Bifurcation analysis of nonlinear retinal horizontal cell models. II. Network properties

1. We have previously presented a model of horizontal-cell soma isolated from fish retina. The model consists of a synaptic conductance representing input from photoreceptors in parallel with voltage-dependent membrane currents. Membrane-current models are based on I-V curves measured in isolated fish horizontal cells. Bifurcation theory was used to analyze model properties. The major findings of this study were 1) the inward Ca²⁺ current must be inactivated to account for horizontal-cell resting potentials and hyperpolarizing responses to light stimuli in a background of dark, and 2) the synaptic conductance controls the bifurcation structure of the model, with bistable behavior occurring at small and monostable behavior occurring at large values of the synaptic conductance. The synaptic conductance at the point of transition from bistable to monostable behavior corresponds to the activation of as few as 100 synaptic channels. Thus tonic synaptic input from photoreceptors and inactivation of the inward Ca²⁺ current act to 'linearize' responses of isolated horizontal-cell models. 2. The model described in this paper extends these analyses to large networks of horizontal cells in which each cell is coupled resistively to its nearest neighbors and is modeled with the use of the full complement of nonlinear membrane currents. Network responses to arbitrary patterns of conductance change (simulating inputs from photoreceptors), current-, or voltage-clamp stimuli are computed using the Newton iteration. The Newton descent direction is computed using either conjugate gradient (CG) or preconditioned CG algorithms. 3. An analysis of network stability properties is performed. Network I-V curves are computed by voltage-clamping the center node and computing the current required to maintain the clamp voltage. Computations are performed on networks of model cells in which the Ca²⁺ current is fully activated and the synaptic conductance is zero, thus making each cell as nonlinear as possible. Coupling conductance values slightly greater than 100 pS provide a current shunt sufficient to prevent the generation of Ca²⁺ action potentials in the network. This coupling conductance corresponds to the conductance of as few as two gap-junction channels and is more than two orders of magnitude less than the coupling known to exist between pairs of cultured horizontal cells. It therefore seems unlikely that any pharmacologic manipulation, save total block of gap-junction channel production, can decouple the network sufficiently to enable generation of Ca²⁺ plateau action potentials in response to localized current injection. 4. Coupling conductances of 30-50 nS yield a model space constant (156-200 μm) and DC transfer function slope resistance (271-350 KΩ) in excellent agreement with experimental data from fish horizontal-cell somas (mean space constant 170 μm; mean slope resistance 270 KΩ). These coupling conductances are within the range mesured in cultured pairs of horizontal cells by Lasater and Dowling (17-50 nS). Slope resistance (at the rest potential) of a model isolated cell is 715 MΩ. Predicted network input resistance at the dark resting potential is 9-14 MΩ for coupling conductances of 30-50 nS. 5. Short-term effects of retinal perfusion with dopamine on area-response curves are modeled by assuming that dopamine decouples cells by a factor of 10 and increases synaptic conductance by a factor of 3. The powerful suppression of the peak magnitude of area-response curves seen after long-term perfusion with dopamine is mimicked by slightly larger increases in the synaptic conductance (roughly a factor of 10). Depolarizations in response to simulated dopamine application are similar to those observed experimentally. The occasional hyperpolarizations in response to dopamine that have been reported are not predicted by this model. 6. Considered with previous modeling results, analyses show that inactivation of the Ca²⁺ current at the dark resting potential, synaptic input from photoreceptors, and the powerful current shunting effect of gap junctions are factors contributing to the linearization of horizontal cell steady-state responses to stimuli presented in a background of dark.