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Analytical and numerical studies of thalamo-cortical neural populationmodels during general anesthesia

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Meysam Hashemi at Aix-Marseille University
Meysam Hashemi
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Bien que l’anesthésie générale est un outil indispensable dans la chirurgie médicale d’aujourd’hui,ses mécanismes sous-jacents précis sont encore inconnus. Au cours de la sédaction induite parle propofol les actions anesthésiques à l’échelle microscopique du neurone isolé conduisent àdes changements spécifiques à l’échelle macroscopique qui sont observables comme les signauxélectroencép-halogrammes (EEG). Pour une concentration faible en propofol, ces changementscaractéristiques comprennent une augmentation de l’activité dans les bandes de fréquence delta (0.5-4Hz) et alpha (8-13 Hz) dans la région frontal, une l’activité augmentée de delta et une l’activité diminuéede alpha dans la région occipital. Dans cette thèse, nous utilisons des modèles de populations neuronalesthalamo-corticales basés sur des données expérimentales. Les effets de propofol sur lessynapses et sur les récepteurs extra-synaptiques GABAergiques situés dans le cortex et le thalamussont modélisés afin de comprendre les mécanismes sous-jacents aux changements observésdans certains puissance de l’EEG-spectrale. Il est démontré que les modèles reproduisent bien lesspectrales caractéristiques observées expérimentalement. Une des conclusions principales de cetravail est que l’origine des delta rythmes est fondamentalement différente de celle des alpha rythmes.Nos résultats indiquent qu’en fonction des valeurs moyennes des potentiels de l’état du systèmeau repos, une augmentation ou une diminution des fonctions de gain thalamo-corticale résulte respectivementen une augmentation ou une diminution de alpha puissance. En revanche, l’évolutionde la delta puissance est plutôt indépendant de l’état du système au repos; la amélioration dela puissance spectrale de delta bande résulte de l’inhibition GABAergique synaptique ou extrasynaptiquepour les fonctions de gain non linéaire à la fois croissante et décroissante. De plus,nous cherchons à identifier les paramètres d’un modèle de thalamo-corticale en ajustant le spectrede puissance de modèle pour les enregistrements EEG. Pour ce faire, nous considérons la tâchede l’estimation des paramètres dans les modèles qui sont décrits par un ensemble d’équationsdifférentielles ordinaires ou bien stochastiques avec retard. Deux études de cas portant sur desdonnées pseudo-expérimentale bruyants sont d’abord effectuées pour comparer les performancesdes différentes méthodes d’optimisation. Les résultats de cette élaboration montrent que laméthode utilisée dans cette étude est capable d’estimer avec précision les paramètres indépendantsdu modèle et cela nous permet d’éviter les coûts de calcul des intégrations numériques.En considérant l’ensemble, les conclusions de cette thèse apportent de nouveaux éclairages surles mécanismes responsables des changements spécifiques qui sont observées pendant la sédationpropofol-induite dans les modèles de EEG.
Time evolution of membrane potential of cortical pyramidal neurons V E governed by Eq. (2.31) simulated with the parameter set displayed in Table 2.1. The numerical simulations are started with different initial values for the system variables. The solid red lines indicate the stable resting states whereas the dashed black line shows the unstable solution. The gray and blues trajectories converge to the larger and smaller stable resting state, respectively, since they are initialized in the basin of attraction of the respective solution. The green trajectories started on the unstable resting state head toward one of the stable solutions.
Time evolution of membrane potential of cortical pyramidal neurons V E governed by Eq. (2.31) simulated with the parameter set displayed in Table 2.1. The numerical simulations are started with different initial values for the system variables. The solid red lines indicate the stable resting states whereas the dashed black line shows the unstable solution. The gray and blues trajectories converge to the larger and smaller stable resting state, respectively, since they are initialized in the basin of attraction of the respective solution. The green trajectories started on the unstable resting state head toward one of the stable solutions.
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Power spectra of φ E computed from linearization about different resting states (c.f. Eq. (2.44)). (A) The spectral power density obtained by linearizing system equations around the smaller resting state reproduces specific human EEG features. (B) No peak can be observed when the spectrum is computed by linearization about the larger resting state.
Power spectra of φ E computed from linearization about different resting states (c.f. Eq. (2.44)). (A) The spectral power density obtained by linearizing system equations around the smaller resting state reproduces specific human EEG features. (B) No peak can be observed when the spectrum is computed by linearization about the larger resting state.
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Characteristic roots of a linear scalar DDE given by ˙ y(t) = ay(t) + by(t − τ ), where a = 0.5, b = −1, and τ = 1. Since all the roots have negative real parts, the system is stable. The characteristic roots obtained by using Lambert function given by λ = 1 τ W k (bτ e −aτ ) + a for k ∈ −50, ..., 0, ..., 50 are represented as open red circles, whereas the blue asterisks show the roots simulated by the spectral discretization of infinitesimal generator based on the Chebyshev nodes.
Characteristic roots of a linear scalar DDE given by ˙ y(t) = ay(t) + by(t − τ ), where a = 0.5, b = −1, and τ = 1. Since all the roots have negative real parts, the system is stable. The characteristic roots obtained by using Lambert function given by λ = 1 τ W k (bτ e −aτ ) + a for k ∈ −50, ..., 0, ..., 50 are represented as open red circles, whereas the blue asterisks show the roots simulated by the spectral discretization of infinitesimal generator based on the Chebyshev nodes.
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The stability region for a linear scalar DDE given by ˙ y(t) = ay(t) + by(t − τ ), where τ = 1. The shaded region presents the exact condition driven by Hayes theorem [172], which depends on the delay value (c.f. Theorem 2), whereas the hatched area shows the approximated condition given by the cone a+ | b |< 0, which is independent of the delay value (c.f. Theorem 1).
The stability region for a linear scalar DDE given by ˙ y(t) = ay(t) + by(t − τ ), where τ = 1. The shaded region presents the exact condition driven by Hayes theorem [172], which depends on the delay value (c.f. Theorem 2), whereas the hatched area shows the approximated condition given by the cone a+ | b |< 0, which is independent of the delay value (c.f. Theorem 1).
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Dynamics of damped harmonic oscillator driven by a random stochastic force. The panel (A) shows the theoretical and numerical power spectra in blue and red, respectively, for different values of damping coefficients; γ = 0, 0.5, 2, 15 Hz. The corresponding simulated time series are presented in panel (B). For smaller values of damping coefficients, the power spectral peak is sharper whereas for large value of γ, there is no peak in the computed spectrum. The intrinsic angular frequency of the oscillator is chosen as ω 0 = 2π, thus f 0 = 1Hz.
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Dynamics of damped harmonic oscillator driven by a random stochastic force. The panel (A) shows the theoretical and numerical power spectra in blue and red, respectively, for different values of damping coefficients; γ = 0, 0.5, 2, 15 Hz. The corresponding simulated time series are presented in panel (B). For smaller values of damping coefficients, the power spectral peak is sharper whereas for large value of γ, there is no peak in the computed spectrum. The intrinsic angular frequency of the oscillator is chosen as ω 0 = 2π, thus f 0 = 1Hz.
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... Responses to stochastic forcing takes a form similar to the response to forcing of a single frequency as shown in Eq. (9) (Hashemi 2016;Masoliver and Porrà 1993). We demonstrate the damping effect on periods of the harmonic oscillator in the presence of Gaussian white noise forcing in Fig. 10. ...
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