Bifurcation theory is used to study properties of nonlinear analytical and computational models of isolated retinal horizontal cells. The analytical model is based on the published data of Shingai and Christensen describing steady-state I-V characteristics of horizontal cells isolated from catfish (Ictalurus punctatus) retina. The computational model is based on I-V characteristics of distinct macroscopic membrane currents observed in horizontal cells isolated from goldfish (Carassius auratus) retina. Slow-model dynamics are analyzed assuming that excitatory processes occur rapidly with respect to the time course of inactivation of the inward Ca2+ and outward K+ currents. A global bifurcation diagram plotting the location and stability properties of critical points as a function of photoreceptor-evoked horizontal-cell postsynaptic membrane conductance G(syn) is derived for the analytical model. The automated bifurcation analysis software AUTO is used to compute global bifurcation diagrams for the computational model. Bifurcation diagrams exhibit a bistable regime at small G(syn) values characterized by two stable and one unstable critical point and a monostable regime at larger G(syn) values characterized by a single attracting stable critical point. The transition between bistable and monostable behavior occurs at a G(syn) value of roughly 0.9 nS for the computational model and 1.7 nS for the analytical model. Estimates of horizontal-cell glutamate-channel conductance suggest that this transition corresponds to the activation of as few as 400-700 glutamate channels. Dark-evoked release of neurotransmitter from photoreceptors may therefore set horizontal-cell synaptic conductance G(syn) to a value within the monostable regime. Photoreceptor-evoked horizontal-cell membrane conductance, total Ca2+ channel conductance, and inactivation of the inward Ca2+ current shown to be the major factors controlling the bifurcation structure of the computational model. Inactivation of the inward Ca2+ currents is required to account for the dark resting potential of horizontal cells as well as light-evoked hyperpolarizing responses. Inactivation of the outward K+ current has little effect on model properties. Isolated horizontal cells generate Ca2+ action potentials, whereas cells in the intact retina normally do not. Simple procedures for modeling the slow dynamics of isolated horizontal-cell Ca2+ action potentials are described. Model analysis indicate that reported differences between the dynamics of isolated horizontal cells versus horizontal cells in the intact retina are in part a consequence of the loss of photoreceptor input (G(syn)) on dissociation.
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