Sums of the degrees of the multiplex network grows linearly with the network size.

<p><i>S</i>_<i>in</i> is the sum of the degrees of the adjacency matrices of the starting networks and <i>S</i>_<i>fn</i> the sum of the degrees of the adjacency matrices of the finally evolved multiplex networks of the evolution process.</p

Extensivity in neural network evolution by maximising <i>H</i>_<i>KS</i>.

<p>Panel a): An example of the parameter space of chemical <i>γ</i> and electrical coupling <i>ϵ</i> for <i>N</i> = 48 neurons arranged in two equally-sized small-world layers. The <b>X</b> point corresponds to the coupling pair for which <i>H</i>_<i>KS</i> is maximal in the parameter space. Panel b): Plot of the Lyapunov spectra for different network sizes <i>N</i> and Panel c): The linear relation between <i>H</i>_<i>KS</i> and network size <i>N</i>, where <i>σ</i> is the slope of the linear fitting to the data.</p

A point in this figure in the coordinate <i>k</i>×<i>l</i> means that the elements <i>S_k</i> and <i>S_l</i> are connected with equal couplings in a bidirectional fashion.

<p>In (A), a 32 elements network, constructed by maximizing the cost function B₁ in Eq. (7) and in (B), 32 elements network, constructed by maximizing the cost function B₂ in Eq. (8). In (A), the network has the topology of a perturbed star, a hub of neurons connected to all the other neurons, where each outer neuron is sparsely connected to other neurons. The arrow points to the hub. In (B),the network has the topology of a perturbed all-to-all network, where elements are almost all-to-all connected. Note that there is one element, the neuron <i>S</i>₃₂, which is only connected to one neuron, the <i>S</i>₁. This isolated neuron is responsible to produce the large spectral gap between the eigenvalues γ₃ and γ₂. In (A), the relevant eigenvalues are γ₃₁ = 4.97272, γ₃₂ = 32, which produce a cost function equal to B₁ = 5.43478. In (B), the relevant eigenvalues are γ₂ = 0.99761, γ₃ = 27.09788, which produce a cost function equal to B₂ = 26.1628.</p

Schematic representation of the time evolution after one time step of two deviation vectors (arrows) corresponding to the direction along the Lyapunov exponent on the 1-dimensional subspace on the space.

<p> and are two trajectories in the phase space of Hamiltonian (3) that drive the dynamics along this line. We denote with and the lengths of the two deviation vectors initially and after one time step, respectively.</p

Plot of the quantities: as defined by Eq. (18) in red dashed line with points, as defined by Eq. (1) in green dashed line with rectangles, as defined by Eq. (19) in black solid line with lower triangles and as defined by Eq. (21) in blue dashed line with upper triangles as a function of for initial conditions located in the neighborhood of SPO2 of the FPU system.

<p>Note that both axes are logarithmic.</p

The average value of the upper bound MIR, 〈<i>I_P</i>〉 [as defined in Eq. (10)] for active networks composed of 8 elements using one of the many topologies obtained by evolving the network maximizing B₁ (circles), all-to-all topology (squares), star topology (diamonds), nearest-neighbor (upper triangle), and maximizing B₂ (down triangle).

<p>The values of indicated by the starts are (evolving 1), (all-to-all), (star), (nearest-neighbor), and (evolving 2). The evolving 1 network has a Laplacian with relevant eigenvalues γ₇ = 3.0000, γ₈ = 6.1004, which produces a cost function equal to B₁ = 1.033. The evolving 2 network has a Laplacian with relevant eigenvalues γ₂ = 0.2243 and γ₃ = 1.4107, which produces a cost function equal to B₂ = 5.2893.</p

〈<i>I_P</i>〉 for the networks shown in Fig. 6(A–B) by circles and squares, respectively.

<p>〈<i>I_P</i>〉 for the networks shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0003479#pone-0003479-g006" target="_blank">Fig. 6(A–B)</a> by circles and squares, respectively.</p

Plot of quantities: of Eq. (18) in red dashed line with points and of Eq. (1) in green dashed line with rectangles (panel A).

<p>Plot of quantities with red points with the power-law fitting of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0089585#pone.0089585.e264" target="_blank">Eq. (27)</a> in green line (panel B). Plot of with red points with the power-law fitting of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0089585#pone.0089585.e271" target="_blank">Eq. (29)</a> in green line (panel C). Power-law dependence of to in red points, in the interval that corresponds to the energy interval of panels A, B and C and of the power-law fitting of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0089585#pone.0089585.e271" target="_blank">Eq. (29)</a> in green dashed line (panel D). Note that all axes are logarithmic.</p

The quantities <i>I_C</i> (black circles), <i>I_P</i> (red squares), <i>I_S</i> (green diamonds), and <i>H_KS</i> (blue diamonds), for two (A) and four (B) coupled neurons, in an all-to-all topology.

<p>Notice that since there are only two different eigenvalues, there is only one channel of communication whose upper bound for the MIR is given by <i>I_P</i> = |λ¹−λ²|. Also, <i>I_S</i> and <i>I_C</i> represent the mutual information exchanged between any two pairs of elements in the system. In (A), σ^2* = 0.092, σ<i>_BPS</i>≅0.2, , σ<i>_PS</i> = 0.47, and σ<i>_CS</i> = 0.5. In (B), σ^2* = 0.046, σ<i>_BPS</i>≅0.1, , σ<i>_PS</i> = 0.24, and σ<i>_CS</i> = 0.25. CS indicates the coupling interval σ≥σ<i>_CS</i> for which there exists complete synchronization.</p

MIR between the central neuron and an outer one (black circles), , (resp. <i>I_S</i> (1, <i>k</i>), in green line), and between two outer ones (red squares), , (resp. <i>I_S</i> (<i>k</i>, <i>l</i>), in blue line).

<p> Blue diamonds represents the KS-entropy. Other quantities are σ^4* = 0.181, σ^2* = 0.044, , , , σ<i>_BPS</i> = 0.265, σ<i>_PS</i> = 0.92, and σ<i>_CS</i> = 1.0. The star indicates the parameter for which BPS first appears.</p