arXiv subjects

B. A. Zambrano-Luna

Publications and source records attributed to B. A. Zambrano-Luna.

10 recordsLinked to original sources

arXiv2026

Pattern Formation in Quantum Hierarchical Cellular Neural Networks

We present a new class of quantum neural networks (QNNs) whose states are solutions of $p$-adic Schrödinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of $p$-adic cellular neural networks (CNNs). The CNNs are continuous limits of discrete hierarchical neural networks (NNs). The CNNs are bio-inspired by the Wilson-Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new $p$-adic Schrödinger equations, which allows the construction of new QNNs on simple graphs. We also conduct detailed numerical simulations, offering a clear insight into the functioning of the new QNNs. At a mathematical level, we show the existence of local solutions for the new $p$ -adic Schrödinger equations.

quant-ph↗

arXiv2026

Pseudo-Traveling Waves and Bumps in Quantum and Classical Hierarchical Cellular Neural Networks

We study the existence of pseudo-traveling waves and bump solutions for two classes of hierarchical cellular neural networks (CNNs) defined over the ring of $p$-adic integers $\mathbb{Z}_{p}$. The first type is a $p$-adic CNN described by a reaction-diffusion equation, while the second type is its quantum analog obtained via Wick rotation. The $p$-adic CNNs are hierarchical versions of the classical Chua-Yang CNNs; these networks have a tree-like hierarchical architecture with infinitely many cells and hidden layers. The states are governed by integro-differential equations on $% \mathbb{Z}_{p}$. The $p$-adic traveling waves behave fundamentally differently from their Archimedean counterparts. A traveling wave restricted to a $p$-adic sphere yields a countably infinite collection of independent patterns. We introduce the notion of pseudo-traveling waves as finite truncations of this structure and prove their existence for both the classical and quantum networks. We further establish the existence of time-independent solutions (bumps) for both models. Our theoretical results are complemented by numerical simulations that approximate pseudo-traveling-wave solutions for quantum CNNs.

math-ph↗

arXiv2025

Deciphering culprits for cyanobacterial blooms and lake vulnerability in north-temperate lakes

Harmful cyanobacterial blooms (CBs) are increasingly prevalent worldwide, posing significant environmental and health concerns. We derive a stoichiometric model describing the population dynamics and toxicity of cyanobacteria in north-temperate freshwater ecosystems. Our model quantifies the hypoxic effects of CBs on fish mortality and evaluates the impact of microcystin-LR (MC-LR) on aquatic macro-invertebrates, phytoplankton, and fish species. Analyzing data from diverse north-temperate lakes with varying physical characteristics, we identify eutrophication as a pivotal catalyst in bloom proliferation. Under predicted warming scenarios coupled with increased eutrophication, peak MC-LR concentrations will surge dramatically, and blooms will occur earlier in the year. We uncover severe bioaccumulation of MC-LR in higher trophic species; the response to CBs among fish at intermediate trophic levels was heterogeneous across lakes. We compare our model against observations from several north-temperate lakes, demonstrating its robustness and applicability. Our insights are critical for informing targeted interventions to mitigate CBs.

math.DS↗

arXiv2025

Decoding the spatial spread of cyanobacterial blooms in an epilimnion

Cyanobacterial blooms (CBs) pose significant global challenges due to their harmful toxins and socio-economic impacts, with nutrient availability playing a key role in their growth, as described by ecological stoichiometry (ES). However, real-world ecosystems exhibit spatial heterogeneity, limiting the applicability of simpler, spatially uniform models. To address this, we develop a spatially explicit partial differential equation model based on ES to study cyanobacteria in the epilimnion of freshwater systems. We establish the well-posedness of the model and perform a stability analysis, showing that it admits two linearly stable steady states, leading to either extinction or saturation. We use the finite elements method to numerically solve our system on a real lake domain derived from Geographic Information System (GIS) data and realistic wind conditions extrapolated from ERA5-Land. Our numerical results highlight the importance of lake shape and size in CB monitoring, while global sensitivity analysis using Sobol Indices identifies light attenuation and intensity as primary drivers of bloom variation, with water movement influencing early bloom stages and nutrient input becoming critical over time. This model supports continuous water-quality monitoring, informing agricultural, recreational, economic, and public health strategies for mitigating CBs.

math.AP↗

arXiv2024

Hierarchical Neural Networks, p-Adic PDEs, and Applications to Image Processing

The first goal of this article is to introduce a new type of p-adic reaction-diffusion cellular neural network with delay. We study the stability of these networks and provide numerical simulations of their responses. The second goal is to provide a quick review of the state of the art of p-adic cellular neural networks and their applications to image processing.

cs.NE↗

arXiv2023

Hierarchical Wilson-Cowan Models and Connection Matrices

This work aims to study the interplay between the Wilson-Cowan model and the connection matrices. These matrices describe the cortical neural wiring, while the Wilson-Cowan equations provide a dynamical description of neural interaction. We formulate the Wilson-Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well-posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson-Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson-Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson-Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson-Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex.

q-bio.NC↗

arXiv2023

p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines

Understanding how deep learning architectures work is a central scientific problem. Recently, a correspondence between neural networks (NNs) and Euclidean quantum field theories (QFTs) has been proposed. This work investigates this correspondence in the framework of p-adic statistical field theories (SFTs) and neural networks (NNs). In this case, the fields are real-valued functions defined on an infinite regular rooted tree with valence p, a fixed prime number. This infinite tree provides the topology for a continuous deep Boltzmann machine (DBM), which is identified with a statistical field theory (SFT) on this infinite tree. In the p-adic framework, there is a natural method to discretize SFTs. Each discrete SFT corresponds to a Boltzmann machine (BM) with a tree-like topology. This method allows us to recover the standard DBMs and gives new convolutional DBMs. The new networks use O(N) parameters while the classical ones use O(N^{2}) parameters.

hep-th↗

arXiv2023

p-adic Cellular Neural Networks: Applications to Image Processing

The p-adic cellular neural networks (CNNs) are mathematical generalizations of the neural networks introduced by Chua and Yang in the 80s. In this work we present two new types of CNNs that can perform computations with real data, and whose dynamics can be understood almost completely. The first type of networks are edge detectors for grayscale images. The stationary states of these networks are organized hierarchically in a lattice structure. The dynamics of any of these networks consists of transitions toward some minimal state in the lattice. The second type is a new class of reaction-diffusion networks. We investigate the stability of these networks and show that they can be used as filters to reduce noise, preserving the edges, in grayscale images polluted with additive Gaussian noise. The networks introduced here were found experimentally. They are abstract evolution equations on spaces of real-valued functions defined in the p-adic unit ball for some prime number p. In practical applications the prime p is determined by the size of image, and thus, only small primes are used. We provide several numerical simulations showing how these networks work.

nlin.CG↗

arXiv2022

Graphs, local zeta functions, Log-Coulomb Gases, and phase transitions at finite temperature

We study a log-gas on a network (a finite, simple graph) confined in a bounded subset of a local field (i.e. R, C, Q_{p} the field of p-adic numbers). In this gas, a log-Coulomb interaction between two charged particles occurs only when the sites of the particles are connected by an edge of the network. The partition functions of such gases turn out to be a particular class of multivariate local zeta functions attached to the network and a positive test function which is determined by the confining potential. The methods and results of the theory of local zeta functions allow us to establish that the partition functions admit meromorphic continuations in the parameter \b{eta} (the inverse of the absolute temperature). We give conditions on the charge distributions and the confining potential such that the meromorphic continuations of the partition functions have a pole at a positive value \b{eta}_{UV}, which implies the existence of phase transitions at finite temperature. In the case of p-adic fields the meromorphic continuations of the partition functions are rational functions in the variable p^{-\b{eta}}. We give an algorithm for computing such rational functions. For this reason, we can consider the p-adic log-Coulomb gases as exact solvable models. We expect that all these models for different local fields share common properties, and that they can be described by a uniform theory.

math-ph↗

arXiv2021

p-adic Cellular Neural Networks

In this article we introduce the p-adic cellular neural networks which are mathematical generalizations of the classical cellular neural networks (CNNs) introduced by Chua and Yang. The new networks have infinitely many cells which are organized hierarchically in rooted trees, and also they have infinitely many hidden layers. Intuitively, the p-adic CNNs occur as limits of large hierarchical discrete CNNs. More precisely, the new networks can be very well approximated by hierarchical discrete CNNs. Mathematically speaking, each of the new networks is modeled by one integro-differential equation depending on several p-adic spatial variables and the time. We study the Cauchy problem associated to these integro-differential equations and also provide numerical methods for solving them.

nlin.CG↗