A Statistical View on Calcium Oscillations.

Transient rises and falls of the intracellular calcium concentration have been observed in numerous cell types and under a plethora of conditions. There is now a growing body of evidence that these whole-cell calcium oscillations are stochastic, which poses a significant challenge for modelling. In this review, we take a closer look at recently developed statistical approaches to calcium oscillations. These models describe the timing of whole-cell calcium spikes, yet their parametrisations reflect subcellular processes. We show how non-stationary calcium spike sequences, which e.g. occur during slow depletion of intracellular calcium stores or in the presence of time-dependent stimulation, can be analysed with the help of so-called intensity functions. By utilising Bayesian concepts, we demonstrate how values of key parameters of the statistical model can be inferred from single cell calcium spike sequences and illustrate what information whole-cell statistical models can provide about the subcellular mechanistic processes that drive calcium oscillations. In particular, we find that the interspike interval distribution of HEK293 cells under constant stimulation is captured by a Gamma distribution.

Complex patterns of subcellular cardiac alternans.

Cardiac alternans, in which the membrane potential and the intracellular calcium concentration exhibit alternating durations and peak amplitudes at consecutive beats, constitute a precursor to fatal cardiac arrhythmia such as sudden cardiac death. A crucial question therefore concerns the onset of cardiac alternans. Typically, alternans are only reported when they are fully developed. Here, we present a modelling approach to explore recently discovered microscopic alternans, which represent one of the earliest manifestations of cardiac alternans. In this case, the regular periodic dynamics of the local intracellular calcium concentration is already unstable, while the whole-cell behaviour suggests a healthy cell state. In particular, we use our model to investigate the impact of calcium diffusion in both the cytosol and the sarcoplasmic reticulum on the formation of microscopic calcium alternans. We find that for dominant cytosolic coupling, calcium alternans emerge via the traditional period doubling bifurcation. In contrast, dominant luminal coupling leads to a novel route to calcium alternans through a saddle-node bifurcation at the network level. Combining semi-analytical and computational approaches, we compute areas of stability in parameter space and find that as we cross from stable to unstable regions, the emergent patterns of the intracellular calcium concentration change abruptly in a fashion that is highly dependent upon position along the stability boundary. Our results demonstrate that microscopic calcium alternans may possess a much richer dynamical repertoire than previously thought and further strengthen the role of luminal calcium in shaping cardiac calcium dynamics.

A Bayesian approach to modelling heterogeneous calcium responses in cell populations.

Calcium responses have been observed as spikes of the whole-cell calcium concentration in numerous cell types and are essential for translating extracellular stimuli into cellular responses. While there are several suggestions for how this encoding is achieved, we still lack a comprehensive theory. To achieve this goal it is necessary to reliably predict the temporal evolution of calcium spike sequences for a given stimulus. Here, we propose a modelling framework that allows us to quantitatively describe the timing of calcium spikes. Using a Bayesian approach, we show that Gaussian processes model calcium spike rates with high fidelity and perform better than standard tools such as peri-stimulus time histograms and kernel smoothing. We employ our modelling concept to analyse calcium spike sequences from dynamically-stimulated HEK293T cells. Under these conditions, different cells often experience diverse stimulus time courses, which is a situation likely to occur in vivo. This single cell variability and the concomitant small number of calcium spikes per cell pose a significant modelling challenge, but we demonstrate that Gaussian processes can successfully describe calcium spike rates in these circumstances. Our results therefore pave the way towards a statistical description of heterogeneous calcium oscillations in a dynamic environment.

On the phase space structure of IP3 induced Ca2+ signalling and concepts for predictive modeling.

The correspondence between mathematical structures and experimental systems is the basis of the generalizability of results found with specific systems and is the basis of the predictive power of theoretical physics. While physicists have confidence in this correspondence, it is less recognized in cellular biophysics. On the one hand, the complex organization of cellular dynamics involving a plethora of interacting molecules and the basic observation of cell variability seem to question its possibility. The practical difficulties of deriving the equations describing cellular behaviour from first principles support these doubts. On the other hand, ignoring such a correspondence would severely limit the possibility of predictive quantitative theory in biophysics. Additionally, the existence of functional modules (like pathways) across cell types suggests also the existence of mathematical structures with comparable universality. Only a few cellular systems have been sufficiently investigated in a variety of cell types to follow up these basic questions. IP3 induced Ca2+signalling is one of them, and the mathematical structure corresponding to it is subject of ongoing discussion. We review the system's general properties observed in a variety of cell types. They are captured by a reaction diffusion system. We discuss the phase space structure of its local dynamics. The spiking regime corresponds to noisy excitability. Models focussing on different aspects can be derived starting from this phase space structure. We discuss how the initial assumptions on the set of stochastic variables and phase space structure shape the predictions of parameter dependencies of the mathematical models resulting from the derivation.

Probabilistic encoding of stimulus strength in astrocyte global calcium signals.

Astrocyte calcium signals can range in size from subcellular microdomains to waves that spread through the whole cell (and into connected cells). The differential roles of such local or global calcium signaling are under intense investigation, but the mechanisms by which local signals evolve into global signals in astrocytes are not well understood, nor are the computational rules by which physiological stimuli are transduced into a global signal. To investigate these questions, we transiently applied receptor agonists linked to calcium signaling to primary cultures of cerebellar astrocytes. Astrocytes repetitively tested with the same stimulus responded with global signals intermittently, indicating that each stimulus had a defined probability for triggering a response. The response probability varied between agonists, increased with agonist concentration, and could be positively and negatively modulated by crosstalk with other signaling pathways. To better understand the processes determining the evolution of a global signal, we recorded subcellular calcium "puffs" throughout the whole cell during stimulation. The key requirement for puffs to trigger a global calcium wave following receptor activation appeared to be the synchronous release of calcium from three or more sites, rather than an increasing calcium load accumulating in the cytosol due to increased puff size, amplitude, or frequency. These results suggest that the concentration of transient stimuli will be encoded into a probability of generating a global calcium response, determined by the likelihood of synchronous release from multiple subcellular sites.

Modelling cell cycle synchronisation in networks of coupled radial glial cells.

Radial glial cells play a crucial role in the embryonic mammalian brain. Their proliferation is thought to be controlled, in part, by ATP mediated calcium signals. It has been hypothesised that these signals act to locally synchronise cell cycles, so that clusters of cells proliferate together, shedding daughter cells in uniform sheets. In this paper we investigate this cell cycle synchronisation by taking an ordinary differential equation model that couples the dynamics of intracellular calcium and the cell cycle and extend it to populations of cells coupled via extracellular ATP signals. Through bifurcation analysis we show that although ATP mediated calcium release can lead to cell cycle synchronisation, a number of other asynchronous oscillatory solutions including torus solutions dominate the parameter space and cell cycle synchronisation is far from guaranteed. Despite this, numerical results indicate that the transient and not the asymptotic behaviour of the system is important in accounting for cell cycle synchronisation. In particular, quiescent cells can be entrained on to the cell cycle via ATP mediated calcium signals initiated by a driving cell and crucially will cycle in near synchrony with the driving cell for the duration of neurogenesis. This behaviour is highly sensitive to the timing of ATP release, with release at the G1/S phase transition of the cell cycle far more likely to lead to near synchrony than release during mid G1 phase. This result, which suggests that ATP release timing is critical to radial glia cell cycle synchronisation, may help us to understand normal and pathological brain development.

Subcellular calcium dynamics in a whole-cell model of an atrial myocyte.

In this study, we present an innovative mathematical modeling approach that allows detailed characterization of Ca(2+) movement within the three-dimensional volume of an atrial myocyte. Essential aspects of the model are the geometrically realistic representation of Ca(2+) release sites and physiological Ca(2+) flux parameters, coupled with a computationally inexpensive framework. By translating nonlinear Ca(2+) excitability into threshold dynamics, we avoid the computationally demanding time stepping of the partial differential equations that are often used to model Ca(2+) transport. Our approach successfully reproduces key features of atrial myocyte Ca(2+) signaling observed using confocal imaging. In particular, the model displays the centripetal Ca(2+) waves that occur within atrial myocytes during excitation-contraction coupling, and the effect of positive inotropic stimulation on the spatial profile of the Ca(2+) signals. Beyond this validation of the model, our simulation reveals unexpected observations about the spread of Ca(2+) within an atrial myocyte. In particular, the model describes the movement of Ca(2+) between ryanodine receptor clusters within a specific z disk of an atrial myocyte. Furthermore, we demonstrate that altering the strength of Ca(2+) release, ryanodine receptor refractoriness, the magnitude of initiating stimulus, or the introduction of stochastic Ca(2+) channel activity can cause the nucleation of proarrhythmic traveling Ca(2+) waves. The model provides clinically relevant insights into the initiation and propagation of subcellular Ca(2+) signals that are currently beyond the scope of imaging technology.

Cannabinoid-mediated short-term plasticity in hippocampus.

Endocannabinoids (eCBs) modulate both excitatory and inhibitory neurotransmission in hippocampus via activation of pre-synaptic cannabinoid receptors. Here, we present a model for cannabinoid mediated short-term depression of excitation (DSE) based on our recently developed model for the equivalent phenomenon of suppressing inhibition (DSI). Furthermore, we derive a simplified formulation of the calcium-mediated endocannabinoid synthesis that underlies short-term modulation of neurotransmission in hippocampus. The simplified model describes cannabinoid-mediated short-term modulation of both hippocampal inhibition and excitation and is ideally suited for large network studies. Moreover, the implementation of the simplified DSI/DSE model provides predictions on how both phenomena are modulated by the magnitude of the pre-synaptic cell's activity. In addition we demonstrate the role of DSE in shaping the post-synaptic cell's firing behaviour qualitatively and quantitatively in dependence on eCB availability and the pre-synaptic cell's activity. Finally, we explore under which conditions the combination of DSI and DSE can temporarily shift the fine balance between excitation and inhibition. This highlights a mechanism by which eCBs might act in a neuro-protective manner during high neural activity.

Modelling the coupling between intracellular calcium release and the cell cycle during cortical brain development.

Most neocortical neurons formed during embryonic brain development arise from radial glial cells which communicate, in part, via ATP mediated calcium signals. Although the intercellular signalling mechanisms that regulate radial glia proliferation are not well understood, it has recently been demonstrated that ATP dependent intracellular calcium release leads to an increase of nearly 100% in overall cellular proliferation. It has been hypothesised that cytoplasmic calcium accelerates entry into S phase of the cell cycle and/or acts to recruit otherwise quiescent cells onto the cell cycle. In this paper we study this cell cycle acceleration and recruitment by forming a differential equation model for ATP mediated calcium-cell cycle coupling via Cyclin D in a single radial glial cell. Bifurcation analysis and numerical simulations suggest that the cell cycle period depends only weakly on cytoplasmic calcium. Therefore, the accelerative impact of calcium on the cell cycle can only account for a small fraction of the large increase in proliferation observed experimentally. Crucially however, our bifurcation analysis reveals that stable fixed point and stable limit cycle solutions can coexist, and that calcium dependent Cyclin D dynamics extend the oscillatory region to lower Cyclin D synthesis rates, thus rendering cells more susceptible to cycling. This supports the hypothesis that cycling glial cells recruit quiescent cells (in G0 phase) onto the cell cycle, via a calcium signalling mechanism, and that this may be the primary means by which calcium augments proliferation rates at the population scale. Numerical simulations of two coupled cells demonstrate that such a scenario is indeed feasible.

Fundamental properties of Ca2+ signals.

BACKGROUND: Ca2+ is a ubiquitous and versatile second messenger that transmits information through changes of the cytosolic Ca2+ concentration. Recent investigations changed basic ideas on the dynamic character of Ca2+ signals and challenge traditional ideas on information transmission. SCOPE OF REVIEW: We present recent findings on key characteristics of the cytosolic Ca2+ dynamics and theoretical concepts that explain the wide range of experimentally observed Ca2+ signals. Further, we relate properties of the dynamical regulation of the cytosolic Ca2+ concentration to ideas about information transmission by stochastic signals. MAJOR CONCLUSIONS: We demonstrate the importance of the hierarchal arrangement of Ca2+ release sites on the emergence of cellular Ca2+ spikes. Stochastic Ca2+ signals are functionally robust and adaptive to changing environmental conditions. Fluctuations of interspike intervals (ISIs) and the moment relation derived from ISI distributions contain information on the channel cluster open probability and on pathway properties. GENERAL SIGNIFICANCE: Robust and reliable signal transduction pathways that entail Ca2+ dynamics are essential for eukaryotic organisms. Moreover, we expect that the design of a stochastic mechanism which provides robustness and adaptivity will be found also in other biological systems. Ca2+ dynamics demonstrate that the fluctuations of cellular signals contain information on molecular behavior. This article is part of a Special Issue entitled Biochemical, biophysical and genetic approaches to intracellular calcium signaling.

Atrial cardiomyocyte calcium signalling.

Whereas Ca(2+) signalling in ventricular cardiomyocytes is well described, much less is known regarding the Ca(2+) signals within atrial cells. This is surprising given that atrial cardiomyocytes make an important contribution to the refilling of ventricles with blood, which enhances the subsequent ejection of blood from the heart. The dependence of cardiac function on the contribution of atria becomes increasingly important with age and exercise. Disruption of the rhythmic beating of atrial cardiomyocytes can lead to life-threatening conditions such as atrial fibrillation. Atrial and ventricular myocytes have many structural and functional similarities. However, one key structural difference, the lack of transverse tubules ("T-tubules") in atrial myocytes, make these two cell types display vastly different calcium patterns in response to electrical excitation. The lack of T-tubules in atrial myocytes means that depolarisation provokes calcium signals that originate around the periphery of the cells. Under resting conditions, such Ca(2+) signals do not propagate towards the centre of the atrial cells and so do not fully engage the contractile machinery. Consequently, contraction of atrial myocytes under resting conditions is modest. However, when atrial myocytes are stimulated with a positive inotropic agonist, such as isoproterenol, the peripheral Ca(2+) signals trigger a global wave of Ca(2+) that propagates in a centripetal manner into the cells. Enhanced centripetal movement of Ca(2+) in atrial myocytes leads to increased contraction and a more substantial contribution to blood pumping. This article is part of a Special Issue entitled: 11th European Symposium on Calcium.

How does the ryanodine receptor in the ventricular myocyte wake up: by a single or by multiple open L-type Ca2+ channels?

We study here the early stage of Ca(2+)-induced Ca(2+) release (CICR) in the diadic cleft of cardiac ventricular myocytes. A crucial question for this mechanism is whether the activation of the ryanodine receptors (RyRs) is triggered by one or by multiple open L-type Ca(2+) channels (LCCs). We address the problem through a modelling approach that allows us to investigate both possibilities. The model is based on a spatially resolved description of a Ca(2+) release unit (CaRU), consisting of the junctional sarcoplasmic reticulum and the diadic cleft, with well-defined channel placement. By taking advantage of largely varying time scales of the Ca(2+) dynamics in the diadic cleft, the governing equations can be reduced to one ordinary differential equation that describes the Ca(2+) fluxes, the electric field due to surface charges and diffusion. Our study shows that the mechanisms of the early stage of CICR shape measurable properties of CICR in a characteristic way. From here we conclude that the activation of RyRs requires multiple open LCCs.

Three-dimensional spatio-temporal modelling of store operated Ca2+ entry: Insights into ER refilling and the spatial signature of Ca2+ signals.

The spatial organisation of Orai channels and SERCA pumps within ER-PM junctions is important for enhancing the versatility and specificity of sub-cellular Ca2+ signals generated during store operated Ca2+ entry (SOCE). In this paper, we present a novel three dimensional spatio-temporal model describing Ca2+ dynamics in the ER-PM junction and sub-PM ER during SOCE. We investigate the role of Orai channel and SERCA pump location to provide insights into how these components shape the Ca2+ signals generated and affect ER refilling. We find that the organisation of Orai channels within the ER-PM junction controls the amplitude and shape of the Ca2+ profile but does not enhance ER refilling. The model shows that ER refilling is only weakly affected by the location of SERCA2b pumps within the ER-PM junction and that the placement of SERCA2a pumps within the ER-PM junction has much greater control over ER refilling.

Sign changes as a universal concept in first-passage-time calculations.

First-passage-time problems are ubiquitous across many fields of study, including transport processes in semiconductors and biological synapses, evolutionary game theory and percolation. Despite their prominence, first-passage-time calculations have proven to be particularly challenging. Analytical results to date have often been obtained under strong conditions, leaving most of the exploration of first-passage-time problems to direct numerical computations. Here we present an analytical approach that allows the derivation of first-passage-time distributions for the wide class of nondifferentiable Gaussian processes. We demonstrate that the concept of sign changes naturally generalizes the common practice of counting crossings to determine first-passage events. Our method works across a wide range of time-dependent boundaries and noise strengths, thus alleviating common hurdles in first-passage-time calculations.

Evolution of moments and correlations in nonrenewal escape-time processes.

The theoretical description of nonrenewal stochastic systems is a challenge. Analytical results are often not available or can be obtained only under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad hoc Monte Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker-Planck equation (FPE) to describe the statistics of nonrenewal stochastic systems. We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a nontrivial time scale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and time-scale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allow for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCCs) and discuss the challenge of reliably computing the SCCs, which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of nonrenewal dynamics in a wide range of stochastic systems exhibiting short- and long-range correlations.

Neural Field Models with Threshold Noise.

The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the classic sigmoidal population firing-rate function so prevalent in large scale neuronal modelling. In this paper we explore the effects of such threshold noise without recourse to averaging and show that spatial correlations can have a strong effect on the behaviour of waves and patterns in continuum models. Moreover, for a prescribed spatial covariance function we explore the differences in behaviour that can emerge when the underlying stationary distribution is changed from Gaussian to non-Gaussian. For travelling front solutions, in a system with exponentially decaying spatial interactions, we make use of an interface approach to calculate the instantaneous wave speed analytically as a series expansion in the noise strength. From this we find that, for weak noise, the spatially averaged speed depends only on the choice of covariance function and not on the shape of the stationary distribution. For a system with a Mexican-hat spatial connectivity we further find that noise can induce localised bump solutions, and using an interface stability argument show that there can be multiple stable solution branches.

First-passage times in integrate-and-fire neurons with stochastic thresholds.

We consider a leaky integrate-and-fire neuron with deterministic subthreshold dynamics and a firing threshold that evolves as an Ornstein-Uhlenbeck process. The formulation of this minimal model is motivated by the experimentally observed widespread variation of neural firing thresholds. We show numerically that the mean first-passage time can depend nonmonotonically on the noise amplitude. For sufficiently large values of the correlation time of the stochastic threshold the mean first-passage time is maximal for nonvanishing noise. We provide an explanation for this effect by analytically transforming the original model into a first-passage-time problem for Brownian motion. This transformation also allows for a perturbative calculation of the first-passage-time histograms. In turn this provides quantitative insights into the mechanisms that lead to the nonmonotonic behavior of the mean first-passage time. The perturbation expansion is in excellent agreement with direct numerical simulations. The approach developed here can be applied to any deterministic subthreshold dynamics and any Gauss-Markov processes for the firing threshold. This opens up the possibility to incorporate biophysically detailed components into the subthreshold dynamics, rendering our approach a powerful framework that sits between traditional integrate-and-fire models and complex mechanistic descriptions of neural dynamics.

Translating intracellular calcium signaling into models.

The rich experimental data on intracellular calcium has put theoreticians in an ideal position to derive models of intracellular calcium signaling. Over the last 25 years, a large number of modeling frameworks have been suggested. Here, I will review some of the milestones of intracellular calcium modeling with a special emphasis on calcium-induced calcium release (CICR) through inositol-1,4,5-trisphosphate and ryanodine receptors. I will highlight key features of CICR and how they are represented in models as well as the challenges that theoreticians face when translating our current understanding of calcium signals into equations. The selected examples demonstrate that a successful model provides mechanistic insights into the molecular machinery of the Ca²âº signaling toolbox and determines the contribution of local Ca²âº release to global Ca²âº patterns, which at the moment cannot be resolved experimentally.

Exploring oscillations in a point model of the intracellular Ca²âº concentration.

Intracellular calcium (Ca²âº) oscillations are a key signaling mechanism in most cell types. A prominent approach to modeling intracellular Ca²âº oscillations is the use of ordinary differential equations (ODEs), which treat the intracellular Ca²âº concentration as spatially homogenous. Although ODEs cannot account for the interaction of Ca²âº microdomains to form cell-wide Ca²âº patterns, modelers still choose ODEs because (a) the study of ODEs is computationally cheap, and a large body of techniques is available to investigate ODEs in great detail, or (b) sufficient experimental data are not available to develop a spatially extended model. Irrespective of the reason, analyzing ODEs is a key instrument in the toolbox of modelers. In this protocol, we look at a well-known model for Ca²âº oscillations, the De Young-Keizer model, along with the Li-Rinzel approximation of the De Young-Keizer model. The main emphasis of this protocol is the use of the open source software package XPPAUT to numerically study ODEs. The knowledge gained here can be directly transferred to other ODE systems and therefore may serve as a template for future studies.

Time to blip--stochastic simulation of single channel opening.

The stochastic dynamics of the inositol-1,4,5-trisphosphate (IP3) receptor (IP3R) is key to understanding a wide range of observed calcium (Ca²âº) signals. The stochastic nature results from the constant binding and unbinding of Ca²âº and IP3 to and from their respective binding sites and is especially important in the initiation of a Ca²âº puff (i.e., the release of Ca²âº through a cluster of IP3Rs). Once the first IP3R opens, the Ca²âº concentration rises significantly around the ion channel, increasing the open probability for neighboring IP3Rs. This opening may trigger the activation of further receptors, giving rise to a Ca²âº puff. In this protocol, we determine the time that it takes for a single IP3R to open from rest. We explicitly take into account the tetrameric structure of the IP3R and the fact that multiple subunits must be active before the channel opens. We develop code for a stochastic simulation of the IP3R and simulate it using the software package MATLAB. This protocol shows the basic form of a stochastic simulation algorithm and may serve as a starting point to investigate more complex gating dynamics.