to a description of membrane currents and intracellular processes. The spatial representation of calcium was based on dividing the intracellular space into eighteen different compartments in the transversal direction, each with separate systems for internal calcium storage and release, and tracking ionic fluxes between compartments in addition to the dynamics driven by membrane currents and calcium release. The model was parameterized employing a population-of-models approach using experimental data from different sources. The parameterization of this novel model resulted in a reduced population of models with inherent variability in calcium dynamics and electrophysiological properties, all of which fall within the range of observed experimental values. As such, the population of models may represent natural variability in cardiomyocyte electrophysiology or inherent uncertainty in the underlying experimental data. The ionic model population was also able to reproduce the U-shaped waveform observed in line-scans of triggered calcium waves in atrial cardiomyocytes, characteristic of the absence of T-tubules, resulting in a centripetal calcium wave due to subcellular calcium diffusion. This novel spatial model of the rabbit atrial cardiomyocyte can be used to integrate experimental findings, offering the potential to enhance our understanding of the pathophysiological role of calcium-handling abnormalities under diseased conditions, such as atrial fibrillation.

Computational assessment of the functional role of sinoatrial node exit pathways in the human heart

AIM

The human right atrium and sinoatrial node (SAN) anatomy is complex. Optical mapping experiments suggest that the SAN is functionally insulated from atrial tissue except at discrete SAN-atrial electrical junctions called SAN exit pathways, SEPs. Additionally, histological imaging suggests the presence of a secondary pacemaker close to the SAN. We hypothesise that a) an insulating border-SEP anatomical configuration is related to SAN arrhythmia; and b) a secondary pacemaker, the paranodal area, is an alternate pacemaker but accentuates tachycardia. A 3D electro-anatomical computational model was used to test these hypotheses.

METHODS

A detailed 3D human SAN electro-anatomical mathematical model was developed based on our previous anatomical reconstruction. Electrical activity was simulated using tissue specific variants of the Fenton-Karma action potential equations. Simulation experiments were designed to deploy this complex electro-anatomical system to assess the roles of border-SEPs and paranodal area by mimicking experimentally observed SAN arrhythmia. Robust and accurate numerical algorithms were implemented for solving the mono domain reaction-diffusion equation implicitly, calculating 3D filament traces, and computing dominant frequency among other quantitative measurements.

RESULTS

A centre to periphery gradient of increasing diffusion was sufficient to permit initiation of pacemaking at the centre of the 3D SAN. Re-entry within the SAN, micro re-entry, was possible by imposing significant SAN fibrosis in the presence of the insulating border. SEPs promoted the micro re-entry to generate more complex SAN-atrial tachycardia. Simulation of macro re-entry, i.e. re-entry around the SAN, was possible by inclusion of atrial fibrosis in the presence of the insulating border. The border shielded the SAN from atrial tachycardia. However, SAN micro-structure intercellular gap junctional coupling and the paranodal area contributed to prolonged atrial fibrillation. Finally, the micro-structure was found to be sufficient to explain shifts of leading pacemaker site location.

CONCLUSIONS

The simulations establish a relationship between anatomy and SAN electrical function. Microstructure, in the form of intercellular gap junction coupling, was found to regulate SAN function and arrhythmia.

The Autonomic Nervous System Regulates the Heart Rate through cAMP-PKA Dependent and Independent Coupled-Clock Pacemaker Cell Mechanisms

Sinoatrial nodal cells (SANCs) generate spontaneous action potentials (APs) that control the cardiac rate. The brain modulates SANC automaticity, via the autonomic nervous system, by stimulating membrane receptors that activate (adrenergic) or inactivate (cholinergic) adenylyl cyclase (AC). However, these opposing afferents are not simply additive. We showed that activation of adrenergic signaling increases AC-cAMP/PKA signaling, which mediates the increase in the SANC AP firing rate (i.e., positive chronotropic modulation). However, there is a limited understanding of the underlying internal pacemaker mechanisms involved in the crosstalk between cholinergic receptors and the decrease in the SANC AP firing rate (i.e., negative chronotropic modulation). We hypothesize that changes in AC-cAMP/PKA activity are crucial for mediating either decrease or increase in the AP firing rate and that the change in rate is due to both internal and membrane mechanisms. In cultured adult rabbit pacemaker cells infected with an adenovirus expressing the FRET sensor AKAR3, PKA activity and AP firing rate were tightly linked in response to either adrenergic receptor stimulation (by isoproterenol, ISO) or cholinergic stimulation (by carbachol, CCh). To identify the main molecular targets that mediate between PKA signaling and pacemaker function, we developed a mechanistic computational model. The model includes a description of autonomic-nervous receptors, post- translation signaling cascades, membrane molecules, and internal pacemaker mechanisms. Yielding results similar to those of the experiments, the model simulations faithfully reproduce the changes in AP firing rate in response to CCh or ISO or a combination of both (i.e., accentuated antagonism). Eliminating AC-cAMP-PKA signaling abolished the core effect of autonomic receptor stimulation on the AP firing rate. Specifically, disabling the phospholamban modulation of the SERCA activity resulted in a significantly reduced effect of CCh and a failure to increase the AP firing rate under ISO stimulation. Directly activating internal pacemaker mechanisms led to a similar extent of changes in the AP firing rate with respect to brain receptor stimulation. Thus, Ca²⁺ and cAMP/PKA-dependent phosphorylation limits the rate and magnitude of chronotropic changes in the spontaneous AP firing rate.

Myokit: A simple interface to cardiac cellular electrophysiology

Myokit is a new powerful and versatile software tool for modeling and simulation of cardiac cellular electrophysiology. Myokit consists of an easy-to-read modeling language, a graphical user interface, single and multi-cell simulation engines and a library of advanced analysis tools accessible through a Python interface. Models can be loaded from Myokit's native file format or imported from CellML. Model export is provided to C, MATLAB, CellML, CUDA and OpenCL. Patch-clamp data can be imported and used to estimate model parameters. In this paper, we review existing tools to simulate the cardiac cellular action potential to find that current tools do not cater specifically to model development and that there is a gap between easy-to-use but limited software and powerful tools that require strong programming skills from their users. We then describe Myokit's capabilities, focusing on its model description language, simulation engines and import/export facilities in detail. Using three examples, we show how Myokit can be used for clinically relevant investigations, multi-model testing and parameter estimation in Markov models, all with minimal programming effort from the user. This way, Myokit bridges a gap between performance, versatility and user-friendliness.

Three-dimensional computer model of the right atrium including the sinoatrial and atrioventricular nodes predicts classical nodal behaviours

The aim of the study was to develop a three-dimensional (3D) anatomically-detailed model of the rabbit right atrium containing the sinoatrial and atrioventricular nodes to study the electrophysiology of the nodes. A model was generated based on 3D images of a rabbit heart (atria and part of ventricles), obtained using high-resolution magnetic resonance imaging. Segmentation was carried out semi-manually. A 3D right atrium array model (∼3.16 million elements), including eighteen objects, was constructed. For description of cellular electrophysiology, the Rogers-modified FitzHugh-Nagumo model was further modified to allow control of the major characteristics of the action potential with relatively low computational resource requirements. Model parameters were chosen to simulate the action potentials in the sinoatrial node, atrial muscle, inferior nodal extension and penetrating bundle. The block zone was simulated as passive tissue. The sinoatrial node, crista terminalis, main branch and roof bundle were considered as anisotropic. We have simulated normal and abnormal electrophysiology of the two nodes. In accordance with experimental findings: (i) during sinus rhythm, conduction occurs down the interatrial septum and into the atrioventricular node via the fast pathway (conduction down the crista terminalis and into the atrioventricular node via the slow pathway is slower); (ii) during atrial fibrillation, the sinoatrial node is protected from overdrive by its long refractory period; and (iii) during atrial fibrillation, the atrioventricular node reduces the frequency of action potentials reaching the ventricles. The model is able to simulate ventricular echo beats. In summary, a 3D anatomical model of the right atrium containing the cardiac conduction system is able to simulate a wide range of classical nodal behaviours.

Modern perspectives on numerical modeling of cardiac pacemaker cell

Cardiac pacemaking is a complex phenomenon that is still not completely understood. Together with experimental studies, numerical modeling has been traditionally used to acquire mechanistic insights in this research area. This review summarizes the present state of numerical modeling of the cardiac pacemaker, including approaches to resolve present paradoxes and controversies. Specifically we discuss the requirement for realistic modeling to consider symmetrical importance of both intracellular and cell membrane processes (within a recent "coupled-clock" theory). Promising future developments of the complex pacemaker system models include the introduction of local calcium control, mitochondria function, and biochemical regulation of protein phosphorylation and cAMP production. Modern numerical and theoretical methods such as multi-parameter sensitivity analyses within extended populations of models and bifurcation analyses are also important for the definition of the most realistic parameters that describe a robust, yet simultaneously flexible operation of the coupled-clock pacemaker cell system. The systems approach to exploring cardiac pacemaker function will guide development of new therapies such as biological pacemakers for treating insufficient cardiac pacemaker function that becomes especially prevalent with advancing age.

Short-term desensitization of muscarinic K+ current in the heart

Acetylcholine (ACh) rapidly increases cardiac K(+) currents (IKACh) by activating muscarinic K(+) (KACh) channels followed by a gradual amplitude decrease within seconds. This phenomenon is called short-term desensitization and its precise mechanism and physiological role are still unclear. We constructed a mathematical model for IKACh to examine the conditions required to reconstitute short-term desensitization. Two conditions were crucial: two distinct muscarinic receptors (m2Rs) with different affinities for ACh, which conferred an IKACh response over a wide range of ACh concentrations, and two distinct KACh channels with different affinities for the G-protein βγ subunits, which contributed to reconstitution of the temporal behavior of IKACh. Under these conditions, the model quantitatively reproduced several unique properties of short-term desensitization observed in myocytes: 1), the peak and quasi-steady states with 0.01-100 μM [ACh]; 2), effects of ACh preperfusion; and 3), recovery from short-term desensitization. In the presence of 10 μM ACh, the IKACh model conferred recurring spontaneous firing after asystole of 8.9 s and 10.7 s for the Demir and Kurata sinoatrial node models, respectively. Therefore, two different populations of KACh channels and m2Rs may participate in short-term desensitization of IKACh in native myocytes, and may be responsible for vagal escape at nodal cells.

Rate-dependent Ca2+ signalling underlying the force-frequency response in rat ventricular myocytes: a coupled electromechanical modeling study

BACKGROUND

Rate-dependent effects on the Ca2+ sub-system in a rat ventricular myocyte are investigated. Here, we employ a deterministic mathematical model describing various Ca2+ signalling pathways under voltage clamp (VC) conditions, to better understand the important role of calmodulin (CaM) in modulating the key control variables Ca2+/calmodulin-dependent protein kinase-II (CaMKII), calcineurin (CaN), and cyclic adenosine monophosphate (cAMP) as they affect various intracellular targets. In particular, we study the frequency dependence of the peak force generated by the myofilaments, the force-frequency response (FFR).

METHODS

Our cell model incorporates frequency-dependent CaM-mediated spatially heterogenous interaction of CaMKII and CaN with their principal targets (dihydropyridine (DHPR) and ryanodine (RyR) receptors and the SERCA pump). It also accounts for the rate-dependent effects of phospholamban (PLB) on the SERCA pump; the rate-dependent role of cAMP in up-regulation of the L-type Ca2+ channel (ICa,L); and the enhancement in SERCA pump activity via phosphorylation of PLB.

RESULTS

Our model reproduces positive peak FFR observed in rat ventricular myocytes during voltage-clamp studies both in the presence/absence of cAMP mediated β-adrenergic stimulation. This study provides quantitative insight into the rate-dependence of Ca2+-induced Ca2+-release (CICR) by investigating the frequency-dependence of the trigger current (ICa,L) and RyR-release. It also highlights the relative role of the sodium-calcium exchanger (NCX) and the SERCA pump at higher frequencies, as well as the rate-dependence of sarcoplasmic reticulum (SR) Ca2+ content. A rigorous Ca2+ balance imposed on our investigation of these Ca2+ signalling pathways clarifies their individual roles. Here, we present a coupled electromechanical study emphasizing the rate-dependence of isometric force developed and also investigate the temperature-dependence of FFR.

CONCLUSIONS

Our model provides mechanistic biophysically based explanations for the rate-dependence of CICR, generating useful and testable hypotheses. Although rat ventricular myocytes exhibit a positive peak FFR in the presence/absence of beta-adrenergic stimulation, they show a characteristic increase in the positive slope in FFR due to the presence of Norepinephrine or Isoproterenol. Our study identifies cAMP-mediated stimulation, and rate-dependent CaMKII-mediated up-regulation of ICa,L as the key mechanisms underlying the aforementioned positive FFR.

Hyperpolarization-activated current, If, in mathematical models of rabbit sinoatrial node pacemaker cells

A typical feature of sinoatrial (SA) node pacemaker cells is the presence of an ionic current that activates upon hyperpolarization. The role of this hyperpolarization-activated current, I_f, which is also known as the “funny current” or “pacemaker current,” in the spontaneous pacemaker activity of SA nodal cells remains a matter of intense debate. Whereas some conclude that I_f plays a fundamental role in the generation of pacemaker activity and its rate control, others conclude that the role of I_f is limited to a modest contribution to rate control. The ongoing debate is often accompanied with arguments from computer simulations, either to support one's personal view or to invalidate that of the antagonist. In the present paper, we review the various mathematical descriptions of I_f that have been used in computer simulations and compare their strikingly different characteristics with our experimental data. We identify caveats and propose a novel model for I_f based on our experimental data.

Effect of hyperpolarization-activated current I(f) on robustness of sinoatrial node pacemaking: theoretical study on influence of intracellular Na(+) concentration

To elucidate the effects of hyperpolarization-activated current I(f) on robustness of sinoatrial node (SAN) pacemaking in connection with intracellular Na(+) concentration (Na(i)) changes, we theoretically investigated 1) the impacts of I(f) on dynamical properties of SAN model cells during inhibition of L-type Ca(2+) channel currents (I(CaL)) or hyperpolarizing loads and 2) I(f)-dependent changes in Na(i) and their effects on dynamical properties of model cells. Bifurcation analyses were performed for Na(i)-variable and Na(i)-fixed versions of mathematical models for rabbit SAN cells; equilibrium points (EPs), limit cycles (LCs), and their stability were determined as functions of model parameters. Increasing I(f) conductance (g(f)) shrank I(CaL) conductance (g(CaL)) regions of unstable EPs and stable LCs (rhythmic firings) in the Na(i)-variable system but slightly broadened that of rhythmic firings at lower g(f) in the Na(i)-fixed system. In the Na(i)-variable system, increased g(f) yielded elevations in Na(i) at EPs and during spontaneous oscillations, which caused EP stabilization and shrinkage in the parameter regions of unstable EPs and rhythmic firings. As g(f) increased, parameter regions of unstable EPs and stable LCs determined for hyperpolarizing loads shrank in the Na(i)-variable system but were enlarged in the Na(i)-fixed system. These findings suggest that 1) I(f) does not enhance but rather attenuates robustness of rabbit SAN cells via facilitating EP stabilization and LC destabilization even in physiological g(f) ranges; and 2) the enhancing effect of I(f) on robustness of pacemaker activity, which could be observed at lower g(f) when Na(i) was fixed, is actually reversed by I(f)-dependent changes in Na(i).

How the Hodgkin-Huxley equations inspired the Cardiac Physiome Project

Early modelling of cardiac cells (1960-1980) was based on extensions of the Hodgkin-Huxley nerve axon equations with additional channels incorporated, but after 1980 it became clear that processes other than ion channel gating were also critical in generating electrical activity. This article reviews the development of models representing almost all cell types in the heart, many different species, and the software tools that have been created to facilitate the cardiac Physiome Project.

Onset of atrial arrhythmias elicited by autonomic modulation of rabbit sinoatrial node activity: a modeling study

Neuronal modulation of the sinoatrial node (SAN) plays a crucial role in the initiation and maintenance of atrial arrhythmias (AF), although the exact mechanisms remain unclear. We used a computer model of a rabbit right atrium (RA) with a heterogeneous SAN and detailed ionic current descriptions for atrial and SAN myocytes to explore reentry initiation associated with autonomic activity. Heterogeneous acetylcholine (ACh)-dependent ionic responses along with L-type Ca current (I(Ca,L)) upregulation were incorporated in the SAN only. During control, activation was typical with the leading pacemaker site located close to the superior vena cava or the intercaval region. With cholinergic stimulation, activation patterns frequently included caudal shifts of the leading pacemaker site and occasional double breakouts. The model became increasingly arrhythmogenic for the ACh concentration >20 nM and for large I(Ca,L) conductance. Reentries obtained included counterclockwise rotors in the free wall, clockwise reentry circulating between the SAN and free wall, and typical flutter. The SAN was the cause of reentry with a common leading sequence of events: a bradycardic beat with shifting in the caudal direction, followed by a premature beat or unidirectional block within the SAN. Electrotonic loading, and not just overdrive pacing, squelches competing pacemaker sites in the SAN. Cholinergic stimulation concomitant with I(Ca,L) upregulation shifts leading pacemaker site and can lead to reentry. A heterogeneous response to autonomic innervation, a large myocardial load, and an extensive SAN in the intercaval region are required for neurally induced SAN-triggered reentry.

Effect of clonidine on cardiac baroreflex delay in humans and rats

The delay τ between rising systolic blood pressure (SBP) and baroreflex bradycardia has been found to increase when vagal tone is low. The α(2)-agonist clonidine increases cardiac vagal tone, and this study tested how it affects τ. In eight conscious supine human volunteers clonidine (6 μg/kg po) reduced τ, assessed both by cross correlation baroreflex sensitivity and sequence methods (both P < 0.05). Experiments on urethane-anaesthetized rats reproduced the phenomenon and investigated the underlying mechanism. Heart rate (HR) responses to increasing SBP produced with an arterial balloon catheter showed reduced τ (P < 0.05) after clonidine (100 μg/kg iv). The central latency of the reflex was unaltered, however, as shown by the unchanged timing with which antidromically identified cardiac vagal motoneurons (CVM) responded to the arterial pulse. Testing the latency of the HR response to brief electrical stimuli to the right vagus showed that this was also unchanged by clonidine. Nevertheless, vagal stimuli delivered at a fixed time in the cardiac cycle (triggered from the ECG R-wave) slowed HR with a 1-beat delay in the baseline state but a 0-beat delay after clonidine (n = 5, P < 0.05). This was because clonidine lengthened the diastolic period, allowing the vagal volleys to arrive at the heart just in time to postpone the next beat. Calculations indicate that naturally generated CVM volleys in both humans and rats arrive around this critical time. Clonidine thus reduces τ not by changing central or efferent latencies but simply by slowing the heart.

Roles of hyperpolarization-activated current If in sinoatrial node pacemaking: insights from bifurcation analysis of mathematical models

To elucidate the roles of hyperpolarization-activated current (I(f)) in sinoatrial node (SAN) pacemaking, we theoretically investigated 1) the effects of I(f) on stability and bifurcation during hyperpolarization of SAN cells; 2) combined effects of I(f) and the sustained inward current (I(st)) or Na(+) channel current (I(Na)) on robustness of pacemaking against hyperpolarization; and 3) whether blocking I(f) abolishes pacemaker activity under certain conditions. Bifurcation analyses were performed for mathematical models of rabbit SAN cells; equilibrium points (EPs), periodic orbits, and their stability were determined as functions of parameters. Unstable steady-state potential region determined with applications of constant bias currents shrunk as I(f) density increased. In the central SAN cell, the critical acetylcholine concentration at which bifurcations, to yield a stable EP and quiescence, occur was increased by smaller I(f), but decreased by larger I(f). In contrast, the critical acetylcholine concentration and conductance of gap junctions between SAN and atrial cells at bifurcations progressively increased with enhancing I(f) in the peripheral SAN cell. These effects of I(f) were significantly attenuated by eliminating I(st) or I(Na), or by accelerating their inactivation. Under hyperpolarized conditions, blocking I(f) abolished SAN pacemaking via bifurcations. These results suggest that 1) I(f) itself cannot destabilize EPs; 2) I(f) improves SAN cell robustness against parasympathetic stimulation via preventing bifurcations in the presence of I(st) or I(Na); 3) I(f) dramatically enhances peripheral cell robustness against electrotonic loads of the atrium in combination with I(Na); and 4) pacemaker activity of hyperpolarized SAN cells could be abolished by blocking I(f).

A novel quantitative explanation for the autonomic modulation of cardiac pacemaker cell automaticity via a dynamic system of sarcolemmal and intracellular proteins

Classical numerical models have attributed the regulation of normal cardiac automaticity in sinoatrial node cells (SANCs) largely to G protein-coupled receptor (GPCR) modulation of sarcolemmal ion currents. More recent experimental evidence, however, has indicated that GPCR modulation of SANCs automaticity involves spontaneous, rhythmic, local Ca(2+) releases (LCRs) from the sarcoplasmic reticulum (SR). We explored the GPCR rate modulation of SANCs using a unique and novel numerical model of SANCs in which Ca(2+)-release characteristics are graded by variations in the SR Ca(2+) pumping capability, mimicking the modulation by phospholamban regulated by cAMP-mediated, PKA-activated signaling. The model faithfully predicted the entire range of physiological chronotropic modulation of SANCs by the activation of beta-adrenergic receptors or cholinergic receptors only when experimentally documented changes of sarcolemmal ion channels are combined with a simultaneous increase/decrease in SR Ca(2+) pumping capability. The novel numerical mechanism of GPCR rate modulation is based on numerous complex synergistic interactions between sarcolemmal and intracellular processes via membrane voltage and Ca(2+). Major interactions include changes of diastolic Na(+)/Ca(2+) exchanger current that couple earlier/later diastolic Ca(2+) releases (predicting the experimentally defined LCR period shift) of increased/decreased amplitude (predicting changes in LCR signal mass, i.e., the product of LCR spatial size, amplitude, and number per cycle) to the diastolic depolarization and ultimately to the spontaneous action potential firing rate. Concomitantly, larger/smaller and more/less frequent activation of L-type Ca(2+) current shifts the cellular Ca(2+) balance to support the respective Ca(2+) cycling changes. In conclusion, our model simulations corroborate recent experimental results in rabbit SANCs pointing to a new paradigm for GPCR heart rate modulation by a complex system of dynamically coupled sarcolemmal and intracellular proteins.

A coupled SYSTEM of intracellular Ca2+ clocks and surface membrane voltage clocks controls the timekeeping mechanism of the heart's pacemaker

Ion channels on the surface membrane of sinoatrial nodal pacemaker cells (SANCs) are the proximal cause of an action potential. Each individual channel type has been thoroughly characterized under voltage clamp, and the ensemble of the ion channel currents reconstructed in silico generates rhythmic action potentials. Thus, this ensemble can be envisioned as a surface "membrane clock" (M clock). Localized subsarcolemmal Ca(2+) releases are generated by the sarcoplasmic reticulum via ryanodine receptors during late diastolic depolarization and are referred to as an intracellular "Ca(2+) clock," because their spontaneous occurrence is periodic during voltage clamp or in detergent-permeabilized SANCs, and in silico as well. In spontaneously firing SANCs, the M and Ca(2+) clocks do not operate in isolation but work together via numerous interactions modulated by membrane voltage, subsarcolemmal Ca(2+), and protein kinase A and CaMKII-dependent protein phosphorylation. Through these interactions, the 2 subsystem clocks become mutually entrained to form a robust, stable, coupled-clock system that drives normal cardiac pacemaker cell automaticity. G protein-coupled receptors signaling creates pacemaker flexibility, ie, effects changes in the rhythmic action potential firing rate, by impacting on these very same factors that regulate robust basal coupled-clock system function. This review examines evidence that forms the basis of this coupled-clock system concept in cardiac SANCs.

Concealed conduction effects in the atrium

A one-dimensional (1-D) model of the atrium together with the sinoatrial (SA) and atrioventricular (AV) nodes is presented in this article. The two nodes are each modeled by 15-element, diffusively coupled, modified van der Pol oscillator chains, while the atrium tissue is represented by a 90-element chain of diffusively coupled FitzHugh-Nagumo (FHN) equations. The modified van der Pol oscillators are able to reproduce physiologically important properties, such as the refraction period, phase-sensitivity properties, and modes of change of the action potential frequency. The activity of both branches of the autonomous nervous system may be introduced into the model in a simplified way. The model enables the study of the effect of the magnitude of the action potential conduction rate in the nodes on interspike intervals (ISIs; equivalent of RR intervals) and explains the occurrence of RR-interval alternans in certain patients. The effect of breathing modulation of heart rate and of a single deep breath can also be modeled. Finally, concealed conduction effects in the atrium are studied, yielding results comparable with recorded heart rate variability data.

Cholinergic receptor signaling modulates spontaneous firing of sinoatrial nodal cells via integrated effects on PKA-dependent Ca(2+) cycling and I(KACh)

Prior studies indicate that cholinergic receptor (ChR) activation is linked to beating rate reduction (BRR) in sinoatrial nodal cells (SANC) via 1) a G(i)-coupled reduction in adenylyl cyclase (AC) activity, leading to a reduction of cAMP or protein kinase A (PKA) modulation of hyperpolarization-activated current (I(f)) or L-type Ca(2+) currents (I(Ca,L)), respectively; and 2) direct G(i)-coupled activation of ACh-activated potassium current (I(KACh)). More recent studies, however, have indicated that Ca(2+) cycling by the sarcoplasmic reticulum within SANC (referred to as a Ca(2+) clock) generates rhythmic, spontaneous local Ca(2+) releases (LCR) that are AC-PKA dependent. LCRs activate Na(+)-Ca(2+) exchange (NCX) current, which ignites the surface membrane ion channels to effect an AP. The purpose of the present study was to determine how ChR signaling initiated by a cholinergic agonist, carbachol (CCh), affects AC, cAMP, and PKA or sarcolemmal ion channels and LCRs and how these effects become integrated to generate the net response to a given intensity of ChR stimulation in single, isolated rabbit SANC. The threshold CCh concentration ([CCh]) for BRR was approximately 10 nM, half maximal inhibition (IC(50)) was achieved at 100 nM, and 1,000 nM stopped spontaneous beating. G(i) inhibition by pertussis toxin blocked all CCh effects on BRR. Using specific ion channel blockers, we established that I(f) blockade did not affect BRR at any [CCh] and that I(KACh) activation, evidenced by hyperpolarization, first became apparent at [CCh] > 30 nM. At IC(50), CCh reduced cAMP and reduced PKA-dependent phospholamban (PLB) phosphorylation by approximately 50%. The dose response of BRR to CCh in the presence of I(KACh) blockade by a specific inhibitor, tertiapin Q, mirrored that of CCh to reduced PLB phosphorylation. At IC(50), CCh caused a time-dependent reduction in the number and size of LCRs and a time dependent increase in LCR period that paralleled coincident BRR. The phosphatase inhibitor calyculin A reversed the effect of IC(50) CCh on SANC LCRs and BRR. Numerical model simulations demonstrated that Ca(2+) cycling is integrated into the cholinergic modulation of BRR via LCR-induced activation of NCX current, providing theoretical support for the experimental findings. Thus ChR stimulation-induced BRR is entirely dependent on G(i) activation and the extent of G(i) coupling to Ca(2+) cycling via PKA signaling or to I(KACh): at low [CCh], I(KACh) activation is not evident and BRR is attributable to a suppression of cAMP-mediated, PKA-dependent Ca(2+) signaling; as [CCh] increases beyond 30 nM, a tight coupling between suppression of PKA-dependent Ca(2+) signaling and I(KACh) activation underlies a more pronounced BRR.

Tertiapin-Q removes a large and rapidly acting component of vagal slowing of the guinea-pig cardiac pacemaker

The participation of acetylcholine-activated potassium current (I(K,ACh)) and hyperpolarization-activated pacemaker current (I(f)) in vagal bradycardia were examined using vagally-innervated preparations of guinea-pig atria. Preparations were maintained in Krebs-Henseleit solution (36 degrees C). Before treatment, trains of vagal stimuli (10 s at 2, 5 and 10 Hz) produced graded bradycardias displaying rapid onset and offset. Tertiapin-Q (300 nM), which blocks I(K,ACh), had no effect on baseline atrial rate. In tertiapin-Q, vagal bradycardia displayed a gradual onset and offset, with a peak response ~50% of that recorded in control conditions. Cumulative addition of 1 mM ZD7288 (blocker of I(f)) caused atrial rate to fall by ~60%, but had no further effect on the amplitude of the vagal bradycardia, while response onset and offset became slightly faster. From these observations, we argue that (i) vagal bradycardia was attributable primarily to activation of I(K,ACh), (ii) vagal modulation of I(f) had a minor influence on the rate of onset and offset of bradycardia, and (iii) removal of the influence of I(K,ACh) unmasked a slow response, of undetermined origin, to vagal stimulation. In a separate set of experiments we compared the effects of 1 mM Ba(2+) and 300 nM tertiapin-Q on vagal bradycardia. Ba(2+) reduced baseline atrial rate and the response to vagal stimulation. Subsequent cumulative addition of tertiapin-Q had no additional effect on baseline atrial rate, but caused further reduction in the amplitude of vagal bradycardia, suggesting that 1 mM Ba(2+) did not achieve a complete block of I(K,ACh) in this preparation.

Mathematical simulations of ligand-gated and cell-type specific effects on the action potential of human atrium

In the mammalian heart, myocytes and fibroblasts can communicate via gap junction, or connexin-mediated current flow. Some of the effects of this electrotonic coupling on the action potential waveform of the human ventricular myocyte have been analyzed in detail. The present study employs a recently developed mathematical model of the human atrial myocyte to investigate the consequences of this heterogeneous cell-cell interaction on the action potential of the human atrium. Two independent physiological processes which alter the physiology of the human atrium have been studied. i) The effects of the autonomic transmitter acetylcholine on the atrial action potential have been investigated by inclusion of a time-independent, acetylcholine-activated K(+) current in this mathematical model of the atrial myocyte. ii) A non-selective cation current which is activated by natriuretic peptides has been incorporated into a previously published mathematical model of the cardiac fibroblast. These results identify subtle effects of acetylcholine, which arise from the nonlinear interactions between ionic currents in the human atrial myocyte. They also illustrate marked alterations in the action potential waveform arising from fibroblast-myocyte source-sink principles when the natriuretic peptide-mediated cation conductance is activated. Additional calculations also illustrate the effects of simultaneous activation of both of these cell-type specific conductances within the atrial myocardium. This study provides a basis for beginning to assess the utility of mathematical modeling in understanding detailed cell-cell interactions within the complex paracrine environment of the human atrial myocardium.

Synergism of coupled subsarcolemmal Ca2+ clocks and sarcolemmal voltage clocks confers robust and flexible pacemaker function in a novel pacemaker cell model

Recent experimental studies have demonstrated that sinoatrial node cells (SANC) generate spontaneous, rhythmic, local subsarcolemmal Ca(2+) releases (Ca(2+) clock), which occur during late diastolic depolarization (DD) and interact with the classic sarcolemmal voltage oscillator (membrane clock) by activating Na(+)-Ca(2+) exchanger current (I(NCX)). This and other interactions between clocks, however, are not captured by existing essentially membrane-delimited cardiac pacemaker cell numerical models. Using wide-scale parametric analysis of classic formulations of membrane clock and Ca(2+) cycling, we have constructed and initially explored a prototype rabbit SANC model featuring both clocks. Our coupled oscillator system exhibits greater robustness and flexibility than membrane clock operating alone. Rhythmic spontaneous Ca(2+) releases of sarcoplasmic reticulum (SR)-based Ca(2+) clock ignite rhythmic action potentials via late DD I(NCX) over much broader ranges of membrane clock parameters [e.g., L-type Ca(2+) current (I(CaL)) and/or hyperpolarization-activated ("funny") current (I(f)) conductances]. The system Ca(2+) clock includes SR and sarcolemmal Ca(2+) fluxes, which optimize cell Ca(2+) balance to increase amplitudes of both SR Ca(2+) release and late DD I(NCX) as SR Ca(2+) pumping rate increases, resulting in a broad pacemaker rate modulation (1.8-4.6 Hz). In contrast, the rate modulation range via membrane clock parameters is substantially smaller when Ca(2+) clock is unchanged or lacking. When Ca(2+) clock is disabled, the system parametric space for fail-safe SANC operation considerably shrinks: without rhythmic late DD I(NCX) ignition signals membrane clock substantially slows, becomes dysrhythmic, or halts. In conclusion, the Ca(2+) clock is a new critical dimension in SANC function. A synergism of the coupled function of Ca(2+) and membrane clocks confers fail-safe SANC operation at greatly varying rates.

Genesis and regulation of the heart automaticity

The heart automaticity is a fundamental physiological function in higher organisms. The spontaneous activity is initiated by specialized populations of cardiac cells generating periodical electrical oscillations. The exact cascade of steps initiating the pacemaker cycle in automatic cells has not yet been entirely elucidated. Nevertheless, ion channels and intracellular Ca(2+) signaling are necessary for the proper setting of the pacemaker mechanism. Here, we review the current knowledge on the cellular mechanisms underlying the generation and regulation of cardiac automaticity. We discuss evidence on the functional role of different families of ion channels in cardiac pacemaking and review recent results obtained on genetically engineered mouse strains displaying dysfunction in heart automaticity. Beside ion channels, intracellular Ca(2+) release has been indicated as an important mechanism for promoting automaticity at rest as well as for acceleration of the heart rate under sympathetic nerve input. The potential links between the activity of ion channels and Ca(2+) release will be discussed with the aim to propose an integrated framework of the mechanism of automaticity.

Regional difference in dynamical property of sinoatrial node pacemaking: role of na+ channel current

To elucidate the regional differences in sinoatrial node pacemaking mechanisms, we investigated 1), bifurcation structures during current blocks or hyperpolarization of the central and peripheral cells, 2), ionic bases of regional differences in bifurcation structures, and 3), the role of Na(+) channel current (I(Na)) in peripheral cell pacemaking. Bifurcation analyses were performed for mathematical models of the rabbit sinoatrial node central and peripheral cells; equilibrium points, periodic orbits, and their stability were determined as functions of parameters. Structural stability against applications of acetylcholine or electrotonic modulations of the atrium was also evaluated. Blocking L-type Ca(2+) channel current (I(Ca,L)) stabilized equilibrium points and abolished pacemaking in both the center and periphery. Critical acetylcholine concentration and gap junction conductance for pacemaker cessation were higher in the periphery than in the center, being dramatically reduced by blocking I(Na). Under hyperpolarized conditions, blocking I(Na), but not eliminating I(Ca,L), abolished peripheral cell pacemaking. These results suggest that 1), I(Ca,L) is responsible for basal pacemaking in both the central and peripheral cells, 2), the peripheral cell is more robust in withstanding hyperpolarizing loads than the central cell, 3), I(Na) improves the structural stability to hyperpolarizing loads, and 4), I(Na)-dependent pacemaking is possible in hyperpolarized peripheral cells.

The effects of tertiapin-Q on responses of the sinoatrial pacemaker of the guinea-pig heart to vagal nerve stimulation and muscarinic agonists

Using Langendorff preparations of the guinea-pig heart, we have examined the participation of the acetylcholine (ACh)-activated potassium channel, IK,ACh, in the bradycardia produced by electrical stimulation of the vagus (parasympathetic) nerve and muscarinic agonists (ACh and bethanecol, bolus i.a.). Hearts from young animals (160-250 g) were perfused with Krebs-Henseleit solution, and pacemaker frequency was determined from the P wave of an ECG. Tertiapin-Q was used to block IK,ACh. Vagal stimulation (10 s trains at 2, 5 and 10 Hz) produced graded reductions in atrial rate that were substantially attenuated, and to a similar extent, by 300 nm and 1 microm tertiapin-Q (to 0.42 +/- 0.12, mean +/- s.d., of the control values; P < 0.001). Acetylcholine (3 nmol) produced brief graded bradycardias that were also attenuated by tertiapin-Q (0.24 +/- 0.24; P = 0.006). Similar results were obtained when experiments were repeated in 2 mm Cs+ (to block the hyperpolarization-activated pacemaker current). Bethanecol (30, 50 and 70 nmol), a muscarinic agonist with no appreciable nicotinic activity, produced sustained bradycardias that were attenuated by 300 nm tertiapin-Q (0.36 +/- 0.21; P < 0.0001). The responses to vagal stimulation and ACh developed more slowly in tertiapin-Q, indicating that a rapidly acting mechanism had been blocked. Responses to vagal stimulation were faster in 2 mm Cs+. Together, these observations show that ACh released from parasympathetic nerve varicosities exerts a considerable part of its effect on the pacemaker by activating IK,ACh and acts in a manner not readily distinguishable from that of directly applied muscarinic agonists.

Computer modelling of the sinoatrial node

Over the past decades patch-clamp experiments have provided us with detailed information on the different types of ion channels that are present in the cardiac cell membrane. Sophisticated cardiac cell models based on these data can help us understand how the different types of ion channels act together to produce the cardiac action potential. In the field of biological pacemaker engineering, such models provide important instruments for the assessment of the functional implications of changes in density of specific ion channels aimed at producing stable pacemaker activity. In this review, an overview is given of the progress made in cardiac cell modelling, with particular emphasis on the development of sinoatrial (SA) nodal cell models. Also, attention is given to the increasing number of publicly available tools for non-experts in computer modelling to run cardiac cell models.

Nonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the heart

A dedicated nonlinear oscillator model able to reproduce the pulse shape, refractory time, and phase sensitivity of the action potential of a natural pacemaker of the heart is developed. The phase space of the oscillator contains a stable node, a hyperbolic saddle, and an unstable focus. The model reproduces several phenomena well known in cardiology, such as certain properties of the sinus rhythm and heart block. In particular, the model reproduces the decrease of heart rate variability with an increase in sympathetic activity. A sinus pause occurs in the model due to a single, well-timed, external pulse just as it occurs in the heart, for example due to a single supraventricular ectopy. Several ways by which the oscillations cease in the system are obtained (models of the asystole). The model simulates properly the way vagal activity modulates the heart rate and reproduces the vagal paradox. Two such oscillators, coupled unidirectionally and asymmetrically, allow us to reproduce the properties of heart rate variability obtained from patients with different kinds of heart block including sino-atrial blocks of different degree and a complete AV block (third degree). Finally, we demonstrate the possibility of introducing into the model a spatial dimension that creates exciting possibilities of simulating in the future the SA the AV nodes and the atrium including their true anatomical structure.

Phase response characteristics of sinoatrial node cells

In this work, the dynamic response of the sinoatrial node (SAN), the natural pacemaker of the heart, to short external stimuli is investigated using the Zhang et al. model. The model equations are solved twice for the central cell and for the peripheral cell. A short current pulse is applied to reset the spontaneous rhythmic activity of the single sinoatrial node cell. Depending on the stimulus timing either a delay or an advance in the occurrence of next action potential is produced. This resetting behavior is quantified in terms of phase transition curves (PTCs) for short electrical current pulses of varying amplitude which span the whole period. For low stimulus amplitudes the transition from advance to delay is smooth, while at higher amplitudes abrupt changes and discontinuities are observed in PTCs. Such discontinuities reveal critical stimuli, the application of which can result in annihilation of activity in central SAN cells. The detailed analysis of the ionic mechanisms involved in its resetting behavior of sinoatrial node cell models provides new insight into the dynamics and physiology of excitation of the sinoatrial node of the heart.

Postganglionic nerve stimulation induces temporal inhibition of excitability in rabbit sinoatrial node

Vagal stimulation results in complex changes of pacemaker excitability in the sinoatrial node (SAN). To investigate the vagal effects in the rabbit SAN, we used optical mapping, which is the only technology that allows resolving simultaneous changes in the activation pattern and action potentials morphologies. With the use of immunolabeling, we identified the SAN as a neurofilament 160-positive but connexin 43-negative region ( n = 5). Normal excitation originated in the SAN center with a cycle length (CL) of 405 ± 14 ms ( n = 14), spread anisotropically along the crista terminalis (CT), and failed to conduct toward the septum. Postganglionic nerve stimulation (PNS, 400–800 ms) reduced CL by 74 ± 7% transiently and shifted the leading pacemaker inferiorly (78%) or superiorly (22%) from the SAN center by 2–10 mm. In the intercaval region between the SAN center and the septal block zone, PNS produced an 8 ± 1-mm2 region of transient hyperpolarization and inexcitability. The first spontaneous or paced excitation following PNS could not enter this region for 500–1,500 ms. Immunolabeling revealed that the PNS-induced inexcitable region is located between the SAN center and the block zone and has a 2.5-fold higher density of choline acetyltransferase than CT but is threefold lower than the SAN center. The fact that the inexcitability region does not coincide with the most innervated area indicates that the properties of the myocytes themselves, as well as intercellular coupling, must play a role in the inexcitability induction. Optically mapping revealed that PNS resulted in transient loss of pacemaker cell excitability and unidirectional entrance conduction block in the periphery of SAN.

Dynamical mechanisms of pacemaker generation in IK1-downregulated human ventricular myocytes: insights from bifurcation analyses of a mathematical model

Dynamical mechanisms of the biological pacemaker (BP) generation in human ventricular myocytes were investigated by bifurcation analyses of a mathematical model. Equilibrium points (EPs), periodic orbits, stability of EPs, and bifurcation points were determined as functions of bifurcation parameters, such as the maximum conductance of inward-rectifier K+ current (I(K1)), for constructing bifurcation diagrams. Stable limit cycles (BP activity) abruptly appeared around an unstable EP via a saddle-node bifurcation when I(K1) was suppressed by 84.6%. After the bifurcation at which a stable EP disappears, the I(K1)-reduced system has an unstable EP only, which is essentially important for stable pacemaking. To elucidate how individual sarcolemmal currents contribute to EP instability and BP generation, we further explored the bifurcation structures of the system during changes in L-type Ca2+ channel current (I(Ca,L)), delayed-rectifier K+ currents (I(K)), or Na(+)/Ca2+ exchanger current (I(NaCa)). Our results suggest that 1), I(Ca,L) is, but I(K) or I(NaCa) is not, responsible for EP instability as a requisite to stable BP generation; 2), I(K) is indispensable for robust pacemaking with large amplitude, high upstroke velocity, and stable frequency; and 3), I(NaCa) is the dominant pacemaker current but is not necessarily required for the generation of spontaneous oscillations.

The cardiac cycle time effect revisited: temporal dynamics of the central-vagal modulation of heart rate in human reaction time tasks

Lacey and Lacey (1974) suggested that during reaction time tasks higher brain centers dynamically adjust efferent vagal nerve pulses to the sino-atrial node of the heart, inducing phase-dependent heart rate changes. Since then, animal and human neuro-physiological results have provided evidence for this hypothesis. Higher subcortical and cortical brain centers may have reciprocal interactive pathways relating to autonomic control comparable to those at the level of peripheral autonomic changes and brain stem reflexes. In humans such central effects may be observed in the short latency vagal control of heart rate that has been studied mostly in reaction time (RT) tasks. RT task parameters modulate vagal pulses to the cardiac sino-atrial node (SAN), which in turn exerts a phase-dependent change in the ongoing cardiac interbeat interval. Simulations of human RT task effects in an animal model of heart rate change support this hypothesis. The current study examined evidence for vagal control of three human phasic heart rate responses in RT tasks. The evidence indicates that the initiation of an RT response triggers a reflexive shift from vagal activation to vagal inhibition. This shift is cardiac cycle phase dependent. Graded anticipatory cardiac deceleration during the warning interval of an RT task varies with task relevance and time uncertainty. This response may be part of a control process engaged in time keeping. Hence, temporal variables mediate the central-autonomic-vagal modulation of heart rate.

Roles of L-type Ca2+ and delayed-rectifier K+ currents in sinoatrial node pacemaking: insights from stability and bifurcation analyses of a mathematical model

To elucidate the dynamical mechanisms of the sinoatrial (SA) node pacemaker activity, we investigated the roles of L-type Ca2+ (ICa,L) and delayed-rectifier K+ (IKr) currents in pacemaking by stability and bifurcation analyses of our rabbit SA node model (Kurata Y, Hisatome I, Imanishi S, and Shibamoto T. Am J Physiol Heart Circ Physiol 283: H2074-H2101, 2002). Equilibrium points (EPs), periodic orbits, stability of EPs, and Hopf bifurcation points were calculated as functions of conductance or gating time constants of the currents for constructing bifurcation diagrams. Structural stability (robustness) of the system was also evaluated by computing stability and dynamics during applications of constant bias currents (Ibias). Blocking ICa,L or IKr caused stabilization of an EP and cessation of pacemaking via a Hopf bifurcation. The unstable zero-current potential region determined with Ibias applications, where spontaneous oscillations appear, shrunk and finally disappeared as ICa,L diminished, but shrunk little when IKr was eliminated. The reduced system, including no time-dependent current except ICa,L, exhibited pacemaker activity. These results suggest that ICa,L is responsible for EP instability and pacemaker generation, whereas IKr is not necessarily required for constructing a pacemaker cell system. We further explored the effects of various K+ currents with different kinetics on stability and dynamics of the model cell. The original IKr of delayed activation and inward rectification appeared to be most favorable for generating large-amplitude oscillations with stable frequency, suggesting that IKr acts as an oscillation amplifier and frequency stabilizer. IKr may also play an important role in preventing bifurcation to quiescence of the system.

A mathematical model of the electrophysiological alterations in rat ventricular myocytes in type-I diabetes

Our mathematical model of the rat ventricular myocyte (Pandit et al., 2001) was utilized to explore the ionic mechanism(s) that underlie the altered electrophysiological characteristics associated with the short-term model of streptozotocin-induced, type-I diabetes. The simulations show that the observed reductions in the Ca(2+)-independent transient outward K(+) current (I(t)) and the steady-state outward K(+) current (I(ss)), along with slowed inactivation of the L-type Ca(2+) current (I(CaL)), can result in the prolongation of the action potential duration, a well-known experimental finding. In addition, the model demonstrates that the slowed reactivation kinetics of I(t) in diabetic myocytes can account for the more pronounced rate-dependent action potential duration prolongation in diabetes, and that a decrease in the electrogenic Na(+)-K(+) pump current (I(NaK)) results in a small depolarization in the resting membrane potential (V(rest)). This depolarization reduces the availability of the Na(+) channels (I(Na)), thereby resulting in a slower upstroke (dV/dt(max)) of the diabetic action potential. Additional simulations suggest that a reduction in the magnitude of I(CaL), in combination with impaired sarcoplasmic reticulum uptake can lead to a decreased sarcoplasmic reticulum Ca(2+) load. These factors contribute to characteristic abnormal [Ca(2+)](i) homeostasis (reduced peak systolic value and rate of decay) in myocytes from diabetic animals. In combination, these simulation results provide novel information and integrative insights concerning plausible ionic mechanisms for the observed changes in cardiac repolarization and excitation-contraction coupling in rat ventricular myocytes in the setting of streptozotocin-induced, type-I diabetes.

Dynamical description of sinoatrial node pacemaking: improved mathematical model for primary pacemaker cell

We developed an improved mathematical model for a single primary pacemaker cell of the rabbit sinoatrial node. Original features of our model include 1) incorporation of the sustained inward current (I(st)) recently identified in primary pacemaker cells, 2) reformulation of voltage- and Ca(2+)-dependent inactivation of the L-type Ca(2+) channel current (I(Ca,L)), 3) new expressions for activation kinetics of the rapidly activating delayed rectifier K(+) channel current (I(Kr)), and 4) incorporation of the subsarcolemmal space as a diffusion barrier for Ca(2+). We compared the simulated dynamics of our model with those of previous models, as well as with experimental data, and examined whether the models could accurately simulate the effects of modulating sarcolemmal ionic currents or intracellular Ca(2+) dynamics on pacemaker activity. Our model represents significant improvements over the previous models, because it can 1) simulate whole cell voltage-clamp data for I(Ca,L), I(Kr), and I(st); 2) reproduce the waveshapes of spontaneous action potentials and ionic currents during action potential clamp recordings; and 3) mimic the effects of channel blockers or Ca(2+) buffers on pacemaker activity more accurately than the previous models.

A mathematical model of action potential heterogeneity in adult rat left ventricular myocytes

Mathematical models were developed to reconstruct the action potentials (AP) recorded in epicardial and endocardial myocytes isolated from the adult rat left ventricle. The main goal was to obtain additional insight into the ionic mechanisms responsible for the transmural AP heterogeneity. The simulation results support the hypothesis that the smaller density and the slower reactivation kinetics of the Ca(2+)-independent transient outward K(+) current (I(t)) in the endocardial myocytes can account for the longer action potential duration (APD), and more prominent rate dependence in that cell type. The larger density of the Na(+) current (I(Na)) in the endocardial myocytes results in a faster upstroke (dV/dt(max)). This, in addition to the smaller magnitude of I(t), is responsible for the larger peak overshoot of the simulated endocardial AP. The prolonged APD in the endocardial cell also leads to an enhanced amplitude of the sustained K(+) current (I(ss)), and a larger influx of Ca(2+) ions via the L-type Ca(2+) current (I(CaL)). The latter results in an increased sarcoplasmic reticulum (SR) load, which is mainly responsible for the higher peak systolic value of the Ca(2+) transient [Ca(2+)](i), and the resultant increase in the Na(+)-Ca(2+) exchanger (I(NaCa)) activity, associated with the simulated endocardial AP. In combination, these calculations provide novel, quantitative insights into the repolarization process and its naturally occurring transmural variations in the rat left ventricle.