ANALYSIS OF AN AUTONOMOUS PHASE MODEL FOR NEURONAL PARABOLIC BURSTING

An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tiss ue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is develope d to study endogenous parabolic bursting, a class of burst activity ob served experimentally in excitable membrane. The phase model is motiva ted by Rinzel and Lee's dissection of a model for neuronal parabolic b ursting (J. Math. Biol. 25, 653-675 (1987)). Rapid spiking is represen ted canonically by a one-variable phase equation that is coupled bi-di rectionally to a two-variable slow system, The model is analyzed in th e slow-variable phase plane? using quasi steady-state assumptions and formal averaging, We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spik ing. The relative speed of activation and inactivation of the slow var iables strongly influences the burst pattern as well as other dynamics . We find conditions of the bistability of solutions between continuou s spiking and bursting. Although the phase model is simple, we demonst rate that it captures many dynamical features of more complex biophysi cal models.