Browsing by Author "Courdurier Bettancourt, Matías Alejandro"

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    A partially averaged system to model neuron responses to interferential current stimulation
    (SPRINGER HEIDELBERG, 2023) Cerpa Jeria, Eduardo Esteban; Courdurier Bettancourt, Matías Alejandro; Hernandez, Esteban; Medina, Leonel E.; Paduro Williamson, Esteban Andrés
    The interferential current (IFC) therapy is a noninvasive electrical neurostimulation technique intended to activate deep neurons using surface electrodes. In IFC, two independent kilohertz-frequency currents purportedly intersect where an interference field is generated. However, the effects of IFC on neurons within and outside the interference field are not completely understood, and it is unclear whether this technique can reliable activate deep target neurons without side effects. In recent years, realistic computational models of IFC have been introduced to quantify the effects of IFC on brain cells, but they are often complex and computationally costly. Here, we introduce a simplified model of IFC based on the FitzHugh-Nagumo (FHN) model of a neuron. By considering a modified averaging method, we obtain a non-autonomous approximated system, with explicit representation of relevant IFC parameters. For this approximated system we determine conditions under which it reliably approximates the complete FHN system under IFC stimulation, and we mathematically prove its ability to predict nonspiking states. In addition, we perform numerical simulations that show that the interference effect is observed only for a narrow set of IFC parameters and, in particular, for a beat frequency no higher than about 100 [Hz]. Our novel model tailored to the IFC technique contributes to the understanding of neurostimulation modalities using this type of signals, and can have implications in the design of noninvasive electrical stimulation therapies.
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    Boundary control of elliptic solutions to enforce local constraints
    (2013) Bal, Guillaume; Courdurier Bettancourt, Matías Alejandro
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    Construction of solutions for some localized nonlinear schrodinger equations
    (2019) Bourget, Olivier; Courdurier Bettancourt, Matías Alejandro; Fernández Jaña, Claudio Alonso
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    Lipschitz stability for backward heat equation with application to fluorescence microscopy
    (SIAM PUBLICATIONS, 2021) Arratia, Pablo; Courdurier Bettancourt, Matías Alejandro; Cueva, Evelyn; Osses, Axel; Palacios Farias, Benjamin Pablo
    In this work we study a Lipschitz stability result in the reconstruction of a compactly supported initial temperature for the heat equation in R-n, from measurements along a positive time interval and over an open set containing its support. We employ a nonconstructive method which ensures the existence of the stability constant, but it is not explicit in terms of the parameters of the problem. The main ingredients in our method are the compactness of support of the initial condition and the explicit dependency of solutions to the heat equation with respect to it. By means of Carleman estimates we obtain an analogous result for the case when the observation is made along an exterior region omega x (t, T), such that the unobserved part R-n\omega is bounded. In the latter setting, the method of Carleman estimates gives a general conditional logarithmic stability result when initial temperatures belong to a certain admissible set, without the assumption of compactness of support and allowing an explicit stability constant. Furthermore, we apply these results to deduce similar stability inequalities for the heat equation in R and with measurements available on a curve contained in Rx[0,infinity), leading to the derivation of stability estimates for an inverse problem arising in 2D fluorescence microscopy. In order to further understand this Lipschitz stability, in particular, the magnitude of its stability constant with respect to the parameters of the problem, a numerical reconstruction is presented based on the construction of a linear system for the inverse problem in fluorescence microscopy. We investigate the stability constant by analyzing the condition number of the corresponding matrix.
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    Mathematical modeling for 2D light-sheet fluorescence microscopy image reconstruction
    (2020) Cueva, E.; Courdurier Bettancourt, Matías Alejandro; Osses, A.; Castaneda, V.; Palacios, B.; Hartel, S.
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    PET Reconstruction With Non-Negativity Constraint in Projection Space: Optimization Through Hypo-Convergence
    (IEEE, 2020) Bousse, Alexandre; Courdurier Bettancourt, Matías Alejandro; Émond, Élise; Thielemans, Kris; Hutton, Brian F.; Irarrazaval Mena, Pablo; Visvikis, Dimitris
    Standard positron emission tomography (PET) reconstruction techniques are based on maximum-likelihood (ML) optimization methods, such as the maximum-likelihood expectation-maximization (MLEM) algorithm and its variations. Most methodologies rely on a positivity constraint on the activity distribution image. Although this constraint is meaningful from a physical point of view, it can be a source of bias for low-count/high-background PET, which can compromise accurate quantification. Existing methods that allow for negative values in the estimated image usually utilize a modified log-likelihood, and therefore break the data statistics. In this paper, we propose to incorporate the positivity constraint on the projections only, by approximating the (penalized) log-likelihood function by an adequate sequence of objective functions that are easily maximized without constraint. This sequence is constructed such that there is hypo-convergence (a type of convergence that allows the convergence of the maximizers under some conditions) to the original log-likelihood, hence allowing us to achieve maximization with positivity constraint on the projections using simple settings. A complete proof of convergence under weak assumptions is given. We provide results of experiments on simulated data where we compare our methodology with the alternative direction method of multipliers (ADMM) method, showing that our algorithm converges to a maximizer, which stays in the desired feasibility set, with faster convergence than ADMM. We also show that this approach reduces the bias, as compared with MLEM images, in necrotic tumors-which are characterized by cold regions surrounded by hot structures-while reconstructing similar activity values in hot regions.
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    Semi-local inversion of the geodesic ray transform in the hyperbolic plane
    (2013) Courdurier Bettancourt, Matías Alejandro; Sáez Trumper, Mariel Inés Aura
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    Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data
    (2015) Courdurier Bettancourt, Matías Alejandro; Monard, F.; Osses, A.; Romero, F.
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    The impact of high frequency based stability on the onset of action potentials in neuron models
    (2024) Cerpa Jeria, Eduardo Esteban; Corrales, Nathaly; Courdurier Bettancourt, Matías Alejandro; Medina, Leonel E.; Paduro, Esteban
    This paper studies the phenomenon of conduction block in model neurons using high-frequency biphasic stimulation (HFBS). The focus is investigating the triggering of undesired onset action potentials when the HFBS is turned on. The approach analyzes the transient behavior of an averaged system corresponding to the FitzHugh-Nagumo neuron model using Lyapunov and quasi-static methods. The first result provides a more comprehensive understanding of the onset activation through a mathematical proof of how to avoid it using a ramp in the amplitude of the oscillatory source. The second result tests the response of the blocked system to a piecewise linear stimulus, providing a quantitative description of how the HFBS strength translates into conduction block robustness. The results of this work can provide insights for the design of electrical neurostimulation therapies
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    Unsupervised reconstruction of accelerated cardiac cine MRI using neural fields
    (2025) Catalán, T.; Courdurier Bettancourt, Matías Alejandro; Osses, A.; Fotaki, A.; Botnar, René Michael; Sahli Costabal, F.; Prieto, C.
    Background: Cardiac cine MRI is the gold standard for cardiac functional assessment, but the inherently slow acquisition process creates the necessity of reconstruction approaches for accelerated undersampled acquisitions. Several regularization approaches that exploit spatial–temporal redundancy have been proposed to reconstruct undersampled cardiac cine MRI. More recently, methods based on supervised deep learning have been also proposed to further accelerate acquisition and reconstruction. However, these techniques rely on usually large dataset for training, which are not always available and might introduce biases. Methods: In this work we propose NF-cMRI, an unsupervised approach based on implicit neural field representations for cardiac cine MRI. We evaluate our method in in-vivo undersampled golden-angle radial multi-coil acquisitions for undersampling factors of 13x, 17x and 26x. Results: The proposed method achieves excellent scores in sharpness and robustness to artifacts and comparable or improved spatial–temporal depiction than state-of-the-art conventional and unsupervised deep learning reconstruction techniques. Conclusions: We have demonstrated NF-cMRI potential for cardiac cine MRI reconstruction with highly undersampled data.