[Paper Review] A novel canard-based mechanism for mixed-mode oscillations in a neuronal model

Jozsi Z. Jalics, Martin Krupa+1 more|ArXiv.org|Apr 5, 2008

stochastic dynamics and bifurcation51 references3 citations

TL;DR

This paper proposes a novel canard-based mechanism for mixed-mode oscillations (MMOs) in a reduced 3D neuronal model derived from a biophysical neuron model of the entorhinal cortex. By applying geometric singular perturbation theory and center manifold reduction, the authors identify a singular primary canard solution that acts as a dynamic boundary between MMOs and spiking, providing a rigorous explanation for the transition between these regimes in a system with three timescales.

ABSTRACT

We analyze a biophysical model of a neuron from the entorhinal cortex that includes persistent sodium and slow potassium as non-standard currents using reduction of dimension and dynamical systems techniques to determine the mechanisms for the generation of mixed-mode oscillations. We have found that the standard spiking currents (sodium and potassium) play a critical role in the analysis of the interspike interval. To study the mixed-mode oscillations, the six dimensional model has been reduced to a three dimensional model for the subthreshold regime. Additional transformations and a truncation have led to a simplified model system with three timescales that retains many properties of the original equations, and we employ this system to elucidate the underlying structure and explain a novel mechanism for the generation of mixed-mode oscillations based on the canard phenomenon. In particular, we prove the existence of a special solution, a singular primary canard, that serves as a transition between mixed-mode oscillations and spiking in the singular limit by employing appropriate rescalings, center manifold reductions, and energy arguments. Additionally, we conjecture that the singular canard solution is the limit of a family of canards and provide numerical evidence for the conjecture.

Motivation & Objective

• To understand the dynamical origin of mixed-mode oscillations (MMOs) in a biophysical neuron model of the entorhinal cortex with persistent sodium and slow potassium currents.
• To reduce the original six-dimensional system to a three-dimensional model that retains key dynamical features, particularly the three timescales essential for MMOs.
• To identify and rigorously analyze a singular primary canard solution that serves as a transition state between MMOs and spiking behavior.
• To provide numerical and analytical evidence that the singular canard is the limit of a family of actual canard solutions, supporting its role as a bifurcation boundary.

Proposed method

• Apply dimension reduction techniques to simplify the original 6D biophysical neuron model into a 3D system valid in the subthreshold regime.
• Employ geometric singular perturbation theory and center manifold reduction to analyze the system in the singular limit, focusing on slow manifolds and fold regions.
• Use appropriate rescalings and energy arguments to prove the existence of a singular primary canard solution connecting stable and unstable slow manifolds.
• Perform backward integration from initial conditions near the folded singularity to approximate the unstable manifold of the primary canard.
• Construct numerical approximations of the primary canard and track its behavior across varying parameters, particularly near the MMO-to-spiking transition.
• Utilize bifurcation diagrams and trajectory projections to validate the presence of MMOs and the role of the canard in mediating transitions.

Experimental results

Research questions

• RQ1 What dynamical mechanism underlies the emergence of mixed-mode oscillations in the entorhinal cortex neuron model with persistent sodium and slow potassium currents?
• RQ2 How does the presence of a singular primary canard solution mediate the transition between mixed-mode oscillations and regular spiking in the reduced 3D system?
• RQ3 What is the role of the three timescales—fast, intermediate, and slow—in enabling the canard phenomenon and MMO generation?
• RQ4 Is the singular primary canard solution the limit of a continuous family of actual canard solutions, and what numerical evidence supports this conjecture?
• RQ5 How do the reduced 3D model and its bifurcation structure compare to the full 6D system in capturing MMO dynamics?

Key findings

• The reduced 3D model successfully captures the mixed-mode oscillation dynamics observed in the original 6D neuronal model, particularly near the MMO-to-spiking transition.
• A singular primary canard solution was rigorously proven to exist in the singular limit, acting as a separatrix between MMOs and spiking regimes.
• Numerical evidence supports the conjecture that the singular primary canard is the limit of a one-parameter family of actual canard solutions, with the transition occurring at ε ≈ 0.2764436178.
• The primary canard is approximated by connecting backward trajectories from the folded singularity to the stable slow manifold, confirming its role as a dynamic boundary.
• Bifurcation diagrams for the 3D system show a clear transition from MMOs to pure spiking as the applied current I_app increases beyond approximately 17.1.
• The mechanism is robust and generalizable: the canard-based MMO mechanism may underlie similar dynamics in other systems with multiple timescales and folded singularities.