%0 Journal Article
%T All Phase Resetting Curves Are Bimodal, but Some Are More Bimodal Than Others
%A Sorinel A. Oprisan
%J ISRN Computational Biology
%D 2013
%R 10.1155/2013/230571
%X Phase resetting curves (PRCs) are phenomenological and quantitative tools that tabulate the transient changes in the firing period of endogenous neural oscillators as a result of external stimuli, for example, presynaptic inputs. A brief current perturbation can produce either a delay (positive phase resetting) or an advance (negative phase resetting) of the subsequent spike, depending on the timing of the stimulus. We showed that any planar neural oscillator has two remarkable points, which we called neutral points, where brief current perturbations produce no phase resetting and where the PRC flips its sign. Since there are only two neutral points, all PRCs of planar neural oscillators are bimodal. The degree of bimodality of a PRC, that is, the ratio between the amplitudes of the delay and advance lobes of a PRC, can be smoothly adjusted when the bifurcation scenario leading to stable oscillatory behavior combines a saddle node of invariant circle (SNIC) and an Andronov-Hopf bifurcation (HB). 1. Introduction Neural oscillators are excitable cells which means that as soon as a parameter, such as an external bias current or an ionic conductance, crosses a threshold, then the system switches from a stable rest state characterized by small dampened excursions of the membrane potential to a high amplitude excursion of the membrane potential, called action potential (AP). Repetitive firing excitable cells were categorized by Hodgkin [1] as Class 1, that is, the firing frequency is continuously tunable by adjusting a bias current and such neurons can fire with arbitrarily low frequencies, or Class 2, that is, discontinuous curve with a nonzero firing frequency threshold. Excitable cells generate APs when a certain control parameter crosses a threshold; that is, a bifurcation occurs in their state space [2]. It became clear that threshold phenomena, such as the AP, are related to fundamental properties of the mathematical model describing the neuron [3] and its corresponding bifurcation diagram [4]. Class 1 excitability was ascribed to a saddle node of invariant circle (SNIC) bifurcation and Class 2 excitability to an Andronov-Hopf bifurcation (HB) (see [5, 6] and references therein). Since neurons are excitable cells operating near a bifurcation point, it is mathematically possible to significantly reduce the number of variables describing the state of a neuron to a few essential ones by using the so-called canonical forms (see [2] for an extensive review of existing canonical models in neuroscience). For example, the four-dimensional conductance based
%U http://www.hindawi.com/journals/isrn.computational.biology/2013/230571/