Systems Biology (Methods in Molecular Biology)

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ðÞnðÞn 1 = 2

q

,

wheren¼19, number of rows or columns of the Adjacency Matrix

in panel (a).

3.2.2 The Model An abstract Ising model [29, 30] considers each siteiin a lattice as
associated with a discrete variable,si, with valueþ1or1 standing
for up or down spin, respectively. In the absence of external forces
the total energy of the lattice is given by

HsðÞ¼J

X

<i,j>

si,sj,

where the notation<i,j>indicates that sitesiandjare nearest

neighbors andJ is a coupling constant. A competition arises

between thermal fluctuations (reflecting the interaction with the

environment), which induce the system to get disordered, and the

opposite tendency to get organized in some specific way depending

on the interaction or coupling (J) between the sites. The spins flip if

flipping decreases their energy, but sometimes also flip into a higher

energy state. The flipping probability is calculated by a Metropolis

algorithm, based on the formula

e

Ediff=T

,

where Ediff is the potential gain in energy andTis the temperature.

Thus, flipping to a higher energy state directly depends upon

temperature and inversely upon Ediff. The Ising model is lying in

the calculation of the energy at each site as the negative of the sum

of the products of its spin with each of its neighbors’ spins.

We took advantage of the 2D Ising algorithm in the software

library of the Netlogo environment [31] which uses a Metropolis/

Monte Carlo method for the probabilistic time evolution of the

spins in conjunction with space periodic boundary conditions and

four nearest neighbors for each node. First, we traced, according to

the Ising model, each element of the square lattice (SL) depicted in

Fig.9d in its evolution along time windows of different lengths:

thanks to the association to brain areas in the (AM) of Fig.9a, this

allows following at each time step the evolution dynamics of each

ROI in the original (AM) through its correlations with all other

ROIs. Thus, by summing up for each rowi(or columnj) the values

of the corresponding columnj(or rowi), the global activation

dynamics of each (ROI) can be reconstructed in the considered

time window.

The implementation of the above procedure was tested

through the temperature dependence of activity trends of the type

shown in Fig.10 concerning three randomly selected areas. The

activity trend observed at a reference temperature of 2.27 a.u.,

(Fig.10, upper panel) disappears by about doubling the value of

Multi-agent Simulations of Population Behavior... 321