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The neocortex, a complex system driving multi-region interactions, remains a core puzzle in neuroscience. Despite quantitative insights across brain scales, understanding the mechanisms underlying neural activities is challenging. Advances…
For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are…
Several approaches to cognition and intelligence research rely on statistics-based models testing, namely factor analysis. In the present work we exploit the emerging dynamical systems perspective putting the focus on the role of the…
An important field of research in functional neuroimaging is the discovery of integrated, distributed brain systems and networks, whose different regions need to work in unison for normal functioning. The EEG is a non-invasive technique…
Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis,…
The behavior of many complex systems is determined by a core of densely interconnected units. While many methods are available to identify the core of a network when connections between nodes are all of the same type, a principled approach…
The standard approach to representation learning on attributed graphs -- i.e., simultaneously reconstructing node attributes and graph structure -- is geometrically flawed, as it merges two potentially incompatible metric spaces. This…
Humans readily recognize objects from sparse line drawings, a capacity that appears early in development and persists across cultures, suggesting neural rather than purely learned origins. Yet the computational mechanism by which the brain…
Neuronal circuits of the cerebral cortex are the structural basis of mammalian cognition. The same qualitative components and connectivity motifs are repeated across functionally specialized cortical areas and mammalian species, suggesting…
Spin textures are ubiquitous in antiferromagnets, yet their consequences for altermagnets remain largely unexplored. We show that smooth spatial variations of the N\'eel order act on itinerant electrons as emergent gauge fields, producing…
Connectomics is a field of neuroscience that analyzes neuronal connections. A connectome is a complete map of a neuronal system, comprising all neuronal connections between its structures. The term "connectome" is close to the word "genome"…
The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of…
Grid cells in the entorhinal cortex, together with head direction, place, speed and border cells, are major contributors to the organization of spatial representations in the brain. In this work we introduce a novel theoretical and…
We present a continuous model for structural brain connectivity based on the Poisson point process. The model treats each streamline curve in a tractography as an observed event in connectome space, here a product space of cortical white…
This dissertation explores applications of discrete geometry in mathematical neuroscience. We begin with convex neural codes, which model the activity of hippocampal place cells and other neurons with convex receptive fields. In Chapter 4,…
On the largest scales, the universe appears to be almost homogeneous and isotropic, adhering to the cosmological principle. In contrast, on smaller scales inhomogeneities and anisotropy become increasingly prominent, reflecting the origin,…
Due to the fact that the numbers of annually published papers have witnessed a linear growth in some citation networks, a geometric model is thus proposed to predict some statistical features of those networks, in which the academic…
Sparse autoencoders have recently produced dictionaries of high-dimensional vectors corresponding to the universe of concepts represented by large language models. We find that this concept universe has interesting structure at three…
The spectral properties of traditional (dyadic) graphs, where an edge connects exactly two vertices, are widely studied in different applications. These spectral properties are closely connected to the structural properties of dyadic…
Recent advancements in measurement techniques have resulted in an increasing amount of data on neural activities recorded in parallel, revealing largely heterogeneous correlation patterns across neurons. Yet, the mechanistic origin of this…