Passive Dendritic Tree Research Articles

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree extending Rall's equivalent cylinder model is presented. It is obtained from a combination of cable theory with the continuum approach. Replacing the fine structure of the branching dendrites by an equivalent, conductive medium characterized by averaged electrical parameters, the one-dimensional cable equations with spatially varying parameters are derived. While these equations can be solved in general only numerically, we were able to formulate a general branching condition (comprising Rall's 3/2 power relationship as a special case) under which analytical solutions can be deduced from those of the equivalent cylinder model. This model allows dendritic trees with a greater variety of branching patterns than before to be analytically treated.

A continuous cable method for determining the transient potential in passive dendritic trees of known geometry.

Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment delta t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.

Nonlinear interactions in a dendritic tree: localization, timing, and role in information processing.

In a passive dendritic tree, inhibitory synaptic inputs activating ionic conductances with an equilibrium potential near the resting potential can effectively veto excitatory inputs. Analog interactions of this type can be very powerful if the inputs are appropriately timed and occur at certain locations. We examine with computer simulations the precise conditions required for strong and specific interactions in the case of a delta-like ganglion cell of the cat retina. We find some critical conditions to be that (i) the peak inhibitory conductance changes must be sufficiently large (i.e., approximately equal to 50 nS or more), (ii) inhibition must be on the direct path from the location of excitation to the soma, and (iii) the time course of excitation and inhibition must substantially overlap. Analog AND-NOT operations realized by satisfying these conditions may underlie direction selectivity in ganglion cells.

Retinal ganglion cells: a functional interpretation of dendritic morphology.

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 (R1= 70 Ω cm, Cm= 2 μF cm-2,Rm= 2500 Ω cm2). 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 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. TheFvalues 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 Rail’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) α cells have spatially inhomogeneous electrical properties, with many subunits. Within each subunit nonlinear effects may take place; between subunits good linear summation is expected.Fvalues are relatively low. (b) β cells at small eccentricities have rather homogeneous electrical properties. Even distal inputs are weighted rather uniformly. Electrical inhomogeneities of the a type appear for P cells at larger eccentricities.Fvalues are low. (c) γ-like cells have few subunits, each with high input resistance underlying nonlinear saturation effects possibly related to a sluggish character.Fvalues are high: inhibition of the shunting type can interact in a strongly nonlinear way with excitatory conductance inputs. (d) δ-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 δ 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. ForRm= 8000 Ω cm2subunits essentially disappear in all types of ganglion cells (for steady state inputs). Our results concerning nonlinear interaction of excitation and inhibition ( values) would, however, remain valid even for much larger values ofRmand for any value ofR1larger than 30-50 Ω cm. The critical requirement is that peak inhibitory conductance changes must be sufficiently large (around 5 x 10-8S) with an equilibrium potential close to the resting potential. Underestimation of the diameters of the dendritic branches may affect these conclusions (Fcould be significantly lower).

The response of a spatially distributed neuron to white noise current injection

The depolarization of passive nerve cylinder or dendritic tree in the equivalent cylinder representation is assumed to satisfy the cable equation. We consider in detail the effects of white noise current injection at a given location for the case of sealed end boundary conditions and for an initial resting state. The depolarization at a point is a Gaussian random process but is not Markovian. Expression (infinite series) are obtained for the expectation, variance, spatial and temporal covariances of the depolarization. We examine the steady state expectation and variance and investigate how these are approached in time over the whole neuronal surface. We consider the relative contributions of various terms in the series for the expectation and variance of the depolarization at x = 0 (soma, trigger zone, recording electrode) for various positions of the input process. It is found that different numbers of terms must be taken to obtain a reasonable approximation depending on whether the stimulus is at proximal, central or distal parts of the dendritic tree. We consider briefly the interspike time problem and see in an approximate way how spatial effects are important in determining the mean time between impulses.