Analysis and reduction of a modeled bursting neuron
Butera, Robert John, Jr
Clark, John W., Jr.
Doctor of Philosophy
A modeled bursting neuron was analyzed using methods based upon geometric singular perturbation theory. The mathematical mechanism of bursting in the model consisted of two slow variables traversing the state-space across a saddle-node bifurcation that defined an interface between quiescence and periodic spiking of the subsystem of fast variables. The parabolic nature of the burst was due to the homoclinicity of the saddle-node bifurcation. The response to the model to perturbatory current pulses was analyzed in the state-space of the slow variables. To aid in the analysis nullclines were calculated, including "average" nullclines, which are the nullclines of the slow-variables averaged over one period of the oscillation of the fast subsystem. The averaged nullclines predicted the existence of beating solutions. The relative stability of beating solutions at different parameter sets was related to specific biophysical mechanisms within the model. In addition, the manner in which the equilibrium and average nullclines for a given variable approached each other provided an indication as to which slow variables' dynamics were significantly perturbed by the firing of action potentials. The results of the numerical analysis were applied to develop a reduced model of the underlying subthreshold oscillations (slow-wave) in membrane potential. Two different low-order models were developed: a 3-variable model, which mimicked the slow-wave of the full model in the absence of action potentials and a second 4-variable model, which included expressions accounting for the perturbatory effects of action potentials on the slow-wave. The 4-variable model more accurately predicted the activity mode (bursting, beating, or silence) in response to application of extrinsic stimulus current or modulatory agents. The 4-variable model also possessed a phase-response curve that was very similar to that of the original 11-variable model. The results suggest that low-order models of bursting cells which do not consider the effects of action potentials may erroneously predict modes of activity and transient responses of the full model upon which the reductions are based. These results also show that it is possible to develop low-order models that retain many of the characteristics of the activity of the higher-order system.
Neurosciences; Mathematics; Biomedical engineering