Ising-like model replicating time-averaged spiking behaviour of in vitro neuronal networks.

We analyze time-averaged experimental data from in vitro activities of neuronal networks. Through a Pairwise Maximum-Entropy method, we identify through an inverse binary Ising-like model the local fields and interaction couplings which best reproduce the average activities of each neuron as well as the statistical correlations between the activities of each pair of neurons in the system. The specific information about the type of neurons is mainly stored in the local fields, while a symmetric distribution of interaction constants seems generic. Our findings demonstrate that, despite not being directly incorporated into the inference approach, the experimentally observed correlations among groups of three neurons are accurately captured by the derived Ising-like model. Within the context of the thermodynamic analogy inherent to the Ising-like models developed in this study, our findings additionally indicate that these models demonstrate characteristics of second-order phase transitions between ferromagnetic and paramagnetic states at temperatures above, but close to, unity. Considering that the operating temperature utilized in the Maximum-Entropy method is T o = 1 , this observation further expands the thermodynamic conceptual parallelism postulated in this work for the manifestation of criticality in neuronal network behavior.

Elastic Backbone Defines a New Transition in the Percolation Model.

The elastic backbone is the set of all shortest paths. We found a new phase transition at p_{eb} above the classical percolation threshold at which the elastic backbone becomes dense. At this transition in 2D, its fractal dimension is 1.750±0.003, and one obtains a novel set of critical exponents ß_{eb}=0.50±0.02, γ_{eb}=1.97±0.05, and ν_{eb}=2.00±0.02, fulfilling consistent critical scaling laws. Interestingly, however, the hyperscaling relation is violated. Using Binder's cumulant, we determine, with high precision, the critical probabilities p_{eb} for the triangular and tilted square lattice for site and bond percolation. This transition describes a sudden rigidification as a function of density when stretching a damaged tissue.

Mandala networks: ultra-small-world and highly sparse graphs.

The increasing demands in security and reliability of infrastructures call for the optimal design of their embedded complex networks topologies. The following question then arises: what is the optimal layout to fulfill best all the demands? Here we present a general solution for this problem with scale-free networks, like the Internet and airline networks. Precisely, we disclose a way to systematically construct networks which are robust against random failures. Furthermore, as the size of the network increases, its shortest path becomes asymptotically invariant and the density of links goes to zero, making it ultra-small world and highly sparse, respectively. The first property is ideal for communication and navigation purposes, while the second is interesting economically. Finally, we show that some simple changes on the original network formulation can lead to an improved topology against malicious attacks.