Population Density Approach to Neural Network Modeling

Population Density Methods (PDM) have gained
prominence in recent
years in Theoretical Neuroscience as an analytical
and time-saving
computational tool. The method involves solving a
density equation
(aka, Fokker-Planck) instead of simulating many
individual neurons.
Simplifying assumptions of the underlying neuron
model are often
made so that the resulting PDM equations have low
dimension for
tractability. Thus, dimension reduction techniques
are vital for
physiological modeling. An introduction to PDM and
the relevant issues are discussed in Chapter 2. A
''moment closure''
dimension reduction technique is analyzed in Chapter

3. We show
   the
   equations are
   ill-posed in the fluctuation-driven regime with
   realistic parameters
   despite several contrary reports in the literature.
   The dimension
   reduction
   method is even worse for the more physiological
   ''theta'' model
   (Chapter 4). A robust and accurate alternative
   reduction technique
   using a moving eigenvector basis is developed and
   implemented in
   Chapter 5. The stochastic firing rate dynamics of
   various neural
   models are analyzed in Chapter 6 with the tools we
   have developed.

   Autorentext

   Cheng Ly received his Ph.D. in Mathematics from The Courant
   Institute (NYU) in 2007
   with Daniel Tranchina. Currently, he is an NSF
   postdoc (MSPRF) in the mathematics department at the University
   of Pittsburgh with
   mentor Bard Ermentrout. Cheng Ly's research involves analyzing
   stochastic neural
   networks in
   Theoretical Neuroscience.

   Klappentext

   Population Density Methods (PDM) have gained
   prominence in recent
   years in Theoretical Neuroscience as an analytical
   and time-saving
   computational tool. The method involves solving a
   density equation
   (aka, Fokker-Planck) instead of simulating many
   individual neurons.
   Simplifying assumptions of the underlying neuron
   model are often
   made so that the resulting PDM equations have low
   dimension for
   tractability. Thus, dimension reduction techniques
   are vital for
   physiological modeling. An introduction to PDM and
   the relevant issues are discussed in Chapter 2. A
   'moment closure'
   dimension reduction technique is analyzed in Chapter

4. We show
   the
   equations are
   ill-posed in the fluctuation-driven regime with
   realistic parameters
   despite several contrary reports in the literature.
   The dimension
   reduction
   method is even worse for the more physiological
   'theta' model
   (Chapter 4). A robust and accurate alternative
   reduction technique
   using a moving eigenvector basis is developed and
   implemented in
   Chapter 5. The stochastic firing rate dynamics of
   various neural
   models are analyzed in Chapter 6 with the tools we
   have developed.