Neural Dynamics: Criticality, Cooperation, Avalanches and Entrainment between Complex Networks Page: 14
25 p.: ill.View a full description of this chapter.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
xx: XX - Chap. 1 - 2012/5/22 - 11:43 - page 14
14
neuronal avalanches. However, this condition inhibits the realization of an impor-
tant aspect of cooperation, namely, locality breakdown. For this reason in addition
to the ATA condition, we also study the case of a regular, two-dimensional (2D)
network, where each node has four nearest neighbors and consequently four links.
It is important to stress that to make our model as realistic as possible, we should
introduce a delay time between the firing of a neuron and the abrupt step ahead of
all its nearest neighbors. This delay should be assumed to be proportional to the
Euclidean distance between the two neurons, and it is expected to be a property of
great importance to prove the breakdown of locality when the scale-free condition
is adopted. The two simplified conditions studied in this chapter, ATA and 2D,
would not be affected by a time delay that should be the same for all the links. For
this reason, we do not further consider time delay.
For the cooperation strength we must assume the condition
K < 1. (1.43)
When K is of the order of magnitude of the potential threshold VT = 1, the col-
lective nature of cooperation is lost because the firing of a few neurons causes an
abrupt cascade in which all the other neurons fire. Thus, we do not consider to
be important the non-monotonic behavior of network efficiency that our numeri-
cal calculations show to emerge by assigning K values of the same order as the
potential threshold.
We also note that in the case of this model the breakdown of the Mittag-Leffler
structure, at large times, is not caused by a lack of cooperation, but by the excess
of cooperation. To shed light on this fact, keep in mind that this model has been
solved exactly by Mirollo and Strogatz when a = 0 [37]. In this case, even if
we adopt initial random conditions, after a few steps, all the neurons fire at the
same time, and the time distance between two consecutive firings is given by Tp
of Eq. (1.42). As an effect of noise the neurons can also fire at times t < Tp,
and consequently, setting a > 0, a new, and much shorter time scale is generated.
When we refer to this as the time scale of interest, the Mirollo and Strogatz time
Tp plays the role of a truncation time and
F 1 (1.44)
Tp
To examine this condition let us assign to K a value very close to K = 0. In this
case even if we assign to all the neurons the same initial condition, x = 0, due
to the presence of stochastic fluctuations the neurons fire at different times thereby
creating a spreading on the initial condition that tends to increase in time, even if
initially the firing occurs mainly at times t = nTp. The network eventually reaches
a stationary condition with a constant firing rate G given by
N
G = T, (1.45)
(T'
where (7) denotes the mean time between two consecutive firings of the same
neuron. For a < 1, (7) = TMS. From the condition of a constant rate G we
Upcoming Pages
Here’s what’s next.
Search Inside
This chapter can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Chapter.
Grigolini, Paolo; Zare, Marzieh; Svenkeson, Adam & West, Bruce J. Neural Dynamics: Criticality, Cooperation, Avalanches and Entrainment between Complex Networks, chapter, May 22, 2012; [Hoboken, New Jersey]. (https://digital.library.unt.edu/ark:/67531/metadc177272/m1/14/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.