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We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since the last discharge, i.e., the refractory period. In this extension we incorporate the elapsed since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblins theory in the case of weak non-linearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical simulations to observe how different firing rates can give different types of behaviors and to contrast them with theoretical results of both classical and extended models.
The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differe
Neural network dynamics emerge from the interaction of spiking cells. One way to formulate the problem is through a theoretical framework inspired by ideas coming from statistical physics, the so-called mean-field theory. In this document, we investi
We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a proce
We consider the nonlinear Schrodinger equation [ u_t = i Delta u + | u |^alpha u quad mbox{on ${mathbb R}^N $, $alpha>0$,} ] for $H^1$-subcritical or critical nonlinearities: $(N-2) alpha le 4$. Under the additional technical assumptions $alphageq 2$
We investigate the dynamical role of inhibitory and highly connected nodes (hub) in synchronization and input processing of leaky-integrate-and-fire neural networks with short term synaptic plasticity. We take advantage of a heterogeneous mean-field