The small world coefficient 4.8 ± 1 optimizes information processing in 2D neuronal networks

www.nature.com/npjsba ARTICLE OPEN The small world coefficient 4.8 ± 1 optimizes information processing in 2D neuronal networks 1234567890():,; F. Aprile 1 , V. Onesto 2 and F. Gentile 3✉ Small world networks have recently attracted much attention because of their unique properties. Mounting evidence suggests that communication is optimized in networks with a small world topology. However, despite the relevance of the argument, little is known about the effective enhancement of information in similar graphs. Here, we provide a quantitative estimate of the efficiency of small world networks. We used a model of the brain in which neurons are described as agents that integrate the signals from other neurons and generate an output that spreads in the system. We then used the Shannon Information Entropy to decode those signals and compute the information transported in the grid as a function of its small-world-ness (SW), of the length (4t) and frequency (f ) of the originating stimulus. In numerical simulations in which SW was varied between 0 and 14 we found that, for certain values of 4t and f , communication is enhanced up to 30 times compared to unstructured systems of the same size. Moreover, we found that the information processing capacity of a system steadily increases with SW until the value SW ¼ 4:8 ± 1, independently on 4t and f . After this threshold, the performance degrades with SW and there is no convenience in increasing indefinitely the number of active links in the system. Supported by the findings of the work and in analogy with the exergy in thermodynamics, we introduce the concept of exordic systems: a system is exordic if it is topologically biased to transmit information efficiently. npj Systems Biology and Applications (2022)8:4 ; https://doi.org/10.1038/s41540-022-00215-y INTRODUCTION In biological systems, in tissues and organs, and the brain, the performance of a system depends less on the characteristics of a single cell and more on how those cells interact collectively to transport signals, information, or nutrients. The emergent properties of these systems arise from the cooperation of a great many elements and cannot be obtained or explained as the sum of the behaviors of each of their parts taken individually1–6. Practically, we can gain access to the precious understanding of those systems by representing them as networks, in which the cells are the nodes and the interactions between cells are the links of the network. Since networks are measurable, one can then establish a relationship between the topology and the emergent properties of a system derived from the collective function of its many parts. Among the numerous variables that can describe the networks’ topology, the small-world coefficient has particular relevance because both experimental evidence and numerical simulations suggest that small-world systems can transport information more efficiently than periodic or random networks of the same size7–12. A network has a small-world topology if nodes of the network are separated from each other by a small number of steps, and very often small-world networks are characterized by a certain number of clusters with many node-node intracluster interactions and less intercluster connections13–15. Nonetheless, while a variety of studies have examined how systems with a specific small-world topology behave under certain conditions, none of them illustrates how the information content of a system varies as a function of its small-world-ness. In this work, using numerical simulations we show how the efficiency of a 2-dimensional network of neurons changes as a function of its small-world characteristics. To do so, firstly we generated a great many configurations (  1000) with different topological characteristics. We placed in a fixed domain 500 points randomly sampled from Gaussian distributions where the number, mean, and standard deviation of the distributions were varied over large intervals. Then, we connected the points of the distributions proportionally to the inverse of their distance and to the local density of other points in their neighbors. This wiring algorithm, developed by one of the authors of this work16, guarantees local and global connectivity, typical of small-world networks. By varying the parameters of the algorithm as described in the Methods of the work, we obtained networks with values of small-world-ness falling in the 0–14 interval. The small-world-ness or small-world coefficient (SW) is a quantitative measure of the topological characteristics of a network relative to an equivalent random graph of that graph. It is defined in terms of the clustering coefficient (cc) and characteristic path length (cpl) as17: SW ¼ ccgraph =ccrand ; cplgraph =cplrand (1) thus small-world networks have high clustering and short paths compared to random graphs of the same size. For a random graph, SW ¼ 1. After having generated configurations with different values of SW, we evaluated how a signal is transported in those networks where the elements of the networks are artificial neurons that receive as an input the signal from other neurons and pass it to the grid upon integration over space and time. This scheme is 1 Department of Electric Engineering and Information Technology, University Federico II, 80125 Naples, Italy. 2Institute of Nanotechnology, National Research Council (CNR‐ NANOTEC), Campus Ecotekne, via Monteroni, Lecce 73100, Italy. 3Nanotechnology Research Center, Department of Experimental and Clinical Medicine, University of Magna Graecia, 88100 Catanzaro, Italy. ✉email: Published in partnership with the Systems Biology Institute F. Aprile et al. 1234567890():,; 2 Fig. 1 The neuronal network model. We simulated the transport of information in the networks, with their nodes represented as artificial neurons that integrate over time the current-pulse trains received as an input and produce, as an output, a discrete pattern of action potentials: in the model, the signals are represented as arrays of 0/1 (bits) (a). Information is decoded in the patterns of actions potentials: using information theory approaches and the Shannon information entropy, we obtained the information received and transmitted by each of the nodes of the networks (b). based on the repetition of a leaky integrate-and-fire model in the network as exhaustively reported in refs. 18,19. In analogy with the behavior of real neurons, the model generates as an output for each node of the grid a sequence of action potentials (train of spikes) that is encoded in the system as a binary sequence of 0 and 1 (Fig. 1a). Then, we used an information theory approach to decode the information stored in each node of the networks20–23. We computed for each sequence of 0 and 1 the associated value of Shannon Information Entropy H: H¼ S X P logðPÞ; (2) i where the index i runs over all the possible substates s of the system, and P is the probability of finding a (...truncated)