of Heterogeneous Network Modules via Maximization of Bidirectional Information Transmission

or $4p^{c}q^{c}(1 - r^{c})$ , respectively. Here, p ^cregulates the total number of connections in the network, q ^ccontrols the fraction of inter-modular connections among all connections in the network, and r ^ccontrols the fraction of inter-modular connections from module 1 to 2 among all inter-modular connections. Four types of oscillator-to-oscillator coupling are introduced to the network. These types of couplings lead to phase locking with 0 (in-phase), π∕2, π (anti-phase), or 3π∕2 phase lag between the two oscillators in a two-oscillator system.

The dynamics of the k-th oscillator in the i-th module ( $i = 1,2,k = 1,\ldots,N$ ) is described by

$\displaystyle\begin{array}{rcl} & & \theta _{t+1}^{(i,k)} =\theta _{ t}^{(i,k)} +\omega ^{(i,k)}1 {}\\ & & \qquad \quad \, + \frac{\alpha } {Np^{c}}\sum _{(j,l)\in G^{(i,k)}}\sin \left (\theta _{t}^{(j,l)} -\theta _{ t}^{(i,k)} -\psi _{ kl}^{ij}\right ) +\beta _{ t}^{(i,k)}, {}\\ \end{array}$

where ω ^(i, k) is a natural frequency, α is a coupling strength, and $\beta _{t}^{(i,k)}$ represents additive noise that affects each oscillator independently. G ^(i, k) represents a set of labels for those oscillators that connect to the oscillator (i, k). Each ψ _kl^ijis randomly assigned to one of four possible values $(m - 1)\pi /2\,m = 1\ldots 4$ , according to probabilities $p_{m}^{ij}$ . Note that $p_{m}^{ij} \geq 0$ and $\sum _{m=1}^{4}p_{m}^{ij} = 1$ . In the case of a two-oscillator system, these couplings lead to phase locking with 0 (in-phase), π∕2, π (anti-phase), or 3π∕2 phase lag between the two oscillators. To characterize the macroscopic states of the modules, we define a phase coherence R ⁽ⁱ⁾(t) and a mean phase Θ ⁽ⁱ⁾(t) for each module as the absolute value and the argument of $(1/N)\sum _{k=1}^{N}\exp (\sqrt{-1}\theta ^{(i,k)}(t))$ :

$\displaystyle{ R^{(i)}(t)\exp \left (\sqrt{-1}\varTheta ^{(i)}(t)\right ) = \frac{1} {N}\sum _{k=1}^{N}\exp (\sqrt{-1}\theta ^{(i,k)}(t)). }$