Browsing by Author "Courdurier, Matías"

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    Approximation and stability results for the parabolic FitzHugh-Nagumo system with combined rapidly oscillating sources
    (2025) Cerpa Jeria, Eduardo Esteban; Courdurier, Matías; Hernandez, Esteban; Medina, Leonel E.; Paduro Williamson, Esteban Andres
    The use of high-frequency currents in neurostimulation has received increased attention in recent years due to its varied effects on tissues and cells. Neurons are commonly modeled as nonlinear systems, and questions such as stability can thus be addressed with well-known averaging methods. A recent strategy called interferential currents uses electrodes delivering sinusoidal signals of slightly different frequencies, and thus classical averaging (well-adapted to deal with a single frequency) cannot be directly applied. In this paper, we consider the one-dimensional FitzHugh-Nagumo system under the effects of a source composed of two terms that are sinusoidal in time and quadratically decaying in space. To study this setting we develop a new averaging strategy to prove that, when the frequencies involved are sufficiently high, the full system can be approximated by an explicit highly-oscillatory term plus the solution of a simpler -- albeit non-autonomous -- system. This decomposition can be seen as a stability result around a varying trajectory. One of the main novelties of the proofs presented here is an extension of the contracting rectangles method to the case of parabolic equations with space and time-depending coefficients.
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    Mathematical analysis and applications of neural networks, with applications to image reconstruction
    (2025) Molina Mejía, Juan José; Courdurier, Matías; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    This thesis explores two fundamental aspects of neural networks: their frequency learning behavior and their application to quantitative Magnetic Resonance Imaging (MRI) reconstruction. The first part investigates the phenomenon of frequency bias, the empirical observation that neural networks tend to learn low-frequency components of a target function more rapidly than high-frequency ones. To provide a rigorous understanding of this behavior, we develop a theoretical framework based on Fourier analysis. Specifically, we derive a partial differential equation that governs the evolution of the error spectrum during training in the Neural Tangent Kernel regime, focusing on two-layer neural networks. Our analysis centers on Fourier Feature networks, a class of architectures where the first layer applies sine and cosine activations using pre-defined frequency distributions. We demonstrate that the network's initialization, particularly the initial density distribution of first-layer weights, plays a crucial role in shaping the frequency learning dynamics. This insight provides a principled way to control or even eliminate frequency bias during training. Theoretical predictions are validated through numerical experiments, which further illustrate the impact of initialization on the inductive biases of neural networks.The second part of the thesis applies neural network techniques to the reconstruction of quantitative MRI data. Quantitative MRI enables the estimation of tissue-specific parameters (e.g., T1, T2, and T2*) that are vital for clinical diagnosis and disease monitoring. However, these methods typically require long acquisition times, which are often mitigated through aggressive undersampling of k-space data. Undersampling, in turn, introduces reconstruction artifacts that must be addressed through regularization. To this end, we propose CConnect, a novel iterative reconstruction method that incorporates convolutional neural networks into the regularization term. CConnect connects multiple CNNs through a shared latent space, allowing the model to capture common structures across different image contrasts. This design enables the effective suppression of aliasing artifacts and improves image quality, even in highly undersampled scenarios. We evaluate CConnect on in-vivo brain T2*-weighted MRI data, demonstrating its superiority over classical low-rank and total variation methods, as well as standard deep learning baselines.