  Cells continuously interact with their environment mediating their responses
through signaling cascades. Very often, external stimuli induce pulsatile
behaviors in intermediaries of the cascade of increasing frequency with the
stimulus strength. This is characteristic of intracellular Ca$^{2+}$ signals
involving Ca$^{2+}$ release through Inositol Trisphosphate Receptors
(IP$_3$Rs). The mean frequency of IP$_3$R-mediated Ca$^{2+}$ pulses has been
observed to scale exponentially with the stimulus strength in many cell types.
In this paper we use a simple ODE model of the intracellular Ca$^{2+}$ dynamics
for parameters for which there is one excitable fixed point. Including
fluctuations through an additive noise term, we derive the mean escape rate
from the stationary state and, thus, the mean interpulse time, as a function of
the fraction, $\beta$, of readily openable IP$_3$Rs. Using an IP$_3$R kinetic
model, experimental observations of spatially resolved Ca$^{2+}$ signals and
previous estimates of the IP$_3$ produced upon stimulation we quantify the
fluctuations and relate $\beta$ to [IP$_3$] and the stimulus strength. In this
way we determine that the mean interpulse time can be approximated by an
exponential function of the latter for ranges such that the covered mean time
intervals are similar or larger than those observed experimentally. The study
thus provides an easily interpretable explanation, applicable to other
pulsatile signaling intermediaries, of the observed exponential dependence
between frequency and stimulus, a key feature that makes frequency encoding
qualitatively different from other ways commonly used by cells to "read" their
environment.
