We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x--y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e.p.) is separated from the outer stable and unstable e.ps by two saddle points. If the system is started at rest at the middle stable e.p. it has the following complex properties: (i) A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. (ii) A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency--current relationship. (iii) Hyperpolarization from rest is followed by post-inhibitory rebound. (iv) The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x--y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.