Spike propagation in dendrites with stochastic ion channels

We investigate the effects of the stochastic nature of ion channels on the faithfulness, precision and reproducibility of electrical signal transmission in weakly active, dendritic membrane under in vitro conditions. The properties of forward and backpropagating action potentials (BPAPs) in the dendritic tree of pyramidal cells are the subject of intense empirical work and theoretical speculation (Larkum et al., 1999; Zhu, 2000; Larkum et al., 2001; Larkum and Zhu, 2002; Schaefer et al., 2003; Williams, 2004; Waters et al., 2005). We numerically simulate the effects of stochastic ion channels on the forward and backward propagation of dendritic spikes in Monte-Carlo simulations on a reconstructed layer 5 pyramidal neuron. We report that in most instances there is little variation in timing or amplitude for a single BPAP, while variable backpropagation can occur for trains of action potentials. Additionally, we find that the generation and forward propagation of dendritic Ca²⁺ spikes are susceptible to channel variability. This indicates limitations on computations that depend on the precise timing of Ca²⁺ spikes. Action Editor : Alain Destexhe     

A simple vector implementation of the Laplace-transformed cable equations in passive dendritic trees

Transient potentials in dendritic trees can be calculated by approximating the dendrite by a set of connected cylinders. The profiles for the currents and potentials in the whole system can then be obtained by imposing the proper boundary conditions and calculating these profiles along each individual cylinder. An elegant implementation of this method has been described by Holmes (1986), and is based on the Laplace transform of the cable equation. By calculating the currents and potentials only at the ends of the cylinders, the whole system of connected cylinders can be described by a set of n equations, where n denotes the number of internal and external nodes (points of connection and endpoints of the cylinders). The present study shows that the set of equations can be formulated by a simple vector equation which is essentially a generalization of Ohm's law for the whole system. The current and potential n-vectors are coupled by a n × n conductance matrix whose structure immediately reflects the connectivity pattern of the connected cylinders. The vector equation accounts for conductances, associated with driving potentials, which may be local or distributed over the membrane. It is shown that the vector equation can easily be adapted for the calculation of transients over a period in which stepwise changes in system parameters have occurred. In this adaptation it is assumed that the initial conditions for the potential profiles at the start of a new period after a stepwise change can be approximated by steady-state solutions. The vector representation of the Laplace-transformed equations is attractive because of its simplicity and because the structure of the conductance matrix directly corresponds to the connectivity pattern of the dendritic tree. Therefore it will facilitate the automatic generation of the equations once the geometry of the branching structure is known.     

Effects of rectification on synaptic efficacy.

We have investigated the effects of postsynaptic membrane properties on the shape of synaptic potentials generated by time-varying synaptic conductances. We used numerical simulation techniques to model cells of several different geometrical forms, from an isopotential sphere to a neuron with a soma and a dendritic tree. A variety of postsynaptic membrane properties were tested: (a) a passive resistance-capacitance membrane, (b) a membrane represented by the Hodgkin and Huxley (HH) equations, and (c) a membrane that was passive except for a delayed rectification represented by a voltage- and time-dependent increase in GK. In all cases we investigated the effects of these postsynaptic membrane properties on synaptic potentials produced by synaptic conductances that were fast or slow compared with the membrane time constant. In all cases the effects of postsynaptic rectification occurred on postsynaptic potentials of amplitudes as low as 1 mV. The HH model (compared with the passive model) produced an increased peak amplitude (from the increase in GNa) but a decreased half-width and a decreased time integral (from the increase in GK). These effects of the HH GK change were duplicated by a simple analytical rectifier model.     

Dendritic sodium spikes endow neurons with inverse firing rate response to correlated synaptic activity

Many neurons possess dendrites enriched with sodium channels and are capable of generating action potentials. However, the role of dendritic sodium spikes remain unclear. Here, we study computational models of neurons to investigate the functional effects of dendritic spikes. In agreement with previous studies, we found that point neurons or neurons with passive dendrites increase their somatic firing rate in response to the correlation of synaptic bombardment for a wide range of input conditions, i.e. input firing rates, synaptic conductances, or refractory periods. However, neurons with active dendrites show the opposite behavior: for a wide range of conditions the firing rate decreases as a function of correlation. We found this property in three types of models of dendritic excitability: a Hodgkin-Huxley model of dendritic spikes, a model with integrate and fire dendrites, and a discrete-state dendritic model. We conclude that fast dendritic spikes confer much broader computational properties to neurons, sometimes opposite to that of point neurons.     

Dendritic fibroblasts in three-dimensional collagen matrices

Cell motility determines form and function of multicellular organisms. Most studies on fibroblast motility have been carried out using cells on the surfaces of culture dishes. In situ, however, the environment for fibroblasts is the three-dimensional extracellular matrix. In the current research, we studied the morphology and motility of human fibroblasts embedded in floating collagen matrices at a cell density below that required for global matrix remodeling (i.e., contraction). Under these conditions, cells were observed to project and retract a dendritic network of extensions. These extensions contained microtubule cores with actin concentrated at the tips resembling growth cones. Platelet-derived growth factor promoted formation of the network; lysophosphatidic acid stimulated its retraction in a Rho and Rho kinase-dependent manner. The dendritic network also supported metabolic coupling between cells. We suggest that the dendritic network provides a mechanism by which fibroblasts explore and become interconnected to each other in three-dimensional space.     

Spatial Localization of Synapses Required for Supralinear Summation of Action Potentials and EPSPs

Although the supralinear summation of synchronizing excitatory postsynaptic potentials (EPSPs) and backpropagating action potentials (APs) is important for spike-timing-dependent synaptic plasticity (STDP), the spatial conditions of the amplification in the divergent dendritic structure have yet to be analyzed. In the present study, we simulated the coincidence of APs with EPSPs at randomly determined synaptic sites of a morphologically reconstructed hippocampal CA1 pyramidal model neuron and clarified the spatial condition of the amplifying synapses. In the case of uniform conductance inputs, the amplifying synapses were localized in the middle apical dendrites and distal basal dendrites with small diameters, and the ratio of synapses was unexpectedly small: 8-16% in both apical and basal dendrites. This was because the appearance of strong amplification requires the coincidence of both APs of 3-30 mV and EPSPs of over 6 mV, both of which depend on the dendritic location of synaptic sites. We found that the localization of amplifying synapses depends on A-type K+ channel distribution because backpropagating APs depend on the A-type K+ channel distribution, and that the localizations of amplifying synapses were similar within a range of physiological synaptic conductances. We also quantified the spread of membrane amplification in dendrites, indicating that the neighboring synapses can also show the amplification. These findings allowed us to computationally illustrate the spatial localization of synapses for supralinear summation of APs and EPSPs within thin dendritic branches where patch clamp experiments cannot be easily conducted.     

Ansätze zur quantitativen Analyse der Nachrichtenaufnahme und -verarbeitung in biologischen Rezeptoren

The flux equilibrium theory, used for interpretating active and passive ion transport, can explain the generation of receptor potentials. In a model, driving forces and velocity coefficients are represented by the parameters of electric circuits. From these membrane models ionic fluxes can be calculated quantitatively on the basis of transport equations. These equations are derived from the theory of irreversible thermodynamic processes. Receptor models allow a simulation and prediction of the bioelectric potentials which were recorded by other authors in neuro-physiological experiments under various stimulus conditions. The information capacity of a single receptor channel is determined by the ionic flux and the stimulus parameters. In combination with the network of neuron models, receptor models can be used in a perception. The problems of on-off-activation and lateral inhibition were investigated with such a network.     

Modeling the attenuation and failure of action potentials in the dendrites of hippocampal neurons.

We modeled two different mechanisms, a shunting conductance and a slow sodium inactivation, to test whether they could modulate the active propagation of a train of action potentials in a dendritic tree. Computer simulations, using a compartmental model of a pyramidal neuron, suggest that each of these two mechanisms could account for the activity-dependent attenuation and failure of the action potentials in the dendrites during the train. Each mechanism is shown to be in good qualitative agreement with experimental findings on somatic or dendritic stimulation and on the effects of hyperpolarization. The conditions under which branch point failures can be observed, and a few experimentally testable predictions, are presented and discussed.     

Role of active dendritic conductances in subthreshold input integration

Dendrites of many types of neurons contain voltage-dependent conductances that are active at subthreshold membrane potentials. To understand the computations neurons perform it is key to understand the role of active dendrites in the subthreshold processing of synaptic inputs. We examine systematically how active dendritic conductances affect the time course of postsynaptic potentials propagating along dendrites, and how they affect the interaction between such signals. Voltage-dependent currents can be classified into two types that have qualitatively different effects on subthreshold input responses: regenerative dendritic currents boost and broaden EPSPs, while restorative currents attenuate and narrow EPSPs. Importantly, the effects of active dendritic currents on EPSP shape increase as the EPSP travels along the dendrite. The effectiveness of active currents in modulating the EPSP shape is determined by their activation time constant: the faster it is, the stronger the effect on EPSP amplitude, while the largest effects on EPSP width occur when it is comparable to the membrane time constant. We finally demonstrate that the two current types can differentially improve precision and robustness of neural computations: restorative currents enhance coincidence detection of dendritic inputs, whereas direction selectivity to sequences of dendritic inputs is enhanced by regenerative dendritic currents.     

Dendritic action potentials of pyramidal tract neurons in the cat sensorimotor cortex

The intracellular activity of pyramidal tract neurons was studied during electrical stimulation of ventrolateral and ventroposterolateral thalamic nuclei in acute experiments on cats immobilized by myorelaxants. Somatic action potentials were observed and spontaneous spikes were also produced by single and rhythmic stimulation of the thalamic nuclei at the rate of 8–14 Hz, by iontophoretic application of strychnine, and by intracellular depolarizing current pulses. These potentials had a relatively low and variable amplitude of 5–60 mV and are presumed to be dendritic action potentials. It is postulated that these variable potentials arise in the dendrites of pyramidal neurons with multiple zones generating such activity. No interaction was observed where somatic and dendritic action potentials occur simultaneously. The possible functional role of dendritic action potentials is discussed.I. S. Beritashvili Institute of Physiology, Academy of Sciences of the Georgian SSR, Tbilisi. Translated from Neirofiziologiya, Vol. 18, No. 4, pp. 435–443, July–August, 1986.     

COP9 Limits Dendritic Branching via Cullin3-Dependent Degradation of the Actin-Crosslinking BTB-Domain Protein Kelch

Components of the COP9 signalosome (CSN), a key member of the conserved 26S proteasome degradation pathway, have been detected to be altered in patients of several debilitating syndromes. These findings suggest that CSN acts in neural circuits, but the exact function of CSN in brain remains unidentified. Previously, using Drosophila peripheral nervous system (PNS) as a model system, we determined that CSN is a critical regulator of dendritic morphogenesis. We found that defects in CSN led to the strikingly contrast phenotype of either reducing or stimulating dendritic branching. In particular, we have reported that CSN stimulates dendritic branching via Cullin1-mediated proteolysis. Here we describe that CSN inhibits dendritic arborization in PNS neurons acting via control of Cullin3 function: loss of Cullin3 causes excessive dendritic branching. We also identified a downstream target for Cullin3-dependent degradation in neurons – the actin-crosslinking BTB-domain protein Kelch. Inappropriate accumulation of Kelch, either due to the impaired Cullin3-dependent turnover, or ectopic expression of Kelch, leads to uncontrolled dendritic branching. These findings indicate that the CSN pathway modulates neuronal network in a multilayer manner, providing the foundation for new insight into the CSN role in human mental retardation disorders and neurodegenerative disease.     

Reticular activation and the dynamics of neuronal networks

It is postulated that during arousal the cortical system is driven by a spatially and temporally noisy signal arising from non-specific reticulo-cortical pathways. An elementary unit of cortical neuroanatomy is assumed, which permits non-linear dynamics to be represented by stochastic linear equations. Under these assumptions the resonant modes of the system of cortical dendrites approach thermodynamic equilibrium. Specific sensory signals perturb the dendritic system about equilibrium, generate low frequency, linear, non-dispersive waves corresponding to the EEG, which in turn regulate action potential sequences, and instantiate internal inputs to the dendritic field. A large and distributed memory capacity in axo-synaptic couplings, resistance to interference between functionally separate logical operations, and a very large next-state function set emerge as properties of the network. The model is able to explain the close association of the EEG with cognition, the channel of low capacity corresponding to the field of immediate attention, the low overall correlation of action potentials with EEG, and specificity of action potentials in some neurons during particular cognitive activity. Predictions made from hypothesis include features of thermal equilibrium in EEG (determinable by autoregression) and expectation that the cortical evoked response can be accounted for as the response to a sensory impulse of specific time characteristics.     

Bistability in cerebellar Purkinje cell dendrites modelled with high-threshold calcium and delayed-rectifier potassium channels

 Phase-plane analysis of the ionic currents underlying dendritic plateau potentials was carried out to study the nonlinear dynamics and steady-state transfer properties of the dendritic tree in cerebellar Purkinje cells. The results of an analysis of the P-type calcium and delayed rectifier potassium channel system are presented in this study. These channels constitute a simple system that can support bistability and plateau potentials. By requiring both the steady-state current-voltage curve and nullclines to mimic basic plateau potential properties, we obtained well-defined ranges of specific conductance that can support bistability. Hysteresis was found to be surprisingly prevalent in this simple ion-channel system. Using the steady-state current voltage relationship, we derive concise, algebraic expressions for the voltage and current thresholds of state transitions as functions of specific conductance. The significance of bistability in this ion-channel system is discussed with respect to the generation of plateau potentials in Purkinje cells dendrites and the role of the cerebellum in motor control. Received: 13 October 1993/Accepted in revised form: 21 March 1995     

Building a dendritic actin filament network branch by branch: models of filament orientation pattern and force generation in lamellipodia

We review mathematical and computational models of the structure, dynamics, and force generation properties of dendritic actin networks. These models have been motivated by the dendritic nucleation model, which provided a mechanistic picture of how the actin cytoskeleton system powers cell motility. We describe how they aimed to explain the self-organization of the branched network into a bimodal distribution of filament orientations peaked at 35° and ??35° with respect to the direction of membrane protrusion, as well as other patterns. Concave and convex force–velocity relationships were derived, depending on network organization, filament, and membrane elasticity and accounting for actin polymerization at the barbed end as a Brownian ratchet. This review also describes models that considered the kinetics and transport of actin and diffuse regulators and mechanical coupling to a substrate, together with explicit modeling of dendritic networks.     

The radial decline of nerve impulses in a restricted cylindrical extracellular space

In order to increase the potentials recorded extracellularly from nerve fibres, peripheral nerves are often placed in restricted space with cylindrical geometry. Equations are derived for computing the potentials expected at the surface of the cylinder, based on the potentials at the external surface of a small nerve fibre located on the long axis of the cylinder. These equations are evaluated numerically, using two formulae for a nerve impulse given in the literature. In both cases there is little attenuation for cylinders with radii less than 0.5 mm, but the potential declines approximately as a power of radius b for 1<b<10 mm. Various factors which might affect these results under different experimental conditions are discussed.     

Visual Stimuli Evoked Action Potentials Trigger Rapidly Propagating Dendritic Calcium Transients in the Frog Optic Tectum Layer 6 Neurons

The superior colliculus in mammals or the optic tectum in amphibians is a major visual information processing center responsible for generation of orientating responses such as saccades in monkeys or prey catching avoidance behavior in frogs. The conserved structure function of the superior colliculus the optic tectum across distant species such as frogs, birds monkeys permits to draw rather general conclusions after studying a single species. We chose the frog optic tectum because we are able to perform whole-cell voltage-clamp recordings fluorescence imaging of tectal neurons while they respond to a visual stimulus. In the optic tectum of amphibians most visual information is processed by pear-shaped neurons possessing long dendritic branches, which receive the majority of synapses originating from the retinal ganglion cells. Since the first step of the retinal input integration is performed on these dendrites, it is important to know whether this integration is enhanced by active dendritic properties. We demonstrate that rapid calcium transients coinciding with the visual stimulus evoked action potentials in the somatic recordings can be readily detected up to the fine branches of these dendrites. These transients were blocked by calcium channel blockers nifedipine CdCl₂ indicating that calcium entered dendrites via voltage-activated L-type calcium channels. The high speed of calcium transient propagation, >300 μm in <10 ms, is consistent with the notion that action potentials, actively propagating along dendrites, open voltage-gated L-type calcium channels causing rapid calcium concentration transients in the dendrites. We conclude that such activation by somatic action potentials of the dendritic voltage gated calcium channels in the close vicinity to the synapses formed by axons of the retinal ganglion cells may facilitate visual information processing in the principal neurons of the frog optic tectum.     

Electrical advantages of dendritic spines

Many neurons receive excitatory glutamatergic input almost exclusively onto dendritic spines. In the absence of spines, the amplitudes and kinetics of excitatory postsynaptic potentials (EPSPs) at the site of synaptic input are highly variable and depend on dendritic location. We hypothesized that dendritic spines standardize the local geometry at the site of synaptic input, thereby reducing location-dependent variability of local EPSP properties. We tested this hypothesis using computational models of simplified and morphologically realistic spiny neurons that allow direct comparison of EPSPs generated on spine heads with EPSPs generated on dendritic shafts at the same dendritic locations. In all morphologies tested, spines greatly reduced location-dependent variability of local EPSP amplitude and kinetics, while having minimal impact on EPSPs measured at the soma. Spine-dependent standardization of local EPSP properties persisted across a range of physiologically relevant spine neck resistances, and in models with variable neck resistances. By reducing the variability of local EPSPs, spines standardized synaptic activation of NMDA receptors and voltage-gated calcium channels. Furthermore, spines enhanced activation of NMDA receptors and facilitated the generation of NMDA spikes and axonal action potentials in response to synaptic input. Finally, we show that dynamic regulation of spine neck geometry can preserve local EPSP properties following plasticity-driven changes in synaptic strength, but is inefficient in modifying the amplitude of EPSPs in other cellular compartments. These observations suggest that one function of dendritic spines is to standardize local EPSP properties throughout the dendritic tree, thereby allowing neurons to use similar voltage-sensitive postsynaptic mechanisms at all dendritic locations.     

Contribution of extrasynaptic N-methyl-d-aspartate and adenosine A1 receptors in the generation of dendritic glutamate-mediated plateau potentials

Thin basal dendrites can strongly influence neuronal output via generation of dendritic spikes. It was recently postulated that glial processes actively support dendritic spikes by either ceasing glutamate uptake or by actively releasing glutamate and adenosine triphosphate (ATP). We used calcium imaging to study the role of NR2C/D-containing N-methyl-d-aspartate (NMDA) receptors and adenosine A1 receptors in the generation of dendritic NMDA spikes and plateau potentials in basal dendrites of layer 5 pyramidal neurons in the mouse prefrontal cortex. We found that NR2C/D glutamate receptor subunits contribute to the amplitude of synaptically evoked NMDA spikes. Dendritic calcium signals associated with glutamate-evoked dendritic plateau potentials were significantly shortened upon application of the NR2C/D receptor antagonist PPDA, suggesting that NR2C/D receptors prolong the duration of calcium influx during dendritic spiking. In contrast to NR2C/D receptors, adenosine A1 receptors act to abbreviate dendritic and somatic signals via the activation of dendritic K⁺ current. This current is characterized as a slow-activating outward-rectifying voltage- and adenosine-gated current, insensitive to 4-aminopyridine but sensitive to TEA. Our data support the hypothesis that the release of glutamate and ATP from neurons or glia contribute to initiation, maintenance and termination of local dendritic glutamate-mediated regenerative potentials.     

Somato-dendritic Synaptic Plasticity and Error-backpropagation in Active Dendrites

In the last decade dendrites of cortical neurons have been shown to nonlinearly combine synaptic inputs by evoking local dendritic spikes. It has been suggested that these nonlinearities raise the computational power of a single neuron, making it comparable to a 2-layer network of point neurons. But how these nonlinearities can be incorporated into the synaptic plasticity to optimally support learning remains unclear. We present a theoretically derived synaptic plasticity rule for supervised and reinforcement learning that depends on the timing of the presynaptic, the dendritic and the postsynaptic spikes. For supervised learning, the rule can be seen as a biological version of the classical error-backpropagation algorithm applied to the dendritic case. When modulated by a delayed reward signal, the same plasticity is shown to maximize the expected reward in reinforcement learning for various coding scenarios. Our framework makes specific experimental predictions and highlights the unique advantage of active dendrites for implementing powerful synaptic plasticity rules that have access to downstream information via backpropagation of action potentials.     

Spatial and temporal analysis of calcium-dependent electrical activity in guinea pig Purkinje cell dendrites

We have used the calcium indicator dye arsenazo III, together with a photodiode array, to record intracellular calcium changes simultaneously from all regions of individual guinea pig cerebellar Purkinje cells in slices. The optical signals, recorded with millisecond time resolution, are good indicators of calcium-dependent electrical events. For many cells the sensitivity of the recordings was high enough to detect signals from each array element without averaging. Consequently, it was possible to use these signals to follow the complex spatial and temporal patterns of plateau and spike potentials. Calcium entry corresponding to action potentials was detected from all parts of the dendritic field including the fine spiny branchlets, demonstrating that calcium action potentials spread over the entire arbor. Usually, the entire dendritic tree fired at once. But sometimes only restricted areas had signals at any one moment with transients detected in different regions at other times. In one cell, six separate zones were distinguished. These results show that calcium action potentials could be regenerative in some dendrites and could fail to propagate into others. Signals from plateau potentials were also detected from extensive areas in the dendritic field but were always smaller than those caused by a burst of action potentials.