Acetylcholine (ACh), one of the brain’s most potent neuromodulators, can affect intrinsic neuron properties through blockade of an M-type potassium current. The effect of ACh on excitatory and inhibitory cells with this potassium channel modulates their membrane excitability, which in turn affects their tendency to synchronize in networks. Here, we study the resulting changes in dynamics in networks with inter-connected excitatory and inhibitory populations (E–I networks), which are ubiquitous in the brain. Utilizing biophysical models of E–I networks, we analyze how the network connectivity structure in terms of synaptic connectivity alters the influence of ACh on the generation of synchronous excitatory bursting. We investigate networks containing all combinations of excitatory and inhibitory cells with high (Type I properties) or low (Type II properties) modulatory tone. To vary network connectivity structure, we focus on the effects of the strengths of inter-connections between excitatory and inhibitory cells (E–I synapses and I–E synapses), and the strengths of intra-connections among excitatory cells (E–E synapses) and among inhibitory cells (I-I synapses). We show that the presence of ACh may or may not affect the generation of network synchrony depending on the network connectivity. Specifically, strong network inter-connectivity induces synchronous excitatory bursting regardless of the cellular propensity for synchronization, which aligns with predictions of the PING model. However, when a network’s intra-connectivity dominates its inter-connectivity, the propensity for synchrony of either inhibitory or excitatory cells can determine the generation of network-wide bursting.