Laboratoire de Neurobiologie des Reseaux Sensorimoteurs, Centre
National de la Recherche Scientifique-Unité Propre de Recherche
de l'Enseignement Supérieur Associée-7060, 75270 Paris Cedex 06, France
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INTRODUCTION |
The Xenopus embryo and larvae have
become extremely useful preparations for investigations of locomotor
neural networks. The pioneering studies of Roberts and his coworkers
(Roberts 1989
) have not only identified the few
principal neurons of the network with their synaptic connections but
also introduced minimal neural network models that simulate an
impressive amount of behavior observed in this preparation (Dale
1995b
; Roberts and Tunstall 1990
; Roberts
et al. 1995
). Furthermore whole cell voltage-clamp measurements
have been done on cultured (O'Dowd et al. 1988
) or isolated cells (Dale 1991
, 1995a
) as well as intact
surface neurons (Desarmenien et al. 1993
; Prime
et al. 1998
, 1999
). Quantitative models describing these
measurements have been restricted to a single somatic compartment
(Dale 1995a
; Lockery and Spitzer 1988
) because dendritic structures are minimal at early developmental stages.
Although the dendritic structure at stage 37/38 is minimal, it clearly
exists as shown by both morphology and electrophysiological estimates
of the electrotonic structure (Soffe 1990
; Van
Mier et al. 1985
). This is in keeping with the original empiric
model of Roberts and Tunstall (1990)
that has three
compartments representing the dendrite, axon, and the soma. Thus it
would appear useful to understand the dendritic properties as they
begin to develop especially with regard to the appearance of NMDA
receptors (Prime et al. 1999
) because they appear to be
correlated directly with the presence of dendritic trees (Prime
1994
; Prime et al. 1998
).
The whole cell clamp experiments reported here were done at larval
stages 42-47 where the dendritic structure is clearly more developed
(Van Mier et al. 1985
) and likely to play a definitive role in locomotor behavior. The major advantages of our approach are
the normal electrical activity of functional neurons can be measured as
demonstrated by patterned network behavior (Fig.
1), the neurons are not isolated from
their normal milieu, and thus minimal distortions in structure are
likely to have occurred because of measurement procedures as is
demonstrated by the maintenance of synaptic events during the
experiments.

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Fig. 1.
Rhythmic activity of isolated Xenopus spinal cords.
A: extracellular recording of bilateral activity from an
isolated Xenopus embryo spinal cord (stage 37/38) using
suction electrodes at the rostral and caudal ends. Rhythmic activity
was evoked by a 1-ms pulse of current. Combined frequency is ~40 Hz;
however, an alternation of large and small spikes at 20 Hz can be seen
in part of the record, suggesting a fictive locomotor pattern typical
of the intact preparation. B: simultaneous extracellular
recordings of rhythmic bursting activity from both sides of an intact
stage 46 Xenopus larvae. Alternating bursts of ~1 Hz
occurred spontaneously. C: extracellular recording of
bilateral activity from an isolated Xenopus larvae whole
spinal cord (stage 46) using suction electrodes. Rhythmic activity was
evoked by 50 µM N-methyl-D-aspartate
(NMDA) in the perfusion fluid. Burst frequency was ~1 Hz.
D: extracellular recording of bilateral activity from an
isolated Xenopus larvae half spinal cord (stage 46) that
was obtained by cutting the spinal cord of C at the
midline. Bursting frequency evoked by 50 µM NMDA was nearly the same
as observed from the whole spinal cord; however the burst duration
decreased two- to threefold. One-second calibration bar applies to
B-D.
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In addition, both real-time and frequency domain measurements were used
to determine the electrotonic behavior and take into account inevitable
electrode properties (Moore and Christensen 1985
;
Wright et al. 1996
). These experiments suggest that
there is a correlation between the electrotonic structure and the
excitability properties elicited from the somatic region. The action
potentials of the neurons that showed strong accommodation also had
different electrotonic parameters compared with nonaccommodating
neurons. Thus in addition to the effects of the voltage-dependent
conductances on firing behavior (Dale and Kuenzi 1997
),
the structure in which these conductances are expressed is correlated
with the rhythmic behavior of the neuron, perhaps as some function of
development from the embryo to later larval stages.
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METHODS |
Experimental preparation
The developmental stages 42-47 of Xenopus larvae
were obtained after hormonally induced fertilization. Embryos and
larvae were developed at room temperature and the stages were
morphologically selected as described by NieuwenKoop and Faber
(1994)
and Van Mier et al. (1985)
. In
accordance with the European Communities Council directive of November
24, 1986 and following the procedures issued by the French
Ministère de l'Agriculture, the larvae were anesthetized in
tricaïne methanesulfate (MS222 Sigma) and placed in Ringer
solution [composed of (in mM) 110 NaCl, 3 KCl, 1.0 MgCl2, 1.0 CaCl2, and 10 HEPES; pH = 7.4] containing 0.5 mg/ml dispase (Boehringer, Mannheim). The notochord, spinal cord, and
overlying musculature were dissected, and the preparation was agitated
for 30-35 min at room temperature, after which the spinal cord was removed easily from all surrounding tissues. Figure 1 illustrates rhythmic locomotor or bursting behavior (Fig. 1A) of an
isolated, stage 37-38, spinal cord similar to that observed in the
intact preparation (Roberts 1989
). Similarly, isolated
larval spinal cords (Fig. 1, C and D)
show network patterns like those observed in the intact preparation
(Fig. 1B). Figure 1D also illustrates bursting activity induced by 50 µM NMDA recorded with suction electrodes.
At the larval stage of development, the connective tissues
eventually making up the meninges had to be removed before a patch electrode seal was possible. Although some recordings were made from
neurons on the external surface (Desarmenien et al.
1993
), the success rate was improved considerably by
further dissection to reach the inner regions of the spinal cord and to
visualize the neurons. The cells of the spinal cord cut between the
otic capsule and the 10th myotome were exposed for intracellular
recording by carefully splitting at the midline and mounting the half
cord with the inner face up (Fig. 2),
revealing groups of neurons: dorsal sensory neurons, presumed medial
interneurons, and presumed ventral motoneurons. The positioning of the
half cord was done with two micromanipulators attached with suction
electrodes or sharp glass rods. All recordings were done from neurons
between the 4th and 6th myotome levels. The Rohon-Beard sensory cells (Spitzer 1982) were identified easily on the dorsal
surface. Longitudinally lined up large motoneurons in the ventral part
of the cord could be distinguished; however, other less visible
motoneurons could be confused with presumed interneurons. Intermediate
neurons near the inner surface of the half cord in dorsal and ventral
medial positions were provisionally identified as interneurons and
selected for analysis from a total of 200 recorded neurons of all
groups. Twenty presumed interneurons, which showed stable recordings
for a minimum of 30 min, were fully analyzed.

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Fig. 2.
Photograph of isolated half spinal cord. Half spinal cord was
maintained in a open position with 2 sharp glass electrodes. Top and
bottom white lines indicate the borders of the dorsal and ventral
regions of the cord. Indicated neurons represent presumed interneurons
and the motoneurons. Large round cells on the dorsal surface were
identified as Rohan Beard sensory neurons. Calibration bar is 50 µm.
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Electrophysiological recording
Figure 3 illustrates the
combination current or voltage clamp and sum of sines (white noise)
method that was used to obtain real-time (Fig. 3, B and
C) and frequency (Fig. 3, D and E)
domain responses (Moore and Christensen 1985
;
Moore et al. 1993
). The experimental procedures were
designed to measure both nonlinear responses evoked by a constant
current, I(t), or voltage clamp in real time, and
steady-state linear behavior in the frequency domain for both clamp
modes. The stimulus protocol used for the two domains is illustrated in
Fig. 3A by a command step representing either a current or
voltage, which is followed by a superimposed steady-state small white
noise signal that evokes a linear response (Wright et al.
1996
). The corresponding transient voltage (Fig. 3B)
or current (Fig. 3C) responses are schematic representations of the influence of active conductances.

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Fig. 3.
Combination of current or voltage clamp with white noise analysis.
Similarity of the current- and voltage-clamp protocols is indicated by
diagram of a command stimulus that consists of a step function on which
is superimposed a small signal, sum of sines, called white noise.
A: fast Fourier transform (FFT) of the white noise
stimulus was done during the last second of the command stimulus. This
transform provided current or voltage references,
Iref(f) or
Vref(f),
respectively, which are magnitude and phase functions that then were
used in the computation of the point impedance functions for the 2 experimental conditions. B: real-time damped oscillatory
voltage response followed by a filtered white noise signal shows the
typical constant or command current behavior. FFT of the filtered
response to the white noise current provides voltage magnitude and
phase functions. C: similarly, in a separate experiment,
the current response to a somatic voltage-clamp command provides
real-time transient currents followed by small signal sum of sine
currents that are responses to the white noise command voltage inputs.
FFT of the steady-state current gives the corresponding magnitude and
phase functions associated with the voltage reference signals. In a
linear system, the voltage- and current-clamp data are equivalent and
satisfy the relation,
Z(f) = 1/Y(f). This was verified
experimentally to show that the voltage clamp instrumentation is
adequate. D: impedance,
Z(f), magnitude plots are
shown for 2 passive neuronal models, a soma and 1 with an electrotonic
structure (Rall neuron). Low-frequency impedance of the Rall neuron is
less than the soma alone because the total surface area and size are
correspondingly larger. Electrotonic structure shows a small inflection
in the mid frequency range. E: corresponding phase
functions show that the Rall neuron has marked inflections that are
easily distinguished from the asymptotic behavior of a simple soma.
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The data were obtained with an Axoclamp 2B (Axon Instruments, Foster
City, CA), filtered at 500 Hz with an 115 db/octave elliptical filter
(Krohn-Hite, Model 3900, Avon, MA) and digitized at 12 bits. The values
of both the current and the voltage were measured during voltage- and
current-clamp measurements. This point is of some importance because
the value of the current measured in current clamp was not identical to
the command input. The linear responses were analyzed in the frequency
domain to obtain magnitude and phase functions of either the impedance
or admittance corresponding to current or voltage clamp, respectively.
A fast Fourier transform (FFT) of 1,024 points (0.8 s) of the voltage,
V(t), or current, I(t),
response provided the corresponding output functions of frequency,
namely V(f) and
I(f), as well as the stimulus or
command input,
I(f)ref or
V(f)ref, for
current and voltage clamp, respectively. The corresponding impedance,
Z(f) and admittance functions,
Y(f), then were computed as
Z(f) = V(f)/I(f)ref
and Y(f) = I(f)/V(f)ref.
A step-by-step description of experimental procedure is as follows:
1) 400-ms step constant currents are injected into the soma,
immediately followed by superimposed low amplitude white noise for
1 s. The steady-state responses for both the current and the
voltage during the last 0.8 s are analyzed in the frequency domain
to obtain impedance magnitude and phase functions, namely the output
voltage with respect to the input current. 2) Similar voltage-clamp steps then are done for a range of membrane potentials to
obtain the admittance magnitude and phase functions, i.e., output
current with respect to input voltage. 3) The resulting real-time transient responses and frequency domain functions then are
fitted using parameter estimation techniques with an electrotonic model
having voltage-dependent conductances to obtain quantitative descriptions of each neuron, i.e., a complete neuronal model with its
electrotonic structure and kinetic behavior at all membrane potentials
for both constant current- and voltage-clamp conditions.
In principle, small-signal current- and voltage-clamp measurements
should lead to reciprocal frequency domain functions,
Z(f) = 1/Y(f). This property is a striking
example of one of the advantages of transform functions, namely two
types of data can be compared independently of a theoretical model. The
equivalence of the two functions provides a test of the current- and
voltage-clamp instrumentation (Magistretti et al. 1996
),
where the latter requires stable electronic circuitry to control the
membrane potential at all measured frequencies. Current-clamp responses
have the advantage that errors from a control amplifier are minimal
compared with the voltage clamp and in general tend to be more
reliable. For stable conditions, small signal measurements of the two
modes should be consistent, independently of any particular model. Thus
the linear frequency domain measurements in voltage- and current-clamp
modes provide a minimal test of the voltage clamp and should be
compared before any comparisons of real-time voltage- and current-clamp
responses are meaningful. Because of this equivalence the data always
was plotted as impedance functions, however, all voltage-clamp data are
actually admittance measurements. A difference in the impedance functions may occur due to changes in the leakage conductance or
electrode properties; however, the electrotonic parameters, A and L (see definitions in the following text),
were required to be constant.
Because the impedance or its reciprocal, the admittance, is a ratio of
response and stimulus, the final form of the data were corrected for
the effect of the antialiasing filter. However, the real-time responses
are not ratios and consequently contain a damped antialiasing filter
response. The 500-Hz band was chosen because it provides the best
compromise for the determination of electrotonic and voltage-dependent
conductance properties. Because the sampling rate was 0.78125 ms, the
determination of transient real-time kinetics is limited to a few
milliseconds with some rapid ringing because of the sharp antialiasing
filters. This limitation is less severe for the frequency domain
because a 1-ms relaxation time has a corner frequency of 160 Hz that is in the middle of the frequency band measured. It is for these reasons
that the frequency domain is extremely useful for the estimation of the
kinetic behavior.
Figure 3, D and E, illustrates typical magnitude
and phase functions for passive neurons similar to those obtained in
these experiments. The superimposed plots show that the dendrites
impose marked inflections on the phase function that are not present with isolated somatic structures. Magnitude functions are less sensitive; however, they do exhibit small inflections as well. The
frequency domain functions thus provide an accurate method to determine
the electrotonic structure that is essential for the subsequent steps
in the analysis of the active membrane properties.
Electrode properties and compensation methods
Electrodes were pulled with a laser heated puller (Sutter P2000,
Sutter Instruments, Novato, CA) from 1.5 mm glass (GC150F, Electromedical Insutruments, Pangbourne, UK). The electrodes were filled with either (in mM) 90 K-gluconate, 20 KCl, 2 MgCl2, 10 HEPES, 10 EGTA, 3 ATP, and 0.05 GTP or
55 Cs2SO4, 55 sucrose, 2 MgCl2, 10 HEPES, 10 EGTA, 3 ATP, and 0.05 GTP. In
both solutions, the pH was adjusted to 7.4 with 10 mM KOH. Thus the
internal perfusion fluids contained either potassium or cesium as the
principle internal cation. Because different internal solutions were
used no correction for liquid junction potentials was made. This
correction is likely to be ~10 mV based on the considerations of
previous measurements (Neher 1992
).
The electrode impedance was measured at the depth in the solution of
the selected neuron and just before making the gigaseal of the patch.
Because the electrode impedance is low and not well matched to the
Axoclamp headstage, it was measured in series with a parallel RC
electronic model that was used to provide a calibration of the method.
Measurements and analyses then were done on the combination electrode
and electronic model just as with an actual neuron. The electronic
model impedance is comparable with measured neurons and provides a
method to simulate the contribution of the electrode in the typical
recording situation. Figure 4 illustrates that a change of solution levels of 1 mm led to changes in the electrode capacitance of 2 pF. The data and superimposed model fits
indicate that real-time data are relatively insensitive to these small
changes; however, the differences in the phase function are quite
clear, as indicated by Ce1 and
Ce2 for the phase impedance of Fig.
4C. Under these conditions, the use of the Axoclamp
electronic compensation capabilities for the electrode capacitance and
series resistance led to phase functions that could not be reliably
estimated (Wilson and Park 1989
). Capacitance
compensation alone was partially effective. Using quartz glass or
coating the electrodes with silicone elastomer (Sylgard) reduced the
value of the capacitance; however, it was still not possible to achieve
adequate compensation. Because these procedures distort the measured
data in an uncalibrated manner, no electronic compensation was done,
and the electrode was modeled as part of the measurement system. This
had the additional advantage that any changes in the series electrode
impedance always would be taken into account because each voltage-clamp
record contained high-frequency data that are sensitive to electrode properties. Thus fitting recorded data over a reasonably wide frequency
range requires accurate electrode parameters. If the electrode
increases its resistance, this will become apparent at high frequencies
in contrast to most voltage-dependent conductance effects that are more
sensitive to lower frequency ranges. This approach is also more exact
than traditional series resistance compensation methods, which are
always partial because of stability problems in the voltage clamp.

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Fig. 4.
Effect of solution level on electrode capacitance. A whole cell
electrode tip was placed in the recording chamber with its connection
to ground through an electronic model consisting of a parallel
resistance, R, and capacitance, C.
Electrode was modeled as a resistance, Re,
and capacitance, Ce, as discussed in the
APPENDIX. A: superimposed real-time
responses to 10 pA of current were measured and fitted for 2 solution
levels. Four curves cannot be distinguished showing that the real-time
response is insensitive to the solution level. B:
impedance magnitude plots for the same conditions as A.
C: phase functions as in B in which
differences in the responses and fitted curves can be seen at high
frequencies. Smooth lines are model (D) fits for the 2 measurements as follows: C = 80 pF,
R = 526 M , Re = 13.5 M , Ce1 = 3.5 pF, and
Ce2 = 5.6 pF, for the low and high
solution levels (bottom and top curves),
respectively. D: schematic diagram of electronic and
electrode model.
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The mean parameter values of the electrode impedance in the solution
just before making the seal were Re = 10.5 M
and Ce = 5.69 pF (Table
1). All measurements were done with the
electrode within a few micrometers of the recorded neuron. This
procedure avoided errors in the electrode parameters that were due to
different solution levels. The range of
Ce corresponding to the lowest and the
highest levels of solution, 0.1 and 1 mm, was 2 to 6 pF, respectively.
After making the seal and establishing the whole cell clamp recording,
the electrode was refitted using a neuronal model in series with the
electrode. The new fit included both the electrode and neuron; however,
the electrode properties were estimated over the entire frequency
range. The effect of the electrode on the frequency domain functions is
shown in Fig. 7, B and D (see following text). As
mentioned in the following, the passive electrotonic parameters were
determined with one-half the frequency range and fixed electrode
parameters. The electrode parameter values during the whole cell
recording are given in Table 1. The average increase of
Re from 10 to 17 M
is likely due to
plugging of the electrode by membrane fragments and cytoplasmic
material during the breakthrough of the membrane. The decrease in the
electrode capacitance may be related partially to different factors,
namely, the attachment of membrane fragments on the inner cell wall at
the tip, variations in solution levels, or a need for a distributed
capacitance to accurately model the electrode. Because parameter
estimation done with more complicated electrode models (Major et
al. 1994
) did not significantly alter the electrotonic
parameters, all of our analyses were done with an electrode modeled as
a single resistance, Re, and a
nondistributed capacitance, Ce (see
APPENDIX). It is worth noting that the determination of the
electrode capacitance is a critical part of an accurate estimation of
the soma capacitance. In this regard, electronic compensation methods
are especially subject to error because overcompensation can reduce
part of the observed dendritic capacitance, which is seen from the soma
through a series resistance like the electrode itself.
Thus once the electrode properties are known, it is possible to
separate the properties of the attached electrode from those of the neuron and evaluate the electrotonic structure. This itself requires a valid theoretical formalism for the dendritic tree to obtain
the excitability properties of both the soma and dendrites.
Data analysis and rationale
The goal of these experiments is to analyze dendritic membrane
excitability. The rationale of the analysis is to use different types
of measurements that allow the determination of the physiological electrotonic structure, both active and passive. The basic steps in the
analysis consist of the following: 1) determination of passive electrotonic structure at hyperpolarized membrane potentials with a linear analytic model, i.e., having perfect spatial resolution. 2) Evaluation of compartmental models at different membrane
potentials to evaluate the required number of compartments needed to
achieve adequate spatial resolution during the activation of ionic
conductances. 3) The analysis of real-time kinetic responses
with compartmental models that are constrained by the passive
electrotonic structure.
THEORETICAL CONSIDERATIONS
ANALYTIC VERSUS COMPARTMENTAL
MODELS.
Linear-analytical models
resting neurons.
The experimental protocol described in the preceding text involves an
analysis of both large step nonlinear data and small-signal linearized
responses. Our goal is to obtain a minimal model that accurately
describes the soma and dendritic behavior observed in the measured
neurons. It was found that the dendritic tree could be well described
by a single Rall equivalent cylinder, thus avoiding the need to use
more complex models based on detailed morphology. Models with two
dendritic cables connected to the soma did not significantly improve
the error of the fit. This result supports the hypothesis that the
branching dendritic structure of Xenopus neurons follows
the impedance matching criteria at branch points developed by Rall
(1960)
. More complex morphological models are only
needed when the branch point matching criteria are not met. Although
anatomic measurements on fixed tissue can indicate such a discrepancy,
it is not clear if these estimates always apply to living tissue.
Electrophysiological measurements, in both the time and frequency
domains from neurons in their normal physiological state, can provide
data to evaluate the accuracy of collapsed dendritic models and as such
are likely to be more suitable for determining the adequacy of
simplified dendritic models to describe data than anatomic measurements
from fixed tissues. The ability of a single uncoupled model to
adequately describe our data also demonstrates that the neurons are not
likely to be electrotonically coupled (Perrins and Roberts
1995a
). Thus because interneurons do not show electrototonic
coupling (Perrins and Roberts 1995b
), our analysis
further supports the presumption that the selected cells are from this
class of neurons.
The implementation of the collapsed dendritic cylindrical model has
been done with analytic (Major 1993a
,b
) and
compartmental models (Rapp et al. 1994
). However, linear
analytic models (see APPENDIX) are advantageous because
they have perfect spatial resolution and are computationally efficient.
The applicability of these models in different neurons was determined
by comparing the analytic and compartmental formulations as a function
of number of compartments. Near the resting potential there was good
agreement using from three to five compartments.
The passive electrotonic structure always was determined near the
resting potential with both passive and active
linearized responses (Borst and Haag
1996
; Major et al. 1994
; Surkis et al. 1998
). Under these conditions, an analytic model equivalent to an infinite number of compartments was used (see APPENDIX).
Although the linear analytic model avoids errors due to spatial
resolution, it is only valid near the resting potential or under
experimental conditions such that the neuron is entirely passive. It
should be emphasized that linearized models are not necessarily passive and are capable of describing the kinetic behavior of the
voltage-dependent conductances (Moore and Buchanan 1993
)
over a limited potential range if the steady-state potentials
throughout the cable are relatively constant (Murphey et al.
1995
).
LINEAR COMPARTMENTAL AND ANALYTIC MODELS
DEPOLARIZED NEURONS
IN STEADY STATE.
At depolarized potentials the steady-state potential profile inherent
in the dendritic structure leads to variable activation of the ionic
conductances. This requires a compartmental model to correctly
determine voltage-dependent admittance functions at each dendritic
location (see APPENDIX). Nevertheless the analytic model is
a good approximation at moderate depolarizations and can provide an
excellent initial estimate of the final impedance. This point is of
computational importance because calculation of the analytic impedance
can be orders of magnitude faster than compartmental estimates.
The number of compartments necessary to assure adequate spatial
resolution is clearly a function of both the electrotonic structure and
the nature of the ionic conductances (Bush and Sejnowski 1993
). At rest or with a passive neuron, the best test of the compartmental model is a comparison with the analytic solution. However, active neurons at depolarized potentials require increasing the number of compartments until no significant change occurs. Under
these depolarized conditions, the compartmental model then can be used
to evaluate the adequacy of the analytic model. For many neurons, the
electrotonic structure was sufficiently compact that very little error
occurred with the analytic model at depolarized potentials.
Significantly larger errors occurred with compartmental models having
too few segments than ever observed due to a steady-state potential
profile error.
NONLINEAR COMPARTMENTAL MODELS
LARGE SIGNAL KINETIC ANALYSIS OF
IONIC CURRENTS.
The analysis of the constant current responses clearly requires
nonlinear kinetic equations; however, this is also true for a somatic
voltage clamp because of the effects of the dendritic membrane
potential transients. These kinetic models have been described
extensively by numerous authors (Koch and Segev 1998
), and our specific implementation is given in the APPENDIX.
Our initial analysis was done with the minimum number of three
dendritic compartments in which the passive electrotonic parameters (ratio of dendritic to somatic surface areas, A, and
electrotonic length, L) were identical to those of the
analytic model (see APPENDIX). The spatial resolution of
the compartmental model then was evaluated by increasing the number of
compartments until minimal differences were obtained. In general, the
resting neuron was described adequately by a three-compartmental
dendritic model; however,
10 compartments were essential to describe
voltage-clamp currents at depolarized potentials when the ionic
conductances are significantly activated (Bush and Sejnowski
1993
; D'Aguano et al. 1989
). A dendritic model
with 30 compartments was usually indistinguishable from the analytic model.
In brief, the parameter estimation methods, as previously described
(Murphey et al. 1995
, 1996
), were used in the following order: The passive electrotonic and electrode parameters initially were
estimated at resting or hyperpolarized potentials with a linear
analytic model (see APPENDIX) over the entire
frequency range. Parameter estimation was done by an iterative gradient
descent method. Fits in a particular minimum were considered adequate if the minimal error change was <0.1%. In addition local minima were
avoided by the use of both the frequency and time domain data under
voltage and current clamp. Afterward, the electrode parameters were
fixed and the electrotonic parameters were estimated over one-half the
frequency range. The validity of a three-compartmental dendritic model
also was confirmed for the passive membrane by demonstrating an
adequate fit of the small signal real-time data. The data then were
fitted simultaneously in the frequency and time domains with the linear
analytic model and a three-dendritic compartmental model,
respectively. The adequacy of both the frequency and time domain fits
were compared directly with a 30-compartmental model. If the fitting
criteria were not met, the data were fitted again with the higher
resolution compartmental model that was implemented using the
FindMinimum and FindRoot procedures of Mathematica (Wolfram Research,
Champaign, IL). These procedures then were applied to the active
conductances after fixing the passive parameters. Multiple records at
different membrane potentials were analyzed simultaneously in both the
time and frequency domains. If necessary, the electrode parameters,
Re and
Ce, were refitted during the course of
the experiment; however, the electrotonic parameters remained at their
original estimated values. In this case, constraining limits based on
measured electrode parameters (Fig. 4) were placed on
Ce and
Re, as follows:
Ce (2-6 pF) and a minimum for
Re of 6-10 M
, which was measured
in the bath before the patch was made.
In summary, the approach developed in this paper is to analyze both
whole cell voltage- and current-clamp data with a real-time three-dendritic compartment nonlinear model that is constrained simultaneously by the linear analytic frequency domain form of the same
model. These two forms of the neuronal model are computationally efficient because of the small number of compartments in the former and
the analytic representation of the latter. The number of compartments then is increased to evaluate the errors due to inadequate spatial resolution for the real-time transients and the dendritic potential profile in the steady state.
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RESULTS |
Whole cell action potentials
All neurons perfused with the potassium electrode solution
had resting potentials more negative than
55 mV and action potential magnitudes that reached overshooting positive values. The action potential behavior varied between a Type I repetitive firing behavior to a Type II single action potential in response to a maintained constant current. Figure 5A
shows a single action potential from a Type I neuron responding to a
just threshold stimulus. Increasing the stimulus strength increased the
number of action potentials (Fig. 5B) and the average firing
rate. In general, the cells responded repetitively with minimal
accommodation to a maintained constant current that was twofold or
greater than threshold (Fig. 5C). Figure 5D
illustrates superimposed constant current steady-state current voltage
(I-V) curves for nine neurons showing this type of action
potential behavior. A marked rectification is observed where the
maximal slope conductance is reached at about
40 mV.

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Fig. 5.
Current voltage (I-V) plots and action potential
behavior of Type I neurons. Current-clamp responses for different
constant current levels for a Type I neuron, namely, A:
10 and +10 pA, B: +20 pA, and C: +30
pA. D: constant current-voltage curves from 9 neurons
that show repetitive firing and significant rectification. Each neuron
is represented by a different symbol. Neuron 97H05A of
A-C is represented by in
D. I-V curves were made from difference
in the average values of the voltage 100 ms before the step and the
last 100 ms of the constant current step, just before the beginning of
the white noise signal.
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About 1/4 of the neurons, referred to as Type II neurons, showed marked
accommodation in their firing responses. Figure
6, A-C, illustrates two
examples of this type of behavior. The most extreme form is seen in
Fig. 6A in which a single action potential is evoked by
constant current stimulation
10 times threshold. An alternative type
of adaptation is seen in Fig. 6, B and C, in
which a constant current evokes a series of action potentials that
decrease in size and finally cease. These neurons show increased frequencies and numbers of action potentials as the current increases (Fig. 6C); however, the responses are not maintained. The
constant current voltage curves of Type II neurons show a significant
rectification over a wide potential range (Fig. 6D) leading
to a maximum slope conductance at a more positive potential than seen
in Type I neurons.

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Fig. 6.
I-V curves and firing behavior of Type II neurons.
A: single or no action potential response from a Type II
neuron during 10-, +10-, +20-, +30-, +40-, and +200-pA current
injections, neuron 97F05C. B: multiple,
but not sustained, action potential responses from other Type II neuron
during a sustained 10-pA current and, in C, 20-pA
stimulation, neuron 97J30A. D: constant current-voltage
curves from 8 neurons show marked rectification approaching a 0 slope.
Neurons of A and B are identified by the
same symbols in D.
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Passive electrotonic or structural parameters
The analysis of electrotonic behavior provides information about
the passive structure of a neuron on which active properties are built.
The excitable properties of neurons are inextricably linked to their
electrotonic structure. Although morphological measurements provide a
good indication of dendritic cable properties, these must be measured
in the intact cell to ascertain the actual electrotonic behavior.
Ideally morphology could provide the basis of the model used for
interpreting electrophysiological measurements; however, such models
can require large numbers of compartments and are computationally
cumbersome. Our approach is to obtain a minimal model that is
consistent with the morphology and rigorously satisfies the spatial
resolution requirements imposed by the electrophysiologically measured
cable properties.
To measure passive neuronal properties, data at hyperpolarized membrane
potentials were obtained in the presence of TTX during internal
perfusion of potassium gluconate or cesium sulfate. The electrode
properties were analyzed in conjunction with those of essentially
passive neurons in a sequential manner such that fitted electrode
parameters were fixed before finally evaluating the passive
electrotonic structure (see METHODS). Neurons perfused with
cesium sulfate had unstable resting potentials and survived better if
voltage clamped at
60 mV as rapidly as possible after establishing
the whole cell recording. Removal of the potassium conductance was
presumably responsible for the instability of the resting potential and
often led to maintained depolarized potentials that damaged the
neurons. The d.c. impedance of these neurons often exceeded 10 G
.
Figure 7A illustrates the
potential time course in response to a hyperpolarizing current for a
virtually passive cesium perfused neuron. A multiexponential analysis
of this response to obtain the membrane time constant and associated
electrotonic properties of the dendritic tree is notoriously difficult
(Spruston and Johnston 1992
). In addition to
problems associated with separating exponential functions, the
properties of the electrode are difficult to assess. The corresponding
measurements done in the frequency domain (Fig. 7B)
provide more sensitivity than real-time measurements for the estimation
of electrode and electrotonic parameters. The fits shown in Fig. 7
indicate good agreement for both the real-time and frequency domain
current-clamp data. Figure 7, C and D,
shows voltage-clamp data and model fits for the same neuron using the model parameters that describe the current-clamp data (Fig. 7, A and B) with the notable exception of
gsoma, which was reduced to one-half its
value. This difference is likely due to the incomplete exchange of
cesium and potassium ions during the constant current measurement that
was done at the beginning of the experiment. Otherwise the parameters
are identical and show that the two measurements are equivalent as is
demonstrated in subsequent figures from other neurons. The effect of
the electrode on the impedance functions also is illustrated for both
the current- (Fig. 7B at
70 mV) and voltage-clamp
(Fig. 7D at
63 mV) frequency domain experiments. The
only significant changes were in the phase functions showing a
deviation in the phase at high frequencies when the electrode was
removed.

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Fig. 7.
Passive electrotonic structure of single neurons. Four types of data
were used to test the ability of a Rall model neuron to quantitatively
describe the passive electrotonic behavior in a cesium perfused neuron.
The dashed lines represent the model without an electrode, which is
just visible in A and completely superimposed in
C. A: hyperpolarizing constant current
response to 0.01 nA. B: impedance magnitudes in
megohms (M ) and phase responses in radians (rad) at 70 and 130
mV. Effect of the electrode is only illustrated at 70 mV.
C: voltage-clamp responses to a 10-mV depolarization and
hyperpolarization from a holding level of 63 mV. D:
impedance magnitude and phase plots of admittance data obtained during
the same voltage-clamp steps. Effect of the electrode is only
illustrated at 63 mV. Same model is used for all the superimposed
smooth curves and all 4 data sets. Parameter values for the passive
neuronal model were as follows: Csoma = 2.39 pF; L = 0.133; A = 6.03;
gsoma = 0.013 nS; and
Vleak = 25.6 mV. Electrode model
parameters were Ce = 2.85 pF and
Re = 17 M . A small bias current in
C was required because of the depolarization caused by
cesium perfusion. Cell 97K17C, Type B.
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The mean values (Table 2) of the passive
electrotonic parameters of cesium perfused neurons show a soma
capacitance of ~3 pF and a dendritic structure that was approximated
by a single equivalent cylinder having an electrotonic length of 0.25 and a dendritic to soma area ratio of 3.6. Assuming that the neuron is
a perfect sphere with a specific capacitance of 0.5-1
µF/cm2, then its diameter would be 10-14 µm,
respectively. These estimates are consistent with visual observations
of these neurons during the experiments (see Fig. 2). The majority of
experiments were done at stage 47; however, some measurements at stages
42-46 showed that the electrotonic parameters between these stages are
not radically different (Table 4). The results of this study should apply to all stages between 42 and 47; however, it is clear that dendritic structures are in different states of development during these various stages.
Voltage-gated potassium conductances
Because part of the resting conductance is due to the potassium
conductance, it is often difficult to determine the passive electrotonic structure with real-time hyperpolarizing pulses
(Spruston and Johnston 1992
; Surkis et al.
1998
). Thus the effects of voltage-dependent conductances must
be generally taken into account in resting neurons. We have been able
to demonstrate that hyperpolarized interneurons perfused with cesium or
potassium ions are essentially passive; however at the resting
potential most voltage-dependent conductances can contribute to the
linear properties. Therefore passive electrotonic parameters of a
normal resting neuron must be estimated in the presence of active
conductances. Thus it is necessary to assess the distribution of ionic
channels between the soma and dendritic tree. Although morphological
labeling methods can indicate receptor distributions, it is important
to functionally estimate these distributions. This is not generally
possible by direct measurements in intact tissues; however, in our
analysis, a homogenous distribution of ionic channels was shown to be
adequate for the experimental conditions explored. This result is of
some significance because it has been demonstrated previously that the
frequency domain method is capable of detecting differences in the
spatial distributions of activated receptors (Murphey et al.
1995
); however, extremely accurate frequency domain
measurements with averaging would be required before spatial effects
can be seen easily. This was not done in these experiments.
Figure 8 illustrates data obtained in the
presence of TTX for both current- and voltage-clamp modes and provides
a stringent test of the full nonlinear model containing a single
voltage-dependent potassium conductance throughout the dendritic tree.
The time course of the hyperpolarizing potential response during
K-gluconate perfusion is more rapid than observed in the corresponding
cesium perfused neuron of Fig. 7. These results support the hypothesis that part of the conductance at rest is due to the voltage-dependent potassium conductance. The depolarizing potential response was lower in
magnitude and also faster due to the increased activation of the
potassium conductance (Table 3). Thus the
normal resting conductance has both voltage-dependent and -independent
(leakage) components (Surkis et al. 1998
).

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Fig. 8.
Voltage-dependent potassium conductance. A-D show the
responses of the potassium conductance for 4 different stimulus
conditions measured in 1 µM TTX. A: hyper- and
depolarizing currents show a decreasing time constant as the
conductance is activated. Current levels were 0.01, 0.01, 0.02, and
0.03 nA. B: corresponding impedance plots are given for
each current-clamp step indicating a decrease in the low-frequency
impedance with depolarization. C: voltage-clamp currents
for 4 step potentials show rapid activation kinetics that are too fast
to be resolved in the real-time records. Inward synaptic currents are
shown decreasing in amplitude with depolarization. D:
impedance plots of the admittance data are shown for 3 of the
voltage-clamp steps. Superimposed smooth fits of both frequency and
time domain data show fits for a 3-compartment dendritic model. A
nearly twofold increase in the conductance occurred at 30 mV.
Electrotonic parameters are Csoma = 3.67 pF; L = 0.247; A = 1.77;
gsoma = 0.13 nS; and
Vleak = 59.8 mV. Electrode parameters
are Ce = 2 pF and
Re = 12.5 M . Active potassium
conductance parameters are: gK = 0.36 nS; vn = 4.2 mV;
sn = 0.047 mV 1;
tn = 2.4 ms;
rn = 0.001; and
VK = 90 mV (this
vK was used for all figures and tables).
Neuron 97K03D, Type A.
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The corresponding frequency domain data and fitted curves (Fig.
8. B and D) show that the model parameters
provide a good description of both the passive and active parameters.
As with the passive case of Fig. 7, the voltage-clamp responses (Fig. 8, C and D) are shown to be well predicted by the
current-clamp model of Fig. 8, A and B. These
results confirm that the voltage clamp is adequately controlling the
soma potential and provides verification that the dendritic model used
for both the current- and voltage-clamp experiments is an accurate
description. The electrotonic length of this neuron was ~0.25 and was
sufficiently small to allow the three-compartment dendritic model to
accurately describe the voltage-clamp currents. The compactness of the
dendritic tree also allowed observation of putative excitatory inward
synaptic currents that decrease in amplitude during depolarizing
voltage-clamp steps.
The steady-state electrotonic structure of the dendrite is given by a
minimum of three parameters, the soma conductance
(gsoma), electrotonic length
(L), and the ratio of the dendritic to somatic areas
(A). Differences among neurons that might occur due to the dendritic structure should be revealed by a grouping of these neurons
according to these variables (see Table 4).
The three dimensional plot of these variables in Fig.
9 shows that there is a clustering of
neurons (stars) toward the front corner of the A versus
L surface defined by A > 4 and
L < 0.3. This Type B group (stars) has a high-input
impedance and a gsoma < 50 pS. The
remaining Type A (circles) has a broader range of all three parameters;
however, no neurons were found that had A > 4 when L > 0.4. The Type A neurons have a higher average
gsoma and electrotonic length in
potassium gluconate compared with cesium sulfate perfused cells (Table
2 and encircled symbols in Fig. 9). This difference is consistent with
the assumption that removal of potassium ions decreases the passive
dendritic conductance and correspondingly decreases the electrotonic
length. Thus part of the variation observed in the Type A neurons is
probably due to the cesium perfusion, which suggests the average value
of L with normal internal potassium ions is more
physiological. Type B neurons have average
gsoma and electrotonic lengths that
are both lower than Type A neurons and independent of the internal
potassium concentration (Table 2).

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Fig. 9.
Distribution of neuronal types based on electrotonic structure.
Coordinates of the 3-dimensional plot were the d.c. electrotonic
parameters: soma conductance (gsoma),
electrotonic length (L), and the ratio of the dendritic
to soma areas (A). Clustering of the star symbols
provides a way to define Type A (circles) and Type B (stars) neurons.
All neurons in which the firing behavior was measured are indicated by
filled (8 of 19) symbols. One Type I, A neuron was not shown because
one of its coordinates (gsoma), is out of
the range (Table 4, 97K05C) of the graph. In each
instance, there is a strict correlation between Type A and Type I
multiple firing neurons or Type B and Type II accommodating neurons.
Circled symbols represent neurons perfused with cesium sulfate, which
in the case of Type A neurons usually show a lower value of
L.
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The excitability properties of these two classes, Type A and Type B,
appear to be correlated with the Type I and II classification of Figs.
5 and 6. The closed symbols in Fig. 9 indicate neurons in which the
correlation was confirmed. The firing properties of the remaining
neurons were not measured. The means of these two types are given in
Table 2 show that the Type A neurons have an average electrotonic
length of 0.36 and a dendritic to soma area ratio of 3.3; however, with
normal internal potassium ions, L = 0.44 and
A = 3.64. The Type B neurons have a larger dendritic to
soma area ratio of 6.5; however, a smaller electrotonic length of 0.21. These latter neurons represent ~30% of the presumed interneurons that were measured. Because the correlation of electrotonic structure with excitability properties required consideration of both
L and A, it would be unlikely that only the
morphological area of the dendritic tree would distinguish neurons with
different firing properties.
Spatial resolution and dendritic potential profile
The potassium conductance near rest is well described by a
three-compartmental dendritic model in real time and the analytic model
for the frequency domain; however, major discrepancies can occur at
large depolarizations. As discussed in METHODS, as the effective electrotonic length increases because of the activation of
voltage-dependent conductances, the number of compartments must be
increased until there are no changes in the linear and nonlinear
responses of the model equations. Under these conditions, the
compartmental models can be used to obtain an increased spatial resolution for the transient response and the steady-state dendritic potential profile for the frequency domain. Figure
10A illustrates a neuron
having a passive electrotonic length of 0.5 where the time domain fits
for a three-dendritic compartment model (
) are progressively worse
with depolarization. At these potentials, the effective electrotonic
length is considerably >0.5, which means that the level of
depolarization of the peripheral dendrites is considerably less than
the soma. Therefore the spatial resolution of three-compartmental model
for this data are not adequate and cannot quantitatively describe the
voltage-clamp results. Curiously, the predictions of a
three-compartmental model show less current than measured, probably
because the lack of spatial resolution leads to an abnormally low level
of depolarization in the end compartments that produces less current
than observed with more compartments.

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Fig. 10.
Influence of electrotonic structure on potassium currents.
A: voltage-clamp currents at 3 levels of depolarization.
B: corresponding impedance plots for the each
voltage-clamp step fitted with an analytic model. A significant
discrepancy is seen between the fit of the 3-compartmental model in
A, suggesting a need for increased spatial resolution.
Dashed lines show an improved model fit for the increase in number of
compartments from 3 to 30. Other parameters of the model were identical
to those used in the 3-compartmental model. Parameters were
Csoma = 3.95 pF; L = 0.479; A = 2.89;
gsoma = 0.15 nS; and
Vleak = 57.5 mV. Electrode model
parameters are Ce = 2.9 pF and
Re = 25 M . Active parameters were
gK = 4.34 nS;
vK is 90 mV;
vn = 30.2 mV;
sn = 0.031 mV 1;
tn = 2.4 ms;
rn = 0.02 mV 1.
Neuron 97J01A, Type A.
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Figure 10A, - - -, shows a marked improvement of the
model, using the same passive and active parameters, when the number of compartments for the real-time response was increased to 30. The frequency domain fits (Fig. 10B) show that the analytic (
)
and 30 compartmental (- - -) models show better agreement with the data near the resonant peak; however, the low-frequency impedance of
the analytic model matches the data better. The decreased magnitude at
low frequencies of the analytic model compared with the compartmental model is not due to spatial resolution errors but occurs because the
analytic model assumes a uniform potential throughout the neuron. The
increased low-frequency impedance magnitude of the 30-compartmental
model is a consequence of the reduced activation of the potassium
conductance in the peripheral compartments as would be expected because
of the increased electrotonic length. The analytic and 30-compartment
models superimpose at the resting potential; however, a frequency
domain model with only three dendritic compartments is completely
inadequate at all membrane potentials. The relatively good agreement
between the analytic and 30-compartmental model frequency domain fits
suggests that spatial resolution errors are more significant than those
due to the potential profile. Although we used 30 compartments, 10 compartments were generally sufficient for most neuronal structures
(Bush and Sejnowski 1993
; D'Aguano et al.
1989
). Nevertheless, we emphasize that the preceding procedure
is a relatively simple way to evaluate the correct number of
compartments for a particular set of conductances and consequently is
preferred to assuming a fixed number.
The neuron of Fig. 10 also shows a marked linear impedance resonance
that is determined by the interaction of the relaxation time
of the potassium conductance and the passive electrotonic properties. The half-activation potassium conductance time
constant, tn, is of the order of
milliseconds, which leads to transients that cannot be resolved in the
real-time measurements; however, because a 1-ms time constant has the
corner frequency of 160 Hz, it is possible to estimate this parameter
from the impedance data. Thus the activation time constant principally
was determined by the frequency domain resonance, which is sensitive to
its value. Both Type A and B models do show impedance resonances;
however, the Type B model shows a resonance at more depolarized
potentials. This simulation result is consistent with our observation
that, over a limited range of membrane depolarizations, resonance was more frequently observed in Type A than B neurons.
The slow decay seen in the voltage-clamp currents was not modeled;
however, we have obtained essentially identical activation time
constants with an inactivating potassium conductance model. An
additional cause of the slow decay in the current could be a change in
the internal concentration of potassium ions because these neurons are
relatively small compared with other preparations. In general, the
potassium conductance has a positive slope conductance; however, an
inactivating potassium conductance could in principle show a negative
slope conductance. Tables 3 and 5 show
the results for the fast voltage-dependent potassium conductance for
the two groups of neurons, Type A and Type B.
 |
DISCUSSION |
The passive membrane properties of the larval Type A and B neurons
showed input resistances (Rin) of 1-3
G
that was measured at
70 mV (Table 2). These values are slightly
higher than the measured resistances of embryonic neurons, which were
generally 1 G
(Dale 1991
). Embryonic soma are
somewhat larger than those of the larva; however, the dendritic
structure would tend to reduce the effect on the input resistance of
the smaller soma. Estimates of the embryo soma capacitance vary from 10 to 120 pF (Dale 1995b
; Prime et al. 1998
;
Roberts and Tunstall 1990
; Soffe 1990
),
possibly reflecting a large distribution of sizes due to different
developmental states and measurement conditions. A value of 9.5 pF was
used by Dale (1995b)
in model simulations that were
based on measurements of isolated embryonic neurons. Furthermore
isolated larval neurons have been reported to have a lower cell
capacitance of 3.2 pF (Sun and Dale 1998
). This value is
quite comparable to our soma capacitances of 3.3 and 2.6 pF for Type A
and B neurons, respectively (Table 2). In our measurements of intact
neurons, there is a significant additional dendritic capacitance that
was 4-7 times that of the soma. Thus the total capacitance of the
intact larval neuron is considerably larger than the isolated neuron.
The choice of a correct dendritic model is dependent on the
electrotonic structure and how it might change dynamically. If the
assumption of a single equivalent cylinder is reasonable, then the
analytic model provides the best linear description, which can be
passive or active. We have shown that the resting neurons in the
Xenopus larva can be described by an analytic model and that
a three-dendritic compartmental model is usually adequate for real-time
response. This small number of compartments is not adequate for
depolarized neurons because the dynamic electrotonic length increases
and the voltage-clamp currents cannot be described correctly. We have
found that 10-30 compartments is sufficient for describing the
real-time behavior of voltage-clamped Xenopus larval neurons.
The relative contribution of the dendritic cable to the small-signal
passive conductance often is referred to as
, namely, gdendrite/gsoma
where
= (A/L) tanh L in our
terminology (APPENDIX). For the range of L
values found in these experiments,
is nearly equal to A. We were not able to find correlations between active properties and
individual electrotonic parameters, such as A or L; however, the two together do seem to allow the separation
a particular group, Type B, from the remaining Type A neurons. The mean
parameter values of these groups suggest that the larger dendritic area
of the Type B neuron is associated with a lower electrotonic length
compared with Type A neurons. This relationship enhances integrative
mechanisms because the larger dendritic region is electrotonically
closer to the soma. Even Type A neurons do not appear to have
electrotonic lengths >0.4 for dendritic area ratios >4.
The membrane conductance of neurons at rest appears to be different for
Type A and B neurons. In contrast to Type B neurons, Type A neurons
have a component of the resting passive conductance that is dependent
on potassium ions in addition to the voltage-dependent potassium
conductance. Our electrotonic analysis separates active and passive
properties and should show a different value of L for
neurons having similar dendritic areas and different passive gsoma's. Table 2 indicates that
L is lower for Type A neurons when
gsoma is reduced by cesium compared
with control values with normal internal potassium ions. Furthermore
Fig. 9 shows that gsoma increases with
L and decreases with A, as is suggested by the
relationships between gsoma,
A and L given in the APPENDIX. Correspondingly, Type B neurons do not show a change in
gsoma or L. The
demonstration that Type A neurons show a lower L in the
presence of cesium ions supports the hypothesis that the membrane resistivity of the dendritic membrane is similar to that of the soma.
If the cesium perfused neurons were removed from Fig. 9, the Type A
neurons would be more homogenous in their properties and would show a
greater difference in electrotonic length compared with Type B neurons.
In the Type B group, action potentials showed marked adaptation
and occasionally remained depolarized with small levels of injected
current. The action potential behavior was measured in 8 of the 19 neurons of Fig. 9, as indicated by the filled symbols. In each
instance, there was a match between Types A and B and Types I and II,
respectively (Figs. 5 and 6). The Type II accommodating neurons also
have been observed at stage 37/38 (Roberts and Sillar 1990
) for dorsolateral commissural (dlc) neurons in contrast to single impulse responses from ventral interneurons. Because this behavior appears to be correlated with dendritic structure in our
measurements of larvae, it is tempting to speculate that embryonic dlc
interneurons may have a significant dendritic structure.
The differences between Type I and II neurons also are reflected
clearly in the I-V curves. Similar behavior for central
cochlear neurons has been observed in brain slices of the guinea pig
cochlear nucleus (White et al. 1994
). Both the high
passive resistance and steeper potassium conductance activation curve
give Type B neurons more steady-state rectification. The lower
gsoma would allow steady-state
negative conductances due to sodium and calcium ions to influence the
I-V curve, especially for the condition that the potassium
current is not activated strongly at moderate depolarizations. Although
Type A and B neurons have similar half activation potentials, the
steeper slope of the Type B activation curve delays activation of the
potassium conductance. This would be manifested in an I-V
curve by an abrupt increase in current near the half activation
potential, as is observed in the I-V plots for Type II neurons.
In summary, the analysis of these experiments has provided a
quantitative description of the passive electrotonic properties of
putative spinal interneurons of Xenopus larvae that consists of a soma with one equivalent dendritic cable. The models with a
limited number of compartments are remarkably accurate for the different types of neurons; however, active conductances are likely to
require increasing numbers of compartments because of dynamical variations in the space constant. The use of an analytic model provides
electrotonic parameters where spatial resolution is perfect. This
procedure then allows a systematic and accurate way to determine the
number of compartments needed for describing the active properties of
any given neuron.
This work was supported in part by the Centre National de
la Recherche Scientifique, France.
Address for reprint requests: L. E. Moore, Laboratoire de
Neurobiologie des Reseaux Sensorimoteurs, Centre National de la
Recherche ScientifiqueUnité Propre de Recherche de
l'Enseignement Supérieur Associée-7060, 45 Rue des
Saints-Peres, 75270 Paris Cedex 06, France.
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.