Dispersion gap and localized spiral waves in a model for intracellular Ca2+ dynamics

Bistable spiral wave dynamics in electrically excitable media

We show that a positive feedback loop between sodium current inactivation and wave-front ramp-up speed causes a saddle-node bifurcation to result in bistable planar and spiral waves in electrically excitable media, in which both slow and fast waves are triggered by different stimulation protocols. Moreover, the two types of spiral wave conduction may interact to give rise to more complex spiral wave dynamics. The transitions between different spiral wave behaviors via saddle-node bifurcation can be a candidate mechanism for transitions widely seen in cardiac arrhythmias and neural diseases.

Wave failure at strong coupling in intracellular Ca^{2+} signaling system with clustered channels

As an important intracellular signal, Ca^{2+} ions control diverse cellular functions. In this paper, we discuss the Ca^{2+} signaling with a two-dimensional model in which the inositol 1,4,5-trisphosphate (IP_{3}) receptor channels are distributed in clusters on the endoplasmic reticulum membrane. The wave failure at large Ca^{2+} diffusion coupling is discussed in detail in the model. We show that with varying model parameters the wave failure is a robust behavior with either deterministic or stochastic channel dynamics. We suggest that the wave failure should be a general behavior in inhomogeneous diffusing systems with clustered excitable regions and may occur in biological Ca^{2+} signaling systems.

Calcium waves in a grid of clustered channels with synchronous IP3 binding and unbinding

Calcium signals in cells occur at multiple spatial scales and variable temporal duration. However, a physical explanation for transitions between long-lasting global oscillations and localized short-term elevations (puffs) of cytoplasmic Ca²⁺ is still lacking. Here we introduce a phenomenological, coarse-grained model for the calcium variable, which is represented by ordinary differential equations. Due to its small number of parameters, and its simplicity, this model allows us to numerically study the interplay of multi-scale calcium concentrations with stochastic ion channel gating dynamics even in larger systems. We apply this model to a single cluster of inositol trisphosphate (IP ₃) receptor channels and find further evidence for the results presented in earlier work: a single cluster may be capable of producing different calcium release types, where long-lasting events are accompanied by unbinding of IP ₃ from the receptor (Rückl et al., PLoS Comput. Biol. 11, e1003965 (2015)). Finally, we show the practicability of the model in a grid of 64 clusters which is computationally intractable with previous high-resolution models. Here long-lasting events can lead to synchronized oscillations and waves, while short events stay localized. The frequency of calcium releases as well as their coherence can thereby be regulated by the amplitude of IP ₃ stimulation. Finally the model allows for a new explanation of oscillating [IP ₃], which is not based on metabolic production and degradation of IP ₃.

Role of time delay on intracellular calcium dynamics driven by non-Gaussian noises

Effect of time delay (τ) on intracellular calcium dynamics with non-Gaussian noises in transmission processes of intracellular Ca²⁺ is studied by means of second-order stochastic Runge-Kutta type algorithm. By simulating and analyzing time series, normalized autocorrelation function, and characteristic correlation time of cytosolic and calcium store’s Ca²⁺ concentration, the results exhibit: (i) intracellular calcium dynamics’s time coherence disappears and stability strengthens as τ → 0.1s; (ii) for the case of τ < 0.1s, the normalized autocorrelation functions of cytosolic and calcium store’s Ca²⁺ concentration show damped motion when τ is very short, but they trend to a level line as τ → 0.1s, and for the case of τ > 0.1s, they show different variation as τ increases, the former changes from underdamped motion to a level line, but the latter changes from damped motion to underdamped motion; and (iii) at the moderate value of time delay, reverse resonance occurs both in cytosol and calcium store.

Unusual spiral wave dynamics in the Kessler-Levine model of an excitable medium

The Kessler-Levine model is a two-component reaction-diffusion system that describes spatiotemporal dynamics of the messenger molecules in a cell-to-cell signaling process during the aggregation of social amoeba cells. An excitation wave arising in the model has a phase wave at the wave back, which simply follows the wave front after a fixed time interval with the same propagation velocity. Generally speaking, the medium excitability and the refractoriness are two important factors which determine the spiral wave dynamics in any excitable media. The model allows us to separate these two factors relatively easily since the medium refractoriness can be changed independently of the medium excitability. For rigidly rotating waves, the universal relationship has been established by using a modified free-boundary approach, which assumes that the front and the back of a propagating wave are thin in comparison to the wave plateau. By taking a finite thickness of the domain boundary into consideration, the validity of the proposed excitability measure has been essentially improved. A novel method of numerical simulation to suppress the spiral wave instabilities is introduced. The trajectories of the spiral tip observed for a long refractory period have been investigated under a systematic variation of the medium refractoriness.

Origin of finite pulse trains: Homoclinic snaking in excitable media

Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.

Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Targeting and clustering of IP3 receptors: key determinants of spatially organized Ca2+ signals

Inositol 1,4,5-trisphosphate receptors (IP3R) are intracellular Ca2+ channels that are almost ubiquitously expressed in animal cells. The spatiotemporal complexity of the Ca2+ signals evoked by IP3R underlies their versatility in cellular signaling. Here we review the mechanisms that contribute to the subcellular targeting of IP3R and the dynamic interplay between IP3R that underpin their ability to generate complex intracellular Ca2+ signals.

Image analysis of Ca2+ signals as a basis for neurotoxicity assays: promises and challenges

Free intracellular calcium ([Ca(2+)](i)) controls a wide range of cellular functions such as contraction, neurotransmitter and hormone release, metabolism, cell division and differentiation. Cytosolic Ca(2+) levels are abnormal in cells exposed to toxicants and understanding how these levels become altered may improve our ability to design high-throughput methods for the sensitive detection of cellular responses to a toxic exposure. Because Ca(2+) is involved in multiple aspects of cellular function, its role in signaling is complex. It is therefore necessary to identify the individual pathways targeted during toxicant exposure in order to use them as a tool for predictive measurements of toxicity and as targets for prevention or reversal of injury. This review illustrates several methods available for analysis of Ca(2+) responses in vitro and their applicability for understanding mechanisms of toxicity at the molecular and cellular levels. The review will also consider the usefulness of Ca(2+) imaging for predicting a unique signature for classes of toxicants. Towards this end, two methodological approaches for assessment of Ca(2+) responses to toxicants are examined: steady state measurements and complex spatial and/or temporal measurements. Each of the methods described and appropriately used results in reliable and reproducible measurements which may be applied in a high-throughput fashion to individualize in vitro assessment of cellular responses caused by toxicants.

Generation of finite wave trains in excitable media

Spatiotemporal control of excitable media is of paramount importance in the development of new applications, ranging from biology to physics. To this end, we identify and describe a qualitative property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The wave trains are produced by a transient pacemaker generated by a one-time suitably tailored spatially localized finite amplitude stimulus, and belong to a family of fast pulse trains. A second family, of slow pulse trains, is also present. The latter are created through a clumping instability of a traveling wave state (in an excitable regime) and are inaccessible to single localized stimuli of the type we use. The results indicate that the presence of a large multiplicity of stable, accessible, multi-pulse states is a general property of simple models of excitable media.

Modeling Ca2+ feedback on a single inositol 1,4,5-trisphosphate receptor and its modulation by Ca2+ buffers

The inositol 1,4,5-trisphosphate receptor/channel (IP(3)R) is a major regulator of intracellular Ca(2+) signaling, and liberates Ca(2+) ions from the endoplasmic reticulum in response to binding at cytosolic sites for both IP(3) and Ca(2+). Although the steady-state gating properties of the IP(3)R have been extensively studied and modeled under conditions of fixed [IP(3)] and [Ca(2+)], little is known about how Ca(2+) flux through a channel may modulate the gating of that same channel by feedback onto activating and inhibitory Ca(2+) binding sites. We thus simulated the dynamics of Ca(2+) self-feedback on monomeric and tetrameric IP(3)R models. A major conclusion is that self-activation depends crucially on stationary cytosolic Ca(2+) buffers that slow the collapse of the local [Ca(2+)] microdomain after closure. This promotes burst-like reopenings by the rebinding of Ca(2+) to the activating site; whereas inhibitory actions are substantially independent of stationary buffers but are strongly dependent on the location of the inhibitory Ca(2+) binding site on the IP(3)R in relation to the channel pore.

Electric-field-induced wave groupings of spiral waves with oscillatory dispersion relation

The dynamic behavior of spiral-shaped excitation patterns with oscillatory dispersion is investigated under the influence of externally applied direct current or alternating current. For these two types of electric field, wave-grouping phenomena are generally observed. For the direct current field, the spiral wave drifts approximately along a straight line and wave groupings appear in certain ranges of spatial polar angles when the strength of the external field is larger than a threshold. In terms of the Doppler effect induced by the drift of the spiral tip and the oscillatory dispersion, we propose a theory model to predict the spatial distribution of wave grouping and the critical strength of the current. In contrast, for the alternating current field, the spiral wave may stay stationary and wave grouping may appear in the whole space with a different manner. This finding indicates that movement of the spiral tip is not necessary for the appearance of wave grouping.

Anomalous pulse interaction in dissipative media

We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.

Chaotic pulse transmission and spiral formation in a calcium oscillation model

We study a two-dimensional reaction-diffusion equation for calcium oscillation with a pacemaker region. When the pacemaker entrains the whole system, circular waves are observed as a target pattern. However, if the pace of the pacemaker is too fast, the pulse propagation to the outer region sometimes fails in a chaotic manner. We find that spiral waves are spontaneously created at the interface between the pacemaker region and the outer region.

Wave-pinned filaments of scroll waves

Scroll waves are three-dimensional excitation patterns that rotate around one-dimensional space curves. Typically these filaments are closed loops or end at the system boundary. However, in excitable media with anomalous dispersion, filaments can be pinned to the wake of traveling wave pulses. This pinning is studied in experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction and a three-variable reaction-diffusion model. We show that wave-pinned filaments are related to the coexistence of rotating and translating wave defects in two dimensions. Filament pinning causes a continuous expansion of the total filament length. It can be ended by annihilating the pinning pulse in a frontal wave collision. Following such an annihilation, the filament connects itself to the system boundary. Its postannihilation shape that is initially the exposed rim of the scroll wave unwinds continuously over numerous rotation periods.

Scroll Wave Filaments Terminate in the Back of Traveling Fronts

Experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction demonstrate that three-dimensional scroll waves can rotate around filaments that end in the wake of a traveling excitation pulse. The vortex structures nucleate during the collision of three nonrotating excitation pulses. The nucleation process and the wave-termination of filaments are direct consequences of the system's anomalous dispersion relation. Vortex filaments are found to expand with about twice the speed of their anchoring wave fronts. Filament expansion is accompanied by the build-up of phase differences in spiral rotation creating strongly twisted wave structures. Experiments employ optical tomography for the reconstruction of the three-dimensional wave patterns.

Wave trains in an excitable FitzHugh-Nagumo model: bistable dispersion relation and formation of isolas

We investigate the dispersion relations of nonlinear periodic wave trains in excitable systems which describe the dependence of the propagation velocity on the wavelength. Pulse interaction by oscillating pulse tails within a wave train leads to bistable wavelength bands, in which two stable and one unstable wave train coexist for the same wavelength. The essential spectra of the unstable wave trains exhibit a circle of eigenvalues with positive real parts which is detached from the imaginary axis. We describe the destruction of the bistable dispersion curve and the formation of isolas of wave trains in a sequence of transcritical bifurcations unfolding into pairs of saddle-node bifurcations. It turns out that additional dispersion curves of unstable wave trains play an important role in the destruction of the bistable dispersion curve.

A mesoscopic stochastic mechanism of cytosolic calcium oscillations

Based on a model of intracellular calcium (Ca(2+)) oscillation with self-modulation of inositol 1,4,5-trisphosphate signal, the mesoscopic stochastic differential equations for the intracellular Ca(2+) oscillations are theoretically derived by using the chemical Langevin equation method. The effects of the finite biochemical reaction molecule number on both simple and complex cytosolic Ca(2+) oscillations are numerically studied. In the case of simple intracellular Ca(2+) oscillation, it is found that, with the increase of molecule number, the coherence resonance or autonomous resonance phenomena can occur for some external stimulation parameter values. In the cases of complex cytosolic Ca(2+) oscillations, each extremum of concentration of cytosolic Ca(2+) oscillations corresponds to a peak in the histogram of Ca(2+) concentration, and the most probability appeared during the bursting plateau level for bursting, but at the largest minimum of Ca(2+) concentration for chaos. For quasi-periodicity, however, there are only two peaks in the histogram of Ca(2+) concentration, and the most probability is located at low concentration state.

Propagation failures, breathing pulses, and backfiring in an excitable reaction-diffusion system

We report results from experiments with a pseudo-one-dimensional Belousov-Zhabotinsky reaction that employs 1,4-cyclohexanedione as its organic substrate. This excitable system shows traveling oxidation pulses and pulse trains that can undergo complex sequences of propagation failures. Moreover, we present examples for (i) breathing pulses that undergo periodic changes in speed and size and (ii) backfiring pulses that near their back repeatedly generate new pulses propagating in opposite direction.

Propagation failure dynamics of wave trains in excitable systems

We report experimental and numerical results on temporal patterns of propagation failures in reaction-diffusion systems. Experiments employ the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagation failures occur in the frontier region of the wave train and can profoundly affect its expansion speed. The specific rhythms observed vary from simple periodic to highly complex and possibly chaotic sequences. All but the period-1 sequences are found in the transition region between "merging" and "tracking" dynamics, which correspond to wave behavior caused by two qualitatively different types of anomalous dispersion relations.

Multiphoton laser scanning microscopy as a tool for Xenopus oocyte research

Multiphoton laser scanning microscopy (MPLSM) has become an increasingly invaluable tool in fluorescent optical imaging. There are several distinct advantages to implementing MPLSM as a Xenopus oocyte research tool. MPLSM increases signal-to-noise ratio and therefore increases image quality because there is no out-of-focus fluorescence as would be created in conventional or confocal microscopy. All the light that is generated can be collected and used to generate an image because point detection of descanned fluorescence is not required. This is particularly useful when imaging deep into tissue sections, as is necessary for Xenopus oocytes, which are notoriously large (approximately 1-mm diameter). Because multiphoton lasers use pulsed energy in the infrared wavelengths, the energy can also travel further into tissues with much less light scattering. Because there is no out-of-focus excitation, phototoxicity, photodamage, and photobleaching are significantly reduced, which is particularly important for long-term experiments that require the same region to be scanned repeatedly. Finally, multiple fluorophores can be simultaneously excited because of the broader absorption spectra of multiphoton dyes. In this chapter, we describe the advantages and disadvantages of using MPLSM to image Xenopus oocytes as compared to conventional and confocal microscopy. The practical application of imaging oocytes is demonstrated with specific examples.

Front aggregation in multiarmed excitation vortices

Using the Belousov-Zhabotinsky reaction, we study the pinning of multiarmed spiral waves to nonexcitable obstacles. With increasing obstacle size, the individual arms switch from a repulsive to an attractive state. This transition yields densely aggregated spiral arms and is caused by anomalous dispersion. A kinematic model reproduces the measurements quantitatively and identifies the transition as a supercritical pitchfork bifurcation.

Release currents of IP(3) receptor channel clusters and concentration profiles

We simulate currents and concentration profiles generated by Ca(2+) release from the endoplasmic reticulum (ER) to the cytosol through IP(3) receptor channel clusters. Clusters are described as conducting pores in the lumenal membrane with a diameter from 6 nm to 36 nm. The endoplasmic reticulum is modeled as a disc with a radius of 1-12 microm and an inner height of 28 nm. We adapt the dependence of the currents on the trans Ca(2+) concentration (intralumenal) measured in lipid bilayer experiments to the cellular geometry. Simulated currents are compared with signal mass measurements in Xenopus oocytes. We find that release currents depend linearly on the concentration of free Ca(2+) in the lumen. The release current is approximately proportional to the square root of the number of open channels in a cluster. Cytosolic concentrations at the location of the cluster range from 25 microM to 170 microM. Concentration increase due to puffs in a distance of a few micrometers from the puff site is found to be in the nanomolar range. Release currents decay biexponentially with timescales of <1 s and a few seconds. Concentration profiles decay with timescales of 0.125-0.250 s upon termination of release.

Modeling the dependence of the period of intracellular Ca2+ waves on SERCA expression

Contrary to intuitive expectations, overexpression of sarco-endoplasmic reticulum (ER) Ca(2+) ATPases (SERCAs) in Xenopus oocytes leads to a decrease in the period and an increase in the amplitude of intracellular Ca(2+) waves. Here we examine these experimental findings by modeling Ca(2+) release using a modified Othmer-Tang-model. An increase in the period and a reduction in the amplitude of Ca(2+) wave activity are obtained when increases in SERCA density are simulated while keeping all other parameters of the model constant. However, Ca(2+) wave period can be reduced and the wave amplitude and velocity can be significantly increased when an increase in the luminal ER Ca(2+) concentration due to SERCA overexpression is incorporated into the model. Increased luminal Ca(2+) occurs because increased SERCA activity lowers cytosolic Ca(2+), which is partially replenished by Ca(2+) influx across the plasma membrane. These simulations are supported by experimental data demonstrating higher luminal Ca(2+) levels, decreased periods, increased amplitude, and increased velocity of Ca(2+) waves in response to increased SERCA density.

Oscillatory dispersion and coexisting stable pulse trains in an excitable medium

For planar wave trains in excitable media, we found a novel type of anomalous dispersion distinguished by bistable domains in the dependence of the propagation velocity on the wavelength. Within one medium alternative stable pulse trains can coexist having the same wavelength but different velocities. The phenomenon is related to oscillatory recovery of excitations, which causes small amplitude oscillations in the refractory tail of pulses. Crucial for the bistability is that the pulses in the trains are locked into one oscillation maximum in the tail of the preceding pulse in the train.

On the role of stochastic channel behavior in intracellular Ca2+ dynamics

I present a stochastic model for intracellular Ca(2+) oscillations. The model starts from stochastic binding and dissociation of Ca(2+) to binding sites on a single subunit of the IP(3)-receptor channel but is capable of simulating large numbers of clusters for many oscillation periods too. I find oscillations with variable periods ranging from 17 s to 120 s and a standard deviation well in the experimentally observed range. Long period oscillations can be perceived as nucleation phenomenon and can be observed for a large variety of single channel dynamics. Their period depends on the geometric characteristics of the cluster array. Short periods are in the range of the time scale of channel dynamics. Both long and short period oscillations occur for parameters with a nonoscillatory deterministic regime.