Thermal history of the string universe

Thermal history of the string universe based on the Brandenberger and Vafa's scenario is examined. The analysis thereby provides a theoretical foundation of the string universe scenario. Especially the picture of the initial oscillating phase is shown to be natural from the thermodynamical point of view. A new tool is employed to evaluate the multi state density of the string gas. This analysis points out that the well-known functional form of the multi state density is not applicable for the important region $T \leq T_H$, and derives a correct form of it.Comment: 39 pages, no figures, use revtex.sty, aps.sty, aps10.sty & preprint.st

LETTER Communicated by David Horn A Stochastic Method to Predict the Consequence of Arbitrary Forms of Spike-Timing-Dependent Plasticity

Synapses in various neural preparations exhibit spike-timing-dependent plasticity (STDP) with a variety of learning window functions. The window functions determine the magnitude and the polarity of synaptic change according to the time difference of pre- and postsynaptic spikes. Numerical experiments revealed that STDP learning with a single-expo nential window function resulted in a bimodal distribution of synaptic conductances as a consequence of competition between synapses. A slightly modified window function, however, resulted in a unimodal distribution rather than a bimodal distribution. Since various window functions have been observed in neural preparations, we develop a rigorous mathematical method to calculate the conductance distribution for any given window function. Our method is based on the Fokker-Planck equation to determine the conductance distribution and on the Ornstein-Uhlenbeck process to characterize the membrane potential fluctuations. Demonstrating that our method reproduces the known quantitative results of STDP learning, we apply the method to the type of STDP learning found recently in the CA1 region of the rat hippocampus. We find that this learning can result in nearly optimized competition between synapses. Meanwhile, we find that the type of STDP learning found in the cerebellum-like structure of electric fish can result in all-or-none synapses: either all the synaptic conductances are maximized, or none of them becomes significantly large. Our method also determines the window function that optimizes synaptic competition.