Neural Dynamics: Criticality, Cooperation, Avalanches and Entrainment between Complex Networks Page: 5

                                
xx: XX -    Chap. 1 -    2012/5/22 -   11:43 -   page 5

1.0                               0.004
0.5
S0.0                                 0.002
-0.5
-1.0                               0.000          -  -
0         1          2              -1           0           1
K                                   n
Figure 1.1 (Left panel) The equilibrium mean field for different values of the cooperation
parameter K. A bifurcation occurs at the critical point K = Kc = 1. (Right panel)
Potential barriers for K subcritical (dashed line, K = 0.2), critical (solid line, K = 1.0),
and supercritical (dotted line, K = 1.8).
holds true only in the limiting case N -  oc. In the case of a finite network,
N < oc, the mean field fluctuates in time forcing us to adopt
N1 - N2
= II + f(t),                                            (1.12)
N
where f(t) is a random fluctuation, which according to the Law of Large Numbers
has an intensity proportional to 1/N. Inserting Eq. (1.12) into Eq. (1.10) yields
dII =    (eKeKf _ -KHe-K f _ 90 (eKeKf +e-KHe-Kf II (1.13)
dt     2                            2                            .
and in the limiting case N  c  the fluctuations vanish, f = 0, so that Eq. (1.13)
generates the well known phase transition prediction at the critical value of the
control parameter
Kc = 1.                                                          (1.14)
Fig. 1.1 shows that for K < 1 the mean field has only one possible equilibrium
value, Ileq = 0. At the critical point K = Kc = 1 this vanishing equilibrium
value splits into two opposite components, one positive and one negative. To
understand the important role of criticality, we notice that a finite number of units
generates the fluctuation f and, this, in turn, forces fluctuations in the mean field.
At criticality, the fluctuations induced in II(t) have a relatively extended range of
free evolution, as made clear in Fig. 1.1. In fact, the separation between the two
repulsive walls is greatest at criticality. In between them, a free diffusion regime
occurs. The supercritical condition K  > Kc generates a barrier of higher and
higher intensity with the increase of K. At the same time the widths of the two
wells shrink, bounding the free evolution regime of the fluctuating mean field to a
smaller region.