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Abstract Dynamical formulations of optimal transport (OT) frame the task of comparing distributions as a variational problem which searches for a path between distributions minimizing a kinetic energy functional. In applications, it is frequently natural to require paths of distributions to satisfy additional conditions. Inspired by this, we introduce a model for dynamical OT which incorporates constraints on the space of admissible paths into the framework of unbalanced OT, where the source and target measures are allowed to have a different total mass. Our main results establish, for several general families of constraints, the existence of solutions to the variational problem which defines this path constrained unbalanced OT framework. These results are primarily concerned with distributions defined on an Euclidean space, but we extend them to distributions defined over parallelizable Riemannian manifolds as well. We also consider metric properties of our framework, showing that, for certain types of constraints, our model defines a metric on the relevant space of distributions. This metric is shown to arise as a geodesic distance of a Riemannian metric, obtained through an analogue of Otto’s submersion in the classical OT setting. 

AbstractEating behaviours are influenced by the integration of gustatory, olfactory and somatosensory signals, which all contribute to the perception of flavour. Although extensive research has explored the neural correlates of taste in the gustatory cortex (GC), less is known about its role in encoding thermal information. This study investigates the encoding of oral thermal and chemosensory signals by GC neurons compared to the oral somatosensory cortex. In this study we recorded the spiking activity of more than 900 GC neurons and 500 neurons from the oral somatosensory cortex in mice allowed to freely lick small drops of gustatory stimuli or deionized water at varying non‐nociceptive temperatures. We then developed and used a Bayesian‐based analysis technique to assess neural classification scores based on spike rate and phase timing within the lick cycle. Our results indicate that GC neurons rely predominantly on rate information, although phase information is needed to achieve maximum accuracy, to effectively encode both chemosensory and thermosensory signals. GC neurons can effectively differentiate between thermal stimuli, excelling in distinguishing both large contrasts (14vs. 36°C) and, although less effectively, more subtle temperature differences. Finally a direct comparison of the decoding accuracy of thermosensory signals between the two cortices reveals that whereas the somatosensory cortex exhibited higher overall accuracy, the GC still encodes significant thermosensory information. These findings highlight the GC's dual role in processing taste and temperature, emphasizing the importance of considering temperature in future studies of taste processing.image Key pointsFlavour perception relies on gustatory, olfactory and somatosensory integration, with the gustatory cortex (GC) central to taste processing.GC neurons also respond to temperature, but the specifics of how the GC processes taste and oral thermal stimuli remain unclear.The focus of this study is on the role of GC neurons in the encoding of oral thermal information, particularly compared to the coding functions of the oral somatosensory cortex.We found that whereas the somatosensory cortex shows a higher classification accuracy for distinguishing water temperature, the GC still encodes a substantial amount of thermosensory information.These results emphasize the importance of including temperature as a key factor in future studies of cortical taste coding. 

Abstract The growing complexity of biological data has spurred the development of innovative computational techniques to extract meaningful information and uncover hidden patterns within vast datasets. Biological networks, such as gene regulatory networks and protein-protein interaction networks, hold critical insights into biological features’ connections and functions. Integrating and analyzing high-dimensional data, particularly in gene expression studies, stands prominent among the challenges in deciphering these networks. Clustering methods play a crucial role in addressing these challenges, with spectral clustering emerging as a potent unsupervised technique considering intrinsic geometric structures. However, spectral clustering’s user-defined cluster number can lead to inconsistent and sometimes orthogonal clustering regimes. We propose theMulti-layer Bundling (MLB)method to address this limitation, combining multiple prominent clustering regimes to offer a comprehensive data view. We call the outcome clusters “bundles”. This approach refines clustering outcomes, unravels hierarchical organization, and identifies bridge elements mediating communication between network components. By layering clustering results, MLB provides a global-to-local view of biological feature clusters enabling insights into intricate biological systems. Furthermore, the method enhances bundle network predictions by integrating thebundle co-cluster matrixwith the affinity matrix. The versatility of MLB extends beyond biological networks, making it applicable to various domains where understanding complex relationships and patterns is needed. 

Normal matrices, or matrices which commute with their adjoints, are of fundamental importance in pure and applied mathematics. In this paper, we study a natural functional on the space of square complex matrices whose global minimizers are normal matrices. We show that this functional, which we refer to as the non-normal energy, has incredibly well-behaved gradient descent dynamics: despite it being nonconvex, we show that the only critical points of the non-normal energy are the normal matrices, and that its gradient descent trajectories fix matrix spectra and preserve the subset of real matrices. We also show that, even when restricted to the subset of unit Frobenius norm matrices, the gradient flow of the non-normal energy retains many of these useful properties. This is applied to prove that low-dimensional homotopy groups of spaces of unit norm normal matrices vanish; for example, we show that the space of $$d \times d$$ complex unit norm normal matrices is simply connected for all $$d \geq 2$$. Finally, we consider the related problem of balancing a weighted directed graph – that is, readjusting its edge weights so that the weighted in-degree and out-degree are the same at each node. We adapt the non-normal energy to define another natural functional whose global minima are balanced graphs and show that gradient descent of this functional always converges to a balanced graph, while preserving graph spectra and realness of the weights. Our results were inspired by concepts from symplectic geometry and Geometric Invariant Theory, but we mostly avoid invoking this machinery and our proofs are generally self-contained.