Two distinct desynchronization processes caused by lesions in globally coupled neurons

To accomplish a task, the brain works like a synchronized neuronal network where all the involved neurons work together. When a lesion spreads in the brain, depending on its evolution, it can reach a significant portion of relevant area. As a consequence, a phase transition might occur: the neurons desynchronize and cannot perform a certain task anymore. Lesions are responsible for either disrupting the neuronal connections or, in some cases, for killing the neuron. In this work, we will use a simplified model of neuronal network to show that these two types of lesions cause different types of desynchronization.Comment: 5 pages, 3 figure

Synchronization of phase oscillators due to nonlocal coupling mediated by the slow diffusion of a substance

Many systems of physical and biological interest are characterized by assemblies of phase oscillators whose interaction is mediated by a diffusing chemical. The coupling effect results from the fact that the local concentration of the mediating chemical affects both its production and absorption by each oscillator. Since the chemical diffuses through the medium in which the oscillators are embedded, the coupling among oscillators is non-local: it considers all the oscillators depending on their relative spatial distances. We considered a mathematical model for this coupling, when the diffusion time is arbitrary with respect to the characteristic oscillator periods, yielding a system of coupled nonlinear integro-differential equations which can be solved using Green functions for appropriate boundary conditions. In this paper we show numerical solutions of these equations for three finite domains: a linear one-dimensional interval, a rectangular, and a circular region, with absorbing boundary conditions. From the numerical solutions we investigate phase and frequency synchronization of the oscillators, with respect to changes in the coupling parameters for the three considered geometries

Phase synchronization of coupled bursting neurons and the generalized Kuramoto model

Bursting neurons fire rapid sequences of action potential spikes followed by a quiescent period. The basic dynamical mechanism of bursting is the slow currents that modulate a fast spiking activity caused by rapid ionic currents. Minimal models of bursting neurons must include both effects. We considered one of these models and its relation with a generalized Kuramoto model, thanks to the definition of a geometrical phase for bursting and a corresponding frequency. We considered neuronal networks with different connection topologies and investigated the transition from a non-synchronized to a partially phase-synchronized state as the coupling strength is varied. The numerically determined critical coupling strength value for this transition to occur is compared with theoretical results valid for the generalized Kuramoto model.Comment: 31 pages, 5 figure

Hamiltonian description for magnetic field lines: a tutorial

Under certain circumstances, the equations for the magnetic field lines can be recast in a canonical form, after defining a suitable field line Hamiltonian. This analogy is extremely useful for dealing with a variety of problems involving magnetically confined plasmas, like in tokamaks and other toroidal devices, where there is usually one symmetric coordinate which plays the role of time in the canonical equations. In this tutorial paper we review the basics of the Hamiltonian description for magnetic field lines, emphasizing the role of a variational principle and gauge invariance. We present representative applications of the formalism, using cylindrical and magnetic flux coordinates in tokamak plasmas

Intermingled basins in coupled Lorenz systems

We consider a system of two identical linearly coupled Lorenz oscillators, presenting synchro- nization of chaotic motion for a specified range of the coupling strength. We verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction, such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa. We investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins, and also obtained scaling laws that characterize quantitatively the riddling of both basins for this system