## Abstract

We add a slow hyperpolarization-activated inward current I<latex>$_{\text{H}}$</latex> = g<latex>$_{\text{H}}$</latex>m<latex>$_{\text{H}}$</latex>(v - v<latex>$_{\text{H}}$</latex>) to our previous model of rebound bursting (Hindmarsh & Rose Phil. Trans. R. Soc. Lond. B 346, 129-150 (1994a)) to give a four-dimensional physiological model, and a corresponding four-dimensional model of the model. The physiological model generates periodic `bursts of bursts' or `spindles' resembling those recorded experimentally in thalamocortical (TC) neurons. The model of the model is simplified to a two-dimensional system having a limit cycle which corresponds to the slow spindle oscillation of the physiological model. Analysis of the stability of this two-dimensional model allows us to divide the parameter space of the slope (<latex>$\gamma _{m_{\text{H}}}$</latex>) and shift (<latex>$\theta _{m_{\text{H}}}$</latex>) parameters of m<latex>$_{\text{H}_{\infty}}$</latex>(v) into regions in which the model generates spindles or continuous bursting. This enables us to determine the parameter values required for spindling in the physiological model and to explain the experimentally observed effects of noradrenaline. Next we examine whether a cell at a stable equilibrium point can be driven into spindling by applying a sinusoidal input at the resonant frequency. This is done by averaging the equations for the driven model of the model. Analysis of the stability of these averaged equations shows how the regions of the (<latex>$\theta _{m_{\text{H}}}$</latex>, <latex>$\gamma _{m_{\text{H}}}$</latex>) parameter space change when the system is driven by a sinusoidal input. This enables us to choose parameter values for a physiological model of a driven spindle. We show that if the physiological model is modified to include a voltage-dependent time constant for m<latex>$_{\text{H}}$</latex>, spindles, similar to those of TC cells, can be obtained with a small Ca<latex>$^{2+}$</latex>-activated K<latex>$^{+}$</latex> current. Finally our knowledge of the form of the bifurcation diagram and the conditions for resonance leads to a new suggestion for the roles of GABA<latex>$_{\text{A}}$</latex> and GABA<latex>$_{\text{B}}$</latex> inhibitory postsynaptic potentials when TC cells are driven into spindling by neurons of the nucleus reticularis thalami.