Continuation-Based numerical detection of after-depolarization and spike-adding thresholds

The changes in neuronal firing pattern are signatures of brain function, and it is of interest to understandhowsuch 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 neuronmodel 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 showhow to formulate such transient behavior as a two-point boundary value problem (2PBVP), whichcan be solved using well-known continuation methods. The 2PBVPis formulated such that the transient response is represented by a finite orbit segment forwhich onsets ofADPand 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-parametercontinuation of the detected onset points to identify the boundaries between regions with andwithout 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.