Phys. Rev. E 67:021908 (2003)
Piecewise linear differential equations and integrate-and-fire neurons : insights from two-dimensional membrane models,
A. Tonnelier and W. Gerstner
We derive and study two-dimensional generalizations of integrate-and-fire models which can be found from a piecewise linear idealization of the FitzHugh-Nagumo or
Morris-Lecar model. These models give rise to new properties not present in one-dimensional integrate-and-fire models. A detailed analytical study of the models is presented.
In particular, (i) for the piecewise linear FitzHugh-Nagumo model, we determine analytically the bistability regime between stationary solutions and oscillations, that is, typical for
class-II models. (ii) In the piecewise Morris-Lecar model, we find a noncanonical class-I transition from a stationary state to oscillations with logarithmic dependence similar to
that found for leaky integrate-and-fire models. (iii) Furthermore, we establish a relation to the recently proposed resonate-and-fire model and show that a short input current
pulse can trigger several spikes.