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PHYSICAL REVIEW E 86, 051918 (2012)
Cooperation in neural systems: Bridging complexity and periodicity
Marzieh Zare and Paolo Grigolini
Center for Nonlinear Science, University of North Texas, PO. Box 311427, Denton, Texas 76203-1427, USA
(Received 31 July 2012; revised manuscript received 22 October 2012; published 29 November 2012)
Inverse power law distributions are generally interpreted as a manifestation of complexity, and waiting time
distributions with power index < 2 reflect the occurrence of ergodicity-breaking renewal events. In this paper
we show how to combine these properties with the apparently foreign clocklike nature of biological processes.
We use a two-dimensional regular network of leaky integrate-and-fire neurons, each of which is linked to its four
nearest neighbors, to show that both complexity and periodicity are generated by locality breakdown: Links of
increasing strength have the effect of turning local interactions into long-range interactions, thereby generating
time complexity followed by time periodicity. Increasing the density of neuron firings reduces the influence of
periodicity, thus creating a cooperation-induced renewal condition that is distinctly non-Poissonian.
The emergence of scale-free distributions from phase tran-
sitions affords a promising way to define complexity, which is
the subject of a rapidly developing field of investigation, still
in its infancy . According to a widely shared theoretical
interdisciplinary perspective, ranging from neurophysiology
to sociology, from geophysics to economics, heavy tails
and inverse power law distributions [2,3] are thought to
be the signature of complexity. Of special interest are the
scale-free distributions in time, for instance, the waiting
time distributions of the "light on" and "light off" states
of intermittent fluorescence in quantum dots [4-7], which
has been proved [8,9] to be a renewal non-Poisson process.
These scale-free waiting time distribution densities have the
i(t) o -. (1)
Hence the corresponding cumulative distributions, or survival
probabilities, yield the very slow decay
(t)t) t) dt x t_ . (2)
Experimental observation shows that < 2, thereby making
the mean waiting time distribution diverge, with a consequent
violation of ergodicity. The connection between temporal com-
plexity and ergodicity breaking has stimulated the extension
of fundamental theoretical tools such as the invariant density
, the Kolmogorov-Sinai entropy , and the Khinchin
theorem . Other interesting aspects of ergodicity breaking
in condensed matter are illustrated in Refs. [13-15].
Neural avalanches, with scale-free distribution in both size
and time duration, are another significant manifestation of
natural complexity emerging from criticality [16-21]. The
hypothesis of criticality affords an attractive way of explaining
the intelligent global behavior of cooperative systems. The
conversion of local into long-range interactions generated by
criticality, termed cooperation-induced long-range interac-
tion, was recently proved to yield an efficient way to transport
information  and may be used to explain the zero time lag
synchronization among remote cerebral cortical areas .
PACS number(s): 87.16.dj, 05.45.Tp, 05.65.+b, 87.19.1j
Furthermore, criticality-induced fluctuations are non-Poisson
renewal processes , namely, close relatives of the complex
dynamical processes of Refs. [10-15], in accordance with the
earlier work of Refs. [25,26].
As attractive as this interdisciplinary perspective of com-
plexity may be, it is not clear to what extent it is compatible
with the intrinsic periodicity of biological processes . This
is a subject attracting increasing attention, as proved by the
recent remarkable work of Ref. , which does, however
unexplored the periodicity-criticality issue.
The main purpose of this paper is to show, with the help of
a model of cooperating neurons, that periodicity may emerge
naturally from the criticality-induced long-range correlations.
As in Refs. [22,23], criticality breaks the statistical indepen-
dence of two neurons, regardless of their Euclidean distance,
For larger values of the cooperation parameter, this eventually
yields perfect synchronization and periodicity.
The outline of this paper is as follows. Section II describes
the neural model of this paper. It is a rigorously local version
of the model of the earlier work of Ref. . Section III
illustrates the form of temporal complexity generated by
the cooperative interaction between the units of this model.
Section IV focuses on a technique of statistical analysis that
is used to establish the influence of periodicity on temporal
complexity. It shows that in the low-density case, when the
slow decay of the survival probability makes the system
perceive periodicity, the renewal condition is broken and the
neural quakes become predictable. Section V discusses the size
distribution of neural quakes and shows that the adoption of
the distribution density rather than the cumulative distribution
makes the influence of periodicity on the avalanche statistics
visible, although affording less precise information on the
power law indices. We make concluding remarks in Sec. VI.
II. COOPERATIVE NEURAL NETWORK
We use the popular Leaky Integrate-and-Fire Model (LIFM)
, in the presence of a noise of intensity ua2 [31,32]. The
LIFM is described by
-yx(t) + S + a~(t),
2012 American Physical Society
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Zare, Marzieh & Grigolini, Paolo. Cooperation in neural systems: Bridging complexity and periodicity, article, November 29, 2012; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc132986/m1/1/?rotate=270: accessed March 28, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.