Dynamic and spatial features of the inhibitory pallidal GABAergic synapses.

The globus pallidus, one of the basal ganglia nuclei, plays a major role in both basal ganglia physiology and pathophysiology. The globus pallidus is innervated mainly by striatal spiny neurons and globus pallidus collaterals. These GABAergic synapses constitute 90% of the input to globus pallidus cells. Despite the dominance of this inhibitory GABAergic input, globus pallidus cells are spontaneously active and most of them increase their firing rate in a task related manner. To explain this apparent inconsistency, we studied the dynamic and spatial effects of GABAergic inputs to globus pallidus neurons. To this end, we used intra-cellular recording from globus pallidus neurons in rat brain slices, investigating the effect of bath and local GABA application, as well as the responses to electrical stimulation of the striatum. We showed that the properties of the responses to either local or global GABA applications are similar to the responses of globus pallidus cells to GABA release from nerve terminals. Since the stimulus-evoked responses have been shown to be inhibitory in nature, we concluded that GABAergic inputs to globus pallidus both at soma and dendrite level are inhibitory. Furthermore, we showed that GABA can promote globus pallidus synchronization by affecting the timing of globus pallidus spiking, and that the globus pallidus GABAergic synapse undergoes rapid frequency-dependent depression. This prominent synaptic depression can account for the ability of globus pallidus neurons to fire in the presence of a majority of inhibitory inputs and might indicate that globus pallidus neurons are tuned to detect frequency changes. Furthermore, globus pallidus synaptic depression rules out the possibility of activation of GABAeregic afferents as the main mechanisms of high-frequency deep brain stimulation, used for treatment of severe parkinsonian patients.

The role of single neurons in information processing.

Neurons carry out the many operations that extract meaningful information from sensory receptor arrays at the organism's periphery and translate these into action, imagery and memory. Within today's dominant computational paradigm, these operations, involving synapses, membrane ionic channels and changes in membrane potential, are thought of as steps in an algorithm or as computations. The role of neurons in these computations has evolved conceptually from that of a simple integrator of synaptic inputs until a threshold is reached and an output pulse is initiated, to a much more sophisticated processor with mixed analog-digital logic and highly adaptive synaptic elements.

Subthreshold voltage noise due to channel fluctuations in active neuronal membranes.

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na+ and K+ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.

Untangling dendrites with quantitative models.

Our understanding of the function of dendrites has been greatly enriched by an inspiring dialogue between theory and experiments. Rather than functionally ignoring dendrites, representing neurons as single summing points, we have realized that dendrites are electrically and chemically distributed nonlinear units and that this has important consequences for interpreting experimental data and for the role of neurons in information processing. Here, we examine the route to unraveling some of the enigmas of dendrites and highlight the main insights that have been gained. Future directions are discussed that will enable theory and models to keep shedding light on dendrites, where the most fundamental input-output adaptive processes take place.

Axons as computing devices: basic insights gained from models.

Detailed models of single neurons are typically focused on the dendritic tree and ignore the axonal tree, assuming that the axon is a simple transmission line. In the last 40 years, however, several theoretical and experimental studies have suggested that axons could implement information processing tasks by exploiting: 1) the time delay in action potential (AP) propagation along the axon; 2) the differential filtering of APs into the axonal subtrees; and 3) their activity-dependent excitability. Models for axonal trees have attempted to examine the feasibility of these ideas. However, because the physiological and anatomical data on axons are seriously limited, realistic models of axons have not been developed. The present paper summarizes the main insights that were gained from simplified models of axons; it also highlights the stochastic nature of axons, a topic that was largely neglected in classical models of axons. The advance of new experimental techniques makes it now possible to pay a very close experimental visit to axons. Theoretical tools and fast computers enable to go beyond the simplified models and to construct realistic models of axons. When tightly linked, experiments and theory will help to unravel how axons share the information processing tasks that single neurons implement.

Taming time in the olfactory bulb.

The biophysics of an unusual synaptic arrangement within the olfactory bulb suggests a way in which rapidly inactivating potassium channels could modulate the timing of oscillations that underlie odor recognition.

Signal transfer in passive dendrites with nonuniform membrane conductance.

In recent years it became clear that dendrites possess a host of ion channels that may be distributed nonuniformly over their membrane surface. In cortical pyramids, for example, it was demonstrated that the resting membrane conductance G(m)(x) is higher (the membrane is "leakier") at distal dendritic regions than at more proximal sites. How does this spatial nonuniformity in G(m)(x) affect the input-output function of the neuron? The present study aims at providing basic insights into this question. To this end, we have analytically studied the fundamental effects of membrane non-uniformity in passive cable structures. Keeping the total membrane conductance over a given modeled structure fixed (i.e., a constant number of passive ion channels), the classical case of cables with uniform membrane conductance is contrasted with various nonuniform cases with the following general conclusions. (1) For cylindrical cables with "sealed ends," monotonic increase in G(m)(x) improves voltage transfer from the input location to the soma. The steeper the G(m)(x), the larger the improvement. (2) This effect is further enhanced when the stimulation is distal and consists of a synaptic input rather than a current source. (3) Any nonuniformity in G(m)(x) decreases the electrotonic length, L, of the cylinder. (4) The system time constant tau(0) is larger in the nonuniform case than in the corresponding uniform case. (5) When voltage transients relax with tau(0), the dendritic tree is not isopotential in the nonuniform case, at variance with the uniform case. The effect of membrane nonuniformity on signal transfer in reconstructed dendritic trees and on the I/f relation of the neuron is also considered, and experimental methods for assessing membrane nonuniformity in dendrites are discussed.

Dendritic asymmetry cannot account for directional responses of neurons in visual cortex.

A simple model was proposed to account for the direction selectivity of neurons in the primary visual cortex, area V1. In this model, the temporal asymmetries in the summation of inhibition and excitation that produce directionality were generated by structural asymmetries in the tangential organization of the basal dendritic tree of cortical neurons. We reconstructed dendritic trees of neurons with known direction preferences and found no correlation between the small biases of a neuron's dendritic morphology and its direction preference. Detailed simulations indicated that even when the electrotonic asymmetries in the dendrites were extreme, as in cortical Meynert cells, the biophysical properties of single neurons could contribute only partially to the directionality of cortical neurons.

Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations.

Important advances in experimental methods have made it possible to measure the electrical events in dendrites directly and to record optically from dendritic spines. These new techniques allow us to focus on the input region of the neuron and highlight the excitable properties of the dendritic membrane. Interestingly, some of the recent experimental findings were anticipated by earlier theoretical research, for example, the observation that some spines possess excitable channels that might generate local all-or-none events. Computer models were used previously to explore the conditions for initiating an action potential at the dendritic tree, in particular, at the spine head, and for active propagation between excitable spines and excitable dendritic arbors. The consequences for synaptic amplification, for the extent of active spread in the tree and for non-linear discriminations between different patterns of synaptic inputs were also considered. Here we review the biophysical insights gained from the theory and demonstrate how these elucidate the recent experimental results.

Ion channel stochasticity may be critical in determining the reliability and precision of spike timing.

The firing reliability and precision of an isopotential membrane patch consisting of a realistically large number of ion channels is investigated using a stochastic Hodgkin-Huxley (HH) model. In sharp contrast to the deterministic HH model, the biophysically inspired stochastic model reproduces qualitatively the different reliability and precision characteristics of spike firing in response to DC and fluctuating current input in neocortical neurons, as reported by Mainen & Sejnowski (1995). For DC inputs, spike timing is highly unreliable; the reliability and precision are significantly increased for fluctuating current input. This behavior is critically determined by the relatively small number of excitable channels that are opened near threshold for spike firing rather than by the total number of channels that exist in the membrane patch. Channel fluctuations, together with the inherent bistability in the HH equations, give rise to three additional experimentally observed phenomena: subthreshold oscillations in the membrane voltage for DC input, "spontaneous" spikes for subthreshold inputs, and "missing" spikes for suprathreshold inputs. We suggest that the noise inherent in the operation of ion channels enables neurons to act as "smart" encoders. Slowly varying, uncorrelated inputs are coded with low reliability and accuracy and, hence, the information about such inputs is encoded almost exclusively by the spike rate. On the other hand, correlated presynaptic activity produces sharp fluctuations in the input to the postsynaptic cell, which are then encoded with high reliability and accuracy. In this case, information about the input exists in the exact timing of the spikes. We conclude that channel stochasticity should be considered in realistic models of neurons.

Reading neuronal synchrony with depressing synapses.

A recent experiment showed that neurons in the primary auditory cortex of the monkey do not change their mean firing rate during an ongoing tone stimulus. The only change was an enhanced correlation among the individual spike trains during the tone. We show that there is an easy way to extract this coherence information in the cortical cell population by projecting the spike trains through depressing synapses onto a postsynaptic neuron.

Low-amplitude oscillations in the inferior olive: a model based on electrical coupling of neurons with heterogeneous channel densities.

The mechanism underlying subthreshold oscillations in inferior olivary cells is not known. To study this question, we developed a single-compartment, two-variable, Hodgkin-Huxley-like model for inferior olive neurons. The model consists of a leakage current and a low-threshold calcium current, whose kinetics were experimentally measured in slices. Depending on the maximal calcium and leak conductances, we found that a neuron model's response to current injection could be of four qualitatively different types: always stable, spontaneously oscillating, oscillating with injection of current, and bistable with injection of current. By the use of phase plane techniques, numerical integration, and bifurcation analysis, we subdivided the two-parameter space of channel densities into four regions corresponding to these behavioral types. We further developed, with the use of such techniques, an empirical rule of thumb that characterizes whether two cells when coupled electrically can generate sustained, synchronized oscillations like those observed in inferior olivary cells in slices, of low amplitude (0.1-10 mV) in the frequency range 4-10 Hz. We found that it is not necessary for either cell to be a spontaneous oscillator to obtain a sustained oscillation. On the other hand, two spontaneous oscillators always form an oscillating network when electrically coupled with any arbitrary coupling conductance. In the case of an oscillating pair of electrically coupled nonidentical cells, the coupling current varies periodically and is nonzero even for very large coupling values. The coupling current acts as an equalizing current to reconcile the differences between the two cells' ionic currents. It transiently depolarizes one cell and/or hyperpolarizes the other cell to obtain the regenerative response(s) required for the synchronized oscillation. We suggest that the subthreshold oscillations observed in the inferior olive can emerge from the electrical coupling between neurons with different channel densities, even if the inferior olive nucleus contains no or just a small proportion of spontaneously oscillating neurons.

Organization of octopus arm movements: a model system for studying the control of flexible arms.

Octopus arm movements provide an extreme example of controlled movements of a flexible arm with virtually unlimited degrees of freedom. This study aims to identify general principles in the organization of these movements. Video records of the movements of Octopus vulgaris performing the task of reaching toward a target were studied. The octopus extends its arm toward the target by a wave-like propagation of a bend that travels from the base of the arm toward the tip. Similar bend propagation is seen in other octopus arm movements, such as locomotion and searching. The kinematics (position and velocity) of the midpoint of the bend in three-dimensional space were extracted using the direct linear transformation algorithm. This showed that the bend tends to move within a single linear plane in a simple, slightly curved path connecting the center of the animal's body with the target location. Approximately 70% of the reaching movements demonstrated a stereotyped tangential velocity profile. An invariant profile was observed when movements were normalized for velocity and distance. Two arms, extended together in the same behavioral context, demonstrated identical velocity profiles. The stereotyped features of the movements were also observed in spontaneous arm extensions (not toward an external target). The simple and stereotypic appearance of the bend trajectory suggests that the position of the bend in space and time is the controlled variable. We propose that this strategy reduces the immense redundancy of the octopus arm movements and hence simplifies motor control.

Modeling back propagating action potential in weakly excitable dendrites of neocortical pyramidal cells.

Simultaneous recordings from the soma and apical dendrite of layer V neocortical pyramidal cells of young rats show that, for any location of current input, an evoked action potential (AP) always starts at the axon and then propagates actively, but decrementally, backward into the dendrites. This back-propagating AP is supported by a low density (-gNa = approximately 4 mS/cm2) of rapidly inactivating voltage-dependent Na+ channels in the soma and the apical dendrite. Investigation of detailed, biophysically constrained, models of reconstructed pyramidal cells shows the following. (i) The initiation of the AP first in the axon cannot be explained solely by morphological considerations; the axon must be more excitable than the soma and dendrites. (ii) The minimal Na+ channel density in the axon that fully accounts for the experimental results is about 20-times that of the soma. If -gNa in the axon hillock and initial segment is the same as in the soma [as recently suggested by Colbert and Johnston [Colbert, C. M. & Johnston, D. (1995) Soc. Neurosci. Abstr. 21, 684.2]], then -gNa in the more distal axonal regions is required to be about 40-times that of the soma. (iii) A backward propagating AP in weakly excitable dendrites can be modulated in a graded manner by background synaptic activity. The functional role of weakly excitable dendrites and a more excitable axon for forward synaptic integration and for backward, global, communication between the axon and the dendrites is discussed.

A brief history of time (constants).

That the cerebral cortex processes information at prodigious speeds cannot be doubted. Yet the passive time constant, tau(m), of neurons, often thought of as a measure of the neuron's "response time' to synaptic input, is relatively long. In the 1950s, tau(m) was estimated to be only a few milliseconds for mammalian central neurons; with improvement in recording techniques, its estimated value grew over the years and it now stands near 20-100 msec. However, as we will argue here, the functional meaning of tau(m) is ambiguous. On the basis of a newly introduced definition of local delay, we show that the time window for synaptic integration in passive dendritic trees can be much smaller than the time constant. We argue that the voltage response to very brief synaptic inputs is essentially independent of tau(m). We discuss how tau(m) can change dynamically with the global activity of the network, as well as the difficulties of defining a time constant in structures with voltage-dependent elements. We conclude that the classically defined tau(m) only provides a very rough estimate, typically an overestimate, of the response time of neurons and that alternative measures are required to capture the dependency of the time course of the membrane potential on ligand-gated and/or voltage-dependent membrane conductances.

Electrical consequences of spine dimensions in a model of a cortical spiny stellate cell completely reconstructed from serial thin sections.

We built a passive compartmental model of a cortical spiny stellate cell from the barrel cortex of the mouse that had been reconstructed in its entirety from electron microscopic analysis of serial thin sections (White and Rock, 1980). Morphological data included dimensions of soma and all five dendrites, neck lengths and head diameters of all 380 spines (a uniform neck diameter of 0.1 micron was assumed), locations of all symmetrical and asymmetrical (axo-spinous) synapses, and locations of all 43 thalamocortical (TC) synapses (as identified from the consequences of a prior thalamic lesion). In the model, unitary excitatory synaptic inputs had a peak conductance change of 0.5 nS at 0.2 msec; conclusions were robust over a wide range of assumed passive-membrane parameters. When recorded at the soma, all unitary EPSPs, which were initiated at the spine heads, were relatively iso-efficient; each produced about 1 mV somatic depolarization regardless of spine location or geometry. However, in the spine heads there was a twentyfold variation in EPSP amplitudes, largely reflecting the variation in spine neck lengths. Synchronous activation of the TC synapses produced a somatic depolarization probably sufficient to fire the neuron; doubling or halving the TC spine neck diameters had only minimal effect on the amplitude of the composite TC-EPSP. As have others, we also conclude that from a somato-centric viewpoint, changes in spine geometry would have relatively little direct influence on amplitudes of EPSPs recorded at the soma, especially for a distributed, synchronously activated input such as the TC pathway. However, consideration of the detailed morphology of an entire neuron indicates that, from a dendro-centric point of view, changes in spine dimension can have a very significant electrical impact on local processing near the sites of input.

Subthreshold oscillations and resonant frequency in guinea-pig cortical neurons: physiology and modelling.

1. Intracellular recordings were made from neurons in slices from guinea-pig frontal cortex. In 50% of the cells, sustained subthreshold voltage oscillations were evoked by long (> 6 s) depolarizing pulses. The peak-to-peak amplitude of these oscillations was less than 5 mV and the frequency was voltage dependent, increasing with depolarization from 4 (near rest) to 20 Hz (at 30 mV depolarization). 2. The impedance-frequency relationship of both oscillating and non-oscillating cells was studied by intracellular injection of sinusoidal current with linearly changing frequency. In most cells, a peak in the impedance magnitude (resonant behaviour) was observed at depolarized levels. The frequency of the peak impedance (peak frequency) increased with depolarization from 3 (near rest) to 15 Hz (at 30 mV depolarization). 3. Application of TTX (10(-6) M) significantly decreased the impedance magnitude near the peak frequency. The subthreshold oscillations, however, as well as the action potentials, were fully blocked by TTX. On the other hand, TEA (15 mM) and Cs+ (5 mM) abolished both the subthreshold oscillations and the resonant behaviour. Replacing Ca2+ with Co2+ (5 mM) or Ni2+ (1 mM) did not abolish the subthreshold oscillations. The peak in the frequency-response curve was only slightly reduced. 4. An isopotential membrane model, consisting of a leak current, a fast persistent sodium current, a slow non-inactivating potassium current (with the kinetics of the M-current) and membrane capacitance, is sufficient to produce both voltage oscillations and resonant behaviour. The kinetics of the K+ current by itself is sufficient to produce resonance behaviour. The Na+ current amplifies the peak impedance magnitude and is essential for the generation of subthreshold oscillation. The model correctly predicted the behaviour of the frequency response before and after TTX and TEA application, as well as the relation between the expected passive impedance and the experimental impedance. 5. We speculate that the tendency of the neurons to generate voltage signals at a certain frequency (as a result of the subthreshold oscillations) and to preferentially respond to inputs arriving at the same frequency (the resonance behaviour) promotes population activity at that preferred frequency.

The morphoelectrotonic transform: a graphical approach to dendritic function.

Electrotonic structure of dendrites plays a critical role in neuronal computation and plasticity. In this article we develop two novel measures of electrotonic structure that describe intraneuronal signaling in dendrites of arbitrary geometry. The log-attenuation Lij measures the efficacy, and the propagation delay Pij the speed, of signal transfer between any two points i and j. These measures are additive, in the sense that if j lies between i and k, the total distance Lik is just the sum of the partial distances: Lik = Lij + Ljk, and similarly Pik = Pij + Pjk. This property serves as the basis for the morphoelectrotonic transform (MET), a graphical mapping from morphological into electrotonic space. In a MET, either Pij or Lij replace anatomical distance as the fundamental unit and so provide direct functional measures of intraneuronal signaling. The analysis holds for arbitrary transient signals, even those generated by nonlinear conductance changes underlying both synaptic and action potentials. Depending on input location and the measure of interest, a single neuron admits many METs, each emphasizing different functional consequences of the dendritic electrotonic structure. Using a single layer 5 cortical pyramidal neuron, we illustrate a collection of METs that lead to a deeper understanding of the electrical behavior of its dendritic tree. We then compare this cortical cell to representative neurons from other brain regions (cortical layer 2/3 pyramidal, region CA1 hippocampal pyramidal, and cerebellar Purkinje). Finally, we apply the MET to electrical signaling in dendritic spines, and extend this analysis to calcium signaling within spines. Our results demonstrate that the MET provides a powerful tool for obtaining a rapid and intuitive grasp of the functional properties of dendritic trees.