All Phase Resetting Curves Are Bimodal, but Some Are More Bimodal Than Others

Sorinel A Oprisan

doi:10.1155/2013/230571

Abstract

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).

Highlights

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, 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)
In this paper we shown that (1) all Phase resetting curves (PRCs) are bimodal and (2) the degree of bimodality of a PRC can be smoothly adjusted if the bifurcation scenario leading to stable limit cycle oscillations combines a saddle node of invariant circle (SNIC) and a Hopf bifurcation (HB)
Our current study extends previous results [28, 30] by (1) introducing the concept of neutral points, that is, phases at which external perturbations produce no phase resetting, and (2) showing that a SNIC that ends with a HB leads to a PRC that is a linear combination of both Type I and Type II PRCs

Summary

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, 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). The most significant effect of a perturbation occurs during the cycle that contains the perturbation and is quantified by the first order PRC; that is, F1(φ) = P1/Pi − 1, where Pi is the intrinsic period of oscillation of isolated neuron and P1 is the transiently modified duration of the current cycle due to a perturbation applied at the stimulus time ts or phase φ = ts/Pi (see Figure 1(a)). Our current study extends previous results [28, 30] by (1) introducing the concept of neutral points, that is, phases at which external perturbations produce no phase resetting, and (2) showing that a SNIC that ends with a HB leads to a PRC that is a linear combination of both Type I and Type II PRCs

Theory

Results

Discussions