Cable Theory for Finite Length Dendritic Cylinders with Initial and Boundary Conditions

The cable equation is solved in the Laplace transform domain for arbitrary initial and boundary conditions. The cable potential is expressed directly in terms of the impedance of the terminations and the cable electrotonic length. A computer program is given to invert the transform. Numerical solutions may be obtained for any particular model by inserting expressions describing the terminations and parameter values into the program, without further computation by the modeler. For a finite length cable, sealed at one end, the solution is expressed in terms of the ratio of the termination impedance to the impedance of the finite length cable, a generalization of the steady-state conductance ratio. Analysis of a model of a soma with several primary dendrites shows that the dendrites may be lumped into one equivalent cylinder if they have the same electrotonic length, even though they may vary in diameter. Responses obtained under voltage clamp are conceptually predictable from measurements made under current clamp, and vice versa. The equalizing time constants of an infinite series expression of the solution are the negative reciprocals of the roots of the characteristic equation. Examination of computed solutions shows that solutions which differ theoretically may be indistinguishable experimentally.     

Weak Sinusoidal Electric Fields Entrain Spontaneous Ca Transients in the Dendritic Tufts of CA1 Pyramidal Cells in Rat Hippocampal Slice Preparations

Neurons might interact via electric fields and this notion has been referred to as ephaptic interaction. It has been shown that various types of ion channels are distributed along the dendrites and are capable of supporting generation of dendritic spikes. We hypothesized that generation of dendritic spikes play important roles in the ephaptic interactions either by amplifying the impact of electric fields or by providing current source to generate electric fields. To test if dendritic activities can be modulated by electric fields, we developed a method to monitor local Ca-transients in the dendrites of a neuronal population in acute rat hippocampal slices by applying spinning-disk confocal microscopy and multi-cell dye loading technique. In a condition in which the dendrites of CA1 pyramidal neurons show spontaneous Ca-transients due to added 50 μM 4-aminopyridine to the bathing medium and adjusted extracellular potassium concentration, we examined the impact of sinusoidal electric fields on the Ca-transients. We have found that spontaneously occurring fast-Ca-transients in the tufts of the apical dendrites of CA1 pyramidal neurons can be blocked by applying 1 μM tetrodotoxin, and that the timing of the transients become entrained to sub-threshold 1-4 Hz electric fields with an intensity as weak as 0.84 mV/mm applied parallel to the somato-dendritic axis of the neurons. The extent of entrainment increases with intensity below 5 mV/mm, but does not increase further over the range of 5-20 mV/mm. These results suggest that population of pyramidal cells might be able to detect electric fields with biologically relevant intensity by modulating the timing of dendritic spikes.     

Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites

Spinal motor neurons have voltage gated ion channels localized in their dendrites that generate plateau potentials. The physical separation of ion channels for spiking from plateau generating channels can result in nonlinear bistable firing patterns. The physical separation and geometry of the dendrites results in asymmetric coupling between dendrites and soma that has not been addressed in reduced models of nonlinear phenomena in motor neurons. We measured voltage attenuation properties of six anatomically reconstructed and type-identified cat spinal motor neurons to characterize asymmetric coupling between the dendrites and soma. We showed that the voltage attenuation at any distance from the soma was direction-dependent and could be described as a function of the input resistance at the soma. An analytical solution for the lumped cable parameters in a two-compartment model was derived based on this finding. This is the first two-compartment modeling approach that directly derived lumped cable parameters from the geometrical and passive electrical properties of anatomically reconstructed neurons.     

Hippocampal growth cone responses to focally applied electric fields

A wide variety of cell types respond to electric fields in culture. Despite evidence for electric fields existing in the mammalian embryo, there are few studies testing the effects electric fields exert on neurons from the mammalian central nervous system (CNS). The present study demonstrates orientation responses to focally applied electric fields of embryonic rat hippocampal neurons isolated in culture. The most striking result from this study is that different growth cones of the same neuron can show differential responsiveness to focally applied electric fields: growth cones on the short straight processes that are destined to become dendrites, oriented toward the cathode, whereas growth cones on the longest process, the presumptive axon, did not orient. The present experiments bring a significant increase in resolution to the study of neuronal growth cone orientation by applied electric fields: a novel examination of the early events leading to orientation. Growth cones on dendrites displayed a spectrum of orientation responses: directed lamellipodial extension, directed filopodial extension and/or reorientation, cytoplasmic swelling of existing filopodia, consolidation of filopodia, and rapid elongation of the entire process. Individual growth cones displayed only one or two of these responses. Additionally, not all growth cones on these short processes sustained their initial orientation response: 35% adapted within 6 min. © 1993 John Wiley & Sons, Inc.     

Transfer properties of neuronal dendrites with tonically activated conductances

The somatopetal current transfer was studied in the mathematical models of a reconstructed brainstem motoneuron with tonically activated excitatory synaptic inputs uniformly distributed over dendritic arborization. The soma and axon provided a constant passive leak. The extrasynaptic dendritic membrane was either passive or active (of a Hodgkin-Huxley type). The longitudinal membrane current density (per unit path length) was used as an estimate of the current transfer effectiveness of different dendritic paths. Introduction of a steady uniform voltage-independent conductance per unit membrane area simulated such a synaptic activation. This actions always produced a spatially inhomogeneous membrane depolarization decaying from the distal dendritic tips toward the soma. The reason for such an inhomogeneity was the preponderance of somatopetal over somatofugal input conductance at every site in the dendrites with sealed distal ends and a leaky somatic end. In active dendrites, partial voltage-dependent extrasynaptic conductances followed this depolarization according to their activation-inactivation kinetics. The greater the local depolarization, the greater the contribution of the non-inactivating potassium conductance to the total membrane conductance. The contribution of the inactivated sodium conductance was one order of magnitude smaller. Correspondingly, the effective equilibrium potential of the total transmembrane current became spatially inhomogeneous and shifted to the potassium equilibrium potential. In the passive dendrites, the equilibrium potential remained spatially homogeneous. Inhomogeneities of the dendritic geometry (abrupt change in the diameter and, especially, asymmetrical branching) caused characteristic perturbations in the voltage gradient, so that the path profiles of the voltage, conductances, and currents diverged. This indicated a geometry-induced separation of the dendritic paths in their transfer effectiveness. Active dendrites of the same geometry were less effective than passive ones due to the effect of the potassium conductance associated with the hyperpolarizing equilibrium potential.     

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.     

Emergence of temporal-pattern sensitive neurons in the midbrain of weakly electric fish Gymnarchus niloticus.

Sensitivity of neurons in the torus semicircularis of a weakly electric fish, Gymnarchus niloticus, to two stimulus parameters that are critical for its behavior the jamming avoidance response was examined. The first parameter is the sign of frequency difference between discharge frequencies of fish's own electric organ and that of a neighbor's. The second parameter is the spatial orientation of neighbor's electric field. Whereas neuronal ambiguity of frequency coding for different orientations of neighbor's electric field is predicted, unambiguous JAR occurs at the behavioral level. Most neurons in the torus semicircularis showed sensitivity to the sign of frequency difference. Although a small number of neurons showed preference to a consistent sign of the frequency difference, the coding of the sign of frequency differences was found to be ambiguous with a highly variable pattern of responses for different orientations in most of neurons.     

Modelling the electrotonic structure of starburst amacrine cells in the rabbit retina: A functional interpretation of dendritic morphology

A detailed morphometric analysis of a Lucifer yellow-filled Cb amacrine cell was undertaken to provide raw data for the construction of a neuronal cable model. The cable model was employed to determine whether distal input-output regions of dendrites were electrically isolated from the soma and each other. Calculations of steady state electrotonic current spread suggested reasonable electrical communication between cell body and dendrites. In particular, the centripetal voltage attenuation revealed that a synaptic signal introduced at the distal end of the equivalent dendrite could spread passively along the dendrite and reach the soma with little loss in amplitude. A functional interpretation of this results could favour a postsynaptic rather than a presynaptic scheme for the operation of directional selectivity in the rabbit retina. On the other hand, dendrites of starburst amacrine cells process information electrotonically with a bias towards the centrifugal direction and for a restricted range of membrane resistance values the voltage attenuation in the centripetal direction suggests that the action of these dendrites can be confined locally. A functional interpretation of this result favours a presynaptic version of Vaney's cotransmission model which attempts to explain how the neural network of starburst amacrine cells might account for directionally selective responses observed in the rabbit retina.     

Dual synaptic plasticity in the hippocampus: Hebbian and spatiotemporal learning dynamics

We assume that Hebbian learning dynamics (HLD) and spatiotemporal learning dynamics (SLD) are involved in the mechanism of synaptic plasticity in the hippocampal neurons. While HLD is driven by pre- and postsynaptic spike timings through the backpropagating action potential, SLD is evoked by presynaptic spike timings alone. Since the backpropagation attenuates as it nears the distal dendrites, we assume an extreme case as a neuron model where HLD exists only at proximal dendrites and SLD exists only at the distal dendrites. We examined how the synaptic weights change in response to three types of synaptic inputs in computer simulations. First, in response to a Poisson train having a constant mean frequency, the synaptic weights in HLD and SLD are qualitatively similar. Second, SLD responds more rapidly than HLD to synchronous input patterns, while each responds to them. Third, HLD responds more rapidly to more frequent inputs, while SLD shows fluctuating synaptic weights. These results suggest an encoding hypothesis in that a transient synchronous structure in spatiotemporal input patterns will be encoded into distal dendrites through SLD and that persistent synchrony or firing rate information will be encoded into proximal dendrites through HLD.     

Cable Bacteria in Freshwater Sediments

In marine sediments cathodic oxygen reduction at the sediment surface can be coupled to anodic sulfide oxidation in deeper anoxic layers through electrical currents mediated by filamentous, multicellular bacteria of the Desulfobulbaceae family, the so-called cable bacteria. Until now, cable bacteria have only been reported from marine environments. In this study, we demonstrate that cable bacteria also occur in freshwater sediments. In a first step, homogenized sediment collected from the freshwater stream Giber Å, Denmark, was incubated in the laboratory. After 2 weeks, pH signatures and electric fields indicated electron transfer between vertically separated anodic and cathodic half-reactions. Fluorescence in situ hybridization revealed the presence of Desulfobulbaceae filaments. In addition, in situ measurements of oxygen, pH, and electric potential distributions in the waterlogged banks of Giber Å demonstrated the presence of distant electric redox coupling in naturally occurring freshwater sediment. At the same site, filamentous Desulfobulbaceae with cable bacterium morphology were found to be present. Their 16S rRNA gene sequence placed them as a distinct sister group to the known marine cable bacteria, with the genus Desulfobulbus as the closest cultured lineage. The results of the present study indicate that electric currents mediated by cable bacteria could be important for the biogeochemistry in many more environments than anticipated thus far and suggest a common evolutionary origin of the cable phenotype within Desulfobulbaceae with subsequent diversification into a freshwater and a marine lineage.     

The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.     

Long-term potentiation in rat hippocampal neurons is accompanied by spatially widespread changes in intrinsic oscillatory dynamics and excitability

Oscillations in neural activity are a prominent feature of many brain states. Individual hippocampal neurons exhibit intrinsic membrane potential oscillations and intrinsic resonance in the theta frequency range. We found that the subthreshold resonance frequency of CA1 pyramidal neurons was location dependent, varying more than 3-fold between the soma and the distal dendrites. Furthermore, activity- and NMDA-receptor-dependent long-term plasticity increased this resonance frequency through changes in h channel properties. The increase in resonance frequency and an associated reduction in excitability were nearly identical in the soma and the first 300 mum of the apical dendrites. These spatially widespread changes accompanying long-term synaptic potentiation also reduced the neuron's ability to elicit spikes evoked through a nonpotentiated synaptic pathway. Our results suggest that the frequency response of these neurons depends on the dendritic location of their inputs and that activity can regulate their response dynamics within an oscillating neural network.     

Control of dendritic field formation in Drosophila: the roles of flamingo and competition between homologous neurons

Neurons elaborate dendrites with stereotypic branching patterns, thereby defining their receptive fields. These branching patterns may arise from properties intrinsic to the neurons or competition between neighboring neurons. Genetic and laser ablation studies reported here reveal that different multiple dendritic neurons in the same dorsal cluster in the Drosophila embryonic PNS do not compete with one another for dendritic fields. In contrast, when dendrites from homologous neurons in the two hemisegments meet at the dorsal midline in larval stages, they appear to repel each other. The formation of normal dendritic fields and the competition between dendrites of homologous neurons require the proper expression level of Flamingo, a G protein-coupled receptor-like protein, in embryonic neurons. Whereas Flamingo functions downstream of Frizzled in specifying planar polarity, Flamingo-dependent dendritic outgrowth is independent of Frizzled.     

The electrical resistivity of cytoplasm.

The apparent cytoplasmic resistivity of two different giant cells has been measured using an extension of a previously developed single microelectrode technique. Each cell is penetrated by a metal microelectrode whose complex impedance is measured as a function of frequency between 500 kHz and 5.7 MHz. By plotting the measured impedance data on the complex Z plane and extrapolating the data to infinite frequency, the substantial effects of electrode polarization can be overcome. For Aplysia giant neurons and muscle fibers of the giant barnacle, the extrapolated cytoplasmic specific resistivities are 40 and 74 omega-cm, respectively, at infinite frequency. The barnacle data are in excellent agreement with sarcoplasmic resistivity values derived from the measured cable properties of other marine organisms, and from high frequency conductivity cell measurements in intact barnacle muscle tissue. In the Aplysia neurons, the frequency-dependent part of the electrode impedance is larger when the electrode is in a cell than when it is in an electrolyte solution with the same specific resistivity as the aqueous cytoplasm; however, the phase angle of the frequency-dependent component of the electrode impedance is the same in both cases. This suggests that the high apparent values of cytoplasmic resistivity found using the single microelectrode technique at lower frequencies probably reflect an artifact caused by reduction of the effective surface area of the electrode by intracellular membranes, with a corresponding increase in its polarization impedance.     

Microtubules have opposite orientation in axons and dendrites of Drosophila neurons

In vertebrate neurons, axons have a uniform arrangement of microtubules with plus ends distal to the cell body (plus-end-out), and dendrites have equal numbers of plus- and minus-end-out microtubules. To determine whether microtubule orientation is a conserved feature of axons and dendrites, we analyzed microtubule orientation in invertebrate neurons. Using microtubule plus end dynamics, we mapped microtubule orientation in Drosophila sensory neurons, interneurons, and motor neurons. As expected, all axonal microtubules have plus-end-out orientation. However, in proximal dendrites of all classes of neuron, approximately 90% of dendritic microtubules were oriented with minus ends distal to the cell body. This result suggests that minus-end-out, rather than mixed orientation, microtubules are the signature of the dendritic microtubule cytoskeleton. Surprisingly, our map of microtubule orientation predicts that there are no tracks for direct cargo transport between the cell body and dendrites in unipolar neurons. We confirm this prediction, and validate the completeness of our map, by imaging endosome movements in motor neurons. As predicted by our map, endosomes travel smoothly between the cell body and axon, but they cannot move directly between the cell body and dendrites.     

The functional characteristics of electroreceptive central neurons of the sea catfish Plotosus anguillaris

With the use of microelectrode techniques (extracellular recordings) and the method of post-stimulus histograms, the functional characteristics of medulla oblongata neurons of sea catfish Plotosus were investigated under stimulation of electroreceptors by a homogeneous electric field of different duration, intensity, and direction. Two types of the cells possessing, accordingly, tonic or phase activity were registered among 66 neurons investigated. The mode of responses (inhibition or acceleration) of tonic neurons to the direction of the applied electric current is typical for central neurons of fresh-water catfish connected with ampullae's electroreceptors. Neurons showing a substantial response to fields of an intensity less than 1 microV/cm were registered. The reactions were most pronounced with the duration of electric stimuli in the range of 20-200 ms; however, particularly sensitive neurons showed distinct responses to stimuli of duration of 5 and even 2 ms. Thus, for the first time a high sensitivity of ampullae's electroreceptors to high-frequency stimulus was discovered, which allows one to expand the range of studying electric signals used by weakly electric fish for electrolocation and communication.     

Alterations of ultrastructure and physiology of chemoreceptor dendrites in blowfly taste hairs treated with vinblastine and colchicine

Vinblastine (10^?3 M) and colchicine (10^?2 M) dissolved in fly Ringer solution were applied to chemoreceptor dendrites in individual tarsal taste hairs of blowflies (Phaenicia serricata Meigen), and the resultant changes in the morphology and physiological responsiveness of the receptors were examined. While Ringer solution alone did not have any morphological or physiological effects, treatment with drugs dissolved in Ringer solution increased the latency of responses to sucrose and NaCl and decreased spike amplitude and frequency. The drugs also disrupted microtubules and caused the destruction of the distal end of the dendrites. The observations provide evidence that the microtubules have a role in maintaining the structural integrity of the dendrites. Anodal current applied along with a chemical stimulus (NaCl or sucrose) increased the action potential frequency of treated and untreated receptors, and the electrical current elicited spikes only from those receptors which were activated by the chemical stimulus alone. When drug treatment resulted in a complete cessation of response to chemical stimuli, the receptors were also unresponsive to electrical stimulation. The results of this study are discussed in relation to hypotheses of transduction in chemoreception.     

Integrins regulate repulsion-mediated dendritic patterning of drosophila sensory neurons by restricting dendrites in a 2D space

Dendrites of the same neuron usually avoid each other. Some neurons also repel similar neurons through dendrite-dendrite interaction to tile the receptive field. Nonoverlapping coverage based on such contact-dependent repulsion requires dendrites to compete for limited space. Here we show that Drosophila class IV dendritic arborization (da) neurons, which tile the larval body wall, grow their dendrites mainly in a 2D space on the extracellular matrix (ECM) secreted by the epidermis. Removing neuronal integrins or blocking epidermal laminin production causes dendrites to grow into the epidermis, suggesting that integrin-laminin interaction attaches dendrites to the ECM. We further show that some of the previously identified tiling mutants fail to confine dendrites in a 2D plane. Expansion of these mutant dendrites in three dimensions results in overlap of dendritic fields. Moreover, overexpression of integrins in these mutant neurons effectively reduces dendritic crossing and restores tiling, revealing an additional mechanism for tiling.