Related papers: Global solutions with infinitely many blowups in a…

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery…

The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth…

The study of blow-up solution of time-fractional heat equations is of significant and wide-ranging interest for its multitude of applications. These types of equations are used to model several real problems in science and engineering. This…

This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of…

We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through transcritical and pitchfork type singularities in the reaction term, using a novel adaptation of the geometric blow-up method.…

Contemporary modeling approaches to the dynamics of neural networks consider two main classes of models: biologically grounded spiking neurons and functionally inspired rate-based units. The unified simulation framework presented here…

We consider a model of cell motion with boundary signal production which describes some aspects of eukaryotic cell migration. Generic polarity markers located in the cell are transported by actin which they help to polymerize. This leads to…

Mean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural…

We study the mean field game problem for a nervous system consisting of a large number of neurons with mean-field interaction. In this system, each neuron can modulate its spiking activity by controlling its membrane potential to…

We investigate a suspension bridge model described by a nonlinear plate equation incorporating internal fractional damping and infinite memory effects. The system also includes a nonlinear source term that may induce instability. Using…

Mean-field theory links the physiological properties of individual neurons to the emergent dynamics of neural population activity. These models provide an essential tool for studying brain function at different scales; however, for their…

We consider a large network of Izhikevich neurons. Each neuron has a quadratic integrate-and-fire type model with a recovery variable modelling spike frequency adaptation (SFA). We introduce a biologically motivated synaptic current…

We review a recent approach to the mean-field limits in neural networks that takes into account the stochastic nature of input current and the uncertainty in synaptic coupling. This approach was proved to be a rigorous limit of the network…

In this work, exact solutions are derived for an integer- and fractional-order time-delayed diffusion equation with arbitrary initial conditions. The solutions are obtained using Fourier transform methods in conjunction with the known…

In this paper, we propose a shot noise-based leaky integrated and firing neuron model and provide a detailed analysis of the performance of this model compared to the traditional diffusion approximated model. In theoretical neuroscience,…

We study linear pseudoparabolic equations with unbounded and time-dependent coefficients. We solve the case which has remained open in several recent studies of pseudoparabolic equations with unbounded and time-dependent coefficients. In…

The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale…

Low-dimensional descriptions of neural network dynamics are an effective tool for bridging different scales of organization of brain structure and function. Recent advances in deriving mean-field descriptions for networks of coupled…

We study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions.…

In a first step towards the comprehension of neural activity, one should focus on the stability of the various dynamical states. Even the characterization of idealized regimes, such as a perfectly periodic spiking activity, reveals…