53 Letter __________________________________________________________________ Forma, 18, 53–58, 2003 A Stochastic Fire-Diffuse-Fire Model of Ca 2+ Release Yulia TIMOFEEVA* and Stephen COOMBES Department of Mathematical Sciences, Loughborough University, Leicestershire, LE11 3TU, U.K. *i.timofeeva@lboro.ac.uk (Received November 27, 2002; Accepted December 11, 2002) Keywords: Calcium Sparks, Noise, Non-equilibrium Phase-Transition, Directed- Percolation Abstract. Calcium ions are an important second messenger in living cells transmitting signals in the form of waves. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs) from spatially localised calcium stores. Here we develop a mathematical model of calcium release based upon a stochastic generalisation of the fire-diffuse-fire (FDF) threshold model for calcium release. Our model retains the discrete characteristic of the FDF model (spatially localised stores) but also incorporates a notion of release probability, via the introduction of threshold noise. It is possible to identify a critical level of noise defining a non-equilibrium phase- transition between abortive and propagating waves. This transition is shown to belong to the directed percolation universality class. 1. Introduction Calcium signals in the form of sparks and propagating waves are observed in a wide range of cell types. Experiments have shown the stochastic nature of release events both in systems based on the inositol (1,4,5)-trisphosphate (IP 3 ) receptor (MARCHANT and PARKER, 2001) and the ryanodine receptor (RyR) (CHENG et al ., 1996). The importance of stochasticity for the initiation and propagation of calcium waves has been a subject of limited theoretical investigation. Notable exceptions are the work of KEIZER and SMITH (1998) on stochastic RyR release sites in cardiac myocytes, and the work of BÄR et al . (2000) on stochastic IP 3 channels. In this paper we introduce a stochastic version of the FDF threshold model for calcium release of KEIZER et al. (1998). One of the main advantages of our model is that it is biophysically realistic and computationally cheap to solve. Simulation results are presented for both a one and two dimensional cell model. We demonstrate that different noise intensities can lead to a variety of different structures, including noisy travelling circular fronts, spiral waves, target patterns and large scale coherent periodic rhythms. Moreover, a statistical analysis shows that the model exhibits a non-equilibrium phase transition belonging to the directed percolation universality class.