For more information about this meeting, contact Kris Jenssen, Yuxi Zheng.

Title: | Fokker-Planck Description for Noisy Neuronal Network Dynamics |

Seminar: | Computational and Applied Mathematics Colloquium |

Speaker: | Gregor Kovacic, Rensselaer Polytechnic Institute |

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 and a mean-field
equation can be derived in the limit of small and vanishing
conductance fluctuations, respectively. The talk will present
detailed solutions to these equations, describing both the steady
asynchronous and synchronously-oscillating states of the network, and
will also discuss the effects of the network architecture. The mean-
field provides exact solutions for the steady asynchronous state.
For scale-free neuronal networks, it can be used to argue that the
distributions of the firing rates and neuronal activity correlations
are also scale free. The steady asynchronous state is also 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. Differences between oscillations in all-to-
all coupled and scale-free networks will also be discussed. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 09 / 25 / 2009 |

Time: | 03:35pm - 04:25pm |