Figure 1

Examples of multistage subnetworks used for expanding ${\rho }_{\mathrm{full}}$ and [$b$]. Stages 1, 2, and 3 are blue, red and green, respectively. Left: A sample network. Neurons are indicated by circles, and the connectivity matrix is indicated by solid black arrows. We consider the firing rate of the central neuron, denoted by [$b$]. Center: Displayed on the top is the topological structure of the particular two-stage 3rd order subnetwork described in the text. Each stage $\gamma$ of this subnetwork involves a set of receivers and transmitters (denoted by $\{Z_{\alpha }^{\gamma }{\}}_{\alpha }$ and $\{W_{\beta }^{\gamma ,\alpha }{\}}_{\alpha ,\beta }$, respectively). The degree $p_{\alpha }^{\gamma }$ of every receiver $Z_{\alpha }^{\gamma }$ is equal to the number of arrowheads pointing into $Z_{\alpha }^{\gamma }$, and the degree $q_{\beta }^{\gamma ,\alpha }$ of every transmitter-receiver pair $W_{\beta }^{\gamma ,\alpha }$, $Z_{\alpha }^{\gamma }$ is equal to the number of arrowheads on the arrow connecting the pair. The stages span time in the sense of “causality”, since the contribution to ${\rho }^{[3]}$ coming from stage 1 involves the correction associated with stage 2. Displayed on the bottom is one of the four topologically equivalent 3rd order subnetworks within the full network which has this two-stage structure. Right: Displayed on the top is the topological structure of the three-stage “loop” subnetwork described in the text. Displayed on the bottom is the only subnetwork within the full network with this same structure which contributes to $[b{]}^{[3]}$.Reuse & Permissions

Figure 2

Numerical examples. Left panel: Here we consider a network of discrete state conductance-based integrate-and-fire neurons [8]. We calculate (to first order) the effects of moderate coupling on a single-neuron’s autocorrelation (defined as $[a\underset{\theta }{\to }a]-[a][a]$), and compare the results (solid) with measurements from a 32-neuron simulation (dotted). Right panels: Here we consider a class of random networks on $N=8$ current-based integrate-and-fire neurons (with ${\tau }_V=20\mathrm{ms}$), constructed with sparsity coefficient 0.5. For each random network the neurons are driven by independent Poisson spike trains (strength 0.05 and rate 125 Hz), and the nonzero coupling strengths are fixed constant at $\Delta$. The various panels show the histograms of time-averaged neuronal firing rates produced by this class of random networks as calculated to 2nd-order using our subnetwork expansion (solid) and via simulation (dotted). The red (blue) arrowheads below the firing-rate axis indicate the calculated (simulated) time-averaged neuronal firing rates produced by an all-to-all coupled network with failure coefficient 0.5. Note that the theory matches the simulations very well, even for large $\Delta$ (the maximum possible $\Delta$ is 1).Reuse & Permissions