Saddle Slow Manifolds and Canard Orbits in R 4 $\mathbb{R}^{4}$ and Application to the Full Hodgkin–Huxley Model

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 . 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.

Canard analysis reveals why a large Ca2+ window current promotes early afterdepolarizations in cardiac myocytes

“…Parameter values for these behaviors are marked with ⋄ in Fig 5 labeled 7a, 7b, and 7c. Each panel shows the critical manifold and its stability properties along with the first three maximal canards ( γ 0 , magenta; γ 1 , cyan; γ 2 , orange), computed using numerical continuation and bifurcation software AUTO [ 40 ] and methods developed in [ 41 ] which are described for this system in [ 21 ]. Also superimposed are portions of the solution segment of the full system (Γ, black) following an impulse-producing stimulus.…”

Canard Mechanism and Rhythm Dynamics of Neuron Models

“…A recent review summarizes the pioneering work of Rinzel and Izhikevich on bursting patterns' classification of excitatory systems and proposes a canard mechanism for folded-node singularities, explaining that conductance-based MMOs consist of subthreshold and superthreshold oscillations [100]. Canards have been the subject of active study since it was proposed in the 1980s, and they have been used to understand the firing patterns of neurons [101][102][103][104]. Canards are closely related to synchronization, spiking-adding, and complex oscillations of excitation networks [105][106][107].…”