AbstractComplexity and limited knowledge render it impractical to write down the equations describing a cellular system completely. Cellular biophysics uses hypotheses-based modelling instead. How can we set up models with predictive power beyond the experimental examples used to develop them? The two textbook systems of cellular biophysics, $$\hbox {Ca}^{2+}$$
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signalling and neuronal membrane potential dynamics, both face this question. Both systems also have a non-equilibrium feature in common: on different time scales and for different observables, they exhibit stochastic spiking, i.e., sequences of stereotypical events that are separated by statistically distributed intervals, the interspike intervals (ISI). Here we review recent progress on the description of $$\hbox {Ca}^{2+}$$
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spikes in terms of blips, puffs and cellular $$\hbox {Ca}^{2+}$$
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spikes and focus on stochastic models that can explain the statistics of the single ISIs, in particular its mean and variance and the cell-to-cell variability of these statistics. We also review models of the stochastic integrate-and-fire type and measures like the spike-train power spectrum or the serial correlation coefficient that are used to describe neuronal spike trains. These concepts from computational neuroscience might be applicable for understanding long-term memory effects in $$\hbox {Ca}^{2+}$$
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spiking that extend beyond a single ISI, such as cumulative refractoriness.
The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case
