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Describing the collective dynamics of large neural populations usinglow-dimensional models for averaged variables has long been an attractive taskin theoretical neuroscience. Recently developed reduction methods make itpossible to derive such models directly from the microscopic dynamics ofindividual neurons. To simplify the reduction, the Cauchy distribution isusually assumed for heterogeneous network parameters. Here we extend thereduction method for a wider class of heterogeneities defined by the$q$-Gaussian distribution. The shape of this distribution depends on theTsallis index $q$ and gradually changes from the Cauchy distribution to thenormal Gaussian distribution as this index changes. We derive the mean-fieldequations for an inhibitory network of quadratic integrate-and-fire neuronswith a $q$-Gaussian distributed excitability parameter. It is shown that thedynamic modes of the network significantly depend on the form of thedistribution determined by the Tsallis index. The results obtained from themean-field equations are confirmed by numerical simulation of the microscopicmodel.

Mean-field equations for neural populations with $q$-Gaussian heterogeneities