In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory model neurons. It was developed by Hugh R. Wilson and Jack D. Cowan and extensions of the model have been widely used in modeling neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit cycle behavior, i.e. neural oscillations, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response. Mathematical description The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes. All cells receive the same number of excitatory and inhibitory afferents, that is, all cells receive the same average excitation, x(t). The target is to analyze the evolution in time of number of excitatory and inhibitory cells firing at time t, and respectively. The equations that describes this evolution are the Wilson-Cowan model: where: and are functions of sigmoid form that depends on the distribution of the trigger thresholds (see below) is the stimulus decay function and are respectively the connectivity coefficient giving the average number of excitatory and inhibitory synapses per excitatory cell; and its counterparts for inhibitory cells and are the external input to the excitatory/inhibitory populations. If denotes a cell's threshold potential and is the distribution of thresholds in all cells, then the expected proportion of neurons receiving an excitation at or above threshold level per unit time is: , that is a function of sigmoid form if is unimodal. If, instead of all cells receiving same excitatory inputs and different threshold, we consider that all cells have same threshold but different number of afferent synapses per cell, being the distribution of the number of afferent synapses, a variant of function must be used: Derivation of the model If we denote by the refractory period after a trigger, the proportion of cells in refractory period is and the proportion of sensitive (able to trigger) cells is . The average excitation level of an excitatory cell at time is: Thus, the number of cells that triggers at some time is the number of cells not in refractory interval, AND that have reached the excitatory level, , obtaining in this way the product at right side of the