| Abstract: |
| According to their electrophysiological properties, neurons can be classified as Type I, Type II and Type III, based on their firing properties in response to sustained injected currents. Type III neurons are the least studied class of neurons in this classification. They may exhibit transient spiking to current injection, but they will not fire continuously no matter how much excitatory current is applied. In [Meng, Huguet, Rinzel; DCDS 2012] a range of interesting input processing features is describe as being associated with Type III neurons, including {\em post-inhibitory facilitation (PIF)} and {\em slope detection}. For instance, PIF refers to the phenomenon in which an excitatory (positive) input that fails to induce a spike in a resting neuron can induce a spike when applied with some lag following an inhibitory (negative) input. However, the conditions needed for these properties to arise have not yet been established analytically.
We consider a hybrid neuronal model that combines continuous evolution of trajectories up to a spiking event defined by the finite-time blow-up of the voltage variable together with a discrete jump condition that resets positions of trajectories after spiking occurs. By employing phase-space methods such as, for example, asymptotic rivers and slow-fast analysis we give the first rigorous mathematical treatment of PIF and slope detection in a neuron model. |
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