Related papers: Some remarks about trunks and morphisms of neural …
Neuromorphology is crucial to identifying neuronal subtypes and understanding learning. It is also implicated in neurological disease. However, standard morphological analysis focuses on macroscopic features such as branching frequency and…
This article focuses on the problem of studying shared- and individual-specific structure in replicated networks or graph-valued data. In particular, the observed data consist of $n$ graphs, $G_i, i=1,\ldots,n$, with each graph consisting…
Formal verification has become increasingly important because of the kinds of guarantees that it can provide for software systems. Verification of models of biological and medical systems is a promising application of formal verification.…
The emergence of a variety of graph-based meaning representations (MRs) has sparked an important conversation about how to adequately represent semantic structure. These MRs exhibit structural differences that reflect different theoretical…
In recent work by Johnson et al. (2022), a framework was described for the study of graph problems over classes specified by omitting each of a finite set of graphs as subgraphs. If a problem falls into the framework then its computational…
It has been studied by Curto et al. (SIAM J. on App. Alg. and Geom., 1(1) : 222 $\unicode{x2013}$ 238, 2017) that a neural code that has an open convex realization does not have any local obstruction relative to the neural code. Further, a…
We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an…
This is a survey of our results on the theory of $n$-homomorphisms of Buchstaber--Rees and its generalization that we obtained. In short, we are concerned with classes of linear maps between commutative rings that can be described the "next…
Complex networks can often be decomposed into less complex sub-networks whose structures can give hints about the functional organization of the network as a whole. However, these structural motifs can only tell one part of the functional…
This PhD thesis is focused on the central idea that single neurons in the brain should be regarded as temporally precise and highly complex spatio-temporal pattern recognizers. This is opposed to the prevalent view of biological neurons as…
Neuroscience is undergoing a period of rapid experimental progress and expansion. New mathematical tools, previously unknown in the neuroscience community, are now being used to tackle fundamental questions and analyze emerging data sets.…
We present in this paper a systematic study on how to morph a well-trained neural network to a new one so that its network function can be completely preserved. We define this as \emph{network morphism} in this research. After morphing a…
As a generalization of cyclic codes of length p^s over F_{p^a}, we study n-dimensional cyclic codes of length p^{s_1} X ... X p^{s_n} over F_{p^a} generated by a single "monomial". Namely, we study multi-variable cyclic codes of the form…
Neural network training is typically viewed as gradient descent on a loss surface. We propose a fundamentally different perspective: learning is a structure-preserving transformation (a functor L) between the space of network parameters…
In the past two decades, significant advances have been made in understanding the structural and functional properties of biological networks, via graph-theoretic analysis. In general, most graph-theoretic studies are conducted in the…
Real-world networks often benefit from capturing both local and global interactions. Inspired by multi-modal analysis in brain imaging, where structural and functional connectivity offer complementary views of network organization, we…
The paper introduces a biologically and evolutionarily plausible neural architecture that allows a single group of neurons, or an entire cortical pathway, to be dynamically reconfigured to perform multiple, potentially very different…
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…
This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…
We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and…