Related papers: Neural Field Equations with random data
A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously…
A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which…
Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated…
The nonrelativistic limit of a semilinear field equation is considered in a uniform and isotropic space.The scale-function of the space is constructed based on the Einstein equation.The Cauchy problem of the limit-equation is considered,and…
Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. More recently, as deep…
We investigate the Cauchy problem for the Einstein - scalar field equations in asymptotically flat spherically symmetric spacetimes, in the standard 1+3 formulation. We prove the local existence and uniqueness of solutions for initial data…
In this paper, the regularity properties of Cauchy problem for linear and nonlinear nonlocal wave equations are studied.The equation involves a convolution integral operators with a general kernel operator functions whose Fourier transform…
We present a new modeling paradigm for optimization that we call random field optimization. Random fields are a powerful modeling abstraction that aims to capture the behavior of random variables that live on infinite-dimensional spaces…
Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with…
A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the…
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy…
In the second part of this series of papers, we address the same Cauchy problem that was considered in part 1, namely the nonlocal Fisher-KPP equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-\phi*u), \] where $\phi*u$ is a spatial…
We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the…
Recently, neuronal avalanches have been observed to display oscillations, a phenomenon regarded as the co-existence of a scale-free behaviour (the avalanches close to criticality) and scale-dependent dynamics (the oscillations). Ordinary…
This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple…
Brain-computer interfaces surged extraordinary developments in recent years, and a significant discrepancy now exists between the abundance of available data and the limited headway made in achieving a unified theoretical framework. This…
We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…
In spite of the large amount of existing neural models in the literature, there is a lack of a systematic review of the possible effect of choosing different initial conditions on the dynamic evolution of neural systems. In this short…
We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform…
Evolution and its intelligence element present thrill and challenges in its exploration. Yet, how species have memory, retrieve them and maintain continuity are the fundamental questions. Most of the phenomenon can only be hypothesised by…