Berlin 2012 – scientific programme
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BP: Fachverband Biologische Physik
BP 22: Statistical Physics of Biological Systems III (with DY)
BP 22.9: Talk
Thursday, March 29, 2012, 17:00–17:15, H 1058
Using Branching Processes to Model Critical Neuronal Networks — •Anna Levina1,2, J. Michael Herrmann3, and Theo Geisel1,4 — 1BCCN Göttingen, Germany — 2MPI MIS, Leipzig, Germany — 3University of Edinburgh, UK — 4MPI DS, Göttingen, Germany
Many authors use branching processes (BPs) formalism to model critical neuronal networks. It is indeed very tempting, because BP are a well studied mathematical concept, where it is easy to define what is critical and what is not. Additionally, for BPs it is proved, that a distribution of avalanche sizes follows a power-law with an exponent -3/2. However, only in very few cases does the approximation of activity propagation in a neuronal network by BPs have a rigorous basis. Moreover, a straightforward BPs approximation fails in the presence of delays in the network. Nevertheless, this approach is still used unrestricted to argue about a critical network even for the small system sizes, where the discrepancies are very large.
Here we present analytical and numerical results illustrating reservations in using BPs approximation and ways to overcome them. We show analytically that in the case of a simple neuronal network with a probabilistic synaptic transmission BPs can be used as a valid model in the large network limit. However, for small networks the finite-size corrections are required. We derive these corrections and also discuss how to modify them in the presence of delays. This topic is especially interesting for a growing field of self-organized critical neuronal networks, where branching approximation is used ubiquitously.