Causal and Structural Connectivity of Pulse-Coupled Nonlinear Networks

We study the reconstruction of structural connectivity for a general class of pulse-coupled nonlinear networks and show that the reconstruction can be successfully achieved through linear Granger causality (GC) analysis. Using spike-triggered correlation of whitened signals, we obtain a quadratic relationship between GC and the network couplings, thus establishing a direct link between the causal connectivity and the structural connectivity within these networks. Our work may provide insight into the applicability of GC in the study of the function of general nonlinear networks.

Figure 1

(a) A network with two excitatory neurons and a directed connection from $x_1$ to $x_2$. (b) Effective adjacency matrix $\widehat{\mathbf{A}}$ constructed by GC, which captures the structural connectivity in (a). (c) The coincidence between $\widehat{\mathbf{A}}$ and $\mathbf{A}$ in (a) as a function of rate $\mu$ and magnitude $f$ in the Poisson drive. The white color indicates that $\widehat{\mathbf{A}}\ne \mathbf{A}$ and the black color for $\widehat{\mathbf{A}}=\mathbf{A}$. (d) A network with five excitatory neurons. (e) $\widehat{\mathbf{A}}$ constructed by GC, which captures the topology in (d). (f) Ranked GC in order of magnitude for a network of 100 excitatory neurons with random connectivity (the number of nonzero $A_{ij}$ is $\sim 2000$). The gray (red) line indicates the threshold in the gap of the ranked GC. As for the excitatory neuron case, parameters are chosen as $x_{\mathrm{ex}}=14/3$, ${\sigma }^{\mathrm{ex}}=2\mathrm{ms}$, $\tau =20\mathrm{ms}$, $x_{\mathrm{th}}=1$, $x_r=0$, and ${\tau }_{\mathrm{ref}}=2\mathrm{ms}$ [13]. Here, $\mu =1{\mathrm{ms}}^{-1}$, $f=0.007{\mathrm{ms}}^{-1}$, and $S^{\mathrm{ex}}=0.01{\mathrm{ms}}^{-1}$.

Figure 2

(a) Trajectories of $x_1$ [thin dark line (red)] and $x_2$ [dark line (black)] generated by model (1) and of ${\tilde{x}}_1$ [thick gray line (green)] and ${\tilde{x}}_2$ [thick light gray line (cyan)] generated by regression models. (b) Probability density of ${ϵ}^A$ [thick solid light gray line (cyan)] and ${\eta }^A$ [thin dark dashed line (black)] in the AR model. Inset: Trajectories of ${ϵ}^A$ [thick light gray line (cyan)] and ${\eta }^A$ [thin dark line (black)]. Here, $\mu =0.24{\mathrm{ms}}^{-1}$, $f=0.02{\mathrm{ms}}^{-1}$, and $S^{\mathrm{ex}}=0.01{\mathrm{ms}}^{-1}$.

Figure 3

(a) Spike-triggered correlation (background subtracted): $x_1{|}_2(\tau )$ [thick light gray line (cyan)] and $x_2{|}_1(\tau )$ [thin dark line (black)]. (b) Spike-triggered correlation: ${ϵ}^A{|}_{x_2}(\tau )$ [thick light gray line (cyan)] and ${\eta }^A{|}_{x_1}(\tau )$ [thin dark line (black)]. (c) GC (“dot” symbols) as a function of coupling strength $S^{\mathrm{ex}}$; the thick gray line (cyan) is a quadratic fit.

Figure 4

(a) The coincidence between the structural adjacency matrix $\mathbf{A}$ and the effective adjacency matrix $\widehat{\mathbf{A}}$ for a network with two inhibitory neurons. The network structural connectivity is the same as that in Fig. 1a. The white color indicates that $\widehat{\mathbf{A}}\ne \mathbf{A}$ and the black color that $\widehat{\mathbf{A}}=\mathbf{A}$. (b) Ranked GC in order of magnitude for a network with 80 excitatory and 20 inhibitory neurons. Each matrix entry $A_{ij}$ is randomly chosen as either zero or one (the number of nonzero $A_{ij}$ is $\sim 2000$). The gray (blue) line indicates the GC threshold $F_T$ obtained from the $p$-value ($p=0.001$) test. The accuracy of reconstruction is $\sim 96\%$. For the inhibition, parameters are chosen as $x_{\mathrm{in}}=-2/3$ and ${\sigma }^{\mathrm{in}}=5\mathrm{ms}$ [13]. Here, $\mu =0.24{\mathrm{ms}}^{-1}$, $f=0.02{\mathrm{ms}}^{-1}$, and $S^{\mathrm{in}}=0.01{\mathrm{ms}}^{-1}$.