Impulse encoding across the dendritic morphologies of retinal ganglion cells.

Nerve impulse entrainment and other excitation and passive phenomena are analyzed for a morphologically diverse and exhaustive data set (n = 57) of realistic (3-dimensional computer traced) soma-dendritic tree structures of ganglion cells in the tiger salamander (Ambystoma tigrinum) retina. The neurons, including axon and an anatomically specialized thin axonal segment that is observed in every ganglion cell, were supplied with five voltage- or ligand-gated ion channels (plus leakage), which were distributed in accordance with those found in a recent study that employed an equivalent dendritic cylinder. A wide variety of impulse-entrainment responses was observed, including regular low-frequency firing, impulse doublets, and more complex patterns involving impulse propagation failures (or aborted spikes) within the encoder region, all of which have been observed experimentally. The impulse-frequency response curves of the cells fell into three groups called FAST, MEDIUM, and SLOW in approximate proportion as seen experimentally. In addition to these, a new group was found among the traced cells that exhibited an impulse-frequency response twice that of the FAST category. The total amount of soma-dendritic surface area exhibited by a given cell is decisive in determining its electrophysiological classification. On the other hand, we found only a weak correlation between the electrophysiological group and the morphological classification of a given cell, which is based on the complexity of dendritic branching and the physical reach or "receptive field" area of the cell. Dendritic morphology determines discharge patterns to dendritic (synaptic) stimulation. Orthodromic impulses can be initiated on the axon hillock, the thin axonal segment, the soma, or even the proximal axon beyond the thin segment, depending on stimulus magnitude, soma-dendritic membrane area, channel distribution, and state within the repetitive impulse cycle. Although a sufficiently high dendritic Na-channel density can lead to dendritic impulse initiation, this does not occur with our "standard" channel densities and is not seen experimentally. Even so, impulses initiated elsewhere do invade all except very thin dendritic processes. Impulse-encoding irregularities increase when channel conductances are reduced in the encoder region, and the F/I properties of the cells are a strong function of the calcium- and Ca-activated K-channel densities. Use of equivalent dendritic cylinders requires more soma-dendritic surface area than real dendritic trees, and the source of the discrepancy is discussed.

Impulse encoding mechanisms of ganglion cells in the tiger salamander retina.

A study of nerve impulse generation in ganglion cells of the tiger salamander retina is carried out through a combination of experimental and analytic approaches, including computer simulations based on a single-compartment model. Whole cell recordings from ganglion cells were obtained using a superfused retina-eyecup preparation and studied with pharmacological and electrophysiological techniques, including phase plot analysis. Experimental efforts were guided by computer simulation studies of an excitability model consisting of five voltage- or ion-gated channels, which were identified from earlier voltage-clamp data. The ion channels include sodium, calcium, and three types of potassium channels, namely the A type (IK,A), Ca-activated potassium (IK,Ca), and the delayed rectifier (IK). A leakage channel was included to preserve input resistance continuity between model and experiment. Ion channel densities of Na and Ca currents (INa and ICa) for the single-compartment model were independently determined from phase plot analysis. The IK and IK,A current densities were determined from the measured width of impulses. The IK,Ca was modeled to respond to Ca influx, and a variable-rate Ca-sequestering mechanism was implemented to remove cytoplasmic calcium. Impulse frequency increases when either ICa or IK,Ca is eliminated from the model or blocked pharmacologically in whole cell recording experiments. Faithful simulations of experimental data show that the ionic currents may be grouped into small (IK,Ca, leakage, and stimulus), and large (INa, IK, IA, ICa) on the basis of their peak magnitudes throughout the impulse train. This division of the currents is reflected in their function of controlling the interspike interval (small currents) and impulse generation (large currents). Although the single-compartmental model is qualitatively successful in simulating impulse frequency behavior and its controlling mechanisms, limitations were found that specifically suggest the need to include morphological details. The spike train analysis points to a role for electrotonic currents in the control of the duration of the interspike intervals, which can be compensated by prolonged activation of gK,Ca in the single-compartment model. A detailed, multicompartmental model of the ganglion cell is presented in the companion paper.

Mechanisms by which cell geometry controls repetitive impulse firing in retinal ganglion cells.

Models for generating repetitive impulse activity were developed based on multicompartmental representations of ganglion cell morphology in the amphibian retina. Each model includes five nonlinear ion channels and one linear (leakage) channel. Compartmental distribution of ion channel type and density was designed to simulate whole cell recording experiments carried out in the intact retina-eyecup preparation. Correspondence between the model and physiology emphasized channel-specific details in the impulse waveform, based on phase plot analysis, frequency versus current (F/I) properties, and interspike trajectories for current injected into the soma, as well as the ability to conduct impulses in both orthodromic and antidromic directions. Two general types of model are developed, including equivalent cylinder representations and more realistic compartmentalizations of dendritic morphology. These multicompartmental models include representations for dendritic trees, soma, axon hillock, a thin axonal segment, and axon distal to thin segment. A large number of compartments (

Modeling the repetitive firing of retinal ganglion cells.

A kinetic model for the repetitive firing of retinal ganglion cells was synthesized from voltage-clamp data and evaluated by comparison with whole cell recordings from ganglion cells in the intact tiger salamander retina. Five distinct channels were included in the model and were sufficient to describe the physiologically observed frequency/current relationship in response to various levels of cell depolarization.

Gating current harmonics. IV. Dynamic properties of secondary activation kinetics in sodium channel gating.

The kinetics for sodium channel gating appear to involve three coupled processes: (a) "primary" activation, (b) "secondary" activation, and (c) inactivation. Gating current data obtained in dynamic steady states with sinusoidal voltage-clamp were analyzed to give further details about the secondary activation process in sodium channel gating. Unlike primary activation and inactivation, the secondary activation kinetics involve physical processes that become defined when the data are analyzed as a function of the sinusoid frequency in addition to mean membrane potential. The effects of these processes are described, and a physical interpretation is presented.

Gating current harmonics. III. Dynamic transients and steady states with intact sodium inactivation gating.

Internally perfused squid giant axons with intact sodium inactivation gating were prepared for gating current experiments. Gating current records were obtained in sinusoidally driven dynamic steady states and as dynamic transients as functions of the mean membrane potential and the frequency of the command sinusoid. Controls were obtained after internal protease treatment of the axons that fully removed inactivation. The nonlinear analysis consisted of determining and interpreting the harmonic content in the current records. The results indicate the presence of three kinetic processes, two of which are associated with activation gating (the so-called primary and secondary processes), and the third with inactivation gating. The dynamic steady state data show that inactivation gating does not contribute a component to the gating current, and has no direct voltage-dependence of its own. Rather, the inactivation kinetics appear to be coupled to the primary activation kinetics, and the coupling mechanism appears to be one of reciprocal steric hindrance between two molecular components. The mechanism allows the channel to become inactivated without first entering the conducting state, and will do so in about 40 percent of depolarizing voltage-clamp steps to 0 mV. The derived model kinetics further indicate that the conducting state may flicker between open and closed with the lifetime of either state being 10 microseconds. Dynamic transients generated by the model kinetics (i.e., the behavior of the harmonic components as a function of time after an instantaneous change in the mean membrane potential from a holding potential of -80 mV) match the experimental dynamic transients in all details. These transients have a duration of 7-10 ms (depending on the level of depolarization), and are the result of the developing inactivation following the discontinuous voltage change. A detailed hypothetical molecular model of the channel and gating machinery is presented.

Gating current harmonics. I. Sodium channel activation gating in dynamic steady states.

Internally perfused and pronase-treated giant axons were prepared for gating current measurements. Gating current records were obtained under large-amplitude sinusoidal voltage clamp after allowing for settling times into dynamic steady states. The current records were analyzed as functions of the mean membrane potential of the test sinusoid for which the amplitude and frequency were held constant. The nonlinear analysis consisted of determining the harmonic content (amplitudes and phases) of the distorted periodic current records. The most pronounced feature found in the analysis is a dominant second harmonic centered at Emean = +10 mV. A number of other characteristic harmonic behaviors were also observed. The harmonics tend to die away for very small (less than -60 mV) and very large (greater than +72 mV) values of Emean. The harmonic behavior seen in the axonal data is basically different from that seen in gating current simulations generated by the sodium-activation kinetics of standard models, including the Hodgkin-Huxley model. Some of the differences can be reconciled without requiring fundamental changes in the model kinetic schemes. However, the dominant harmonic feature seen in the axonal data cannot be reconciled with the model kinetics without a fundamental change in the models. The axonal data suggest two moving molecular components with independent degrees of freedom whose properties are outlined on the basis of the data presented herein.

Gating current harmonics. II. Model simulations of axonal gating currents.

A kinetic model of sodium activation gating is presented. The kinetics are based on harmonic analysis of gating current data obtained during large-amplitude sinusoidal voltage clamp in dynamic steady state. The technique classifies gating kinetic schemes into groups based on patterns of the harmonic content in the periodic gating current records. The kinetics that simulate the experimental data contain two independently constrained processes. The model predicts (a) sizable gating currents in response to hyperpolarizing voltage steps from rest; (b) a substantial increase in the initial peak of the gating current following voltage steps from prehyperpolarized potentials; (c) a small delay in the onset of sodium ion current following voltage steps from prehyperpolarized potentials; and (d) flickering during the open state in single channel current records. Although fundamentally different in kinetic structure from the Hodgkin-Huxley model, the present model reproduces the phenomenological development of Na conductance during the initiation and development of action potentials. The implications for possible gating mechanisms are discussed. A model gate is presented.

Excitable channel currents and gating times in the presence of anticonvulsants ethosuximide and valproate.

The effects of the anticonvulsants ethosuximide and valproate on the excitable Na and K channels of the squid giant axon are evaluated and compared. The drugs are highly specific in their effects on channel gating and ion permeability with regard to the membrane side of application. Both drugs when applied internally affect Na activation gating in ways that lead to the conclusion that they do not act as channel blockers. However, external ethosuximide is clearly a voltage-independent Na channel blocker with no effect on channel gating. On the K channel, ethosuximide appears to have a mixed action affecting both gating and the ion flux through open channels. However, valproate slows K channel gating without effect on the flux through open channels. The dose-response curve of the effects has a shape similar to that for ethanol. The implications for paroxysmal discharge and synchronous impulse generation are discussed in a preliminary way.

Rapid sodium channel conductance changes during voltage clamp steps in squid giant axons.

The sodium conductance of the membrane of the giant axon of squid was isolated by the use of potassium-free solutions and voltage-clamped with pulses containing three levels of depolarization. The conductance appears to undergo rapid changes during certain repolarizing clamp steps whose voltage reach at least partially overlaps the gating range. The percentage change in conductance increases with time of depolarization from approximately 0 to approximately 25-30% at 7 ms for a potential step from +70 to -30 mV. Conductance steps were also observed for voltage steps from various depolarized levels to -70 mV. All observed shifts were in the direction of a decreased conductance. The conductance steps appear to be a weak function of the concentration of external calcium, which also acts as a voltage-dependent channel blocker for inwardly directed sodium currents. A number of possible mechanisms are suggested. One of these is discussed in some detail and postulates a voltage- and time-dependent molecular process that does not itself yield open or closed channel conformations, but that affects the magnitude of the rate constants that do connect open and closed state conformations.

On the application of systems analysis to neurophysiological problems.

In this essay we have presented some personal viewpoints on the application of Systems. Analysis to suitable neurophysiological problems and provided reasoned reviews on selected topics (receptor and nerve cell physiology, motor control, vestibular and visual systems) in which new data and ideas were brought about in the past several years by the use of Systems Analysis. Since the nature of the paper precludes the possibility of writing a summary, we simply thank those readers who have endured up to this point. As for those prudent persons who are beginning from this summary, before deciding as to the advisability of taking-up the task, we have a simple advice which runs contrary to the rule stated by the King to Alice in Wonderland (and accepted by one of us; see p. 60): begin from the topic that you like most and eventually go on, using the same criteria, till you are tired: then stop.

Anomalous potassium channel-gating rates as functions of calcium and potassium ion concentrations.

With near normal monovalent ionic concentrations, the rate of increase of the potassium conductance after a depolarizing voltage-clamp step is slowed when the external calcium concentration is increased. This trend in the rise-time with changes in calcium is reversed when the axointernal potassium concentration is reduced (150 mM) and the periaxonal concentration is increased (50 mM); that is, the rise-time decrease with increasing calcium concentration. Furthermore, the degree of sigmoidality of the K-conductance time-course always increase when the rise-times increase for a given test potential. In the case of calcium surface-charge screening, these effects may be caused by a shifted distribution of K-ions within the channels following the altered ion gradient, and by a consequent shift in the reciprocal electrostatic interactions between the ionic charges and channel-gate charges.

Excitation properties of the squid axon membrane and model systems with current stimulation. Statistical evaluation and comparison.

The space-clamped squid axon membrane and two versions of the Hodgkin-Huxley model (the original, and a strongly adapting version) are subjected to a first order dynamic analysis. Stable, repetitive firing is induced by phase-locking nerve impulses to sinusoidal currents. The entrained impulses are then pulse position modulated by additional, small amplitude perturbation sinusoidal currents with respect to which the frequencies response of impulse density functions are measured. (Impulse density is defined as the number of impulses per unit time of an ensemble of membranes with each membrane subject to the same stimulus). Two categories of dynamic response are observed: one shows clear indications of a corner frequency, the other has the corner frequency obscured by dynamics associated with first order conductance perturbations in the interspike interval. The axon membrane responds with first order perturbations whereas the unmodified Hodgkin-Huxley model does not. Quantitative dynamic signatures suggest that the relaxation times of axonal recovery excitation variables are twice as long as those of the corresponding model variables. A number of other quantitative differences between axon and models, including the values of threshold stimuli are also observed.

Excitation parameters of the repetitive firing mechanism from a statistical evaluation of nerve impulse trains.

Repetitive firing of single tonic neurones is modeled to include in detail both membrane excitation kinetics and electrotonic effects due to membrane non-uniformities in the impulse encoder region. The model is evaluated dynamically and compared with similar data obtained from the crayfish stretch receptor neuron. Two dynamic techniques utilizing small amplitude sinusoidal signals are employed. One technique is used to fix the values of two parameters which relate to the electrotonic control of membrane potential in the interspike interval and to the relaxation time of the K-conductance during repetitive firing. The other technique is employed as a consistency check. The dynamics are particularly sensitive to the K-channel relaxation time in the interspike interval.

Repetitive firing: a quantitative study of feedback in model encoders.

Recognition of nonlinearities in the neuronal encoding of repetitive spike trains has generated a number of models to explain this behavior. Here we develop the mathematics and a set of tests for two such models: the leaky integrator and the variable-gamma model. Both of these are nearly sufficient to explain the dynamic behavior of a number of repetitively firing, sensory neurons. Model parameters can be related to possible underlying basic mechanisms. Summed and nonsummed, spike-locked negative feedback are examined in conjunction with the models. Transfer functions are formulated to predict responses to steady state, steps, and sinusoidally varying stimuli in which output data are the times of spike-train events only. An electrical analog model for the leaky integrator is tested to verify predicted responses. Curve fitting and parameter variation techniques are explored for the purpose of extracting basic model parameters from spike train data. Sinusoidal analysis of spike trains appear to be a very accurate method for determining spike-locked feedback parameters, and it is to a large extent a model independent method that may also be applied to neuronal responses.

Repetitive firing: quantitative analysis of encoder behavior of slowly adapting stretch receptor of crayfish and eccentric cell of Limulus.

Techniques developed for determining summed encoder feedback in conjunction with the leaky integrator and variable-gamma models for repetitive firing are applied to spike train data obtained from the slowly adapting crustacean stretch receptor and the eccentric cell of Limulus. Input stimuli were intracellularly applied currents. Analysis of data from cells stringently selected by reproducibility criteria gave a consistent picture for the dynamics of repetitive firing. The variable-gamma model with appropriate summed feedback was most accurate for describing encoding behavior of both cell types. The leaky integrator model, while useful for determining summed feedback parameters, was inadequate to account for underlying mechanisms of encoder activity. For the stretch receptor, two summed feedback processes were detected: one had a short time constant; the other, a long one. Appropriate tests indicated that the short time constant effect was from an electrogenic sodium pump, and the same is presumed for the long time constant summed feedback. Both feedbacks show seasonal and/or species variations. Short hyperpolarizing pulses inhibited the feedback from the long time constant process. The eccentric cell also showed two summed feedback processes: one is due to self inhibition, the other is postulated to be a short time constant electrogenic sodium pump similar to that described in the stretch receptor.