  Neurons are the central biological objects in understanding how the brain
works. The famous Hodgkin-Huxley model, which describes how action potentials
of a neuron are initiated and propagated, consists of four coupled nonlinear
differential equations. Because these equations are difficult to deal with,
there also exist several simplified models, of which many exhibit
polynomial-like non-linearity. Examples of such models are the Fitzhugh-Nagumo
(FHN) model, the Hindmarsh-Rose (HR) model, the Morris-Lecar (ML) model and the
Izhikevich model. In this work, we first prescribe the biologically relevant
parameter ranges for the FHN model and subsequently study the dynamical
behaviour of coupled neurons on small networks of two or three nodes. To do
this, we use a computational real algebraic geometry method called the
Discriminant Variety (DV) method to perform the stability and bifurcation
analysis of these small networks. A time series analysis of the FHN model can
be found elsewhere in related work[15].
