Dominance of Metric Correlations in Two-Dimensional Neuronal Cultures Described through a Random Field Ising Model

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.

Figure 1

Experimental results and model construction. (a) Fraction of responding neurons $\varphi$ at stimulation voltage $V$ for a homogeneous culture ($\rho \simeq 560\mathrm{n}/{\mathrm{mm}}^2$) with only excitatory connections active. From left to right, symbols are experimental data at $[\mathrm{CNQX}]=0$, 100, 300, 500, 700, 1000 nM, and $10\mu \mathrm{M}$. The values of $m=\tilde{m}(1+[\mathrm{CNQX}]/k_d)$ are also indicated, with $\tilde{m}=15$ and $K_d=300\mathrm{nM}$. The grey area illustrates the disintegration of the giant component $g$. The solid lines correspond to the fit of the mean-field approximation to the experiments, and with parameters $H_0=4.2\pm 0.1\mathrm{V}$, ${\sigma }_h=0.80\pm 0.03\mathrm{V}$, and $m_D=39\pm 1$. (b) Bright field image of a region of a neuronal culture. Round objects are neurons. (c) Corresponding fluorescence image. The brightest spots (yellow arrow) are neurons responding to the stimulation. Nonresponding neurons remain darker (red arrow). (d) Disintegration curves $g(m)$ for 3 different neuronal densities $\rho$, each curve showing an average over 4 experimental realizations, and error bars are standard deviation. $m_D$ marks the critical value of $m$ at transition. Lines are a guide to the eye. (e) Sketch of the analytical solution in the mean-field approach and the construction of the giant component $g$.

Figure 2

Comparison between experiments and NNIM. Experimental $g(m/m_D)$ data points for the three densities $\rho$ of Fig. 1. Vertical error bars are standard deviation, and horizontal error bars account for the uncertainty in determining $m_D$. The solid black line shows the analytical mean-field solution, while the dashed one corresponds to the fit between metric simulations and experimental data. Simulations are an average over 5 network realizations, with the light gray area showing the standard deviation.

Figure 3

Simulations of $g(m/m_D)$ curves for contrasting $\mathrm{\Lambda }$. Main plot: Normalized disintegration curves for homogeneous (circles) and aggregated (triangles) metric configurations. The null models for both configurations are practically indistinguishable, were averaged out for clarity (diamonds), and neatly follow the mean field. The curves are obtained from the numerical integration of the general NNIM in $10\times 10{\mathrm{mm}}^2$ networks, and the normalization with $m_D$ is determined through a fit to the mean-field solution (solid line). Each curve is an average over 10 networks. Vertical error bars denote standard deviation, and horizontal bars account for the uncertainty in the determination of $m_D$. The top left and bottom right spatial maps illustrate, respectively, typical aggregated and homogeneous realizations. Scale bars are 1 mm. The top right inset shows the distribution of connections for both realizations, with $\overline{k}=150$.

Figure 4

Deviation from the mean field due to metric correlations. (a) Log-linear representation of $ϵ(\mathrm{\Lambda })$ for different network blueprints. Each simulation corresponds to a particular network realization. Solid triangles correspond to the data shown in Fig. 1 and Fig. 2, each point averaged over 4 cultures. Vertical error bars are standard deviation. Open triangles are individual cultures. Horizontal error bars reflect the uncertainty in measuring $\mathrm{\Lambda }$. Curves are linear fits, with the accompanying value of the slope. Regression coefficients for null models and metric networks are, respectively, $r=0.90$ and $r=0.84$. The $g(m)$ curves for the points marked with * are shown in the Supplemental Material. (b) Highly contrasted image of a homogeneous culture, $\mathrm{\Lambda }\simeq 0.2$. (c) Corresponding image for a strongly aggregated culture, $\mathrm{\Lambda }\simeq 0.6$. Scale bars are $250\mu \mathrm{m}$.