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W. A. Zúñiga-Galindo

Publications and source records attributed to W. A. Zúñiga-Galindo.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2023

p-Adic Statistical Field Theory and Deep Belief Networks

In this work we initiate the study of the correspondence between p-adic statistical field theories (SFTs) and neural networks (NNs). In general quantum field theories over a p-adic spacetime can be formulated in a rigorous way. Nowadays these theories are considered just mathematical toy models for understanding the problems of the true theories. In this work we show these theories are deeply connected with the deep belief networks (DBNs). Hinton et al. constructed DBNs by stacking several restricted Boltzmann machines (RBMs). The purpose of this construction is to obtain a network with a hierarchical structure (a deep learning architecture). An RBM corresponds to a certain spin glass, we argue that a DBN should correspond to an ultrametric spin glass. A model of such a system can be easily constructed by using p-adic numbers. In our approach, a p-adic SFT corresponds to a p-adic continuous DBN, and a discretization of this theory corresponds to a p-adic discrete DBN. We show that these last machines are universal approximators. In the p-adic framework, the correspondence between SFTs and NNs is not fully developed. We point out several open problems.

math-ph↗
arXiv2022

Non-Archimedean Quantum Mechanics via Quantum Groups

We present a new non-Archimedean realization of the Fock representation of the q-oscillator algebras where the creation and annihilation operators act on complex-valued functions, which are defined on a non-Archimedean local field of arbitrary characteristic, for instance, the field of p-adic numbers. This new realization implies that a large number of quantum models constructed using q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum models. In this framework, we select a q-deformation of the Heisenberg uncertainty relation, and construct the corresponding q-deformed Schrödinger equations. In this way we construct a p-adic quantum mechanics which is a p-deformed quantum mechanics. We also solve the time-independent Schrödinger equations for the free particle, and a particle in a non-Archimedean box. In the last case we show the existence of a discrete sequence of energy levels. We determine the eigenvalues of Schrödinger operator for a general radial potential. By choosing the potential in a suitable form we recover the energy levels of the q-hydrogen atom.

math-ph↗
arXiv2022

Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis

Nakao and Mikhailov proposed using continuous models (mean-field models) to study reaction-diffusion systems on networks and the corresponding Turing patterns. This work aims to show that p-adic analysis is the natural tool to carry out this program. This is possible due to the fact that the discrete Laplacian attached to a network turns out to be a particular case of a p-adic Laplacian. By embedding the graph attached to the system into the field of p-adic numbers, we construct a continuous p-adic version of a reaction-diffusion system on a network. The existence and uniqueness of the Cauchy problems for these systems are established. We show that Turing criteria for the p-adic continuous reaction-diffusion system remain essentially the same as in the classical case. However, the properties of the emergent patterns are very different. The classical Turing patterns consisting of alternating domains do not occur in the p-adic continuous case. Instead of this, several domains (clusters) occur. Multistability, that is, coexistence of a number of different patterns with the same parameter values occurs. Clustering and multistability have been observed in the computer simulations of large reaction-diffusion systems on networks. Such behavior can be naturally explained in the p-adic continuous model, in a rigorous mathematical way, but not in the original discrete model.

math.AP↗
arXiv2022

Eigen's Paradox and the Quasispecies Model in a Non-Archimedean Framework

In this article we present a new p-adic generalization of the Eigen-Schuster model where the genomes (sequences) are represented by words written in the alphabet 0,1,,p-1, where p is a prime number, with a time variable length. The time evolution of the concentration of a sequence is controlled by a p-adic evolution equation. The long term behavior of the concentration of a sequence depends on a fitness function f, a mutation measure Q, and an initial concentration distribution. The new model provides essentially two types of asymptotic scenarios for evolution. If the complexity of sequences grows at the right pace, then in the long term the survival is assured. This agrees with the fact that larger genome size improves the replication fidelity. In other case, the sequences cannot copy themselves with sufficiently fidelity, and in the long term they will not survive. Eigen's paradox is one, among the infinitely many, possible scenarios of the evolution in the long term. The mathematical formulation of this fact requires solving the Cauchy problem for the p-adic Eigen-Schuster model in a rigorous mathematical way. This requires imposing restrictions on the fitness function and on the mutation measure, among other conditions. The study of the mentioned initial value problem requires techniques of p-adic wavelets and heat kernels developed in the last 35 years.

q-bio.PE↗
arXiv2022

Non-Archimedean Statistical Field Theory

We construct in a rigorous mathematical way interacting quantum field theories on a p-adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The advantage of the approach presented here is that all the perturbation calculations can be carried out in the standard way using functional derivatives, but in a mathematically rigorous way.

math-ph↗
arXiv2022

Ultrametric Diffusion, Rugged Energy Landscapes and Transition Networks

In this article we introduce the ultrametric networks which are p-adic continuous analogues of the standard Markov state models constructed using master equations. A p-adic transition network (or an ultrametric network) is a model of a complex system consisting of a hierarchical energy landscape, a Markov process on the energy landscape, and a master equation. We focus on networks where the transition rates between two different basins are constant functions, and the jumping process inside of each basin is controlled by a p-adic radial function. We solve explicitly the Cauchy problem for the master equation attached to this type of networks. The solution of this problem is the network response to a given initial concentration. If the Markov process attached to the network is conservative, the long term response of the network is controlled by a Markov chain. If the process is not conservative the network has absorbing states. We define an absorbing time, which depends on the initial concentration, if this time is finite the network reaches an absorbing state in a finite time. We identify in the response of the network the terms responsible for bringing the network to an absorbing state, we call them the fast transition modes. The existence of the fast transition modes is a consequence of the assumption that the energy landscape is ultrametric (hierarchical), and to the best of our understanding this result cannot be obtained using standard methods of Markov state models. Nowadays, it is widely accepted that protein native states are kinetic hubs that can be reached quickly from any other state. The existence of fast transition modes implies that certain states on an ultrametric network work as kinetic hubs.

math-ph↗
arXiv2022

Euclidean Quantum Field Formulation of p-Adic open string amplitudes

We study in a rigorous mathematical way p-adic quantum field theories whose N-point amplitudes are the expectation of products of vertex operators. We show that this type of amplitudes admit a series expansion where each term is an Igusa's local zeta function. The lowest term in this series is a regularized version of the p-adic open Koba-Nielsen string amplitude.

hep-th↗
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

Non-Archimedean Models of Morphogenesis

We study a p-adic reaction-diffusion system and the associated Turing patterns. We establish an instability criteria and show that the Turing patterns are not classical patterns consisting of alternating domains. Instead of this, a Turing pattern consists of several domains (clusters), each of them supporting a different pattern but with the same parameter values. This type of patterns are typically produced by reaction-diffusion equations on large networks.

math.AP↗
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↗
arXiv2021

Local Zeta Functions and Koba-Nielsen String Amplitudes

This article is a survey of our recent work on the connections between Koba-Nielsen amplitudes and local zeta functions (in the sense of Gel'fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. For instance, connections through the adelic relations and through the limit when p tends to 1. Gerasimov and Shatashvili studied the limit p tends to 1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov-Shatashvili Lagrangian are related with the limit p tends to 1 of the p-adic Koba-Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba-Nielsen amplitude is just a new type of local zeta function, which can be studied by using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba-Nielsen amplitudes as functions of the kinematic parameters. The Koba-Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Q_{p}, F_{p}((T)), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba-Nielsen amplitudes give rise to a new amplitudes which we have called Denef-Loeser amplitudes. Along the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

hep-th↗
arXiv2020

Meromorphic Continuation of Koba-Nielsen String Amplitudes

In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics.

math-ph↗
arXiv2019

p-Adic open string amplitudes with Chan-Paton factors coupled to a constant B-field

We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematics parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in the kinematic parameters, this result allows us to regularize the Goshal-Kawano amplitudes, the regularized amplitudes do not have ultraviolet divergences. Due to the need of a certain symmetry, the theory works only for prime numbers which are congruent to 3 modulo 4. We also discuss the limit p tends to 1 in the noncommutative effective field theory and in the Ghoshal-Kawano amplitudes. We show that in the case of four points, the limit p tends to 1 of the regularized Ghoshal-Kawano amplitudes coincides with the Feynman amplitudes attached to the limit p tends to 1 of the noncommutative Gerasimov-Shatashvili Lagrangian.

hep-th↗
arXiv2019

Non-Archimedean Coulomb Gases

This article aims to study the Coulomb gas model over the $d$-dimensional $p$-adic space. We establish the existence of equilibria measures and the $Γ$-limit for the Coulomb energy functional when the number of configurations tends to infinity. For a cloud of charged particles confined into the unit ball, we compute the equilibrium measure and the minimum of its Coulomb energy functional. In the $p$-adic setting the Coulomb energy is the continuum limit of the minus a hierarchical Hamiltonian attached to a spin glass model with a $p$-adic coupling.

math-ph↗