## Abstract

The electrical properties of the different anatomical types of retinal ganglion cells in the cat were calculated on the basis of passive cable theory from measurements made on histological material. Standard values for the electrical parameters were assumed (R<latex>$_i$</latex> = 70 <latex>$\Omega$</latex> cm, C<latex>$_m$</latex> = 2 <latex>$\mu$</latex>F cm<latex>$^{-2}$</latex>, R<latex>$_m$</latex> = 2500 <latex>$\Omega$</latex> cm<latex>$^2$</latex>). We conclude that these neurons need not be equipotential despite their small dimensions, mainly because of their extensive branching. The interactions between excitation and inhibition when the inhibitory battery is near the resting potential can be strongly nonlinear in these cells. To characterize the different types of ganglion cells in terms of this property we introduce F, the factor by which the soma depolarization induced by a given excitatory input is reduced by inhibition. In this framework we analyse some of the integrative properties of an arbitrary passive dendritic tree and we then derive the functional properties that are characteristic for the various types of ganglion cells. Our main results are: (i) Nonlinear saturation at the synapses may be made effectively smaller by spreading the same (conductance) input among several subunits on the dendritic field. Subunits are defined as regions of the dendritic field that are somewhat isolated from each other and roughly equipotential within. (ii) Shunting inhibition can specifically veto an excitatory input, if it is located on the direct path to the soma. The F values can then be very high even when the excitatory inputs are much larger than the inhibitory, as long as the absolute value of inhibition is not too small. Inhibition more distal than excitation is much less effective. (iii) Specific branching patterns coupled with suitable distribution of synapses are potentially able to support complex information processing operations on the incoming excitatory and inhibitory signals. The quantitative analysis of the morphology of cat retinal ganglion cells leads to the following specific conclusions: (i) None of the cells examined satisfies Rall's equivalent cylinder condition. The dendritic tree cannot be satisfactorily approximated by a non-tapering cylinder. (ii) Under the assumption of a passive membrane, the dendritic architecture of the different types of retinal ganglion cells reflects characteristically different electrical properties, which are likely to be relevant for their physiological function and their information processing role: (a) <latex>$\alpha$</latex> cells have spatially inhomogeneous electrical properties, with many subunits. Within each subunit nonlinear effects may take place; between subunits good linear summation is expected. F values are relatively low. (b) <latex>$\beta$</latex> cells at small eccentricities have rather homogeneous electrical properties. Even distal inputs are weighted rather uniformly. Electrical inhomogeneities of the <latex>$\alpha$</latex> type appear for <latex>$\beta$</latex> cells at larger eccentricities. F values are low. (c) <latex>$\gamma$</latex>-like cells have few subunits, each with high input resistance underlying nonlinear saturation effects possibly related to a sluggish character. F values are high: inhibition of the shunting type can interact in a strongly nonlinear way with excitatory conductance inputs. (d) <latex>$\delta$</latex>-like cells show many subunits with a high input resistance, covering well the dendritic area. Within each subunit inhibition on the direct path to the soma can specifically veto a more distal excitation. It is conjectured that such a synaptic organization superimposed on the <latex>$\delta$</latex> cell morphology underlies directional selectivity to motion. (iii) Most of our data refer to steady-state properties. They probably apply, however, to all light evoked signals, since transient inputs with time to peak of 30 ms or more can be treated in terms of steady-state properties of the ganglion cells studied. (iv) All our results are affected only slightly by varying the parameter values within reasonable ranges. If, however, the membrane resistance were very high, all ganglion cells would approach equipotentiality. For R<latex>$_m$</latex> = 8000 <latex>$\Omega$</latex> cm<latex>$^2$</latex> subunits essentially disappear in all types of ganglion cells (for steady state inputs). Our results concerning nonlinear interaction of excitation and inhibition (F values) would, however, remain valid even for much larger values of R<latex>$_m$</latex> and for any value of R<latex>$_i$</latex> larger than 30-50 <latex>$\Omega$</latex> cm. The critical requirement is that peak inhibitory conductance changes must be sufficiently large (around 5 x 10<latex>$^{-8}$</latex> S) with an equilibrium potential close to the resting potential. Underestimation of the diameters of the dendritic branches may affect these conclusions (F could be significantly lower).