How pattern formation in ring networks of excitatory and inhibitoryspiking neurons depends on the input current regime

Pattern formation, i.e., the generation of an inhomogeneous spatial activity distribution in a dynamical system with translation invariant structure, is a well-studied phenomenon in neuronal network dynamics,specifically in neural field models. These are population models to describe the spatio-temporal dynamics of large groups of neurons in terms of macroscopic variables such as population firing rates. Though neural field models are often deduced from and equipped with biophysically meaningfulproperties, a direct mapping to simulations of individual spiking neuron populations is rarely considered. Neurons have a distinct identity defined by their action on their postsynaptic targets. In its simplest form they act either excitatorily or inhibitorily.When the distribution of neuron identities is assumed to be periodic, pattern formation can be observed, given the coupling strength is supercritical, i.e., larger than a critical weight. We find that this critical weight is strongly dependent on the characteristics of the neuronal input, i.e., depends on whether neurons are mean- orfluctuation driven, and different limits in linearizing the full non-linear system apply in order to assess stability.In particular, if neurons are mean-driven, the linearization has a very simple form and becomesindependent of both the fixed point firing rate and the variance of the input current, while in the very strongly fluctuation-driven regime the fixed point rate, as well as the input mean and variance areimportant parameters in the determination of the critical weight.We demonstrate that interestingly even in ``intermediate'' regimes, when the system is technically fluctuation-driven, the simple linearization neglecting the variance of the input can yield the better prediction of the critical couplingstrength. We moreover analyze the effects of structural randomness by rewiring individualsynapses or redistributing weights, as well as coarse-graining on pattern formation.