Canard solutions in neural mass models: consequences on critical regimes

Fabrice Wendling,Elif Köksal Ersöz

doi:10.1186/s13408-021-00109-z

Fabrice Wendling, Elif Köksal Ersöz

Open Access

https://doi.org/10.1186/s13408-021-00109-z

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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.

Highlights

Brain dynamics emerges from neural entities interacting at different levels, from single neurons to large-scale neural networks
In higher dimensional multiple time-scale systems with at least two slow variables, the folded-singularities are generic, they are robust to small parameter perturbations, and canard solutions associated with folded singularities connect stable and unstable branches of a folded critical manifold [36, 53, 59,60,61]
We both investigated canard transitions present in a neurophysiologically-relevant Neural Mass Model (NMM) and analyzed their consequences in terms of subsequent signatures in Local Field Potential (LFP). In this three-time-scale model, the canard transitions occur in the 6-dimensional two-timescale reduced system of slow and super-slow variables. They are associated with degenerate FSN Folded Saddle-Node type II (II) singularities and singular Hopf bifurcations

Summary

Introduction

Brain dynamics emerges from neural entities interacting at different levels, from single neurons to large-scale neural networks. In higher dimensional multiple time-scale systems with at least two slow variables, the folded-singularities are generic, they are robust to small parameter perturbations, and canard solutions associated with folded singularities connect stable and unstable branches of a folded critical manifold [36, 53, 59,60,61]. Degenerate FSN II and singular Hopf bifurcations can lead to canard solutions governing the critical transitions in (8a)–(8f) ( in (2a)–(2h)).

Results

Conclusion