Dynamics of ramping bursts in a respiratory neuron model

TL;DR

This paper introduces a modified mathematical model of respiratory neurons that captures the gradual ramping activity leading to bursts, highlighting the role of extracellular potassium dynamics and bifurcation analysis.

Contribution

It demonstrates that incorporating extracellular potassium dynamics into existing models enables the simulation of ramping bursts and expands the range of burst behaviors.

Findings

Model reproduces pre-inspiratory ramping activity

Burst termination occurs at a homoclinic bifurcation

Enhanced dynamic range of burst parameters compared to previous models

Abstract

Intensive computational and theoretical work has led to the development of mutliple mathematical models for bursting in respiratory neurons in the pre-B\"otzinger Complex (pre-B\"otC) of the mammalian brainstem. Nonetheless, these previous models have not captured the pre-inspiratory ramping aspects of these neurons' activity patterns, in which relatively slow tonic spiking gradually progresses to faster spiking and a full blown burst, with a corresponding gradual development of an underlying plateau potential. In this work, we show that the incorporation of the dynamics of the extracellular potassium ion concentration into an existing model for pre-B\"otC neuron bursting, along with some parameter updates, suffices to induce this ramping behavior. Using fast-slow decomposition, we show that this activity can be considered as a form of parabolic bursting, but with burst termination at a homoclinic bifurcation rather than as a SNIC bifurcation. We also investigate the parameter-dependence of these solutions and show that the proposed model yields a greater dynamic range of burst frequencies, durations, and duty cycles than those produced by other models in the literature.