Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Arbeitsgruppen-Seminar

Tuesday, 21.07.2015 17:15 c.t.

Maximum Pattern Entropy

by Dr. Rob Shaw
from MPIDS Dept. of Nonlinear Dynamics

Contact person: Theo Geisel

Location

Ludwig Prandtl lecture hall

Abstract

Can we define the entropy of a pattern of extended objects, for example, coins randomly placed on a table top? At first the number of possible configurations will grow as objects are added to a fixed domain. But as the situation becomes more crowded, the positions of objects become constrained, and the number of possible patterns decreases. At a special density, the entropy is a maximum. For objects on a fixed lattice, the number of configurations is countable, and the entropy is well-defined. An example is the monomer-dimer gas, random arrangements of dominos on a checkerboard. We show how to extend the definition of this "pattern entropy" to the continuous case. This number is proportional to the average amount of information required to specify, to some resolution, a particular pattern out of the ensemble of possiblities. An example application would be to use this measure to estimate information propagation in a simple neuronal spike model. If we fix the refractory period of a neuron, and action potential spikes otherwise occur on average at random, what is the optimum density of spikes for maximum information transfer down the axon? Too low a firing rate gives a low information transfer, but the spike train for too high a firing rate becomes nearly periodic, with again low information transfer. This situation is analogous to the classical Tonks gas, a collection of hard rods moving in one dimension. The optimum spike rate for the neuron model corresponds to the density giving the peak pattern entropy in the Tonks gas.

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