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Related Experiment Video

Updated: Dec 24, 2025

Optogenetic Entrainment of Hippocampal Theta Oscillations in Behaving Mice

Optogenetic Entrainment of Hippocampal Theta Oscillations in Behaving Mice

Published on: June 29, 2018

Canard-induced complex oscillations in an excitatory network.

Elif Köksal Ersöz^1,2,3, Mathieu Desroches^4,5, Antoni Guillamon⁶

¹MathNeuro Team, Inria Sophia Antipolis Méditerranée, Valbonne, France. elif.koksal@inria.fr.

Journal of Mathematical Biology

|April 9, 2020

Summary

Mathematical models reveal how canard structures define boundaries between complex neural oscillations. This study identifies these boundaries in developing spinal cord models, uncovering mixed-mode bursting oscillations (MMBOs) and their role in neural activity.

Keywords:

Canard solutions Excitability Mixed-mode bursting oscillations Multiple timescale systems Rate models

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Last Updated: Dec 24, 2025

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Published on: June 29, 2018

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Area of Science:

Neuroscience
Computational Neuroscience
Mathematical Biology

Background:

Complex oscillatory activities, including bursting and spiking, are crucial in neuroscience.
Mathematical modeling with multiple time scales is essential for understanding neural dynamics.
Canard structures are known to define boundaries between different dynamical regimes in such models.

Purpose of the Study:

To identify and characterize boundaries between dynamical regimes in multiple-timescale firing rate models of the developing spinal cord.
To investigate the role of canard structures in generating complex oscillations, specifically mixed-mode bursting oscillations (MMBOs).

Main Methods:

Application of geometric singular perturbation theory to analyze multi-dimensional firing rate models (3D and 4D).
Modeling of fast recurrent excitatory networks with synaptic depression and slow variables for firing threshold and synaptic depression.
Identification of canard-mediated transitions and analysis of isolas in parameter space.

Main Results:

Demonstrated canard-induced bursting and mixed-mode oscillations in 3D models.
Revealed canard-mediated slow passage creating MMBOs in a 4D model.
Unveiled complex isolas where MMBOs exist, exhibiting explosive transitions between sub-threshold and bursting regimes.

Conclusions:

Canard structures play a critical role in shaping complex dynamics and transitions in neural models.
MMBOs and their associated transitions are relevant to understanding neural activity patterns, including silent phases and subthreshold fluctuations.
The mathematical framework is applicable to diverse excitable systems with multiple time scales.