Cell information processing via frequency encoding and excitability

Cells continuously interact with their environment and respond to changes accordingly. Very often changes in the concentration of extracellular substances occur which, through receptor binding, give rise to a sequence of intracellular changes in what is called a signaling cascade. Increasing intensities of the external stimulus can result in increasing concentrations or increasing activation of the internal messengers or can induce a pulsatile behavior of increasing frequency with stimulus strength. This last behavior has been observed in intracellular Ca$^{2+}$ signals in which Ca$^{2+}$ is released from the endoplasmic reticulum through Inositol Trisphosphate Receptors (IP$_3$Rs), an ubiquitous signaling mechanism involved in many processes of physiological relevance. A statistical analysis of the time intervals between subsequent IP$_3$R-mediated Ca$^{2+}$ pulses observed experimentally has revealed an exponential dependence with the external stimulus strength in several cell types. This type of dependence, which is reminiscent of Kramers' law for thermally activated barrier crossing, has also been derived for certain excitable systems. Excitable systems have a stable stationary solution and, upon perturbations that surpass a threshold, perform a long excursion in phase space before returning to equilibrium. In this paper we use a very simple mathematical model of IP$_3$R-mediated Ca$^{2+}$ signals and published experimental results to derive the scaling law between the interpulse time and the external stimulus strength.