The classical Hodgkin-Huxley (HH) point-neuron model of action potential
generation is four-dimensional. It consists of four ordinary differential
equations describing the dynamics of the membrane potential and three gating
variables associated to a transient sodium and a delayed-rectifier potassium
ionic currents. Conductance-based models of HH type are higher-dimensional
extensions of the classical HH model. They include a number of supplementary
state variables associated with other ionic current types, and are able to
describe additional phenomena such as sub-threshold oscillations, mixed-mode
oscillations (subthreshold oscillations interspersed with spikes), clustering
and bursting. In this manuscript we discuss biophysically plausible and
phenomenological reduced models that preserve the biophysical and/or dynamic
description of models of HH type and the ability to produce complex phenomena,
but the number of effective dimensions (state variables) is lower. We describe
several representative models. We also describe systematic and heuristic
methods of deriving reduced models from models of HH type