Abstract
Complexity 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+} 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+} spikes in terms of blips, puffs and cellular hbox {Ca}^{2+} 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+} spiking that extend beyond a single ISI, such as cumulative refractoriness.
Highlights
The program of theoretical physics for understanding a given system is to specify first principles to it and to solve the resulting equations
Modelling of Ca2+ signalling has taken place in the tradeoff between models accounting for the randomness of puffs and spikes, cell variability and measured parameter dependencies on one side and rate equation models convenient to simulate time courses on the other side in recent years
We suggest to include higher moment’s dynamics derived from the Master equation to account for spike generation by fluctuations
Summary
The program of theoretical physics for understanding a given system is to specify first principles to it and to solve the resulting equations. The purpose of most models is to simulate cellular behavior, and ordinary differential equations are very convenient to that end Their derivation, has to take the large fluctuations into account, i.e., has to start from stochastic theory as the mathematical structure corresponding to Ca2+ dynamics. An alternative to simulating cellular behavior by differential equations is to determine the distribution of cellular properties generated by the noise inherent to the system [38,54] Such an approach would correspond more to the noisy character of cell dynamics, but will only take hold, if the analysis of experimental results engages into such a view on cellular behavior and measures distributions and/or their moments [54]. We will briefly discuss how concepts from computational neuroscience, such as multidimensional integrate-and-fire models and spike train power spectra could be useful to model and analyze Ca2+ spiking
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