Dominance of Metric Correlations in Two-Dimensional Neuronal Cultures Described through a Random Field Ising Model Lluís Hernández-Navarro, 1,2 Javier G. Orlandi, 1,3 Benedetta Cerruti, 1,4 Eduard Vives, 1,2 and Jordi Soriano 1,2,* 1 Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona 08028, Catalonia, Spain 2 Universitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Catalonia, Spain 3 Complexity Science Group, Department of Physics and Astronomy, University of Calgary, Calgary, Canada T2N 1N4 4 Center for Genomic Science of IIT@SEMM, Istituto Italiano di Tecnologia (IIT), 20139 Milan, Italy (Received 1 June 2016; revised manuscript received 24 March 2017; published 18 May 2017) We introduce a novel random field Ising model, grounded on experimental observations, to assess the importance of metric correlations in cortical circuits in vitro. Metric correlations arise from both the finite axonal length and the heterogeneity in the spatial arrangement of neurons. The experiments consider the response of neuronal cultures to an external electric stimulation for a gradually weaker connectivity strength between neurons, and in cultures with different spatial configurations. The model can be analytically solved in the metric-free, mean-field scenario. The presence of metric correlations precipitates a strong deviation from the mean field. Null models of the same networks that preserve the distribution of connections recover the mean field. Our results show that metric-inherited correlations in spatial networks dominate the connectivity blueprint, mask the actual distribution of connections, and may emerge as the asset that shapes network dynamics. DOI: 10.1103/PhysRevLett.118.208101 Living neuronal circuits exhibit complex collective behavior that is shaped in great measure by the connectivity among neurons [1]. In the process of associating specific network properties to key dynamical traits it was usually assumed that the spatial constraints of the neuronal circuit could be disregarded. In fact, the combination of physical embedding, spatial organization of the neurons, and wiring cost [2,3] not only prevents a neuron from arbitrarily connecting with any other, but naturally shapes strong correlations and spatially inherited features that can be more important in shaping dynamics than the actual dis- tribution of connections [4]. The importance of these elements and their interrelation is the focus of great attention in the context of spatial networks [5], a framework that analyzes those systems in which nodes and links are embedded in space, and where the physical distance among nodes plays a central role, such as in air transportation, social networks, the Internet, and disease spreading [5–7]. In this Letter we explore the importance of spatial embedding using two-dimensional neuronal cultures [8–10] in which we tune the spatial arrangement of neurons. We frame our observations in the context of an Ising model that can be analytically solved in the metric- free, mean-field approach. This solution provides a refer- ence scenario for the quantification of metric effects and their impact on network behavior. Our work is grounded on the percolation experiments in neuronal cultures described in Refs. [11–13]. Cultures were prepared by dissociating rat embryonic cortical tissue and plating the cellular population over glass, giving rise to a de novo network that contained both excitatory and inhibitory neurons. For clarity, we show here excitation- only data. Inhibition was fully blocked with 40 μM bicucul- line. The spatial distribution of neurons varied between a uniform coverage (homogeneous configuration) and a highly localized one (aggregated configuration), and was set by adjusting neuronal adhesion onto the glass substrate [14]. Although neurons covered an area of about 130 mm 2 , only a region of 0.8 × 0.7 mm 2 (W × H) was imaged. Details on experimental procedures, possible artifacts associated with the limited imaging window, and the effect of inhibition are provided in the Supplemental Material. As shown in Fig. 1(a), experiments consisted in obtaining the set of response curves of the neuronal culture upon a series of short, 20 ms biphasic electric stimulations of gradually higher voltages V . Stimulation was delivered through bath electrodes that simultaneously affected all neurons, which responded either according to their sensitivity or in response to other neuronal activations. The neuronal network response to stimulation was quantified through fluorescence calcium imaging [Figs. 1(b)–1(c)], and by counting the fraction of neurons ϕ that responded to a given voltage V [18]. The response curves ϕðV Þ depend on the connectivity between neurons. Synaptic efficacy was reduced by target- ing the α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA)-excitatory receptors with the antagonist 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX) [11,12]. For ½CNQX¼ 0 a neuron requires ~ m ≃ 15 simultaneous inputs to fire and excite other neurons, a condition known as quorum [13,20]. However, more inputs are required as CNQX increases since synaptic strength is weaker, which is expressed as m ¼ ~ mð1 þ½CNQX=K d Þ, with PRL 118, 208101 (2017) PHYSICAL REVIEW LETTERS week ending 19 MAY 2017 0031-9007=17=118(20)=208101(5) 208101-1 © 2017 American Physical Society