One of the most fundamental questions in the field of neuroscience is the emergence of synchronous behaviour in the brain, such as phase, anti-phase, and shift-phase synchronisation. In this work, we investigate how the connectivity between brain areas can influence the phase angle and the neuronal synchronisation. To do this, we consider brain areas connected by means of excitatory and inhibitory synapses, in which the neuron dynamics is given by the adaptive exponential integrate-and-fire model. Our simulations suggest that excitatory and inhibitory connections from one area to another play a crucial role in the emergence of these types of synchronisation. Thus, in the case of unidirectional interaction, we observe that the phase angles of the neurons in the receiver area depend on the excitatory and inhibitory synapses which arrive from the sender area. Moreover, when the neurons in the sender area are synchronised, the phase angle variability of the receiver area can be reduced for some conductance values between the areas. For bidirectional interactions, we find that phase and anti-phase synchronisation can emerge due to excitatory and inhibitory connections. We also verify, for a strong inhibitory-to-excitatory interaction, the existence of silent neuronal activities, namely a large number of excitatory neurons that remain in silence for a long time.
We show that extreme orbits, trajectories that connect local maximum and minimum values of one dimensional maps, play a major role in the parameter space of dissipative systems dictating the organization for the windows of periodicity, hence producing sets of shrimp-like structures. Here we solve three fundamental problems regarding the distribution of these sets and give: (i) their precise localization in the parameter space, even for sets of very high periods; (ii) their local and global distributions along cascades; and (iii) the association of these cascades to complicate sets of periodicity. The extreme orbits are proved to be a powerful indicator to investigate the organization of windows of periodicity in parameter planes. As applications of the theory, we obtain some results for the circle map and perturbed logistic map. The formalism presented here can be extended to many other different nonlinear and dissipative systems.
Statistical properties for the recurrence of particles in an oval billiard with a hole in the boundary are discussed. The hole is allowed to move in the boundary under two different types of motion: (i) counterclockwise periodic circulation with a fixed step length and; (ii) random movement around the boundary. After injecting an ensemble of particles through the hole we show that the surviving probability of the particles without recurring -without escaping -from the billiard is described by an exponential law and that the slope of the decay is proportional to the relative size of the hole. Since the phase space of the system exhibits islands of stability we show that there are preferred regions of escaping in the polar angle, hence given a partial answer to an open problem: Where to place a hole in order to maximize or minimize a suitable defined measure of escaping. PACS numbers: 05.45.-a, 05.45.Pq, 05.45.Tp
The changeover from normal to super diffusion in time dependent billiards is explained analytically. The unlimited energy growth for an ensemble of bouncing particles in time dependent billiards is obtained by means of a two dimensional mapping of the first and second moments of the velocity distribution function. We prove that for low initial velocities the mean velocity of the ensemble grows with exponent ∼ 1/2 of the number of collisions with the border, therefore exhibiting normal diffusion. Eventually, this regime changes to a faster growth characterized by an exponent ∼ 1 corresponding to super diffusion. For larger initial velocities, the temporary symmetry in the diffusion of velocities explains an initial plateau of the average velocity. PACS numbers: 05.45.-a, 05.45.Pq, 05.40.FbAs coined by Enrico Fermi [1] Fermi acceleration (FA) is a phenomenon where an ensemble of classical and non interacting particles acquires energy from repeated elastic collisions with a rigid and time varying boundary. It is typically observed in billiards [2-4] whose boundaries are moving in time [5][6][7][8][9]. If the motion of the boundary is random and the initial velocity is small enough [10], the growth of the average velocity is proportional to n 1/2 , with n denoting the number of collisions. If the initial velocity is larger, a plateau of constant velocity is observed in a plot V vs. n which is explained from the symmetry of the velocity diffusion [11]. The symmetry warrants that part of the ensemble grows and part of it decreases in such a way the growing parcel cancels the portion decreasing. As soon as such symmetry is broken the constant regime is changed to a regime of growth. For deterministic oscillations of the border, the scenario is different. Breathing oscillations preserve the shape but not the area of the billiard. It is known that the average velocity evolves in a sub-diffusive manner with a slope of the order of 1/6 [12,13]. For oscillations preserving the area but not the shape of the billiard there are two regimes of growth. For short time the diffusion of velocities is normal passing to super diffusion regime for large enough number of collisions [14]. This changeover is, so far, not yet explained and our contribution in this letter is to fill up this gap in the theory. This is achieved by studying the momenta of the velocity distribution function, noticing that the dynamical angular/time variables have an inhomogeneous distribution in phase space.The results presented in this letter are illustrated by a time-dependent oval-billiard [15] whose phase space is mixed when the boundary is static. The boundary of the billiard is written as R b (θ, t) = 1 + ǫ [1 + a cos(t)] cos(pθ) where R b is the radius of the boundary in polar coordinates, θ is the polar angle, ǫ controls the circle deformation, p > 0 is an integer number [16] given the shape of the boundary, t is the time and a is the amplitude of oscillation of the boundary. Figure 1 shows a typical scenario of the boundary and three collisions illustr...
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