  Experimental characterization of neuronal dynamics involves recording both of
spontaneous activity patterns and of responses to transient and sustained
inputs. While much theoretical attention has been devoted to the spontaneous
activity of neurons, less is known about the dynamic mechanisms shaping their
responses to transient inputs, although these bear significant physiological
relevance. Here, we study responses to transient inputs in a widely used class
of neuron models (nonlinear adaptive hybrid models) well-known to reproduce a
number of biologically realistic behaviors. We focus on responses to transient
inputs that have been previously associated with Type III neurons, arguably the
least studied category in Hodgkin's classification, which are those neurons
that never exhibit continuous firing in response to sustained excitatory
currents. The two phenomena that we study are post-inhibitory facilitation, in
which an otherwise subthreshold excitatory input can induce a spike if it is
applied with proper timing after an inhibitory pulse, and slope detection, in
which a neuron spikes to a transient input only when the input's rate of change
is in a specific, bounded range. We analyze the origin of these phenomena in
nonlinear hybrid models and provide a geometric characterization of dynamical
structures associated with PIF in the system and an analytical study of slope
detection for tent inputs. While the necessary and sufficient conditions for
these behaviors are easily satisfied in neurons with Type III excitability, our
proofs are quite general and valid for neurons that do not exhibit Type III
excitability as well. This study therefore provides a framework for the
mathematical analysis of these responses to transient inputs associated with
Type III neurons in other systems and for advancing our understanding of these
systems' computational properties.
