Electro-geometrical coupling in non-uniform branching dendrites. Consequences for relative synaptic reflectiveness

Spatial heterogeneity of passive electrical transfer properties of neuronal dendrites due to their metrical asymmetry

The complex and diverse geometry of neuronal dendrites determines the different morphological types of neurons and influences the generation of complex and diverse discharge patterns at the cell output. The recent finding that each temporal pattern has its spatial signature in the form of a combination of high- and low-depolarization states of asymmetrical dendritic branches with active membrane properties raises the question of the nature of such characteristic spatial heterogeneity of electrical states. To answer this, we consider passive dendrites as a conventional reference case using the known current transfer functions, which we complete by corresponding parametric sensitivity functions. These functions for metrically asymmetrical bifurcations of different sizes, as the simplest elements constituting arborizations of arbitrary geometry, are analyzed under different membrane conductivity conditions related to the intensity of activation of ion channels. Characteristic relationships are obtained on the one hand among the size (branch lengths), metrical asymmetry (difference between sister branches in length and/or diameter), and membrane conductivity, and on the other hand, for the difference between the branches in their current transfer effectiveness as an indicator of their electrical asymmetry (heterogeneity). These relationships (i) allow the introduction of a biophysically based criterion for the electrical distinction between metrically asymmetrical branches, (ii) show how the difference first increases and then decreases with increasing membrane conductivity, and (iii) show that the greatest electrical heterogeneity occurs in a lower or higher range of conductivity, corresponding to larger or smaller bifurcation size. As a consequence, the characteristic low-, medium-, and high-conductance states are derived such that metrically asymmetrical parts of simple and complex trees are electrically distinct when the membrane conductivity lies in the size-related medium range, and indistinct otherwise.

Parameter sensitivity of distributed transfer properties of neuronal dendrites: a passive cable approximation

Geometry and membrane properties of the dendrites crucially determine input-output relations in neurons. Unlike geometry often available in detail from computer reconstruction, the membrane resistivity is fragmentarily known if at all. Moreover, it varies during ongoing activity. In this study we address the question: what is the impact of the variation in membrane resistivity on the transfer properties of dendrites? Following a standard approach of the control system theory, we derive and explore the sensitivity functions complementary to the transfer functions of the passive dendrites with arbitrary geometrical parameters (length and diameter) and boundary conditions. We use the location-dependent somatopetal current transfer ratio (the reciprocal of the somatofugal voltage) as the transfer function, and its membrane resistivity derivatives, as the sensitivity functions. In the dendrites, at every path distance from the origin, the sensitivity function in a common form relates the transfer function, membrane resistivity, characteristic input conductance of semi-infinite cable and directional somatofugal input conductances at the given internal site and origin, and the length. Plotted in membrane resistivity versus path distance coordinates, the sensitivity functions display common features: along any coordinate there are low and high ranges, in which the sensitivity, respectively, increases and decreases. The ranges and corresponding rates depend on morphology and boundary conditions in a characteristic manner. These features predict existence of the geometry-dependent range of membrane resistivity (the earlier unattended mid-conductance state), such that the dendrites with a given metrical asymmetry are most distinguished in their transfer properties and electrical states if membrane resistivity is within the range and are not otherwise.

Rallian "equivalent" cylinders reconsidered: comparisons with literal compartments

In Rall's "equivalent" cylinder morphological-to-electrical transformation, neuronal arborizations are reduced to single unbranched core-conductors. The conventional assumption that such an "equivalent" reconstructs the electrical properties of the fibers it represents was tested directly; electrical properties and responses of "equivalent" cylinders were compared with those of their literal branch constituents for fibers with a single symmetrical bifurcation. The numerical solution methods were validated independently by their accurate reconstruction of the responses of an analog circuit configured with compartmental architecture to solve the cable equation for passive fibers with a symmetrical bifurcation. In passive fibers, "equivalent" cylinders misestimated the spatial distribution of voltage amplitudes and steady-state input resistance, partly due to the lack of axial current bifurcation. In active fibers with a single propagating action potential, the spatial distributions of point-to-point conduction velocity values (measured in meters/second) for a literal branch point differed significantly from those of their "equivalent" cylinders. "Equivalent" cylinders also underestimated the diameter-dependent delay in propagation through the branch point and branches, due to the larger "equivalent" diameter. Corrections to the "equivalent" cylinder did not reconcile differences between "equivalent" and literal models. However, "equivalent" and literal branch fibers had the same (a) steady-state resistance "looking into" an isolated symmetrical branch point and (b) geometry-independent point-to-point propagation velocity when measured in space constants per millisecond except within +/-1 space constant from the geometrical inhomogeneity. In summary, Rall's "equivalent" cylinders did not accurately reconstruct all passive or active electrophysiological properties and responses of their literal compartments. For the modeling of individual neurons, the requirement of single-branch resolution is discussed.

The problem of the morphological noise in reconstructed dendritic arborizations

For technical, instrumental and operator-related reasons, three-dimensional (3D) reconstructions of neurons obtained from intracellularly stained neuronal pieces scattered in serial sections are blurred by some morphological noise. This noise may strongly invalidate conclusions drawn from models built using the 3D reconstructions and it must be taken into account when retrieving digitized neurons from available databases. We analyse on several vertebrate neurons examples the main noise-generating sources and the consequences of the noise on the 'quality' of the data. We show how the noise can be detected and evaluated in any database, if sufficient information is presented in this database.

Differential back-invasion of a single complex dendrite of an abducens motoneuron by N-methyl-D-aspartate-induced oscillations: a simulation study

Intracellular recording of abducens motoneurons in vivo has shown that ionophoretic applications of N-methyl-D-aspartate produced long-lasting membrane potential oscillations including a slow depolarization plateau with a burst of fast action potentials. This complex N-methyl-D-aspartate pattern was reproduced in the model of abducens motoneuron in vivo identified, intracellularly stained with horseradish peroxidase and reconstructed at high spatial resolution. The excitable soma of the simulated cell contained voltage-gated Ca, Na and K conductances, N-methyl-D-aspartate-gated voltage-sensitive Ca-Na-K conductance and Ca-dependent K conductance. The dendrite was passive either completely or with the exception of branching nodes containing N-methyl-D-aspartate conductances of the same slow kinetics but of lower values than at the soma. In the completely passive case, the N-methyl-D-aspartate pattern decayed with different rates along different dendritic paths depending on the geometry and topology of the reconstructed dendrite. The branches formed four clusters discriminated in somatofugal attenuations of steady voltages, and were correspondingly discriminated in attenuation of the complex N-methyl-D-aspartate pattern. Fast spikes decayed more than the slow depolarization plateau so that the prevalence of slow over fast components in the transformed pattern increased with somatofugal path distance. As a consequence, the lower the electrotonic effectiveness of a branch in the cluster or in the whole arborization, the lower both the voltage level and the frequency range of its voltage modulation by N-methyl-D-aspartate oscillations. In the case of active branching points, the somatic pattern changed depending on the level of activation of dendritic N-methyl-D-aspartate conductances with slow kinetics of voltage sensitivity. The higher this level, the longer the plateau and burst, and the greater the discharge rate; and the spikes in the burst were smaller. When the pattern spread in the dendrite, the fast spikes decayed and the slow plateau was boosted, with a greater effect along the somatofugal path containing more branching points. These results show how the somatofugal back-invasion along the dendrites by activity patterns generated at the soma can tune voltage-sensitive dendritic conductances. The dendritic back-invasion is geometry- and topology-dependent. It is proposed as a subtle feedback mechanism for the neuron to control its own synaptic inputs.