Applied Mathematics and Computation SeminarFokker-Planck Description for Noisy Neuronal Network DynamicsGregor Kovacic, Rensselaer Polytechnic Institute
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.
Refreshments at 3:45
4:00pm–5:00pm, Tuesday, November 3, 2009 in LGRT 1634
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