Figure 1

Example of a spatial balanced network without translational invariance and with simple sinusoidal external input. (A) Raster plot of excitatory neuron spikes from a simulated network with \(N=5000\) neurons, recurrent connectivity given by Eqs. (2) and (3), and external input given by Eq. (4). (B) External input (green), mean recurrent excitatory input (red), mean recurrent inhibitory input (blue), and mean total input (black) to excitatory neurons as a function of neuron location for the same simulation as A. Currents were averaged over time (500 ms) and over the ten neurons nearest to each plotted location. Currents are computed with \(C_{m}=1\) so are units \(V/s\). (C)–(E) Firing rates of excitatory (red) and inhibitory (blue) neurons as a function of distance for \(N=1000\), 5000, and \(20\text{,}000\) respectively. Light solid curves are from simulations, dotted curves are from Eq. (14), and dashed curves from Eq. (15). Rates were averaged over all neurons in 200 evenly spaced bins and additionally averaged over \(4\times 10^{5}/N\) simulations each with duration 10 s. (F) Firing rate versus mean total input current for all excitatory neurons with \(N=5000\). Dots are from simulations and solid curve is the rectified linear fit used to derive the gain. (G) Same as F, but for inhibitory neurons