Continuation-based numerical detection of after-depolarization and spike-adding thresholds.

PublishedJournal ArticleResearch Support, Non-U.S. Gov'tThe changes in neuronal firing pattern are signatures of brain function, and it is of interest to understand how such changes evolve as a function of neuronal biophysical properties. We address this important problem by the analysis and numerical investigation of a class of mechanistic mathematical models. We focus on a hippocampal pyramidal neuron model and study the occurrence of bursting related to the after-depolarization (ADP) that follows a brief current injection. This type of burst is a transient phenomenon that is not amenable to the classical bifurcation analysis done, for example, for periodic bursting oscillators. In this letter, we show how to formulate such transient behavior as a two-point boundary value problem (2PBVP), which can be solved using well-known continuation methods. The 2PBVP is formulated such that the transient response is represented by a finite orbit segment for which onsets of ADP and additional spikes in a burst can be detected as bifurcations during a one-parameter continuation. This in turn provides us with a direct method to approximate the boundaries of regions in a two-parameter plane where certain model behavior of interest occurs. More precisely, we use two-parameter continuation of the detected onset points to identify the boundaries between regions with and without ADP and bursts with different numbers of spikes. Our 2PBVP formulation is a novel approach to parameter sensitivity analysis that can be applied to a wide range of problems.The research for this letter was done while J.N. was a Ph.D. student at the University of Bristol, supported by grant EP/E032249/1 from the Engineering and Physical Sciences Research Council (EPSRC). The research of K.T-A. was supported by EPSRC grant EP/I018638/1 and that of H.M.O. by grant UOA0718 of the Royal Society of NZ Marsden Fun