scholarly journals Canard solutions in neural mass models: consequences on critical regimes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Elif Köksal Ersöz ◽  
Fabrice Wendling
Keyword(s):  
Temporal Dynamics ◽  
Spatial Scales ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Critical Transitions ◽  
Intracerebral Eeg ◽  
Neural Mass ◽  
Mesoscopic Level ◽  
Eeg Recordings ◽  
Physiological Background

AbstractMathematical 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.

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

Firing patterns of the CA1 pyramidal neuron with geometric singular perturbation: a model study

2020 ◽  
Vol 34 (32) ◽  
pp. 2050316
Author(s):  
Yaru Liu ◽  
Shenquan Liu
Keyword(s):  
Singular Perturbation ◽  
Pyramidal Neurons ◽  
Sodium Current ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
External Stimulation ◽  
Model Study ◽  
Geometric Singular Perturbation ◽  
Current Activation ◽  
Ca1 Pyramidal Neuron

An investigation of CA1 pyramidal model is an important issue for applications, which is intimately related to the composition of ions in the extracellular environment and external stimulation. In this paper, it is demonstrated that the effects of different electrophysiological parameters such as muscarinic-sensitive potassium current activation variable and sustained sodium current inactivation variable on the firing sequence of model by numerical simulations. Furthermore, the paper also discusses that the temperature affects the firing of the CA1 model from direct current (DC) and alternating current (AC) stimuli. It is found that the model exhibits excellent spiking and bursting patterns, even chaotic patterns occur. Meanwhile, generalized mixed oscillations emerge in the model. Additionally, the firing modes are depicted by providing the response curve (RC), inter-spike interval curve (ISI), phase diagram curve (PDC) and the number of spikes per burst curve (NC). Mathematically, the paper elaborates the results which are presented to obtain two lower dimensional subsystems, which govern the fast and slow dynamics for giving insight into the dynamic behaviors of the full 5D system based on the geometric singular perturbation theory (GSPT). Particularly, we analyse the phase diagrams of the CA1 model to understand the properties better. The present results may contribute to further understand the information processing of the CA1 pyramidal neurons.

scholarly journals High similarity between EEG from subcutaneous and proximate scalp electrodes in patients with temporal lobe epilepsy

2018 ◽  
Vol 120 (3) ◽  
pp. 1451-1460 ◽  
Author(s):  
Sigge Weisdorf ◽  
Sirin W. Gangstad ◽  
Jonas Duun-Henriksen ◽  
Karina S. S. Mosholt ◽  
Troels W. Kjær
Keyword(s):  
Temporal Lobe Epilepsy ◽  
Temporal Lobe ◽  
Temporal Dynamics ◽  
Real Life ◽  
High Similarity ◽  
Time Frequency ◽  
Scalp Eeg ◽  
Eeg Recordings ◽  
The Impact

Subcutaneous recording using electroencephalography (EEG) has the potential to enable ultra-long-term epilepsy monitoring in real-life conditions because it allows the patient increased mobility and discreteness. This study is the first to compare physiological and epileptiform EEG signals from subcutaneous and scalp EEG recordings in epilepsy patients. Four patients with probable or definite temporal lobe epilepsy were monitored with simultaneous scalp and subcutaneous EEG recordings. EEG recordings were compared by correlation and time-frequency analysis across an array of clinically relevant waveforms and patterns. We found high similarity between the subcutaneous EEG channels and nearby temporal scalp channels for most investigated electroencephalographic events. In particular, the temporal dynamics of one typical temporal lobe seizure in one patient were similar in scalp and subcutaneous recordings in regard to frequency distribution and morphology. Signal similarity is strongly related to the distance between the subcutaneous and scalp electrodes. On the basis of these limited data, we conclude that subcutaneous EEG recordings are very similar to scalp recordings in both time and time-frequency domains, if the distance between them is small. As many electroencephalographic events are local/regional, the positioning of the subcutaneous electrodes should be considered carefully to reflect the relevant clinical question. The impact of implantation depth of the subcutaneous electrode on recording quality should be investigated further. NEW & NOTEWORTHY This study is the first publication comparing the detection of clinically relevant, pathological EEG features from a subcutaneous recording system designed for out-patient ultra-long-term use to gold standard scalp EEG recordings. Our study shows that subcutaneous channels are very similar to comparable scalp channels, but also point out some issues yet to be resolved.

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

2020 ◽  
pp. 2150003
Author(s):  
Ali Atabaigi
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Limit Cycles ◽  
Heteroclinic Orbits ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Generalist Predator ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Geometric Singular Perturbation

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

Temporal Dynamics of Binocular Disparity Processing in the Central Visual Pathway

2004 ◽  
Vol 91 (4) ◽  
pp. 1782-1793 ◽  
Author(s):  
Michael D. Menz ◽  
Ralph D. Freeman
Keyword(s):  
Striate Cortex ◽  
Temporal Dynamics ◽  
Spatial Scales ◽  
Population Coding ◽  
Dynamic Changes ◽  
Simple Cells ◽  
Specific Feedback ◽  
Simple And Complex Cells ◽  
Dynamic Nonlinearity ◽  
Coarse To Fine

To solve the stereo correspondence problem (i.e., find the matching features of a visual scene in both eyes), it is advantageous to combine information across spatial scales. The details of how this is accomplished are not clear. Psychophysical studies and mathematical models have suggested various types of interactions across spatial scale, including coarse to fine, fine to coarse, averaging, and population coding. In this study, we investigate dynamic changes in disparity tuning of simple and complex cells in the cat's striate cortex over a short time span. We find that disparity frequency increases and disparity ranges decrease while optimal disparity remains constant, and this conforms to a coarse-to-fine mechanism. We explore the origin of this mechanism by examining the frequency and size dynamics exhibited by binocular simple cells and neurons in the lateral geniculate nucleus (LGN). The results suggest a strong role for a feed-forward mechanism, which could originate in the retina. However, we find that the dynamic changes seen in the disparity range of simple cells cannot be predicted from their left and right eye monocular receptive field (RF) size changes. This discrepancy suggests the possibility of a dynamic nonlinearity or disparity specific feedback that alters tuning or a combination of both mechanisms.

scholarly journals A.06 Assessing inter-rater reliability in localizing sleep-related hypermotor seizures: a video-based survey

2019 ◽  
Vol 46 (s1) ◽  
pp. S9
Author(s):  
PM Lobbezoo ◽  
L Nobili ◽  
S Gibbs
Keyword(s):  
Focal Epilepsy ◽  
Small Difference ◽  
Training Session ◽  
Sleep Medicine ◽  
Kappa Statistics ◽  
Clinical Use ◽  
Rater Reliability ◽  
Intracerebral Eeg ◽  
Hypermotor Seizures ◽  
Eeg Recordings

Background: Sleep-related hypermotor epilepsy (SHE) is a focal epilepsy characterized by abrupt sleep-related hypermotor seizures (SRHS) with complex semiology. Although difficult to localize within the frontal lobe recent studies using intracerebral EEG recordings have suggested the existence of four distinct semiology patterns (SP) organized in a rostro-caudal manner. It remains unclear however if these SP are clinically useful. Methods: We aimed to estimate the inter-rater reliability (IR) of classifying SP in SHE amongst epilepsy and sleep medicine experts. Following a short training session, ten experts were asked to review and classify 40 videos of SRHS in patients with confirmed SHE. IR was calculated using Kappa statistics. Results: SP1 and SP4, who are at the opposite ends of the SHE semiology spectrum, had substantial IR (0.82 and 0.67, respectively). Meanwhile, SP2 and SP3 showed fair agreement (0.25 and 0.35, respectively) and represented the major source of variance, with a small difference favouring epilepsy experts. Conclusions: Amongst epilepsy and sleep medicine experts, IR of classifying SRHS into four SP was only mildly satisfactory. SP1 and SP4 were shown to be easily recognizable while SP2 and SP3 were frequently confounded. Improvements in SP recognition are needed before widespread clinical use.

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