## Abstract

The transient potassium A-current is present in almost all neurons and plays an essential role in determining the timing and frequency of action potential generation. We use a three-variable mathematical model to examine the role of the A-current in a rhythmic inhibitory network, as is common in central pattern generation. We focus on a feed-forward architecture consisting of an oscillator neuron inhibiting a follower neuron. We use separation of time scales to demonstrate that the trajectory of the follower neuron within each cycle can be tracked by analyzing the dynamics on a two-dimensional slow manifold that is determined by the two slow model variables: the recovery variable and the inactivation of the A-current. The steady-state trajectory, however, requires tracking the slow variables across multiple cycles. We show that tracking the slow variables, under simplifying assumptions, leads to a one-dimensional map of the unit interval with at most a single discontinuity depending on gA, the maximal conductance of the A-current, or other model parameters. We demonstrate that, as the value of gA is varied, the trajectory of the follower neuron goes through a set of bifurcations to produce n:m periodic solutions, where the follower neuron becomes active m times for each n cycles of the oscillator. Using a generalized Pascal triangle, each n:m trajectory can be constructed as a combination of solutions from a higher level of the triangle.

Original language | English (US) |
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Pages (from-to) | 1564-1590 |

Number of pages | 27 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 8 |

Issue number | 4 |

DOIs | |

State | Published - 2009 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Modeling and Simulation

## Keywords

- 1-d map
- Bifurcation
- Oscillation
- Periodic orbit
- Phase plane
- Slow manifold