Locally contractive dynamics in generalized integrate-and-fire neurons

Integrate-and-fire models of biological neurons combine differential equations with discrete spike events. In the simplest case, the reset of the neuronal voltage to its resting value is the only spike event. The response of such a model to constant input injection is limited to tonic spiking. We here study a generalized model in which two simple spike-induced currents are added. We show that this neuron exhibits not only tonic spiking at various frequencies but also the commonly observed neuronal bursting. Using analytical and numerical approaches, we show that this model can be reduced to a one-dimensional map of the adaptation variable and that this map is locally contractive over a broad set of parameter values. We derive a sufficient analytical condition on the parameters for the map to be globally contractive, in which case all orbits tend to a tonic spiking state determined by the fixed point of the return map. We then show that bursting is caused by a discontinuity in the return map, in which case the map is piecewise contractive. We perform a detailed analysis of a class of piecewise contractive maps that we call bursting maps and show that they robustly generate stable bursting behavior. To the best of our knowledge, this work is the first to point out the intimate connection between bursting dynamics and piecewise contractive maps. Finally, we discuss bifurcations in this return map, which cause transitions between spiking patterns.