Related papers: Global solutions with infinitely many blowups in a…
A novel approach to moment closure problem is used to derive low dimensional laws for the dynamics of the moments of the membrane potential distribution in a population of spiking neurons. Using spectral expansion of the density equation we…
This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a reaction diffusion equation with special diffusion processes. It is shown, under certain conditions on the initial data, that the…
The response time of physical computational elements is finite, and neurons are no exception. In hierarchical models of cortical networks each layer thus introduces a response lag. This inherent property of physical dynamical systems…
We investigate the possible blow-up of strong solutions to a biological network formation model originally introduced by D. Cai and D. Hu \cite{HC}. The model is represented by an initial boundary value problem for an elliptic-parabolic…
The theory of Mean-Field Games is interested in the behaviour of interacting particle systems in which the individual interaction between particles (players) decreases as the size of the population increases. In recent years, it was…
We investigate the expressive power of neural networks from the point of view of descriptive complexity. We study neural networks that use floating-point numbers and piecewise polynomial activation functions from two perspectives: 1) the…
Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical…
The activity of a sparse network of leaky integrate-and-fire neurons is carefully revisited with reference to a regime of a bona-fide asynchronous dynamics. The study is preceded by a finite-size scaling analysis, carried out to identify a…
In this paper we develop a novel method to solve problems involving quantum optical systems coupled to coherent quantum feedback loops featuring time delays. Our method is based on exact mappings of such non-Markovian problems to equivalent…
The usual Langevin approach to describe systems driven by noise fails to describe the long time behavior of systems with multiple attractors. The solution of the associated linear Fokker-Planck equation is always unique, even though it…
Inspired by pioneering work of Ky\^uya Masuda in the 1980s, only much more recent PDE studies address global boundedness versus finite-time blow-up in complex time. The two phenomena are related by passage from real to purely imaginary…
We use probabilistic methods to study properties of mean-field models, arising as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that $n$ particles move forward on the real…
We study decision timing problems on finite horizon with Poissonian information arrivals. In our model, a decision maker wishes to optimally time her action in order to maximize her expected reward. The reward depends on an unobservable…
We report a transition from asynchronous to oscillatory behaviour in balanced inhibitory networks for class I and II neurons with instantaneous synapses. Collective oscillations emerge for sufficiently connected networks. Their origin is…
We present a simple Markov model of spiking neural dynamics that can be analytically solved to characterize the stochastic dynamics of a finite-size spiking neural network. We give closed-form estimates for the equilibrium distribution,…
In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial…
A neural field models the large scale behaviour of large groups of neurons. We extend results of van Gils et al. [2013] and Dijkstra et al. [2015] by including a diffusion term into the neural field, which models direct, electrical…
This work studies nonnegative solutions for the Cauchy, Neumann, and Dirichlet problems of a logistic type reaction-diffusion equation. The finite time blowup results for nonnegative solutions under various restrictions on the coefficients…
Rich out of equilibrium collective dynamics of strongly interacting large assemblies emerge in many areas of science. Some intriguing and not fully understood examples are the glassy arrest in atomic, molecular or colloidal systems,…
Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very…