## Overview of models which contain mechanisms for capturing global coupling.

At any given time, the population activity pattern is defined by neurons which either spike (*s*_{i} = 1, dark discs) or are silent (*s*_{i} = 0, white discs). The probability of spiking is partially determined by an intrinsic firing bias (*α*_{i} for models without local interactions, or the diagonal terms of the coupling matrix *J* for models with local pairwise interactions). When local interactions between neurons are important, they can be parametrized by assigning each pair of neurons a coupling weight. Positive weight (orange) increases the likelihood of the paired neurons spiking together, while negative weight (blue) decreases the likelihood. The negative sum of the intrinsic firing biases of active neurons and the coupling weights of pairs which fire synchronously is referred to as the energy of the population activity pattern. The probability of a given pattern is simply proportional to the exponential of its negative energy. To capture correlations due to global coupling, previous studies considered models which bias the response probabilities with a function of the total network activity (here denoted as *K*, i.e., the sum of the activities of individual neurons). We introduce a different approach (shaded models in the figure) where global coupling is induced by mapping the energy of the activity pattern to its probability with an arbitrary (smooth and increasing) function exp(−*V* (*E*)).