Abstract

Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin–Huxley model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type.In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in mathbb{R}^{4}. We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the model.

Highlights

  • In a wide variety of models, including of semiconductor lasers [2, 3], chemical reactions [4,5,6] and neurons [7,8,9], one finds large differences in time scales

  • Geometric singular perturbation theory (GSPT) exploits the separation of different time scales in order to explain the complex dynamics of slow–fast systems

  • Rubin and Wechselberger [1] applied a center manifold reduction to (32) by setting m = m∞(v) to eliminate m and obtain a three-dimensional reduced model. They applied methods from geometric singular perturbation theory (GSPT) to prove the existence of relaxation oscillations, mixed-mode oscillations (MMOs) and canard orbits in the three-dimensional reduced model

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Summary

Introduction

In a wide variety of models, including of semiconductor lasers [2, 3], chemical reactions [4,5,6] and neurons [7,8,9], one finds large differences in time scales. The first is an extended version of the normal form of a folded node [34, 47], where a fast variable with a stable fast direction is added In this system, we implement and test our 2PBVP-based approach for computing saddle slow manifolds, and their stable and unstable fast manifolds, as well as associated canard orbits. This is followed by a general method for computing two-dimensional saddle slow manifolds, as well as subsets of their three-dimensional stable and unstable manifolds in systems with two fast and two slow variables.

Background on Geometric Singular Perturbation Theory
Computing Two-Dimensional Saddle Slow Manifolds
Two-Point Boundary-Value Problem Setup for Four-Dimensional Slow–Fast Systems
General Approach for Computing Two-Dimensional Slow Manifolds in R4
Computing Attracting and Repelling Slow Manifolds in R4
Computing Saddle Slow Manifolds in R4
Extended Normal Form of a Folded Node
Slow Manifolds of the Extended Normal Form
Computing the Saddle Slow Manifold of the Normal Form
The Overall Geometry of Slow Manifolds
Computing Stable and Unstable Manifolds of Saddle Slow Manifolds
Stable and Unstable Manifolds of a Trajectory on Sεs
Slices of Stable and Unstable Manifolds of Sεs
General Approach for Computing Canard Orbits in R4
Detecting and Continuing Canards of the Normal Form
Continuation of Canard Orbits
Slow Manifolds and Canard Orbits in the Full Hodgkin–Huxley Model
Bifurcation Diagram of the Four-Dimensional Hodgkin–Huxley Model
Slow–Fast Analysis of the Four-Dimensional Hodgkin–Huxley Model
Computing the Attracting Slow Manifold
Computing the Saddle Slow Manifold
Interaction Between Attracting and Saddle Slow Manifolds
Computing Canard Orbits of the Hodgkin–Huxley Model
Ribbons of the Attracting Slow Manifolds
Conclusions and Directions for Future Work
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