asymptotics

Most synaptic excitatory connections are made on dendritic spines. But how the voltage in spines is modulated by its geometry remains unclear. In this talk I shall focus on possible impacts of the synapse geometry on its electrical properties that are surprisingly not well understood on a basic level. I will use as an example the dendritic spine which has a peculiar shape composed of a bulby head connected to a thin neck.

We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the noise inherent to the stochastic approximation scheme dominates any noise from asynchrony.

Recent papers by Feng, Forde and Fouque (SIAM J Financial Math, 2010) and Feng, Fouque and Kumar (Ann. Appl. Prob, 2012) have obtained large deviation results for the small time asymptotic behavior for the log stock price in a fast mean-reverting stochastic volatility model. These results involve specific assumptions about the rate of growth of the speed parameter in the volatility process as the small time decreases to zero.