Examination with a computer of how parameters changes and variabilities influence a model of oscillator entrainment

Abstract

A mathematical model designed originally for pacemaker neurons and post synaptic potentials represents acceptably respiratory driving by a pump through the Hering-Breuer reflex (Vibert et al. 1981). Discrepancies with the respiratory embodiment arose firstly in the zig-zag plots of average driving and driven intervals where the boundaries between positively and negatively-sloped segments differed, and where a hysteresis-like dependence on the order of the observations occurred. Secondly, they arose from the fact that estimates of the slope A and intercept B of the linear dependence of the delay on the arrival phase of the driver differed when based on the model's formulae but using different aspects of the same data. The question thus arose as to which model shortcomings cause discrepancies with this particular embodiment.

This communication presents computer simulations that explore the consequences of modifying, or imposing variabilities, upon several parameters. A reflects how the driver's (i.e. the inflation's) consequences increase as it arrives later in the driven (i.e. phrenic) cycle: when it departs from around 0.7, (as required by the model) and approaches 1. The positively sloped segments broaden at the expense of the negatively-sloped ones which disappear when A reaches 1. The intercept B reflects the effect of the driver when it arrives just after the driven event: its increases shift a relatively unchanged plot up and to the right. When the driving period I exhibits variability or the driver one N exhibits variability or trends, as happens often with the respiratory embodiment, the model-predicted behavior remains acceptably natural-like: in particular, that with a jittery driven period mimics well observations on cats with high CO₂ or superficially anesthetized. For each driving period, the driven intervals pertained to a limited number of classes (e.g. to one, two or three) which occurred in an invariant sequence.