Continuation-based numerical detection of after-depolarisation and spike-adding threshold
Jakub Nowacki, Hinke M. Osinga, and Krasimira T. Tsaneva-Atanasova
The 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 via 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-depolarisation (ADP) that follows after a brief current injection. This type of burst is a transient phenomenon, which is not amenable to the classical bifurcation analysis done, for example, for periodic bursting oscillators. In this paper, we show how to formulate such transient behaviour 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 behaviour 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 different problems.
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