I'm reading a biology article (full text here) that analyzed the variation of a cell parameter over time at different stimulus intensities (calcium concentration). The frequency of peaks increases at increasing calcium concentrations, while their duration remains constant, fitting a gamma distribution. Here are the bursts duration distributions for two different calcium concentrations:


By eye those bursts were classified in isolated bursts (with duration $\tau_{burst}$) and clustered bursts (with duration $\tau_{cluster}$). Here are typical traces of two different cells:


The authors computed the autocorrelation from the traces and then estimated $\tau_{burst}$ and $\tau_{cluster}$. Moreover, they estimated the relative weight of the two types of burst at varying calcium concentrations.

Their method was as follow: (1) they calculated the mean autocorrelation function (for every calcium concentration); (2) they described the curve with an exponential decay function; (3) they estimated the weight of the two types of bursts.

As an example, below are shown two mean autocorrelation functions: one of the tracks obtained at 50mM calcium concentration (blue), the other at 200mM (red). autocorrelation

The first is well fitted by a single decaying exponential with time constant $\tau=70s$; the second is fitted by the sum of two exponentials: $c(t)=a_1e^{-t/\tau_1}+a_2e^{-t/\tau_2}$ with $\tau_1=70s$ and $\tau_2=720s$. They concluded that $\tau=\tau_1=\tau_{burst}$ and $\tau_2=\tau_{cluster}$.

In the supplemental document (here) they say:

The autocorrelation function was computed from the thresholded traces and provided an unbiased estimate of the size of the isolated bursts and clusters, which correspond to the two time scale of the exponential decay in the fit.

My question is: how are the time scales of the exponential decay of the autocorrelations related to the duration of bursts and their relative frequencies?


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