Symbolic dynamics in the study of bursting electrical activity Jorge Duarte, Lu´ ıs Silva, and J. Sousa Ramos Abstract. Many cells exhibit a complex behaviour, characterized by brief bursts of oscillatory activity interspersed with quiescent periods during which the membrane potential changes only slowly. This behaviour is called bursting. The interpretation of bursting in terms of nonlinear dynamics is one of the recent success stories of mathematical physiology and provides an excellent example of how mathematics can be used to understand complex biological dynamical systems. In the present paper we study a map, that replicates the dynamics of bursting cells, presented in [16]. Using symbolic dynamics we characterize the topological entropy of the chaotic bursts and we analyse the variation of this important numerical invariant with the parameters of the system. This procedure allows us to distinguish different chaotic scenarios. 1. Motivation and preliminaries Bursting behavior is ubiquitous in physical and biological systems, specially in neural systems where it plays an important role in information processing (see [12], [6], [7] and [8]). Physiological questions concerning the dynamics of bursting cells lead to chal- lenging mathematical problems. The complex bursting activity of individual bio- logical neurons is the result of high-dimensional dynamics of nonlinear processes responsible for variations in the ionic currents across the membrane. The numeri- cal studies of such neural activity are usually based on either realistic ionic-based models or phenomenological models. The ionic-based models proposed for a single neuron are designed to replicate the physiological processes in the membrane. These models are usually given by many nonlinear equations. The strong nonlinearity and high dimensionality of the phase space is a significant obstacle in understanding the collective behavior of such dynamical systems [5]. Important mechanisms are hid- den behind the complexity of the equations. The phenomenological models are designed to capture the most important features of neural behavior with minimal complexity of the model [4]. Recently a special type of phenomenological models based on a low-dimensional map was proposed. Indeed, the study of dynamical 2000 Mathematics Subject Classification. Primary 34A37, 37B10; Secondary 37E05, 37B40. Key words and phrases. Bursting behavior, difference equations, symbolic dynamics, topo- logical entropy, chaos. Partially supported by Instituto Superior de Engenharia de Lisboa. Partially supported by FCT/POCTI/FEDER. 1