Canards, Clusters, and Synchronization in a Weakly Coupled Interneuron Model

Applying a low current to a recent biophysical model for inhibitory neurons in the cortex leads to mixed mode oscillations (MMOs), a mixture of spikes and subthreshold oscillations. At higher currents, the neurons fire regularly. We show that a specific slow potassium current underlies this behavior. Next we reduce this five-dimensional biophysical model for an inhibitory neuron to three dimensions with a slow/fast time-scale structure. We then show that there is a range of parameters under which the reduced model shows MMOs which can be explained by the canard phenomenon. Many inhibitory interneurons are coupled with electrical tight junctions (gap junctions). We show that such coupling combined with the underlying MMOs produces clustered solutions in large networks. We explain this using the thoery of weakly coupled oscillators and show that the complex dynamics arises from sensitivity near the canard.