Finite-size effects of avalanche dynamics
Christian W. Eurich*
Institut fu¨r Theoretische Physik, Universita¨t Bremen, Otto-Hahn-Allee 1, D-28334 Bremen, Germany
J. Michael Herrmann
Max-Planck-Institut fu¨r Stro¨mungsforschung, Bunsenstrasse 10, D-37073 Go¨ttingen, Germany
Udo A. Ernst
Institut fu¨r Theoretische Physik, Universita¨t Bremen, Otto-Hahn-Allee 1, D-28334 Bremen, Germany
Received 14 September 2000; published 31 December 2002
We study the avalanche dynamics of a system of globally coupled threshold elements receiving random
input. The model belongs to the same universality class as the random-neighbor version of the Olami-Feder-
Christensen stick-slip model. A closed expression for avalanche size distributions is derived for arbitrary
system sizes N using geometrical arguments in the system’s configuration space. For finite systems, approxi-
mate power-law behavior is obtained in the nonconservative regime, whereas for N → , critical behavior with
an exponent of -3/2 is found in the conservative case only. We compare these results to the avalanche
properties found in networks of integrate-and-fire neurons, and relate the different dynamical regimes to the
emergence of synchronization with and without oscillatory components.
DOI: 10.1103/PhysRevE.66.066137 PACS numbers: 05.65.+b, 05.70.Ln, 45.70.Ht, 87.18.Sn
I. INTRODUCTION
In the last decade, a considerable number of publications
have been dedicated to the occurrence of power-law behavior
in systems involving interacting threshold elements driven
by slow external input. The dynamics accounts for phenom-
ena occurring in such diverse systems as piles of granular
matter 1, earthquakes 2, the game of life 3, friction 4,
and sound generated in the lung during breathing 5. An
avalanche of theoretical investigations was triggered by Bak,
Tang, and Wiesenfeld 6 who linked the occurrence of
power laws to the notion of self-organized criticality SOC.
In the so-called sandpile models, locally connected elements
receiving random input self-organize into a critical state
characterized by power-law distributions of avalanches with-
out the explicit tuning of a model parameter e.g., Refs.
7–18. Analytical results were derived for sandpile models
14,15, and it was shown that the existence of a conserva-
tion law is a necessary prerequisite to obtain SOC 16–18.
A second class of models inspired by earthquake dynam-
ics employs continuous driving and nonconservative interac-
tion between the elements of the system 4,19. In the Olami-
Feder-Christensen OFC model 19, where the amount of
dissipation is controlled by a parameter , power-law behav-
ior of avalanches occurs for a wide range of values. Sub-
sequent investigations emphasized the importance of bound-
ary conditions and tied the existence of the observed scaling
behavior to synchronization phenomena induced by spatial
inhomogeneities 20–24. More specifically, Lise and Jensen
25 introduced a random-neighbor interaction in the OFC
model to avoid the buildup of spatial correlations. Further
analysis indeed revealed that the random-neighbor OFC
model does not display SOC in the dissipative regime 26–
28.
In these avalanche models with nonconservative interac-
tion, analytical results have been obtained only for system
size N → so far 26,29. Here we introduce a model that
not only circumvents the problem of system boundaries, but
yields an analytical access also for finite system sizes N. The
elements are globally connected, which makes the system a
mean-field model. Randomness is not introduced through
random neighbors but by providing a random external input.
During an avalanche, the elements become unstable and re-
lax in a fixed order determined by the state of the system
immediately prior to the avalanche. Therefore, the system is
strictly Abelian for dissipation parameters smaller than a
threshold value, which can be readily worked out. In this
case, a geometrical approach in the N-dimensional configu-
ration space yields an exact equation for the distribution of
avalanche sizes.
In Sec. II, the model is specified and compared with other
dissipative avalanche models, in particular, with the random-
neighbor OFC model. In Sec. III, avalanche properties are
presented both numerically and analytically, whereby details
of the analytical calculation of the avalanche size distribu-
tions can be found in Appendixes A–C. Extensions and ap-
plications of the model are formulated in the terminology of
neural networks: The model allows for an interpretation in
terms of a fully connected neural network of nonleaky
integrate-and-fire neurons. Implications of this view such as
the synchronization behavior of local, densely connected
populations of cortical neurons will be discussed in Sec. IV.
The paper concludes with a brief summary and discussion.
II. THEAVALANCHE MODEL
A. Definition
In the model, time is measured in discrete steps, t
=0,1,2, .... Consider a set of N identical threshold ele- *Electronic address: eurich@physik.uni-bremen.de
PHYSICAL REVIEW E 66, 066137 2002
1063-651X/2002/666/06613715/$20.00 ©2002 The American Physical Society 66 066137-1