Department of Physiology, University of Minnesota, Minneapolis, Minnesota 55455
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ABSTRACT |
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Sheasby, Brent W. and Jurgen F. Fohlmeister. 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.
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INTRODUCTION |
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The impulse-encoding mechanism of intact retinal
ganglion cells recently was explored on the basis of a series of models
(Fohlmeister and Miller 1979a,b
), the five nonlinear ion
channels of which were identified from earlier voltage-clamp data
(Kaneda and Kaneko 1991a
,b
; Lasater and Witkovsky
1990
; Lipton and Tauk 1987
; Lukasiewicz and Werblin 1988
). Although the light-evoked impulse responses of individual retinal ganglion cells may be continuous (tonic) or it
may consist of brief bursts (phasic responses) to light ON,
OFF, or both ON/OFF (Baylor and
Fettiplace 1979
; Belgum et al. 1983
), virtually
all ganglion cells respond with tonic repetitive firing to depolarizing
currents injected into the soma (Fig. 1). This relatively uniform response pattern among all ganglion cells allowed a systematic model development with regard to incorporating the
effects of cell morphology on impulse entrainment. The development began with a single compartment model (Fohlmeister and Miller 1997a
; Fohlmeister et al. 1990
) and proceeded to
a series of multicompartment models in which the dendritic tree was
represented by an equivalent cylinder (Fohlmeister and Miller
1997b
). The multicompartment models allowed for nonuniform
distributions of the channels throughout the neuron and clearly showed
that first-order encoding effects are due to the axial or
"longitudinal" currents that naturally arise from the cell
geometry. The present paper extends this analysis to realistic (traced)
dendritic morphology (cf. Arkin and Miller 1988
).
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Experimental spike train records indicated that about half of all
ganglion cells yield a tonic impulse rate of ~1 imp · s1 · pA
1 of constant stimulus
current injected into the soma. The remaining cells responded with
lower impulse rates (0.5-0.7 imp · s
1 · pA
1), with the slowest cell responses showing various
forms of irregularity in the spike train. This variability among
ganglion cells was reproducible by adjusting the geometric parameters
(specifically the diameter) of the dendritic equivalent cylinder, not,
however, by manipulations of ion channel densities or distributions
(cf. Eliasof et al. 1987
). The real dendritic trees of
the present study are not amenable to geometric manipulation. It is
therefore remarkable that our data set of 57 traced cells yields a
distribution in the impulse firing properties that closely matches that
seen experimentally, with however, the notable addition of a new group of cells that respond with high impulse frequencies (called
"superFAST" herein), which were apparently not
recognized as a distinct group in the data collection process. The
early subsections of RESULTS therefore explore details of
the variety of the encoding phenomena on the basis of a single (fixed)
distribution of ion channels, called "standard" herein. This
channel distribution was deemed to represent the healthiest cells in
the equivalent cylinder study. Modifications in the channel
distribution subsequently are considered, leading to the conclusion
that all retinal ganglion cells
irrespective of size or dendritic
branching complexity
may be endowed with a similar distribution of ion channels.
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METHODS |
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Experimental methods for obtaining the impulse train records are
given in Fohlmeister and Miller (1997a,b
). Because
virtually all ganglion cells responded with tonic repetitive firing to
soma stimulation, the cells were classified on the basis of the
steepness of their impulse frequency versus stimulus current
(F/I) response curves (Fohlmeister and Miller
1997a
, Table 2 and Fig. 11). Cells with a response rate of ~1
imp · s
1 · pA
1 (Hz/pA) were
classified as FAST to distinguish them from the somewhat
more rare MEDIUM (~0.7 Hz/pA) and SLOW
(~0.5 Hz/pA) cells. The 95% confidence error bars for F/I
of the MEDIUM and SLOW groups overlap, and this
last SLOW classification was reserved for cells that either
did not respond with impulses to the lower stimulus currents (<20 pA)
or responded only with irregular impulse firing patterns. Temperature
was 22°C.
A primary purpose of this study is to determine the relationship
between the aforementioned physiological response classification and a
morphological classification. Our dendritic morphology data pool
consists of 57 ganglion cells classified into large (L), medium-complex
(MC), medium-simple (MS), small-complex (SC), and small-simple (SS),
examples of which are given in Fig. 2. We
are indebted to Toris et al. (1995) for the neural
tracing (Eutectic Neuronal Reconstruction System) and morphological
classification. This published classification was based on the visual
impression of the size and profuseness of the horseradish peroxidase
(HRP)-stained and traced structures (C. Toris, personal communication).
The cells subsequently were subjected to a cluster analysis, which found them to be consistent with a continuum of dendritic structures rather than discrete classes (Kosta and Velte 1998
).
ASCII versions of the dendritic structures were compartmentalized,
examples of which are given in Fig. 2B. The dendritic
processes are represented by a sufficient number of cylindrical
compartments to faithfully reproduce the variable thickness of those
processes. In addition to the dendritic tree and soma, a
compartmentalized axon of 1 µm diam and 5.5 mm length was connected
to the soma, and this axon contained a narrow segment of
0.3-0.6 µm diam and 90 µm length after an initial
segment of 40 µm length. Except where explicitly stated, all
data presented here were obtained with a narrow segment diameter of 0.4 µm, although the general phenomena described herein occurred also
throughout the range of measured diameters of the narrow segment.
Cytoplasmic resistivity is 110
cm throughout.
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The complement of ionic channels consists of four voltage-gated (Na,
Ca, K, and KA), one calcium-gated (KCa), and
one leakage channel. The Ca channel represents the high-threshold,
"L type," and has no inactivation kinetics (cf. Karschin
and Lipton 1989); the K channel represents the classical
"delayed rectifier," which also is modeled with no inactivation
kinetics; the Na channel and the A-type K channel (Connor and
Stevens 1971
) are modeled with inactivation kinetics. The
Ca-activated K channel is of the "SK" subtypes (Hugues et
al. 1982
) which are apamin sensitive. The ungated leakage
channel was adjusted for input resistance homology with experiment. The
total instantaneous membrane current of every compartment therefore is
given by the sum of the capacitative, plus six ionic currents (i.e., 7 parallel current paths): membrane current = Cm(dV/dt) +
Nam3h(V
VNa) +
Cac3[V
VCa(t)] + (
Kn4 +
Aa3hA +
K, Ca)(V
VK) +
L(V
VL). Details of channel gating kinetics are
given in Fohlmeister and Miller (1997a
,
METHODS/Determining Gating Kinetic Parameters).
Channel densities were distributed throughout the model neurons (Table
1) as in the dendritic equivalent
cylinder models developed in Fohlmeister and Miller
(1997b). The equations of the model neurons were integrated
using the generally applicable computer program NEURON (Hines
1993
). Simulated records were digitized at 5 or 20 kHz (0.2- or
0.05-ms intervals, respectively), which were earlier found to be
adequate rates to resolve, in detail, all features in the phase
plot analysis of the impulse trains; the methods for generating
undistorted phase plots from digitized data records are given in
Fohlmeister and Miller (1997a
, Eqs. 1 and 2).
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RESULTS |
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Impulse frequency versus stimulus current
Impulse trains of the set of realistic model neurons were obtained
by numerical integration with simulated stimulation of 10-50 pA, in
increments of 10 pA (e.g., Fig. 3).
Populations of model neurons were found that correspond to the
physiological classification of FAST (~1 imp · s1 · pA
1), MEDIUM
(~0.7 imp · s
1 · pA
1), and
SLOW (~0.5 imp · s
1 · pA
1) cells (Fohlmeister and Miller 1997a
).
In addition to these groups, a population of cells was encountered with
impulse frequencies that were substantially higher (60-124 Hz at 40 pA) than those of the aforementioned classifications (Tables
2-4).
The slope of the impulse frequency versus stimulus current
(F/I) curve for the equivalent dendritic cylinder models
(Fohlmeister and Miller 1997b
) earlier was found to be a
strong function of soma-dendritic membrane area, and we find that this
group of superFAST" cells have an average soma-dendritic
membrane area of less than half that of the remaining groups (Table
5). Small cells more readily take up HRP
stain, and all stained cells were traced as they appeared (C. Toris,
personal communication). SuperFAST cells therefore probably
are overrepresented in our morphological data set and may actually be
somewhat rare in the Tiger Salamander retina (see DISCUSSION).
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Physiological and morphological classifications
The simulated spike train records were subjected to a phase plot
analysis in which the time rate of change (dV/dt)
is plotted against the membrane potential (V). Phase plot
analysis is a sensitive means of evaluating impulse wave-form in
general (Fitzhugh 1969), and, for our immediate
purposes, of determining the presence and details of an "initial
segment-soma-dendritic (IS-SD) break," in particular. An IS-SD break
sometimes is observed in impulses recorded at the soma, as an
indication that these impulses were initiated on neighboring membrane
with a lower threshold (Coombs et al. 1957
). The IS-SD
break appears as a feature in phase plots corresponding to the early
rising phase of the impulses (Fig. 3, phase plot,
). Although
impulses in retinal ganglion cells do not uniformly exhibit the IS-SD
break, its occurrence increases among the "slower" cells and
becomes characteristically related to the irregular firing of
SLOW cells.
Because impulse frequency response, an IS-SD break, and/or irregular firing are readily apparent features, we established a classification based on the following defining characteristics: group 1, impulse frequency >55 Hz at 40 pA (23 cells); group 2, impulse frequency <55 Hz at 40 pA and no IS-SD break at 10 pA (11 cells); group 3, regular impulse firing, with IS-SD break at 10 pA (14 cells); and group 4, irregular impulse train at 10 pA of constant current stimulus (9 cells).
Group 1 consists of the observed superFAST cells, and the
cutoff criterion of 55 Hz (at 40 pA) was chosen because this value falls in the center of a substantial gap in the observed F/I
slopes among the traced cells of our data set. Note in Table 3 that the
averages of the impulse frequency responses of groups 2-4 correspond
closely to those of FAST, MEDIUM, and
SLOW cells of the earlier physiological classification
(Fohlmeister and Miller 1997a, Fig. 11) and that the
behavior of group 1 cells, with an average F/I slope that is
double that of the group 2 (FAST) cells, suggests that this
is a distinct population with perhaps a distinct signaling function.
The degree of correspondence with the morphological classification
(Toris et al. 1995) is substantially more diffuse with group 1 containing small simple through medium complex cells and group
4 containing medium simple through large cells. Indeed, medium simple
and medium complex cells are represented in every group. On the other
hand, the bulk of small simple cells (13 of 14) fall into group 1, and
groups 3 and 4 contain no small cells of either complexity. Table 2
gives the full distribution by group and morphological classification.
Soma-dendritic membrane area
Contrary to the diffuse correspondence between morphological and
electrophysiological classifications, a much stronger correlation was
found between our physiological classification and the total soma-dendritic surface area of a given cell. The immediate corollary is
that the morphological classificationbased on profuseness of
dendritic branching and on the length of dendrites
is correlated only
weakly with their total membrane surface area. The only substantial exceptions to this rule appear to be the morphologically small simple
cells, which are also predominantly group 1, superFAST cells.
To quantify these statements, we tabulated the surface areas of our data set of cells for the somas, dendrites, and soma-plus-dendrites by both morphological and physiological classifications and computed their averages and standard deviations (Table 5). One immediately striking feature in this tabulation is the great variation in soma and dendritic surface areas within a given morphological group, particularly among the medium simple and medium complex cells, and that soma- and dendritic-membrane areas of individual cells are not correlated. Note, for example, that the soma surfaces of large cells range from 384 to 1,384 µm2, typically smaller than those of medium complex and small complex cells, whereas their dendritic surface areas are typically much larger than those of medium complex and small complex cells.
The primary result of this subsection is, however, the narrow range of surface areas of the dendrites and dendrites-plus-soma found for our electrophysiological group 2 and group 3 cells. This finding shows that the primary determinant for the impulse discharge rate of retinal ganglion cells to soma stimulation is the total amount of membrane present on the dendrites and soma. (Dendritic stimulation, on the other hand, affects the pattern of discharge, see following text.) This result implies that virtually all dendritic membrane is involved in the charging process during interspike intervals and is consistent with the observation that the length of dendrites of retinal ganglion cells is only a fraction of their electrotonic space constants.
To refine this result, we undertook a further subdivision of our group 2, the 11 cells of which contain a subset of 4 that show, in phase plots with 10 pA of stimulus current, a slight tendency toward an IS-SD break. For this subset, the dendritic surface areas range from 3,238 to 3,722 µm2 and the total soma-plus-dendritic areas from 4,507 to 4,871 µm2. The largest surface areas of the remaining seven group 2 cells are 2,936 µm2 (dendrites) and 4,431 µm2 (soma plus dendrites). Thus there is an unequivocal correlation between surface area and electrophysiological impulse rate response.
Because this study is an extension of an analysis done with equivalent
dendritic cylinder models, we present for comparison the membrane areas
used in the construction of the EC2.5 model (Fohlmeister and
Miller 1997b): soma, 1,885 µm2; equivalent
cylinder dendrite, 4,123 µm2; soma plus dendrite, 6,008 µm2. The EC2.5 model was constructed specifically to
represent the (F/I properties of) FAST cells and
did not display IS-SD breaks in their simulated impulse
trains at 10 pA. It is therefore interesting to note that the membrane
surface areas of the EC2.5 model reflect more closely those of our
present group 3 cells, which correspond not to
FAST but to MEDIUM cells. The greater surface
area required of an equivalent cylinder is due to the fact that the
equivalent cylinder cable equation involves a monotonically decreasing
space constant with distance from the soma (Rall 1961
).
Thus electrotonic distance increases more rapidly than the geometric
distance from the soma to the periphery of the tree, and this is
simulated by a longer geometric equivalent cylinder which then yields
the greater surface area.
Resting state and charging parameters
The systematically larger surface area of the equivalent cylinder
requires the lower leakage channel density of
gL = 5 µS/cm2 to yield input
resistance homology with experiment (Coleman and Miller
1989) as well as with the realistic models, which employ gL = 8 µS/cm2 (Fohlmeister
and Miller 1997b
, Table 1). Table 4 lists the averages and
ranges of input resistances (Rn in
gigaOhms) and charging time constants (
in milliseconds)
of the simulated cells as determined by hyperpolarizing current steps
of
1 pA injected into the soma of the resting model neurons.
A noteworthy result of our simulations of realistic models is the
finding that the charging times, , agree with
physiological experiment (Fohlmeister and Miller 1997a
,
Table 2), but both are invariably ~20% smaller (shorter) than those
determined for the equivalent dendritic cylinder models, an observation
already noted earlier (Fohlmeister and Miller 1997b
).
Unlike input resistance, charging times reflect more closely the
membrane RC, and the smaller time constants of the realistic models are
due to the larger leakage conductance (i.e., smaller resistance)
per unit area of membrane, relative to that for the
equivalent cylinder models.
Site of impulse initiation
In general, longitudinal (electrotonic) currents dominate over
membrane current during the interspike intervals, and membrane current
dominates locally during the impulse phase (i.e., where the membrane is
in a state of excitation), thus ensuring the sharpness of spikes
(Fohlmeister and Miller 1997b). Relatively small
stimulus currents (
10 pA) lead to relatively slow subthreshold
depolarization rates, which allow for relatively extensive electrotonic
spread during the latency between current-onset and the first impulse and during the interspike intervals. This spreading of stimulus current
causes action potentials to be initiated on the highly excitable
membrane of the axonal narrow segment (cf. Ringham 1971
) or even on the proximal axon beyond that segment because of geometric effects (Fig. 4, top). As a
result, a pronounced IS-SD break is seen in action potentials recorded
at the soma (Fig. 4, top). As the level of stimulus current
is increased, the local depolarization rate at the site of current
injection (the soma) also is increased, allowing less time for lateral
current spread, with the result that the soma is nearer to threshold
when the axonal narrow segment activates (Fig. 4, middle),
and any IS-SD break becomes less pronounced. Sufficiently large
stimulus currents (
40 pA) can bring the soma membrane to threshold so
rapidly (during the interspike interval), that electrotonic spread of
the stimulus current becomes relatively unimportant, and impulses are
initiated almost simultaneously on the soma membrane, the initial
segment, and the proximal narrow segment (Fig. 4, bottom).
Thus with increasing levels of orthodromic (soma) stimulation, the
impulse initiation site moves to locations more proximal to the soma.
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Axonal doublet spiking and active propagation failure
It has been established for some time that the presence of the
morphologically narrow segment, which follows an initial segment of
some 30-60 µm, can cause impulse propagation difficulties that can
be ameliorated to some extent by increasing the Na-channel density on
the membrane of the narrow segment (Carras et al. 1992; Fohlmeister and Miller 1997b
). Because it is
structurally thin in comparison with its neighboring axon, the narrow
segment also contributes to an electrical impedance mismatch
that occurs at the junction of the axon and the soma. This impedance
mismatch manifests itself in impulse propagation delays as well as
active propagation failures under certain circumstances (cf. also
Antidromic stimulation).
Axial (electrotonic) current flowing from the axon into the soma encounters two additive effects: reduced axial resistance due to the increase in cross-sectional area and increased membrane surface area per unit length in the axial direction. Both effects require the electrical charge carried by the axial current to depolarize a large increase in membrane area on entering the soma. That charge is sometimes insufficient to bring the soma to threshold, or, if sufficient, the soma spike occurs with a latency which is manifested by a pronounced IS-SD break. These effects are most pronounced for large soma-dendritic surface areas and are exacerbated by the presence of the axonal narrow segment that further restricts axial current flow from the axon.
One manifestation of the impedance mismatch between the narrow segment and the soma is the occurrence of more-or-less closely spaced spike doublets propagating on the axon for impulses that are recorded singly in a spike train at the soma (Fig. 5, A and B). The axonal spike doublets may occur for every soma spike, or they may occur for every second or third impulse with the intervening spikes propagating singly on the axon. The corresponding low-frequency impulse trains recorded at the soma show small (1-2 ms) irregularities in the durations of their interspike intervals.
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Axonal spike doublets invariably are initiated with an impulse
generated on the axon some 100-150 µm beyond the narrow segment. This impulse propagates on the axon, whereas its electrotonic (axial)
current in the antidromic direction causes a brief, relatively abrupt
depolarization in the soma-axon hillock region. The abrupt depolarization represents the rising phase of an IS-SD break as seen in
the phase plot of the soma in Fig. 5A' (). This
depolarization, like a threshold shock depolarization, causes an
impulse to be generated in the soma, axon hillock, and proximal narrow
segment almost simultaneously but with a delay (1-2 ms) that allows
the axon to recover sufficiently from the refractoriness of the first impulse to also propagate this second member of the doublet. Because the soma-narrow segment impedance mismatch is more pronounced for the
larger cells, this phenomenon is most prominent for group 4 cells and
occurs more readily at lower stimulus intensities in smaller cells. The
effect therefore enhances the signal by doubling the number of
propagated spikes for the larger cells and at low stimulus levels in general.
A more extreme form of the axonal spike doublet phenomenonseen among
cells with the largest soma-dendritic membrane area
involves one or
more "failed spikes" in the soma followed by a full-blown impulse
recorded in the soma (Fig. 5C). The occurrence of this phenomenon we term "irregular," and it defines our
group 4 (i.e., SLOW cells in Fohlmeister and Miller
1997a
). In these records, all impulses, including the failed
soma spikes, are initiated on the axon some 30-150 µm beyond the
narrow segment and propagate orthodromically on the axon (Fig.
5D). Antidromically, these impulses propagate into the
distal end of the narrow segment (peak to +22 mV), then widen
throughout the narrow segment and decay in amplitude (+18 mV at
midpoint,
20 mV at proximal end), finally reaching peaks of
40 mV
(initial segment) and
55 mV (soma). After one or more failed spikes,
a full-blown impulse occurs in the soma: electrotonic charge from the
repeated axonal discharges, in combination with the maintained stimulus
current, ultimately being sufficient to bring the soma to threshold.
Similar phenomena also have been observed in mammalian (rabbit)
ganglion cells (Fig. 5E), indicating that this effect may
possibly be quite general.
Antidromic stimulation
Antidromically propagating impulses, initiated on the distal axon, can lead to three types of phenomena; including the classical phenomenon of impulse invasion of the soma followed by electrical silence, impulse invasion of the soma followed by an orthodromically propagating "echo-spike," and failure of the impulse to invade the soma.
Except for group 4 (SLOW) cells, the second and third types of phenomena occur only in the presence of the morphological narrow axonal segment, which we conclude to be the proximal cause of these phenomena. As a general trend, in any given cell, a sufficiently thin narrow segment invariably leads to failure of antidromic invasion of the soma. As the diameter of the narrow segment is increased, antidromic spikes begin to invade the soma and to generate an echo-spike in response to that invasion. Both phenomena are related to the propagation delays leading to the failed soma spikes or axonal spike doublets discussed in the last three paragraphs of the previous subsection. Finally, for a narrow segment of sufficiently large diameter, spike invasion of the soma invariably occurs and the echo-spike fails to develop. Group 4 cells form an exception for which antidromic spikes fail to invade the soma even with a "narrow segment" diameter equal to that of the general axon (1 µm).
The range of diameters of the narrow segment for which the echo-spike
occurs shifts to larger values for cells with greater soma-dendritic
surface area or, alternatively, as one advances from
superFAST (group 1) to SLOW (group 4) cells.
Commonly, this diameter range shifts to lower values when the narrow
segment is made more highly excitable. For example, a typical medium
simple group 2 cell with standard channel densities throughout
(Na = 100 mS/cm2 on the
narrow segment) showed no impulse invasion for narrow segment diameters
0.53 µm. The invasion plus echo-spike range occurred for narrow
segment diameters of 0.54-0.63 µm, and invasion with no echo-spike
for narrow segment diameters
0.64 µm. With
Na = 200 mS/cm2 on the narrow
segment, the range of echo spiking in this same cell occurred for
narrow segment diameters of ~0.1 µm smaller and occurred for
diameters of 0.36-0.47 µm when
Na was
increased to 400 mS/cm2 on the narrow segment. With
Na = 400 mS/cm2 on a
sufficiently large diameter narrow segment, antidromic impulses invade
the soma even for group 4 cells, but these invariably are accompanied
by an echo-spike. The sequence of the three panels of Fig.
6 give an indication of the transition
across the three types of antidromic phenomena (noninvasion, invasion
plus echo, invasion alone) that occur with increasing diameter (or
increasing excitability) of the narrow segment.
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On the other hand, we found a much weaker correlation between
Na-channel density on the initial segment and the type of
antidromic invasion (or lack of invasion) phenomenon exhibited by a
given cell. Absence of Na channels on the initial segment (or the
narrow segment) invariably leads to invasion failure. However, soma
invasion does occur with Na = 25 mS/cm2 on the initial segment for the smaller
cells and with
Na
70 mS/cm2 for all cells. It appears therefore that the
excitability properties of the soma and the narrow segment are more
decisive for determining antidromic invasion than those of the initial segment.
The phenomenon of the echo-spike (Fig. 6, middle) arises as
follows: an impulse of normal amplitude (peak to about +27 mV) propagates antidromically on the axon. This impulse increases in
amplitude (peak to about +30 mV) on entering the narrow segment, but
declines in amplitude (peak about 18 mV) in crossing the narrow
segment. The initial segment almost simultaneously peaks at about
40
mV, at which time the rising phase of a pronounced IS-SD break occurs
in the soma. After a delay of 1.5-2 ms, during which time the soma
continues to slowly depolarize, a full-blown spike arises
simultaneously in the soma (to +22 mV), the initial segment (to +20
mV), and the proximal portion of the narrow segment (to +17 mV). This
impulse propagates across the narrow segment and into the axon yielding
the orthodromically propagating echo-spike. The membrane potential
waveform at the location of the junction of the narrow segment and
axon, although clearly an active response sufficiently large to be
considered an impulse (peak about +7 mV), nevertheless shows an
inflection in its rising phase, a kind of axonal IS-SD break, due to a
combination of the low grade refractoriness remaining from the
antidromic impulse and to some impedance-mismatch at that junction.
Modifying the channel densities
The magnitude of the membrane capacitance ultimately sets the
scale for the magnitudes of the ionic conductances (cf.
Fohlmeister and Miller 1997b, Fig. 6); the charging
requirements for the membrane capacitance therefore restricts
modifications of the standard channel density distribution to about
±30%, except for electrotonically coupled regions with little
membrane area such as the axonal narrow segment. The Ca current and the
associated IK, Ca are not essential to spiking
but strongly control the interspike intervals. Their magnitudes are
determined in relation to the spiking current magnitudes. We therefore
evaluate the effects of reducing channel densities ("weakening"
the cells) in the encoder region, modifying channel densities on the
narrow segment of the axon, and varying the calcium channel densities
in the soma and dendritic regions.
Consider the weakened channel densities (Table 1) with the narrow
segment Na = 100 mS/cm2,
which is the minimum value required to insure reasonably faithful propagation of action potentials under most conditions of excitation. This modification results in two principal effects: impulse frequencies are reduced for all stimulus levels and for all classifications of
cells and irregular firing patterns increase substantially among the
cells of our morphological data set.
The percentage change in impulse frequency was least for our group 1 (superFAST) cells (7% at 40 pA) and increased systematically to 17% (group 2) and 31% (group 3), reaching 35% at 40 pA for group 4 (SLOW) cells. The percentage change in the F/I slope for group 1 cells is insufficient for them to be reclassified in a slower category (i.e., group 1) with this reduction in channel densities. Because irregularities in the impulse encoding also increase beyond those observed experimentally, thus making further reductions in the conductances untenable, this supports our new superFAST category as constituting a real population of ganglion cells.
The firing irregularities (i.e., intermittent or periodic spike
failures in the soma) were a defining feature for our group 4 (SLOW) cells and occurred almost exclusively for low
stimulus currents (20 pA). With the reduced channel densities,
irregularities spread to group 3, and occasional group 2 cells for low
stimuli, and to all stimulus levels of the group 4 (SLOW)
cells. This spreading in firing irregularities for weakened cells is
reversed by increasing the Na-channel density on the narrow segment of
the axon alone to
Na = 300 mS/cm2, and impulse firing frequencies also recover the
values determined for the standard channel densities. However,
antidromic invasion does not recover with this modification; a typical
medium simple group 2 cell with weakened channel densities and
Na = 300 mS/cm2 on the narrow
segment requires a minimum narrow segment diameter of 0.7 µm for
invasion. We consider this to be an unacceptable result because the
axonal narrow segments of ganglion cells have diameters in the range of
0.3-0.6 µm, and the majority of cells do show antidromic spike
invasion of the soma experimentally, sometimes followed by an
echo-spike (Fohlmeister and Miller 1997b
).
A fundamental consideration in adjusting the Na-channel density on the
narrow segment is the maintenance of a stable resting state for the
neuron as a whole. With standard channel densities elsewhere, the
Na-channel density on the narrow segment can be increased to
Na = 1,000 mS/cm2 while
maintaining a stable neural resting state. However, such large values
of
Na convert the neuron into a
bistable system in that it will not recover from
the excited state once threshold is crossed. The bifurcation point
leading to a bistable system depends somewhat on the total surface area
of soma and dendrites of a given neuron, but is approximately
Na = 500 mS/cm2. The
corresponding value for the (otherwise) weakened channel densities is
approximately
Na = 400 mS/cm2. Because the model neuron must be
monostable, we consider
Na = 300 mS/cm2 to be a reasonable upper limit for the Na-channel
density on the narrow segment with a sufficient safety margin, although
this value may be as high as ~400 mS/cm2 for the
healthiest cells. The transition from mono- to bistable is quite sharp.
For example, one group 2 neuron with otherwise standard channel
densities is monostable with
Na
498.069 mS/cm2 and bistable with
Na
498.070 mS/cm2 on the
narrow segment. The same neuron weakened is monostable with
Na
396.087 mS/cm2 and
bistable with
Na
396.088 mS/cm2 on the narrow segment.
With the standard channel densities and low stimulus levels (~10 pA),
increasing Na from 100 mS/cm2
to
200 mS/cm2 on the narrow segment moves the orthodromic
impulse initiation point from the proximal axon beyond the
narrow segment onto the narrow segment itself. Propagation delays
between the narrow segment and the soma thereby are reduced, and this
is the principal reason for the reduced degree of firing
irregularities, as noted in the preceding text. The effect is most
pronounced for group 4 and 3 cells and does not apply to the highly
regular firing group 1 (superFAST) cells. On the other
hand, the presence or absence of the potassium currents,
IK and/or IA, on the
narrow segment show little effect on impulse entrainment or other
excitation phenomena in the bulk of cells with standard channel
densities. This picture changes quite dramatically with the weakened
channel densities (and with
Na = 300 mS/cm2 on the narrow segment), as follows.
Spike doublets recorded in the soma
Removing the delayed rectifier and A-type K channels from the
axonal narrow segment introduces spike doublets in impulse trains recorded at the soma for many, but not all, cells (Fig.
7). This occurrence of spike doublets is
more prevalent with lower levels of stimulus current (20 pA). Impulse
bursts may replace the doublet spiking in some cells, and this bursting
can go over to continuous high-frequency firing under certain stimulus
conditions. This last phenomenon is not stable, and we observed no
systematic pattern of it among our grouping of cells. As an example,
one group 4 cell showed bursting with 20 and 50 pA of constant stimulus
current (Fig. 8) and yielded continuous
high-frequency firing with 40 pA of stimulus. A short impulse burst
also can occur immediately after the "break" of current of a
rectangular depolarizing stimulus pulse, during which a regular
low-frequency impulse train was generated.
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During the burst, the membrane potential of the axonal narrow segment
fluctuates erratically within the relatively depolarized range of 30
to 0 mV (Fig. 8, bottom). This fluctuation decouples the
soma firing pattern from that of the axon, with the result that the
number of spikes in the soma, and those propagating on the axon, may
not be equal. Note in Fig. 8 that the fluctuation on the narrow segment
undergoes a depolarizing upswing in response to every spike generated
in either the soma or on the axon. The depth of the subsequent
downswing depends on the degree of synchronization between the soma and
axonal spikes; when the soma and axonal spikes are coincident, the
encoder region undergoes a hard reset, which forces the full
repolarization of the narrow segment, and the burst terminates.
Experimentally, doublet spiking and bursting is sometimes seen toward
the end of a laboratory session, during which the neuron generated
regular impulse trains. In those cases, it may be that the long
experimental protocol has caused a fraction of the channels to become
nonfunctional (i.e., to "weaken" the cell). Nevertheless, it is
interesting to note that impulse trains consisting of spike doublets
are commonly recorded in W ganglion cell somas of the cat retina (Fig.
7C) (Rowe and Palmer 1995
; Rowe, personal
communication; cf. also Caldwell and Daw 1978
). In that
case, doublet spike trains appear to be recorded only in cells with
thin axons, as determined from the impulse propagation velocity on the axon.
Calcium system
The calcium current is the only current modeled with a variable
reversal potential because the cytoplasmic Ca concentration can vary
substantially in response to Ca influx during action potentials. In
addition to its effect on the reversal potential, the cytoplasmic
Ca-concentration also gates the Ca-activated K current (Barrett
et al. 1982; Cannel et al. 1987
; Chad et
al. 1987
; Gorman and Thomas 1978
;
Hernandez-Cruz et al. 1990
; Latorre et al.
1989
; Marty 1981
; Meech 1978
;
Pallota et al. 1995
; Rogawski 1989
). The
Ca-activated K-current IK,Ca acts to strongly
suppress spontaneous excitation and therefore strongly contributes to
the existence of a stable resting state in our model ganglion cells (cf. Fohlmeister and Miller 1997b
). The effect is
uniform across all morphological classes. These model neurons fire
spontaneously in a broad range ~10 imp/s when
IK,Ca is blocked and increase their
impulse-frequency rates by some 30-80% relative to those with
IK,Ca intact (Hugues et al. 1982
)
for all nonzero stimulus levels. Despite these effects in the
F/I properties, IK,Ca contributes little to the shape of action potentials and is too small to be recognized in phase plots.
Unlike IK,Ca, the Ca current,
ICa, can be blocked with virtually no effect on
the resting cell because of its high threshold for activation. The
rising phase of impulses is dominated by INa as
clearly identified and expressed in phase plots (Fig.
9). Thus Ca current comes into
playsubstantially
only as the impulse approaches its peak, at which
point ICa adds some 10 mV to that peak value due
to its large, positive reversal potential. This increased peak voltage
causes a substantial increase in Na inactivation as well as K-channel
activation, leading to a relatively deep afterhyperpolarization and
prolonged subsequent interspike interval. When the Ca-channels are
blocked, impulse amplitude therefore is reduced at both extremes (peak
and afterhyperpolarization), followed by a more rapid return to
threshold. Thus impulse frequencies increase by some 50-100% for all
levels of constant current stimulation, and this agrees quantitatively
with experiment (Fohlmeister and Miller 1997a
,b
).
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The Ca current imparts a unique "signature" to the latter part of
the falling phase of action potentials, clearly expressed in phase
plots, which allows the relatively precise determination of its
magnitude. As Ca is increased, the phase
plot trajectory shows an increasing downward bulge (increasing rate of
repolarization) located about
30 to
40 mV (Fig. 9). For
sufficiently large
Ca (
6
mS/cm2), this bulge is preceded by a decrease in
the rate of repolarization (at ~0 mV, Fig. 9C'), which
indicates a tendency to generate plateau action potentials (similar to
those of cardiac Purkinje fibers). Increasing the Ca current also leads
to strong suppression of impulse frequencies (Fig. 9, top)
and failed spiking in the soma, effects that are only partially
ameliorated by increasing the Na-channel density to 300 or 400 mS/cm2 on the axonal narrow segment. These effects are
uniform across all morphological classes. The calcium signature is
sufficiently large, sensitive and channel specific to allow
the mean Ca-channel density in the dendrite-soma region to be
determined accurately from phase plots (cf. Fohlmeister and
Miller 1997a
,b
).
Dendritic stimulation
The number of active synaptic locations in a given
dendritic tree was determined by dividing the total dendritic membrane area (in square micrometers) by 250, which yields a count that closely
approximates the number of active ribbon-synapses to light stimulation
(R. F. Miller, personal communication). This number ranged from 5 for a group 1 cell to 30 for a group 4 cell. Dendritic sites were
stimulated singly or variously distributed with total excitatory
(depolarizing) currents in the range of 15-320 pA. In all cases, the
resulting impulses or impulse trains were initiated similar to those
with soma stimulation (Fig.
10A), namely in the narrow
segment-initial segment-soma region, with all impulses faithfully
propagated on the axon. With 320 pA of total "synaptic" current
divided equally among the active sites, impulse frequencies ranged from
~50 Hz for the slowest of group 4 (SLOW) cells to ~200
Hz for the group 1 (superFAST) cells. Localized stimulation near a distal end of the dendritic tree increases the latency to first
spike and reduces the impulse frequency relative to localized stimulation near the soma for equal levels of synaptic current. This
effect increases with increasing levels of stimulus is largest for
group 4 and almost nonexistent for group 1 (superFAST)
cells. For example, the latency difference in a relatively slowly
firing group 2 (medium complex) cell stimulated at 20 and 80% along a typical dendrite is 1.4 ms with 30 pA and 8.1 ms with 150 pA of synaptic current. The corresponding increases in impulse frequency are
2.3 and 41.5%, respectively. Substantial though these timing effects
are, a treatment of subunit (Hochstein and Shapley 1976) or transient directional hyperacuity phenomena (Grzywacz et al. 1994
) across morphological classes are beyond the present
scope, primarily because they involve signal processing throughout the retina.
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Impulses initiated as described above invade almost all branches of the
dendritic trees (cf. Stuart and Sakmann 1994); they fail
only to invade very thin branches, analogous to propagation failures
across the axonal narrow segment noted in earlier sections (cf. Fig.
5). Spike failure to invade some dendritic branches appears to be
restricted to a small subset of morphologically medium complex cells
that are also group 1 (superFAST), and therefore of
necessity exhibit many thin dendritic processes. When such thin
processes are stimulated with sufficiently large depolarizing currents
(
10 pA), they remain depolarized (in the range of
30 to
10 mV)
and interact with the spiking in the soma to produce complex
phase-locking patterns (Fig. 10, C and C'). This
phase locking reflects the fact that membranes endowed with both
regenerative (Na) and recovery (K) channels represent oscillators that
are coupled electrotonically through the dendritic morphology. In the
particular neuron shown, only two of five dendritic recording sites
responded with complex oscillations; the remaining three were invaded
by the back-propagating impulses from the soma (like those in Fig.
10A). Although one of the "oscillating" dendrites (Fig. 10C) is phase-locked to the soma, it is evident from
the corresponding phase plot (C') that this dendritic
oscillation is not periodic, and that this is due to small
jitter in the spike train at the soma. The records plotted in Fig. 10,
B-D, are of the 100 ms segment between 400 and 500 ms after
stimulus onset; the jitter is therefore not a transient property.
Instead, the jitter appears to be a manifestation of dynamic chaos
(i.e., deterministic nonperiodic oscillations), and this is
most clearly seen in the second oscillating dendrite (Fig.
10D) and the corresponding phase plot (D').
Chaotic neural responses also were noted in Fohlmeister and
Miller (1997b)
for the dendritic equivalent cylinder models, where channel density ranges for this type of behavior are discussed.
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DISCUSSION |
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Retinal ganglion cells respond with a wide variety of
impulse-encoding phenomena to electrode stimulation in the soma. A
representative set of traced neurons will reproduce the range and
distribution of impulse frequency versus stimulus current
(F/I) responses seen experimentally, as well as certain
special features (e.g., various forms of IS-SD break, spike doublets,
bursting, and other spiking "irregularities") and of passive
charging parameters (input resistance and charging times).
Electrophysiological impulse response rates are correlated strongly
with the total membrane (surface) area of the dendritic processes and
soma but (perhaps surprisingly) (cf. Mainen and Sejnowski
1996) only weakly correlated with morphological complexity of
dendritic branching or the physical reach of the dendritic trees (Table
2). These conclusions are drawn from a modeling study of a set of
ganglion cells (n = 57) traced in three dimensions and
classified by size (small, medium, and large) and complexity (simple
and complex) of the dendritic tree (Toris et al. 1995
).
The traced cells were supplied with five voltage- or ion-gated channels
(plus leakage) that were distributed throughout the model neuron
membrane according to the pattern determined for earlier models of
retinal ganglion cells that employed equivalent dendritic cylinders
(Fohlmeister and Miller 1997b
).
The five-channel model had been developed in connection with a
whole cell recording study, which found that ganglion cells of the
tiger salamander retina could be classified into FAST, MEDIUM, and SLOW cells on the basis of their
frequency versus stimulus discharge rate (Fohlmeister and Miller
1997a). In this connection, perhaps the most important result
is the description of a group of superFAST cells (group 1 in Table 2), which are typically small simple but include small complex
as well as medium simple and medium complex cells morphologically. This
group has an average F/I slope twice (namely 2 imp · s
1 · pA
1) that of FAST
cells and as a group clearly is separated from the FAST
group. These superFAST cells were found to be remarkably robust with regard to both impulse frequency stability as well as
stability in the site of impulse initiation. These neurons tolerate
wide excursions (50% in either direction) from the standard channel
density distribution with only small effects on impulse frequency, and
all impulses are initiated in the axon-hillock region, except for
near-threshold stimulus levels, for which the impulse trigger zone
expands to encompass the axonal narrow segment and even the proximal
axon beyond the narrow segment, a total distance of some 300 µm.
SuperFAST cells (as well as cells of other groups) exhibit
a large dynamic range (<0.2 to >200 imp/s at 22°C). The larger
soma-dendritic surface areas of the FAST, MEDIUM, and SLOW cells (Table 5) cause greater
delays between the spike on the narrow segment and that in the soma
even for stimuli well above threshold (10-20 pA, cf. Fig. 4), and
these large delays are responsible for the various more complex
encoding phenomena noted earlier.
Group 1 (superFAST) cells are probably overrepresented in
our morphological data pool (given in Table 2) due to small cell staining bias and may be more rare in the retina. However, small cells
also are damaged easily by the electrode in the eyecup preparation, which may cause perhaps 50% of the current passed through the electrode to be shunted across the giga-seal, leading to lower impulse
frequencies and measured input resistances. Experimentally they
therefore may have been counted among the FAST cells. Table 2 of Fohlmeister and Miller (1997a) gives the proportion
of 18 FAST cells to 10 MEDIUM cells. If we
combine the number of our present group 1 and group 2 cells, the
corresponding proportion is 34 (group 1 + group 2) to 14 (group 3)
cells. However, if we scale the number of our group 1 cells according
to their reduced soma surface area (cross-sectional area presented to
the electrode) of ~711 µm2 relative to the average area
of ~ 1,200 µm2 for cells of groups 2-4 (Table 5),
the proportions of our group 1 + 2:group 3:group 4 are 25:14:9 cells.
This is in remarkably close agreement with the experimentally found
proportions of FAST:MEDIUM:SLOW = 18:10:7 cells (Fohlmeister and Miller 1997a
, Table 2).
It is reasonable therefore to identify our model groups 1-4 with the electrophysiological groups superFAST through
SLOW, respectively, which renders our numeric
classification redundant.
Viewing all aspects of this study, including the degree of
prevalence of spiking irregularities across our data set of cell morphologies as well as antidromic soma invasion, it appears that Na = 300 mS/cm2 may be the
best (or most typical) value for the membrane of the axonal narrow
segment. It is also possible that the narrow segment contains few or no
delayed rectifier or A-type potassium channels. Reductions (by up to
one-third) in the Na-, Ca-, K-, and KA-channel densities in
the remainder of the encoder region (our weakened model, Table 1) are
permissible; orthodromic impulse trains then are modified somewhat but
remain within observational limits. Antidromic impulses, however, do
not then readily invade the soma. Therefore we do not consider the
weakened model to represent the "healthiest" cells but rather to
represent the lower bound of a safety margin within which the cells
remain viable.
The large variety of observed impulse entrainment phenomena was found
to be due to the process by which some portion of the encoder region
leads neighboring membrane in reaching threshold. Any
instantaneous membrane potential profile along the neural "encoder" region (which extends from the dendrites to the
proximal axon beyond the narrow segment) therefore can be highly
variable and fluctuating in time, which is the primary cause of the
irregularities in impulse frequency seen in some spike train records.
However, impulse initiation never was observed on the dendrites (where Na = 25 mS/cm2), even with
dendritic stimulation, although the dendrites contain the highest
density of calcium channels. This is in agreement with the results of
Velte and Maslund (1997)
, who observed dendritic invasion by impulses in rabbit ganglion cells but not dendritic impulse
initiation (T. Velte, personal communication). As the stimulus current
is increased, impulse frequencies increase and the physical size (or
extent) of the effective encoder region declines and with it the
occurrence of spiking irregularities.
Given the absolute constraints on cell morphology (i.e., absence of free geometric parameters), in conjunction with the diversity of electrophysiological responses and their homology with experimental observation and classifications, and finally their homology with the observed passive responses of the neurons, we conclude that the five-channel model of retinal ganglion cells represents a realistic, first-order encoder for these cells. A single channel-density distribution (standard in Table 1) likely prevails across all morphological and electrophysiological classifications of the cells.
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ACKNOWLEDGMENTS |
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We thank Dr. Paul Coleman for discussions and data and Dr. Robert F. Miller for providing the traced morphological ganglion cell pool.
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FOOTNOTES |
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Address for reprint requests: J. F. Fohlmeister, Dept. Physiology, 6-255 Millard Hall, University of Minnesota, 435 Delaware Street SE, Minneapolis, MN 55455.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 14 January 1998; accepted in final form 15 December 1998.
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REFERENCES |
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