Abstract Mathematical models at multiple temporal and spatial scales can unveil the fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level offered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with different synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-specific oscillations in simulated local field potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets.

Elif Köksal Ersöz

Fabrice Wendling

The Journal of Mathematical Neuroscience

数学神经科学杂志 (JMN) 发表关于神经科学所有领域的数学建模和分析的研究文章，即神经系统及其功能障碍的研究。重点是使用数学作为主要工具来阐明负责在所有相关尺度（从分子世界到认知世界）的神经科学中实验观察到的行为的基本机制。其目的是发表使用先进数学技术来阐明这些问题的作品。它发表完整的原创论文、快速交流和评论文章。特别鼓励结合令人信服的数值实验支持的理论结果的论文。也欢迎介绍和帮助开发可能与未来神经系统特别是人类大脑研究相关的新数学理论的论文.

The Journal of Mathematical Neuroscience (JMN) is dedicated to publishing research articles on mathematical modeling and analysis across all fields of neuroscience, focusing on the study of the nervous system and its disorders. The journal emphasizes the use of mathematics as a primary tool to elucidate the fundamental mechanisms behind experimentally observed behaviors in neuroscience, spanning from the molecular to the cognitive level. We aim to publish original research papers, rapid communications, and review articles that employ advanced mathematical techniques to address these issues. We particularly encourage submissions that present theoretical results supported by compelling numerical experiments. Additionally, we welcome papers that introduce and help develop new mathematical theories that may be relevant to future research on the nervous system, especially the human brain.

《Journal of Mathematical Neuroscience》是一本专注于神经科学各个领域的数学建模与分析研究的期刊。我们致力于使用数学工具揭示神经系统及其功能障碍的基本机制，涵盖从分子层面到认知层面的实验观察。期刊欢迎发表使用先进数学技术的原创研究论文、快速交流和评论文章，特别鼓励那些结合有力数值实验支持的理论成果的研究。同时，我们也欢迎介绍和发展与未来神经系统研究，尤其是人类大脑相关的新数学理论的论文。

Category	Quartile
医学	4
医学, 数学与计算生物学	3
医学, 数学跨学科应用	3
医学, 神经科学	4

Canard solutions in neural mass models: consequences on critical regimes

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