Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris–Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.

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.

Data from each subject in a repeated-measures experiment form a time series, which may include trial-by-trial fluctuations arising from human factors such as practice or fatigue. Concerns about the statistical implications of such effects have increased the popularity of Generalized Additive Mixed Models (GAMMs), a powerful technique for modeling wiggly patterns. We question these statistical concerns and investigate the costs and benefits of using GAMMs relative to linear mixed-effects models (LMEMs). In two sets of Monte Carlo simulations, LMEMs that ignored time-varying effects were no more prone to false positives than GAMMs. Although GAMMs generally boosted power for within-subject effects, they reduced power for between-subject effects, sometimes to a severe degree. Our results signal the importance of proper subject-level randomization as the main defense against statistical artifacts due to by-trial fluctuations.Studies including repeated measurements on individual subjects are extremely common in the social sciences. Because all the data for a single subject cannot be collected simultaneously, the set of observations for that subject will form a time series, and the full dataset a collection of such. By itself, this observation may seem trivial, but its statistical implication-non-independence over time-is not. Human subjects often fluctuate in their performance over the course of an experimental session, reflecting changing environmental, physiological, or psychological factors as a subject completes a task. The psychological

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. IP 3 induced Ca 2þ 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. V

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.

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.

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Main points:1) Transient stimuli trigger astrocyte calcium waves in a probabilistic manner.2) Response probability can be modulated by other signalling pathways.3) Wave initiation follows synchronous calcium release from three or more subcellular "puff" sites.Keywords: Astrocyte; calcium puffs; crosstalk; ATP; glutamate. 2 AbstractAstrocyte 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 signalling 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 signalling 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 signalling 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.3