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The analyses presented herein demonstrate that neuronal electrical activity can be consistently interpreted as a manifestation of murburn redox-mediated electronic dynamics rather than as a process fundamentally driven by transmembrane…
We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a "bridge" between GLMY-theory and group homology theory, which helps to reduce path homology…
Although neural networks are capable of reaching astonishing performances on a wide variety of contexts, properly training networks on complicated tasks requires expertise and can be expensive from a computational perspective. In industrial…
In an enriched setting, we show that higher groupoids and higher categories form categories of fibrant objects. The nerve of a differential graded algebra is a higher category in the category of algebraic varieties, where covers are defined…
There is growing interest in understanding how the structural interconnections among brain regions change with the occurrence of neurological diseases. Diffusion weighted MRI imaging has allowed researchers to non-invasively estimate a…
For constructing neuronal network models computational neuroscientists have access to wide-ranging anatomical data that nevertheless tend to cover only a fraction of the parameters to be determined. Finding and interpreting the most…
Sampling theory has traditionally drawn tools from functional and complex analysis. Past successes, such as the Shannon-Nyquist theorem and recent advances in frame theory, have relied heavily on the application of geometry and analysis.…
We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong…
In this short note, we give an elementary proof of a universal approximation theorem for neural networks with three hidden layers and increasing, continuous, bounded activation function. The result is weaker than the best known results, but…
It will be shown that according to theorems of K. Menger, every neuron grid if identified with a curve is able to preserve the adopted qualitative structure of a data space. Furthermore, if this identification is made, the neuron grid…
The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $\mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $\mathbb{D}^n$; (2)…
The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying…
In certain situations, neural networks are trained upon data that obey underlying symmetries. However, the predictions do not respect the symmetries exactly unless embedded in the network structure. In this work, we introduce architectures…
Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory…
We provide an upper bound on the number of neurons required in a shallow neural network to approximate a continuous function on a compact set with a given accuracy. This method, inspired by a specific proof of the Stone-Weierstrass theorem,…
In this paper, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize…
Task-based modeling with recurrent neural networks (RNNs) has emerged as a popular way to infer the computational function of different brain regions. These models are quantitatively assessed by comparing the low-dimensional neural…
Connectomics and network neuroscience offer quantitative scientific frameworks for modeling and analyzing networks of structurally and functionally interacting neurons, neuronal populations, and macroscopic brain areas. This shift in…
In this paper, we introduce a generalized topological quantum field theory based on the symmetric monoidal category which we call causal network condensation since it can be regarded as a generalization of spin network construction of Baez,…
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…