new links generated in (I) and (II), however, can only relieve the
system from the surplus of energy in the “a” sub-network and
cannot guarantee as table equilibrium (S): soon after brushing
against the equilibrium, in fact, a new cycle is immediately initiated.
Solely in state (III) the global link pattern is stabilizing the
equilibrium.
Thus, the role of restoring a global stability in the network is
assigned to the emergence of a new sub-network in a specific
location with a specific link architecture (pattern(III) in Fig.8). It
is interesting to note that the above picture is strongly reminding
the “negative feedback” mechanism so often invoked to describe
the modulation of physiological equilibria in metabolic cycles.
However, at odds with metabolic cycles, where the chemical nature
(or concentration) of substance(s) flowing through the direct and
inverse pathways is different, in the present case the activation level
of the target node is the only essential trigger of the negative
feedback.3.2 Ising Models
of Brain Areas Activity
3.2.1 The Problem
The areas of human brains active in the “default” mode [25] elicit
peculiar interest since their role under that condition is not fully
understood [26]. Due to the huge complexity of brain physiology
[27] even a crude modeling approach to the problem is welcome,
particularly if—as in the case of the Ising model—it looks able to
describe a phase transition (flipping) of an ensemble based upon the
dynamical behavior of its single components. Such an abstract
model can be extended from the original context of spin alignment
of microscopic magnets to other transition-like phenomena, as the
liquid/gas transition in a fluid or—as in the present case—to the
active/resting transition in the functional states of brain networks.
From experimental evidence of the type in Fig.6, the problem
can be tackled considering the following sequential steps:
l Extracting from fMRI records the time series which describe the
activity levels of the Regions Of Interest (ROIs). In the present
case, for each of 90 ROIs the time series included 150 elements.
l Calculating from the time series a correlation matrix that is a
functional connectivity matrix (Fig.6).
l Filtering the values in the above matrix by some threshold
(including the sign) in order to work out a symmetric Adjacency
Matrix [AM] like that in Fig.9 (A) of 19 selected ROIs in the
present case, whose generic element [AM]i,j represents the
weight of the link connecting agentsiandj. The first three
rows in panel (A) contain the XYZ coordinates of the
corresponding ROI, according to the MRI Atlas of the
Human Brain-Harvard Medical School [28].Multi-agent Simulations of Population Behavior... 319