How does intracellular Ca2+ oscillate: by chance or by the clock?

Amplitude and frequency encoding result in qualitatively distinct informational landscapes in cell signaling

Cells continuously sense their surroundings to detect modifications and generate responses. Very often changes in extracellular concentrations initiate signaling cascades that eventually result in changes in gene expression. Increasing stimulus strengths can be encoded in increasing concentration amplitudes or increasing activation frequencies of intermediaries of the pathway. In this paper we show that the different way in which amplitude and frequency encoding map environmental changes endow cells with qualitatively different information transmission capabilities. While amplitude encoding is optimal for a limited range of stimuli strengths, frequency encoding can transmit information with equal reliability over much broader ranges. The qualitative difference between the two strategies stems from the scale invariant discriminating power of the first transducing step in frequency codification. The apparently redundant combination of both strategies in some cell types may then serve the purpose of expanding the span over which stimulus strengths can be reliably discriminated. In this paper we discuss a possible example of this mechanism in yeast.

An integrate-and-fire approach to Ca2+ signaling. Part II: Cumulative refractoriness

Inositol 1,4,5-trisphosphate-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. The agonist concentration stimulating Ca2+ signals is encoded in the frequency of a Ca2+ concentration spike sequence. When a cell is stimulated, the interspike intervals (ISIs) often show a distinct transient during which they gradually increase, a system property we refer to as cumulative refractoriness. We extend a previously published stochastic model to include the Ca2+ concentration in the intracellular Ca2+ store as a slow adaptation variable. This model can reproduce both stationary and transient statistics of experimentally observed ISI sequences. We derive approximate expressions for the mean and coefficient of variation of the stationary ISIs. We also consider the response to the onset of a constant stimulus and estimate the length of the transient and the strength of the adaptation of the ISI. We show that the adaptation sets the coefficient of variation in agreement with current ideas derived from experiments. Moreover, we explain why, despite a pronounced transient behavior, ISI correlations can be weak, as often observed in experiments. Finally, we fit our model to reproduce the transient statistics of experimentally observed ISI sequences in stimulated HEK cells. The fitted model is able to qualitatively reproduce the relationship between the stationary interval correlations and the number of transient intervals, as well as the strength of the ISI adaptation. We also find positive correlations in the experimental sequence that cannot be explained by our model.

Maintenance of hematopoietic stem cell niche homeostasis requires gap junction-mediated calcium signaling

Regulation of stem cells requires coordination of the cells that make up the stem cell niche. Here, we describe a mechanism that allows communication between niche cells to coordinate their activity and shape the signaling environment surrounding resident stem cells. Using the Drosophila hematopoietic organ, the lymph gland, we show that cells of the hematopoietic niche, the posterior signaling center (PSC), communicate using gap junctions (GJs) and form a signaling network. This network allows PSC cells to exchange Ca2+ signals repetitively which regulate the hematopoietic niche. Disruption of Ca2+ signaling in the PSC or the GJ-mediated network connecting niche cells causes dysregulation of the PSC and blood progenitor differentiation. Analysis of PSC-derived cell signaling shows that the Hedgehog pathway acts downstream of GJ-mediated Ca2+ signaling to modulate the niche microenvironment. These data show that GJ-mediated communication between hematopoietic niche cells maintains their homeostasis and consequently controls blood progenitor behavior.

An efficient reduced-lattice model of IP3R for probing Ca2+ dynamics

Numerous cellular processes are regulated by Ca2+ signals, and the endoplasmic reticulum (ER) membrane's inositol triphosphate receptor (IP3R) is critical for modulating intracellular Ca2+ dynamics. The IP3Rs are seen to be clustered in a variety of cell types. The combination of IP3Rs clustering and IP3Rs-mediated Ca2+-induced Ca2+ release results in the hierarchical organization of the Ca2+ signals, which challenges the numerical simulation given the multiple spatial and temporal scales that must be covered. The previous methods rather ignore the spatial feature of IP3Rs or fail to coordinate the conflicts between the real biological relevance and the computational cost. In this work, a general and efficient reduced-lattice model is presented for the simulation of IP3Rs-mediated multiscale Ca2+ dynamics. The model highlights biological details that make up the majority of the calcium events, including IP3Rs clustering and calcium domains, and it reduces the complexity by approximating the minor details. The model's extensibility provides fresh insights into the function of IP3Rs in producing global Ca2+ events and supports the research under more physiological circumstances. Our work contributes to a novel toolkit for modeling multiscale Ca2+ dynamics and advances knowledge of Ca2+ signals.

Modeling IP3-induced Ca2+ signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. Recent research demonstrated randomness of Ca2+ signaling on all structural levels. We compile eight general properties of Ca2+ spiking common to all cell types investigated and suggest a theory of Ca2+ spiking starting from the random behavior of IP3 receptor channel clusters mediating the release of Ca2+ from the endoplasmic reticulum capturing all general properties and pathway-specific behavior. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces the exponential stimulation response relation of the average interspike interval Tav and its robustness properties, random spike timing with a linear moment relation between Tav and the interspike interval SD and its robustness properties, sensitive dependency of Tav on diffusion properties, and nonoscillatory local dynamics. We explain large cell variability of Tav observed in experiments by variability of channel cluster coupling by Ca2+-induced Ca2+ release, the number of clusters, and IP3 pathway component expression levels. We predict the relation between puff probability and agonist concentration and [IP3] and agonist concentration. Differences of spike behavior between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties.

An integrate-and-fire approach to Ca2+ signaling. Part I: Renewal model

In computational neuroscience integrate-and-fire models capture the spike generation by a subthreshold dynamics supplemented by a simple fire-and-reset rule; they allow for a numerically efficient and analytically tractable description of stochastic single cell as well as network dynamics. Stochastic spiking is also a prominent feature of Ca2+ signaling which suggests to adopt the integrate-and-fire approach for this fundamental biophysical process. The model introduced here consists of two components describing 1) activity of clusters of inositol-trisphosphate receptor channels and 2) dynamics of the global Ca2+ concentrations in the cytosol. The cluster dynamics is given in terms of a cyclic Markov chain, capturing the puff, i.e., the punctuated release of Ca2+ from intracellular stores. The cytosolic Ca2+ concentration is described by an integrate-and-fire dynamics driven by the puff current. For the cyclic Markov chain we derive expressions for the statistics of the interpuff interval, the single-puff strength and the puff current assuming constant cytosolic Ca2+. The latter condition is often well approximated because cytosolic Ca2+ varies much slower than the cluster activity does. Furthermore, because the detailed two-component model is numerically expensive to simulate and difficult to treat analytically, we develop an analytical framework to approximate the driving puff current of the stochastic cytosolic Ca2+ dynamics by a temporally uncorrelated Gaussian noise. This approximation reduces our two-component system to an integrate-and-fire model with a nonlinear drift function and a multiplicative Gaussian white noise, a model that is known to generate a renewal spike train, i.e., a point process with statistically independent interspike intervals. The model allows for fast numerical simulations, permits to derive analytical expressions for the rate of Ca2+ spiking and the coefficient of variation of the interspike interval, and to approximate the interspike interval density and the spike train power spectrum. Comparison of these statistics to experimental data is discussed.

Computational investigation of IP3 diffusion

Inositol 1,4,5-trisphosphate (IP₃) plays a key role in calcium signaling. After stimulation, it diffuses from the plasma membrane where it is produced to the endoplasmic reticulum where its receptors are localized. Based on in vitro measurements, IP₃ was long thought to be a global messenger characterized by a diffusion coefficient of ~ 280 μm²s^-1. However, in vivo observations revealed that this value does not match with the timing of localized Ca²⁺ increases induced by the confined release of a non-metabolizable IP₃ analog. A theoretical analysis of these data concluded that in intact cells diffusion of IP₃ is strongly hindered, leading to a 30-fold reduction of the diffusion coefficient. Here, we performed a new computational analysis of the same observations using a stochastic model of Ca²⁺ puffs. Our simulations concluded that the value of the effective IP₃ diffusion coefficient is close to 100 μm²s^-1. Such moderate reduction with respect to in vitro estimations quantitatively agrees with a buffering effect by non-fully bound inactive IP₃ receptors. The model also reveals that IP₃ spreading is not much affected by the endoplasmic reticulum, which represents an obstacle to the free displacement of molecules, but can be significantly increased in cells displaying elongated, 1-dimensional like geometries.

Purinoreceptors and ectonucleotidases control ATP-induced calcium waveforms and calcium-dependent responses in microglia: Roles of P2 receptors and CD39 in ATP-stimulated microglia

Adenosine triphosphate (ATP) and its metabolites drive microglia migration and cytokine production by activating P2X- and P2Y- class purinergic receptors. Purinergic receptor activation gives rise to diverse intracellular calcium (Ca2+ signals, or waveforms, that differ in amplitude, duration, and frequency. Whether and how these characteristics of diverse waveforms influence microglia function is not well-established. We developed a computational model trained with data from published primary murine microglia studies. We simulate how purinoreceptors influence Ca2+ signaling and migration, as well as, how purinoreceptor expression modifies these processes. Our simulation confirmed that P2 receptors encode the amplitude and duration of the ATP-induced Ca2+ waveforms. Our simulations also implicate CD39, an ectonucleotidase that rapidly degrades ATP, as a regulator of purinergic receptor-induced Ca2+ responses. Namely, it was necessary to account for CD39 metabolism of ATP to align the model's predicted purinoreceptor responses with published experimental data. In addition, our modeling results indicate that small Ca2+ transients accompany migration, while large and sustained transients are needed for cytokine responses. Lastly, as a proof-of-principal, we predict Ca2+ transients and cell membrane displacements in a BV2 microglia cell line using published P2 receptor mRNA data to illustrate how our computer model may be extrapolated to other microglia subtypes. These findings provide important insights into how differences in purinergic receptor expression influence microglial responses to ATP.

Characterizing spontaneous Ca2+ local transients in OPCs using computational modeling

Spontaneous Ca2+ local transients (SCaLTs) in isolated oligodendrocyte precursor cells are largely regulated by the following fluxes: store-operated Ca2+ entry (SOCE), Na+/Ca2+ exchange, Ca2+ pumping through Ca2+-ATPases, and Ca2+-induced Ca2+-release through ryanodine receptors and inositol-trisphosphate receptors. However, the relative contributions of these fluxes in mediating fast spiking and the slow baseline oscillations seen in SCaLTs remain incompletely understood. Here, we developed a stochastic spatiotemporal computational model to simulate SCaLTs in a homogeneous medium with ionic flow between the extracellular, cytoplasmic, and endoplasmic-reticulum compartments. By simulating the model and plotting both the histograms of SCaLTs obtained experimentally and from the model as well as the standard deviation of inter-SCaLT intervals against inter-SCaLT interval averages of multiple model and experimental realizations, we revealed the following: (1) SCaLTs exhibit very similar characteristics between the two data sets, (2) they are mostly random, (3) they encode information in their frequency, and (4) their slow baseline oscillations could be due to the stochastic slow clustering of inositol-trisphosphate receptors (modeled as an Ornstein-Uhlenbeck noise process). Bifurcation analysis of a deterministic temporal version of the model showed that the contribution of fluxes to SCaLTs depends on the parameter regime and that the combination of excitability, stochasticity, and mixed-mode oscillations are responsible for irregular spiking and doublets in SCaLTs. Additionally, our results demonstrated that blocking each flux reduces SCaLTs' frequency and that the reverse (forward) mode of Na+/Ca2+ exchange decreases (increases) SCaLTs. Taken together, these results provide a quantitative framework for SCaLT formation in oligodendrocyte precursor cells.

Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges

Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.

Synchronization of spatially discordant voltage and calcium alternans in cardiac tissue

The heart is an excitable medium which is excited by membrane potential depolarization and propagation. Membrane potential depolarization brings in calcium (Ca) through the Ca channels to trigger intracellular Ca release for contraction of the heart. Ca also affects voltage via Ca-dependent ionic currents, and thus, voltage and Ca are bidirectionally coupled. It has been shown that the voltage subsystem or the Ca subsystem can generate its own dynamical instabilities which are affected by their bidirectional couplings, leading to complex dynamics of action potential and Ca cycling. Moreover, the dynamics become spatiotemporal in tissue in which cells are diffusively coupled through voltage. A widely investigated spatiotemporal dynamics is spatially discordant alternans (SDA) in which action potential duration (APD) or Ca amplitude exhibits temporally period-2 and spatially out-of-phase patterns, i.e., APD-SDA and Ca-SDA patterns, respectively. However, the mechanisms of formation, stability, and synchronization of APD-SDA and Ca-SDA patterns remain incompletely understood. In this paper, we use cardiac tissue models described by an amplitude equation, coupled iterated maps, and reaction-diffusion equations with detailed physiology (the ionic model) to perform analytical and computational investigations. We show that, when the Ca subsystem is stable, the Ca-SDA pattern always follows the APD-SDA pattern, and thus, they are always synchronized. When the Ca subsystem is unstable, synchronization of APD-SDA and Ca-SDA patterns depends on the stabilities of both subsystems, their coupling strengths, and the spatial scales of the initial Ca-SDA patterns. Spontaneous (initial condition-independent) synchronization is promoted by enhancing APD instability and reducing Ca instability as well as stronger Ca-to-APD and APD-to-Ca coupling, a pattern formation caused by dynamical instabilities. When Ca is more unstable and APD is less unstable or APD-to-Ca coupling is weak, synchronization of APD-SDA and Ca-SDA patterns is promoted by larger initially synchronized Ca-SDA clusters, i.e., initial condition-dependent synchronization. The synchronized APD-SDA and Ca-SDA patterns can be locked in-phase, antiphase, or quasiperiodic depending on the coupling relationship between APD and Ca. These theoretical and simulation results provide mechanistic insights into the APD-SDA and Ca-SDA dynamics observed in experimental studies.

Kramers Rate Theory of Pacemaker Dynamics in Noisy Excitable Media

Rhythmic activities, which are usually driven by pacemakers, are common in biological systems. In noisy excitable media, pacemakers are self-organized firing clusters, but the underlying dynamics remains to be elucidated. Here we develop a Kramers rate theory of coupled cells to describe the firing properties of pacemakers and their dependence on coupling strength and system size and dimension. The theory captures accurately the simulation results of tissue models with stochastic Hodgkin-Huxley equations except when transitions from pacemakers to spiral waves occur under weak coupling.

Mean-return-time phase of a stochastic oscillator provides an approximate renewal description for the associated point process

Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer-Schwabedal-Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin-Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.

Analytical approach to the mean-return-time phase of isotropic stochastic oscillators

One notion of phase for stochastic oscillators is based on the mean return-time (MRT): a set of points represents a certain phase if the mean time to return from any point in this set to this set after one rotation is equal to the mean rotation period of the oscillator (irrespective of the starting point). For this so far only algorithmically defined phase, we derive here analytical expressions for the important class of isotropic stochastic oscillators. This allows us to evaluate cases from the literature explicitly and to study the behavior of the MRT phase in the limits of strong noise. We also use the same formalism to show that lines of constant return time variance (instead of constant mean return time) can be defined, and that they in general differ from the MRT isochrons.

Isostables for Stochastic Oscillators

Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.254101], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.

A gap-junction-mediated, calcium-signaling network controls blood progenitor fate decisions in hematopoiesis

Stem cell homeostasis requires coordinated fate decisions among stem cells that are often widely distributed within a tissue at varying distances from their stem cell niche. This requires a mechanism to ensure robust fate decisions within a population of stem cells. Here, we show that, in the Drosophila hematopoietic organ, the lymph gland (LG), gap junctions form a network that coordinates fate decisions between blood progenitors. Using live imaging of calcium signaling in intact LGs, we find that blood progenitors are connected through a signaling network. Blocking gap junction function disrupts this network, alters the pattern of encoded calcium signals, and leads to loss of progenitors and precocious blood cell differentiation. Ectopic and uniform activation of the calcium-signaling mediator CaMKII restores progenitor homeostasis when gap junctions are disrupted. Overall, these data show that gap junctions equilibrate cell signals between blood progenitors to coordinate fate decisions and maintain hematopoietic homeostasis.

A Tale of two receptors

Calcium (Ca²⁺) oscillations in hepatocytes have a wide dynamic range. In particular, recent experimental evidence shows that agonist stimulation of the P2Y family of receptors leads to qualitatively diverse Ca²⁺ oscillations. We present a new model of Ca²⁺ oscillations in hepatocytes based on these experiments to investigate the mechanisms controlling P2Y-activated Ca²⁺ oscillations. The model accounts for Ca²⁺ regulation of the IP₃ receptor (IP₃R), the positive feedback from Ca²⁺ on phospholipase C (PLC) and the P2Y receptor phosphorylation by protein kinase C (PKC). Furthermore, PKC is shown to control multiple cellular substrates. Utilising the model, we suggest the activity and intensity of PLC and PKC necessary to explain the qualitatively diverse Ca²⁺ oscillations in response to P2Y receptor activation.

Unraveling ChR2-driven stochastic Ca2+ dynamics in astrocytes: A call for new interventional paradigms

Optogenetic targeting of astrocytes provides a robust experimental model to differentially induce Ca2+ signals in astrocytes in vivo. However, a systematic study quantifying the response of optogenetically modified astrocytes to light is yet to be performed. Here, we propose a novel stochastic model of Ca2+ dynamics in astrocytes that incorporates a light sensitive component-channelrhodopsin 2 (ChR2). Utilizing this model, we investigated the effect of different light stimulation paradigms on cells expressing select variants of ChR2 (wild type, ChETA, and ChRET/TC). Results predict that depending on paradigm specification, astrocytes might undergo drastic changes in their basal Ca2+ level and spiking probability. Furthermore, we performed a global sensitivity analysis to assess the effect of variation in parameters pertinent to the shape of the ChR2 photocurrent on astrocytic Ca2+ dynamics. Results suggest that directing variants towards the first open state of the ChR2 photocycle (o1) enhances spiking activity in astrocytes during optical stimulation. Evaluation of the effect of Ca2+ buffering and coupling coefficient in a network of ChR2-expressing astrocytes demonstrated basal level elevations in the stimulated region and propagation of calcium activity to unstimulated cells. Buffering reduced the diffusion range of Ca2+ within the network, thereby limiting propagation and influencing the activity of astrocytes. Collectively, the framework presented in this study provides valuable information for the selection of light stimulation paradigms that elicit desired astrocytic activity using existing ChR2 constructs, as well as aids in the engineering of future application-oriented optogenetic variants.

Fundamentals of Cellular Calcium Signaling: A Primer

Ionized calcium (Ca²⁺) is the most versatile cellular messenger. All cells use Ca²⁺ signals to regulate their activities in response to extrinsic and intrinsic stimuli. Alterations in cellular Ca²⁺ signaling and/or Ca²⁺ homeostasis can subvert physiological processes into driving pathological outcomes. Imaging of living cells over the past decades has demonstrated that Ca²⁺ signals encode information in their frequency, kinetics, amplitude, and spatial extent. These parameters alter depending on the type and intensity of stimulation, and cellular context. Moreover, it is evident that different cell types produce widely varying Ca²⁺ signals, with properties that suit their physiological functions. This primer discusses basic principles and mechanisms underlying cellular Ca²⁺ signaling and Ca²⁺ homeostasis. Consequently, we have cited some historical articles in addition to more recent findings. A brief summary of the core features of cellular Ca²⁺ signaling is provided, with particular focus on Ca²⁺ stores and Ca²⁺ transport across cellular membranes, as well as mechanisms by which Ca²⁺ signals activate downstream effector systems.

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.

CaSiAn: a Calcium Signaling Analyzer tool

Summary

Ca²⁺ is a central second messenger in eukaryotic cells that regulates many cellular processes. Recently, we have indicated that typical Ca²⁺ signals are not purely oscillatory as widely assumed, but exhibit stochastic spiking with cell type and pathway specific characteristics. Here, we present the Calcium Signaling Analyzer (CaSiAn), an open source software tool that allows for quantifying these signal characteristics including individual spike properties and time course statistics in a semi-automated manner. CaSiAn provides an intuitive graphical user interface allowing experimentalists to easily process a large amount of Ca²⁺ signals, interactively tune peak detection, revise statistical measures and access the quantified signal properties as excel or text files.

Availability and implementation

CaSiAn is implemented in Java and available on Github (https://github.com/mmahsa/CaSiAn) as well as on the project page (http://r3lab.uni.lu/web/casa).

Supplementary information

Supplementary data are available at Bioinformatics online.

Simulation of calcium signaling in fine astrocytic processes: Effect of spatial properties on spontaneous activity

Astrocytes, a glial cell type of the central nervous system, have emerged as detectors and regulators of neuronal information processing. Astrocyte excitability resides in transient variations of free cytosolic calcium concentration over a range of temporal and spatial scales, from sub-microdomains to waves propagating throughout the cell. Despite extensive experimental approaches, it is not clear how these signals are transmitted to and integrated within an astrocyte. The localization of the main molecular actors and the geometry of the system, including the spatial organization of calcium channels IP3R, are deemed essential. However, as most calcium signals occur in astrocytic ramifications that are too fine to be resolved by conventional light microscopy, most of those spatial data are unknown and computational modeling remains the only methodology to study this issue. Here, we propose an IP3R-mediated calcium signaling model for dynamics in such small sub-cellular volumes. To account for the expected stochasticity and low copy numbers, our model is both spatially explicit and particle-based. Extensive simulations show that spontaneous calcium signals arise in the model via the interplay between excitability and stochasticity. The model reproduces the main forms of calcium signals and indicates that their frequency crucially depends on the spatial organization of the IP3R channels. Importantly, we show that two processes expressing exactly the same calcium channels can display different types of calcium signals depending on the spatial organization of the channels. Our model with realistic process volume and calcium concentrations successfully reproduces spontaneous calcium signals that we measured in calcium micro-domains with confocal microscopy and predicts that local variations of calcium indicators might contribute to the diversity of calcium signals observed in astrocytes. To our knowledge, this model is the first model suited to investigate calcium dynamics in fine astrocytic processes and to propose plausible mechanisms responsible for their variability.

Heterogeneity and Delayed Activation as Hallmarks of Self-Organization and Criticality in Excitable Tissue

Self-organized critical dynamics is assumed to be an attractive mode of functioning for several real-life systems and entails an emergent activity in which the extent of observables follows a power-law distribution. The hallmarks of criticality have recently been observed in a plethora of biological systems, including beta cell populations within pancreatic islets of Langerhans. In the present study, we systematically explored the mechanisms that drive the critical and supercritical behavior in networks of coupled beta cells under different circumstances by means of experimental and computational approaches. Experimentally, we employed high-speed functional multicellular calcium imaging of fluorescently labeled acute mouse pancreas tissue slices to record calcium signals in a large number of beta cells simultaneously, and with a high spatiotemporal resolution. Our experimental results revealed that the cellular responses to stimulation with glucose are biphasic and glucose-dependent. Under physiological as well as under supraphysiological levels of stimulation, an initial activation phase was followed by a supercritical plateau phase with a high number of global intercellular calcium waves. However, the activation phase displayed fingerprints of critical behavior under lower stimulation levels, with a progressive recruitment of cells and a power-law distribution of calcium wave sizes. On the other hand, the activation phase provoked by pathophysiologically high glucose concentrations, differed considerably and was more rapid, less continuous, and supercritical. To gain a deeper insight into the experimentally observed complex dynamical patterns, we built up a phenomenological model of coupled excitable cells and explored empirically the model’s necessities that ensured a good overlap between computational and experimental results. It turned out that such a good agreement between experimental and computational findings was attained when both heterogeneous and stimulus-dependent time lags, variability in excitability levels, as well as a heterogeneous cell-cell coupling were included into the model. Most importantly, since our phenomenological approach involved only a few parameters, it naturally lends itself not only for determining key mechanisms of self-organized criticality at the tissue level, but also points out various features for comprehensive and realistic modeling of different excitable systems in nature.

Deterministic Limit of Intracellular Calcium Spikes

In nonexcitable cells, global Ca^{2+} spikes emerge from the collective dynamics of clusters of Ca^{2+} channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin, while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

Simulation of P2X-mediated calcium signalling in microglia

KEY POINTS

A computational model of P2X channel activation in microglia was developed that includes downfield Ca²⁺ -dependent signalling pathways. This model provides quantitative insights into how diverse signalling pathways in microglia converge to control microglial function.

ABSTRACT

Microglia function is orchestrated through highly coupled signalling pathways that depend on calcium (Ca²⁺ ). In response to extracellular ATP, transient increases in intracellular Ca²⁺ driven through the activation of purinergic receptors, P2X and P2Y, are sufficient to promote cytokine synthesis. Although the steps comprising the pathways bridging purinergic receptor activation with transcriptional responses have been probed in great detail, a quantitative model for how these steps collectively control cytokine production has not been established. Here we developed a minimal computational model that quantitatively links extracellular stimulation of two prominent ionotropic purinergic receptors, P2X4 and P2X7, with the graded production of a gene product, namely the tumour necrosis factor α (TNFα) cytokine. In addition to Ca²⁺ handling mechanisms common to eukaryotic cells, our model includes microglia-specific processes including ATP-dependent P2X4 and P2X7 activation, activation of nuclear factor of activated T-cells (NFAT) transcription factors, and TNFα production. Parameters for this model were optimized to reproduce published data for these processes, where available. With this model, we determined the propensity for TNFα production in microglia, subject to a wide range of ATP exposure amplitudes, frequencies and durations that the cells could encounter in vivo. Furthermore, we have investigated the extent to which modulation of the signal transduction pathways influence TNFα production. Our results suggest that pulsatile stimulation of P2X4 via micromolar ATP may be sufficient to promote TNFα production, whereas high-amplitude ATP exposure is necessary for production via P2X7. Furthermore, under conditions that increase P2X4 expression, for instance, following activation by pathogen-associated molecular factors, P2X4-associated TNFα production is greatly enhanced. Given that Ca²⁺ homeostasis in microglia is profoundly important to its function, this computational model provides a quantitative framework to explore hypotheses pertaining to microglial physiology.

The stretch to stray on time: Resonant length of random walks in a transient

First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time or control might be applied to reduce noise. We investigate moments of the first-passage time distribution under an exponential transient describing relaxation of environmental conditions. We solve the Laplace-transformed (generalized) master equation analytically using a novel method that is applicable to general state schemes. The first-passage time from one end to the other of a linear chain of states is our application for the solutions. The dependence of its average on the relaxation rate obeys a power law for slow transients. The exponent ν depends on the chain length N like ν=-N/(N+1) to leading order. Slow transients substantially reduce the noise of first-passage times expressed as the coefficient of variation (CV), even if the average first-passage time is much longer than the transient. The CV has a pronounced minimum for some lengths, which we call resonant lengths. These results also suggest a simple and efficient noise control strategy and are closely related to the timing of repetitive excitations, coherence resonance, and information transmission by noisy excitable systems. A resonant number of steps from the inhibited state to the excitation threshold and slow recovery from negative feedback provide optimal timing noise reduction and information transmission.

Computational Models for Calcium-Mediated Astrocyte Functions

The computational neuroscience field has heavily concentrated on the modeling of neuronal functions, largely ignoring other brain cells, including one type of glial cell, the astrocytes. Despite the short history of modeling astrocytic functions, we were delighted about the hundreds of models developed so far to study the role of astrocytes, most often in calcium dynamics, synchronization, information transfer, and plasticity in vitro, but also in vascular events, hyperexcitability, and homeostasis. Our goal here is to present the state-of-the-art in computational modeling of astrocytes in order to facilitate better understanding of the functions and dynamics of astrocytes in the brain. Due to the large number of models, we concentrated on a hundred models that include biophysical descriptions for calcium signaling and dynamics in astrocytes. We categorized the models into four groups: single astrocyte models, astrocyte network models, neuron-astrocyte synapse models, and neuron-astrocyte network models to ease their use in future modeling projects. We characterized the models based on which earlier models were used for building the models and which type of biological entities were described in the astrocyte models. Features of the models were compared and contrasted so that similarities and differences were more readily apparent. We discovered that most of the models were basically generated from a small set of previously published models with small variations. However, neither citations to all the previous models with similar core structure nor explanations of what was built on top of the previous models were provided, which made it possible, in some cases, to have the same models published several times without an explicit intention to make new predictions about the roles of astrocytes in brain functions. Furthermore, only a few of the models are available online which makes it difficult to reproduce the simulation results and further develop the models. Thus, we would like to emphasize that only via reproducible research are we able to build better computational models for astrocytes, which truly advance science. Our study is the first to characterize in detail the biophysical and biochemical mechanisms that have been modeled for astrocytes.

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.

Diversity of Evoked Astrocyte Ca2+ Dynamics Quantified through Experimental Measurements and Mathematical Modeling

Astrocytes are a major cell type in the mammalian brain. They are not electrically excitable, but generate prominent Ca²⁺ signals related to a wide variety of critical functions. The mechanisms driving these Ca²⁺ events remain incompletely understood. In this study, we integrate Ca²⁺ imaging, quantitative data analysis, and mechanistic computational modeling to study the spatial and temporal heterogeneity of cortical astrocyte Ca²⁺ transients evoked by focal application of ATP in mouse brain slices. Based on experimental results, we tune a single-compartment mathematical model of IP₃-dependent Ca²⁺ responses in astrocytes and use that model to study response heterogeneity. Using information from the experimental data and the underlying bifurcation structure of our mathematical model, we categorize all astrocyte Ca²⁺ responses into four general types based on their temporal characteristics: Single-Peak, Multi-Peak, Plateau, and Long-Lasting responses. We find that the distribution of experimentally-recorded response types depends on the location within an astrocyte, with somatic responses dominated by Single-Peak (SP) responses and large and small processes generating more Multi-Peak responses. On the other hand, response kinetics differ more between cells and trials than with location within a given astrocyte. We use the computational model to elucidate possible sources of Ca²⁺ response variability: (1) temporal dynamics of IP₃, and (2) relative flux rates through Ca²⁺ channels and pumps. Our model also predicts the effects of blocking Ca²⁺ channels/pumps; for example, blocking store-operated Ca²⁺ (SOC) channels in the model eliminates Plateau and Long-Lasting responses (consistent with previous experimental observations). Finally, we propose that observed differences in response type distributions between astrocyte somas and processes can be attributed to systematic differences in IP₃ rise durations and Ca²⁺ flux rates.

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.

Upon stimulation a large number of cell types respond with transient increases of the intracellular calcium concentration, which often take the form of repetitive spikes. It is therefore believed that calcium spikes play a central role in cellular signal transduction. A critical feature of these calcium spikes is that they occur randomly, which raises the question of how we can predict the timing of calcium spikes. We here show that by using Bayesian ideas and concepts from stochastic processes, we can quantitatively compute the calcium spike rate for a given stimulus. Our analysis also demonstrates that traditional methods for spike rate estimation perform less favourably compared to a Bayesian approach when small numbers of cells are investigated. To test our methodology under conditions that closely mimic those experienced in vivo we challenged cells with agonist concentrations that vary both in space and time. We find that cells that experience similar stimulus profiles are described by similar calcium spike rates. This suggests that calcium spike rates may constitute a quantitative description of whole-cell calcium spiking that reflects both the randomness and the spatiotemporal organisation of the calcium signalling machinery.

IP3 receptor signaling and endothelial barrier function

The endothelium, a monolayer of endothelial cells lining vessel walls, maintains tissue-fluid homeostasis by restricting the passage of the plasma proteins and blood cells into the interstitium. The ion Ca²⁺, a ubiquitous secondary messenger, initiates signal transduction events in endothelial cells that is critical to control of vascular tone and endothelial permeability. The ion Ca²⁺ is stored inside the intracellular organelles and released into the cytosol in response to environmental cues. The inositol 1,4,5-trisphosphate (IP₃) messenger facilitates Ca²⁺ release through IP₃ receptors which are Ca²⁺-selective intracellular channels located within the membrane of the endoplasmic reticulum. Binding of IP₃ to the IP₃Rs initiates assembly of IP₃R clusters, a key event responsible for amplification of Ca²⁺ signals in endothelial cells. This review discusses emerging concepts related to architecture and dynamics of IP₃R clusters, and their specific role in propagation of Ca²⁺ signals in endothelial cells.

Mapping Interpuff Interval Distribution to the Properties of Inositol Trisphosphate Receptors

Tightly clustered inositol trisphosphate receptors (IP₃Rs) control localized Ca²⁺ liberation from the endoplasmic reticulum to generate repetitive Ca²⁺ puffs. Distributions of the interpuff interval (IPI), i.e., the waiting time between successive puffs, are found to be well characterized by a probability density function involving only two parameters, λ and ξ, which represent the basal rate of puff generation and the recovery rate from refractoriness, respectively. However, how the two parameters depend on the kinetic parameters of single IP₃Rs in a cluster is still unclear. In this article, using a stochastic puff model and a single-channel data-based IP₃R model, we establish the dependencies of λ and ξ on two important IP₃R model parameters, IP₃ concentration ([IP₃]) and the recovery rate from Ca²⁺ inhibition (r_low). By varying [IP₃] and r_low in physiologically plausible ranges, we find that the ξ-λ plane is comprised of only two disjoint regions, a biologically impermissible region and a region where each parameter set (ξ, λ) can be caused by using two different combinations of [IP₃] and r_low. The two combinations utilize very different mechanisms to maintain the same IPI distribution, and the mechanistic difference provides a way of identifying IP₃R kinetic parameters by observing properties of the IPI.

Calcium oscillations in wounded fibroblast monolayers are spatially regulated through substrate mechanics

The maintenance of tissue integrity is essential for the life of multicellular organisms. Healing of a skin wound is a paradigm for how various cell types localize and repair tissue perturbations in an orchestrated fashion. To investigate biophysical mechanisms associated with wound localization, we focus on a model system consisting of a fibroblast monolayer on an elastic substrate. We find that the creation of an edge in the monolayer causes cytosolic calcium oscillations throughout the monolayer. The oscillation frequency increases with cell density, which shows that wound-induced calcium oscillations occur collectively. Inhibition of myosin II reduces the number of oscillating cells, demonstrating a coupling between actomyosin activity and calcium response. The spatial distribution of oscillating cells depends on the stiffness of the substrate. For soft substrates with a Young's modulus E ~ 360 Pa, oscillations occur on average within 0.2 mm distance from the wound edge. Increasing substrate stiffness leads to an average localization of oscillations away from the edge (up to ~0.6 mm). In addition, we use traction force microscopy to determine stresses between cells and substrate. We find that an increase of substrate rigidity leads to a higher traction magnitude. For E  <  ~2 kPa, the traction magnitude is strongly concentrated at the monolayer edge, while for E  >  ~8 kPa, traction magnitude is on average almost uniform beneath the monolayer. Thus, the spatial occurrence of calcium oscillations correlates with the cell-substrate traction. Overall, the experiments with fibroblasts demonstrate a collective, chemomechanical localization mechanism at the edge of a wound with a potential physiological role.

Single-cell resolution of intracellular T cell Ca2+ dynamics in response to frequency-based H2O2 stimulation

Adaptive immune cells, such as T cells, integrate information from their extracellular environment through complex signaling networks with exquisite sensitivity in order to direct decisions on proliferation, apoptosis, and cytokine production. These signaling networks are reliant on the interplay between finely tuned secondary messengers, such as Ca2+ and H2O2. Frequency response analysis, originally developed in control engineering, is a tool used for discerning complex networks. This analytical technique has been shown to be useful for understanding biological systems and facilitates identification of the dominant behaviour of the system. We probed intracellular Ca2+ dynamics in the frequency domain to investigate the complex relationship between two second messenger signaling molecules, H2O2 and Ca2+, during T cell activation with single cell resolution. Single-cell analysis provides a unique platform for interrogating and monitoring cellular processes of interest. We utilized a previously developed microfluidic device to monitor individual T cells through time while applying a dynamic input to reveal a natural frequency of the system at approximately 2.78 mHz stimulation. Although our network was much larger with more unknown connections than previous applications, we are able to derive features from our data, observe forced oscillations associated with specific amplitudes and frequencies of stimuli, and arrive at conclusions about potential transfer function fits as well as the underlying population dynamics.

On the dynamical structure of calcium oscillations

Oscillations in the concentration of free cytosolic Ca²⁺ are an important and ubiquitous control mechanism in many cell types. It is thus correspondingly important to understand the mechanisms that underlie the control of these oscillations and how their period is determined. We show that Class I Ca²⁺ oscillations (i.e., oscillations that can occur at a constant concentration of inositol trisphosphate) have a common dynamical structure, irrespective of the oscillation period. This commonality allows the construction of a simple canonical model that incorporates this underlying dynamical behavior. Predictions from the model are tested, and confirmed, in three different cell types, with oscillation periods ranging over an order of magnitude. The model also predicts that Ca²⁺ oscillation period can be controlled by modulation of the rate of activation by Ca²⁺ of the inositol trisphosphate receptor. Preliminary experimental evidence consistent with this hypothesis is presented. Our canonical model has a structure similar to, but not identical to, the classic FitzHugh-Nagumo model. The characterization of variables by speed of evolution, as either fast or slow variables, changes over the course of a typical oscillation, leading to a model without globally defined fast and slow variables.

Stochastic initiation and termination of calcium-mediated triggered activity in cardiac myocytes

Cardiac myocytes normally initiate action potentials in response to a current stimulus that depolarizes the membrane above an excitation threshold. Aberrant excitation can also occur due to spontaneous calcium (Ca²⁺) release (SCR) from intracellular stores after the end of a preceding action potential. SCR drives the Na⁺/Ca²⁺ exchange current inducing a "delayed afterdepolarization" that can in turn trigger an action potential if the excitation threshold is reached. This "triggered activity" is known to cause arrhythmias, but how it is initiated and terminated is not understood. Using computer simulations of a ventricular myocyte model, we show that initiation and termination are inherently random events. We determine the probability of those events from statistical measurements of the number of beats before initiation and before termination, respectively, which follow geometric distributions. Moreover, we elucidate the origin of randomness by a statistical analysis of SCR events, which do not follow a Poisson process observed in other eukaryotic cells. Due to synchronization of Ca²⁺ releases during the action potential upstroke, waiting times of SCR events after the upstroke are narrowly distributed, whereas SCR amplitudes follow a broad normal distribution with a width determined by fluctuations in the number of independent Ca²⁺ wave foci. This distribution enables us to compute the probabilities of initiation and termination of bursts of triggered activity that are maintained by a positive feedback between the action potential upstroke and SCR. Our results establish a theoretical framework for interpreting complex and varied manifestations of triggered activity relevant to cardiac arrhythmias.

Systematic Characterization of Dynamic Parameters of Intracellular Calcium Signals

Dynamic processes, such as intracellular calcium signaling, are hallmark of cellular biology. As real-time imaging modalities become widespread, a need for analytical tools to reliably characterize time-series data without prior knowledge of the nature of the recordings becomes more pressing. The goal of this study is to develop a signal-processing algorithm for MATLAB that autonomously computes the parameters characterizing prominent single transient responses (TR) and/or multi-peaks responses (MPR). The algorithm corrects for signal contamination and decomposes experimental recordings into contributions from drift, TRs, and MPRs. It subsequently provides numerical estimates for the following parameters: time of onset after stimulus application, activation time (time for signal to increase from 10 to 90% of peak), and amplitude of response. It also provides characterization of the (i) TRs by quantifying their area under the curve (AUC), response duration (time between 1/2 amplitude on ascent and descent of the transient), and decay constant of the exponential decay region of the deactivation phase of the response, and (ii) MPRs by quantifying the number of peaks, mean peak magnitude, mean periodicity, standard deviation of periodicity, oscillatory persistence (time between first and last discernable peak), and duty cycle (fraction of period during which system is active) for all the peaks in the signal, as well as coherent oscillations (i.e., deterministic spikes). We demonstrate that the signal detection performance of this algorithm is in agreement with user-mediated detection and that parameter estimates obtained manually and algorithmically are correlated. We then apply this algorithm to study how metabolic acidosis affects purinergic (P2) receptor-mediated calcium signaling in osteoclast precursor cells. Our results reveal that acidosis significantly attenuates the amplitude and AUC calcium responses at high ATP concentrations. Collectively, our data validated this algorithm as a general framework for comprehensively analyzing dynamic time-series.

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.

Ryanodine receptor sensitization results in abnormal calcium signaling in airway smooth muscle cells

Intracellular Ca(2+) dynamics of airway smooth muscle cells (ASMCs) are believed to play a major role in airway hyperresponsiveness and remodeling in asthma. Prior studies have underscored a prominent role for inositol 1,4,5-triphosphate (IP3) receptors in normal agonist-induced Ca(2+) oscillations, whereas ryanodine receptors (RyRs) appear to remain closed during such Ca(2+) oscillations, which mediate ASMC contraction. Nevertheless, RyRs have been hypothesized to play a role in hyperresponsive Ca(2+) signaling. This could be explained by RyRs being "sensitized" to open more frequently by certain compounds. We investigate the implications of RyR sensitization on Ca(2+) dynamics in ASMC using a combination of mathematical modeling and experiments with mouse precision-cut lung slices. Caffeine is used to increase the sensitivity of RyRs to cytosolic Ca(2+) concentration ([Ca(2+)]i) and sarcoplasmic reticulum Ca(2+) ([Ca(2+)]SR). In ASMCs, high caffeine concentrations (>10 mM) induce a sustained elevation of [Ca(2+)]i. Our mathematical model accounts for this by the activation of store-operated Ca(2+) entry that results from a large increase in the RyR sensitivity to [Ca(2+)]SR and the associated Ca(2+) release, which leads to a reduction of [Ca(2+)]SR. Importantly, our model also predicts that: (1) moderate RyR sensitization induces slow Ca(2+) oscillations, a result experimentally confirmed with low concentrations of caffeine; and (2) high RyR sensitization suppresses fast, agonist-induced Ca(2+) oscillations by inducing substantial store-operated Ca(2+) entry and elevated [Ca(2+)]i. These results suggest that RyR sensitization could play a role in ASMC proliferation (by inducing slow Ca(2+) oscillations) and in airway hyperresponsiveness (by inducing greater mean [Ca(2+)]i for similar levels of contractile agonist).

Genetically targeted single-channel optical recording reveals multiple Orai1 gating states and oscillations in calcium influx

Orai1 comprises the pore-forming subunit of the Ca(2+) release-activated Ca(2+) (CRAC) channel. When bound and activated by stromal interacting molecule 1 (STIM1), an endoplasmic reticulum (ER)-resident calcium sensor, Orai1 channels possess high selectivity for calcium but extremely small conductance that has precluded direct recording of single-channel currents. We have developed an approach to visualize Orai1 activity by fusing Orai1 to fluorescent, genetically encoded calcium indicators (GECIs). The GECI-Orai1 probes reveal local Ca(2+) influx at STIM1-Orai1 puncta. By whole cell recording, these fusions are fully functional as CRAC channels. When GECI-Orai1 and the CRAC-activating domain (CAD) of STIM1 were coexpressed at low levels and imaged using a total internal reflectance fluorescence microscope, cells exhibited sporadic fluorescence transients the size of diffraction-limited spots and the brightness of a few activated GECI proteins. Transients typically rose rapidly and fell into two classes according to duration: briefer "flickers" lasting only a few hundred milliseconds, and longer "pulses" lasting one to several seconds. The size, intensity, trace shape, frequency, distribution, physiological characteristics, and association with CAD binding together demonstrate that GECI-Orai1 fluorescence transients correspond to single-channel Orai1 responses. Single Orai1 channels gated by CAD, and small Orai1 puncta gated by STIM1, exhibit repetitive fluctuations in single-channel output. CAD binding supports a role in open state maintenance and reveals a second phase of CAD/STIM1 binding after channel opening. These first recordings of single-channel Orai1 currents reveal unexpected dynamics, and when paired with CAD association, support multiple single-channel states.

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.

Accurate Langevin approaches to simulate Markovian channel dynamics

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

Multiscale Modeling Indicates That Temperature Dependent [Ca2+]i Spiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

Changes in the cytosolic Ca(2+) concentration ([Ca(2+)]i) are the most predominant active signaling mechanism in astrocytes that can modulate neuronal activity and is assumed to influence neuronal plasticity. Although Ca(2+) signaling in astrocytes has been intensively studied in the past, our understanding of the signaling mechanism and its impact on tissue level is still incomplete. Here we revisit our previously published data on the strong temperature dependence of Ca(2+) signals in both cultured primary astrocytes and astrocytes in acute brain slices of mice. We apply multiscale modeling to test the hypothesis that the temperature dependent [Ca(2+)]i spiking is mainly caused by the increased activity of the sarcoendoplasmic reticulum ATPases (SERCAs) that remove Ca(2+) from the cytosol into the endoplasmic reticulum. Quantitative comparison of experimental data with multiscale simulations supports the SERCA activity hypothesis. Further analysis of multiscale modeling and traditional rate equations indicates that the experimental observations are a spatial phenomenon where increasing pump strength leads to a decoupling of Ca(2+) release sites and subsequently to vanishing [Ca(2+)]i spikes.

Plasticity of Neuron-Glial Transmission: Equipping Glia for Long-Term Integration of Network Activity

The capacity of synaptic networks to express activity-dependent changes in strength and connectivity is essential for learning and memory processes. In recent years, glial cells (most notably astrocytes) have been recognized as active participants in the modulation of synaptic transmission and synaptic plasticity, implicating these electrically nonexcitable cells in information processing in the brain. While the concept of bidirectional communication between neurons and glia and the mechanisms by which gliotransmission can modulate neuronal function are well established, less attention has been focussed on the computational potential of neuron-glial transmission itself. In particular, whether neuron-glial transmission is itself subject to activity-dependent plasticity and what the computational properties of such plasticity might be has not been explored in detail. In this review, we summarize current examples of plasticity in neuron-glial transmission, in many brain regions and neurotransmitter pathways. We argue that induction of glial plasticity typically requires repetitive neuronal firing over long time periods (minutes-hours) rather than the short-lived, stereotyped trigger typical of canonical long-term potentiation. We speculate that this equips glia with a mechanism for monitoring average firing rates in the synaptic network, which is suited to the longer term roles proposed for astrocytes in neurophysiology.

T Cell Dynamic Activation and Functional Analysis in Nanoliter Droplet Microarray

Objective

Characterization of the heterogeneity in immune reactions requires assessing dynamic single cell responses as well as interactions between the various immune cell subsets. Maturation and activation of effector cells is regulated by cell contact-dependent and soluble factor-mediated paracrine signalling. Currently there are few methods available that allow dynamic investigation of both processes simultaneously without physically constraining non-adherent cells and eliminating crosstalk from neighboring cell pairs. We describe here a microfluidic droplet microarray platform that permits rapid functional analysis of single cell responses and co-encapsulation of heterotypic cell pairs, thereby allowing us to evaluate the dynamic activation state of primary T cells.

Methods

The microfluidic droplet platform enables generation and docking of monodisperse nanoliter volume (0.523 nl) droplets, with the capacity of monitoring a thousand droplets per experiment. Single human T cells were encapsulated in droplets and stimulated on-chip with the calcium ionophore ionomycin. T cells were also co-encapsulated with dendritic cells activated by ovalbumin peptide, followed by dynamic calcium signal monitoring.

Results

Ionomycin-stimulated cells depicted fluctuation in calcium signalling compared to control. Both cell populations demonstrated marked heterogeneity in responses. Calcium signalling was observed in T cells immediately following contact with DCs, suggesting an early activation signal. T cells further showed non-contact mediated increase in calcium level, although this response was delayed compared to contact-mediated signals.

Conclusions

Our results suggest that this nanoliter droplet array-based microfluidic platform is a promising technique for assessment of heterogeneity in various types of cellular responses, detection of early/delayed signalling events and live cell phenotyping of immune cells.