A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y (i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y (i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

We consider the Gibbs measure on the configurations of N particles on R+ with one fixed particle at one end at 0. The potential includes pair-wise Coulomb interactions between any particle and its 2 K neighbors. Only when K = 1, the model is within the rank-one operators, and it was treated previously. Here, for the case K ≥ 2, exponentially fast convergence of density distribution for the spacings between particles is proved when N → ∞. In addition, we establish the exponential decay of correlations between the spacings when the number of particles between them is increasing. We treat in detail the case K = 2; when K > 2, the proof works in a similar manner.

We consider the system of particles on a finite interval with pair-wise nearest neighbours interaction and external force. This model was introduced by Malyshev to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending recent results on the zero temperature case (ground states) with external force. We derive the asymptotic for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity.

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