Biol. Cybern. 68, 209-214 (1993) Biological Cybernetics © Springer-Verlag1993 Spike initiation and propagation on axons with slow inward currents Thomas B. Kepler *, Eve Marder Department of Biology and Center for Complex Systems, Brandeis University, Waltham, MA 02254, USA Received: 10 June 1992/Accepted in revised form: 17 August 1992 Abstract. We investigate spike initiation and propaga- tion in a model axon that has a slow regenerative conductance as well as the usual Hodgkin-Huxley type sodium and potassium conductances. We study the role of slow conductance in producing repetitive firing, com- pute the dispersion relation for an axon with an addi- tional slow conductance, and show that under appropriate conditions such an axon can produce a traveling zone of secondary spike initiation. This study illustrates some of the complex dynamics shown by excitable membranes with fast and slow conductances. Introduction Many neurons show a large number of voltage depen- dent conductances that give them a complex repertoire of behavior (Baxter and Byrne 1991). Neurons that express plateau and bursting properties play pivotal roles in both vertebrate and invertebrate nervous sys- tems (Harris-Warrick and Marder 1991; Murder 1991, 1992). Despite this, the general assumption is that most axons act merely to carry information between two sites in the nervous system and are well described by systems of equations that describe the inward Na + current, the delayed rectifier K + current (e.g. Hodgkin and Huxley 1952) and in some cases the transient outward current (Connor et al. 1977). However, more complex behavior of axons such as branch point failures or inhomogenei- ties in diameter that can contribute to signal processing or integration are known (Parnas et al. 1976; Ramon et al. 1975; Rinzel 1990). Another example of more complex behavior of ax- ons is seen in the recent work of Meyrand et al. (1992) who found that application of serotonin to the axon of the Lateral Gastric (LG) neuron of the stomatogastric * Present address: Biomathematics Department, North Carolina State University, Campus Box 8203, Raleigh, NC 27695, USA Correspondence to: T. B. Kepler nervous system of the crab Cancer borealis coupled with somatic depolarization can produce a secondary zone of spike initiation. When the serotonin concentration is relatively high the axon produces sustained trains of action potentials. A model of these data (Murder et al. 1992) suggested that a slow inward current sensitive to serotonin and depolarization and localized to a specific axonal region could account for these results. Motivated by the possibility that axons might dis- play slow inward currents, in this study we have pro- duced a model axon in which a slow inward current is found along with a fast inward Na + current, the delayed rectifier outward current and leak current (Connor et al. 1977). The slow inward current we use is distributed uniformly along the axon (as are the other conduc- tances), is activated by depolarization with slow first- order kinetics but shows no time-dependent inactivation. We use this model to explore the behavior of an axon with these conductances and show that the additional slow conductance gives rise to a traveling zone where spikes are spontaneously generated. In this paper we discuss the role of the slow conduc- tance in the sudden transition from stimulus triggered spike transmission to spontaneous spike initiation and the propagation of this spike generation zone along the axon. These phenomena change the dynamics of the neuron. 2 The model The model is described by a current conservation equa- tion and a system of kinetic equations for the conduc- tance gating variables. The former is given by Ot + gN~m3h(V- VN~) -[-gKn4(V- VK) 02V +gL(V-- VL) --ZIs--Y-~x2=O (1) where Vi is the equilibrium potential for ion i and gi is that ion's maximal conductance, C is the capacitance per unit area, V is the membrane potential, and 7 is