University of Massachusetts Amherst
	Department of Mathematics and Statistics
	Applied Mathematics and Computation Seminar

Abstract: Kinetic theory provides a coarse-grained alternative to the integrate-and-fire neuronal network description. In the limit of infinitely short conductance responses, a Boltzmann-type differential- difference equation can be derived for the probability density function of the neuronal voltage. A Fokker-Planck equation can be derived in the limit of small conductance fluctuations,. The talk will present detailed solutions to this equation, describing both the steady asynchronous and synchronously-oscillating states of the network. The steady asynchronous state is described by asymptotic solutions of the Fokker-Planck equation, using the size of the neuronal conductance fluctuations as the small parameter. In addition, the Fokker-Planck equation can also be used to describe the likelihood and temporal period of synchronous network oscillations, in which all the neurons fire in unison. The likelihood of synchrony is computed combinatorially using the network oscillation period and the voltage probability distribution. The oscillation period is found from a first-passage-time problem described by a Fokker-Planck equation, which is solved analyticaly via an eigenfunction expansion. The voltage probability distribution is found using a Central-Limit-Theorem-type argument via a calculation of the voltage cumulants.