Mean First Passage Time in Periodic Attractors

The properties of the mean first passage time in a system characterized by multiple periodic attractors are studied. Using a transformation from a high dimensional space to 1D, the problem is reduced to a stochastic process along the path from the fixed point attractor to a saddle point located between two neighboring attractors. It is found that the time to switch between attractors depends on the effective size of the attractors, $\tau$, the noise, $\epsilon$, and the potential difference between the attractor and an adjacent saddle point as: $~T = {c \over \tau} \exp({\tau \over \epsilon} \Delta {\cal{U}})~$; the ratio between the sizes of the two attractors affects $\Delta {\cal{U}}$. The result is obtained analytically for small $\tau$ and confirmed by numerical simulations. Possible implications that may arise from the model and results are discussed.Comment: 14 pages, 3 figures, submitted to journal of physics

Locking of correlated neural activity to ongoing oscillations

Population-wide oscillations are ubiquitously observed in mesoscopic signals of cortical activity. In these network states a global oscillatory cycle modulates the propensity of neurons to fire. Synchronous activation of neurons has been hypothesized to be a separate channel of signal processing information in the brain. A salient question is therefore if and how oscillations interact with spike synchrony and in how far these channels can be considered separate. Experiments indeed showed that correlated spiking co-modulates with the static firing rate and is also tightly locked to the phase of beta-oscillations. While the dependence of correlations on the mean rate is well understood in feed-forward networks, it remains unclear why and by which mechanisms correlations tightly lock to an oscillatory cycle. We here demonstrate that such correlated activation of pairs of neurons is qualitatively explained by periodically-driven random networks. We identify the mechanisms by which covariances depend on a driving periodic stimulus. Mean-field theory combined with linear response theory yields closed-form expressions for the cyclostationary mean activities and pairwise zero-time-lag covariances of binary recurrent random networks. Two distinct mechanisms cause time-dependent covariances: the modulation of the susceptibility of single neurons (via the external input and network feedback) and the time-varying variances of single unit activities. For some parameters, the effectively inhibitory recurrent feedback leads to resonant covariances even if mean activities show non-resonant behavior. Our analytical results open the question of time-modulated synchronous activity to a quantitative analysis.Comment: 57 pages, 12 figures, published versio

The abelian sandpile and related models

The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The abelian group structure allows an exact calculation of many of its properties. In particular, one can calculate all the critical exponents for the directed model in all dimensions. For the undirected case, the model is related to q= 0 Potts model. This enables exact calculation of some exponents in two dimensions, and there are some conjectures about others. We also discuss a generalization of the model to a network of communicating reactive processors. This includes sandpile models with stochastic toppling rules as a special case. We also consider a non-abelian stochastic variant, which lies in a different universality class, related to directed percolation.Comment: Typos and minor errors fixed and some references adde

Relative-Periodic Elastic Collisions of Water Waves

We compute time-periodic and relative-periodic solutions of the free-surface Euler equations that take the form of overtaking collisions of unidirectional solitary waves of different amplitude on a periodic domain. As a starting guess, we superpose two Stokes waves offset by half the spatial period. Using an overdetermined shooting method, the background radiation generated by collisions of the Stokes waves is tuned to be identical before and after each collision. In some cases, the radiation is effectively eliminated in this procedure, yielding smooth soliton-like solutions that interact elastically forever. We find examples in which the larger wave subsumes the smaller wave each time they collide, and others in which the trailing wave bumps into the leading wave, transferring energy without fully merging. Similarities notwithstanding, these solutions are found quantitatively to lie outside of the Korteweg-de Vries regime. We conclude that quasi-periodic elastic collisions are not unique to integrable model water wave equations when the domain is periodic.Comment: 20 pages, 13 figure