Values of normalized Transfer Entropy.

<p>Values of normalized Transfer Entropy.</p

Glycolytic route to chaos and dynamical effective connectivity.

<p>a–d: The time evolution of the E activity (the normalized concentration , fructose 1,6-bisphospfate) shows a quasi-periodic route to chaos when varying the amplitude of the periodic input-flux from A (top) to A (bottom). (a) Periodic pattern. (b) Quasi-periodic oscillations. (c) Complex quasi-periodic motion indicating the beginning destruction of the periodic behavior. (d) Deterministic chaos. All series are plotted after 10000 seconds. e–h: Effective connectivity of the system for the same values of A in the left panel. The strength of effective connectivity is plotted with arrows width proportional to the Transfer Entropy divided by the maximum value (red arrow), results given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030162#pone-0030162-t001" target="_blank">Table 1</a>. Black dashed circles at the TE from E and E emphasize that the strength of Information flows is not the same, but varies through the quasi-periodic route to chaos.</p

Values of normalized Mutual Information.

<p>Values of normalized Mutual Information.</p

Total information flows and the functional invariant.

<p>Bars represent the total information flow, defined per each enzyme as the total outward TE minus the total inward. For A = 0.021 and E an schematic visualization of the calculation of this flow is shown (bottom graph of the panel). The functionality attributed for each enzyme is an invariant and preserved along the route, ie. E is a source, E is a sink and E has a quasi-zero flow.</p

Time Correlations.

<p>Time Correlations.</p

Matrix of weights connectivity: condition I (only stimulus S1).

<p>Each cell in the table corresponds with a given weight; ith row, jth column is correponding to . Notice that the matrix is symmetric and with the principal diagonal equal to zero. Mean val 2.71; std dev 34.36; min val −150.11; max val 151.23.</p

The metabolic memories are local minima of the DMN dynamics (case ).

<p>For only metabolic memory encoded in the weights, we plotted the time evolution of the distance between the evolving net activity and the initial net activity , fixed to be equal to the activity of the metabolic memory (details in methods). A,B: Time is given in units of Monte Carlo Steps (MCS). Black lines correspond to fixing the initial activity to the memory provided by the LSE solution. Red lines correspond to fixing the activity to an arbitrary memory that has a higher cost than the LSE (i.e. worse than LSE). A: stimulation condition I (only stimulus S1). B: stimulation condition II (both stimuli S1 and S2). In this case of the LSE solution has been found by using an exhaustive method in the search-space and providing the LSE memory (details in the text). Fluctuations in the time-series are originated by a temperature parameter of T = 0.7.</p

The metabolic memories are local minima of the DMN dynamics (case ).

<p>This figure is similar than Fig. 3 but now there are metabolic memory encoded in the weights. A,C: the first metabolic memory in both cases LSE and worse than LSE conditions. B,D: similar than in A,C but for the second metabolic memory. A,B: stimulation condition I (only stimulus S1). C,D: condition II (both stimuli S1 and S2). In this case of , the LSE solution has been found by using an genetic algorithm for minimization of the cost given by Eq. (36), details in the text. The temperature parameter is fixed to T = 0.7.</p

Weights connectivity matrix learned from the DMN by the Boltzmann machine.

<p>For the two stimulation conditions, we plot the two matrices of weights connectivity that are the result of the learning by the Boltzmann machine. Notice that although the mean values in both matrices are small (2.71 in panel A and 0.37 in panel B), the variance in panel A is much higher compared to the one in panel B (A: 34.36, B: 2.83). The tables with these values and their corresponding statistics are given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0058284#pone-0058284-t001" target="_blank">Tables 1</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0058284#pone-0058284-t003" target="_blank">3</a>.</p

Evidence for non-Gaussianity in weights and thresholds obtained through the Boltzmann machine.

<p>Normal probability plot; data deviations from the Gaussian distribution are graphically mapping to the deviations from the straight line (built with a purely Gaussian distribution). A: stimulation condition I in which only the stimulus S1 is presented. B: condition II: both stimuli S1 and S2 are applied to the DMN. For both conditions, weights (colored in red) and thresholds (in blue) are strongly non-Gaussian. This is important as standard Hopfield nets assume that weights are following a Gaussian distribution with mean zero and standard deviation ; this is represented in the inset of panel A with a black line for .</p