No abstract

PrefaceThe idea of writing tbis book came after a course given in Boulder, Colorado, in 1988

This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential 1 N . This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system.We show that, as the system size N diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.

Following Frohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as vertical bar x-y vertical bar(-2+alpha), 0 <=alpha <= 1/2. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well-known result by Dyson about phase transitions at low temperatures. (C) 2005 American Institute of Physics