  We study the dynamics of a low-dimensional system of coupled model neurons as
a step towards understanding the vastly complex network of neurons in the
brain. We analyze the bifurcation structure of a system of two model neurons
with unidirectional coupling as a function of two physiologically relevant
parameters: the external current input only to the first neuron and the
strength of the coupling from the first to the second neuron. Leveraging a
timescale separation, we prove necessary conditions for multiple timescale
phenomena observed in the coupled system, including canard solutions and mixed
mode oscillations. For a larger network of model neurons, we present a
sufficient condition for phase locking when external inputs are heterogeneous.
Finally, we generalize our results to directed trees of model neurons with
heterogeneous inputs.
