Laboratoire de Neurobiologie des Reseaux Sensorimoteurs, Centre National de la Recherche Scientifique-Unité Propre de Recherche de l'Enseignement Supérieur-7060, 75270 Paris Cedex 06, France
![]() |
ABSTRACT |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Mleux, Benoit Saint and L. E. Moore. Active Dendritic Membrane Properties of Xenopus Larval Spinal Neurons Analyzed With a Whole Cell Soma Voltage Clamp. J. Neurophysiol. 83: 1381-1393, 2000. Voltage- and current-clamp measurements of inwardly directed currents were made from the somatic regions of Xenopus laevis spinal neurons. Current-voltage (I-V) curves determined under voltage clamp, but not current clamp, were able to indicate a negative slope conductance in neurons that showed strong accommodating action potential responses to a constant current stimulation. Voltage-clamp I-V curves from repetitive firing neurons did not have a net negative slope conductance and had identical I-V plots under current clamp. Frequency domain responses indicate negative slope conductances with different properties with or without tetrodotoxin, suggesting that both sodium and calcium currents are present in these spinal neurons. The currents obtained from a voltage clamp of the somatic region were analyzed in terms of spatially controlled soma membrane currents and additional currents from dendritic potential responses. Linearized frequency domain analysis in combination with both voltage- and current-clamp responses over a range of membrane potentials was essential for an accurate determination of consistent neuronal model behavior. In essence, the data obtained at resting or hyperpolarized membrane potentials in the frequency domain were used to determine the electrotonic structure, while both the frequency and time domain data at depolarized potentials were required to characterize the voltage-dependent channels. Finally, the dendritic and somatic membrane properties were used to reconstruct the action potential behavior and quantitatively predict the dependence of neuronal firing properties on electrotonic structure. The reconstructed action potentials reproduced the behavior of two broad distributions of interneurons characterized by their degree of accommodation. These studies suggest that in addition to the ionic conductances, electrotonic structure is correlated with the action potential behavior of larval neurons.
![]() |
INTRODUCTION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Knowing the active properties of
Xenopus spinal neurons is essential for an understanding of
the role of voltage-dependent ionic conductances on action potential
behavior. Whole cell patch-clamp measurements of isolated neurons from
the embryo have been used to construct a single-compartment neuronal
model (Dale 1995a) that was used in neural network
simulations to provide a realistic description of locomotor behavior
(Dale 1995b
). These simulations demonstrate the
importance of quantitative measurements for a better understanding of
emergent system behavior and suggest that a single somatic compartment
may be a reasonably sufficient model for the Xenopus embryo
at stage 37/38 (Dale and Kuenzi 1997
; Prime et
al. 1998
; Soffe 1990
; Wolf et al.
1998
). Because Xenopus larvae neurons have
extensive dendritic trees (Van Mier et al. 1985
), it is
necessary to develop a different analysis to quantitatively describe
their active properties, especially those of the dendritic membrane.
Although measurements from small patches or fragments of dendritic
membranes (Kavalali et al. 1997
) provide useful data on
individual channels, it is difficult to determine the overall effect of
membrane conductances on the whole cell behavior from just the kinetic
description of a patch of membrane (see Destexhe et al.
1998
). Whole cell measurements of intact neurons provide important information on the integrative processes and therefore have
been exploited in these experiments in a quantitative manner. This
paper extends the quantitative analysis of electrotonic structure presented in the previous paper (Saint-Mleux and Moore
2000
) to investigate both inward and outward active conductances.
The principal basis of our method is the use of a voltage clamp to
isolate the soma from the dendritic tree (Rall 1960).
This is possible because the voltage clamp controls the soma potential, effectively isolating parallel dendritic branches and allowing their
membrane potentials to respond accordingly. This behavior is
responsible for the well known space-clamp problem of neurons with
dendritic trees (Rall and Segev 1985
; Spruston et
al. 1993
, 1994
; Stuart and Spruston 1998
) and
means that the measured somatic voltage-clamp responses cannot be
analyzed solely with independent parallel ionic currents. The response
of the clamp is composed of both somatic membrane currents and most
importantly, currents caused by the nonclamped dendritic regions. The
channel kinetics of the dendritic membrane determine the unclamped
membrane potential responses that in turn produce a somatic
voltage-clamp current. It is precisely this current that can be
analyzed for dendritic membrane properties. The interpretation of these
data requires detailed knowledge of the electrotonic properties to
evaluate the active conductances in the soma and dendritic cable.
A whole cell clamp in both voltage- and current-clamp modes was used to
constrain more tightly membrane parameters than either method alone. In
contrast to the voltage clamp, the constant current stimulus induces a
"voltage response" from both the somatic and dendritic regions. The potential profiles of the soma and dendritic membranes are radically different in these two cases and provide a
sensitive way to obtain the best parameters for a particular model
(Clements and Redmann 1989). Thus a quantitative
analysis of the two responses provides a way to separate and analyze
the membrane properties of dendritic and somatic regions.
Dendritic and somatic membrane properties were used to reconstruct the
action potential behavior and quantitatively predict the dependence of
neuronal firing properties on electrotonic structure. Finally, this
analytic and theoretical approach provides a new quantitative method to
investigate the pharmacological properties of neurotransmitter
receptors on dendrites (Moore et al. 1999) and their
role in the neural network behavior (Dale 1995b
;
Marder and Calabrese 1996
; Prime et al.
1999
; Roberts et al. 1995
; Tabak and Moore 1998
) of spinal cord circuits that are involved the more complex locomotor patterns of Xenopus larvae.
![]() |
METHODS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Experimental procedure
Current- and voltage-clamp measurements were made in both the
time and frequency domains using methods and analyses identical to
those described in the previous paper (Saint-Mleux and Moore 2000). The photomicrograph of Fig.
1 illustrates presumed interneurons that
have been separated by overstretching the half spinal cord preparation
after the measurements were made. As described previously, both time
and frequency domain data were analyzed simultaneously. To
quantitatively describe these two data types, it is important to
measure them at the same final membrane potential,
V(t). This can be best achieved by obtaining all
the data during the same step-clamp stimulus. Finally, each neuron was
characterized by four types of measurements: real-time constant current
and voltage clamp, each of which had a corresponding steady-state
frequency (f) domain (white noise) determination: impedance,
Z(f), and admittance, Y(f),
respectively.
|
The stimulus protocol for both current- and voltage-clamp measurements was a command step to elicit real time nonlinear kinetic responses followed by a low-amplitude sum of sines, which was superimposed on the preceding step, to obtain frequency domain linear kinetic behavior. The sum of sines stimuli and responses were Fourier analyzed to determine point impedance functions given as the ratio of output to input frequency domain functions. Thus each current- or voltage-clamp step consisted of a nonlinear voltage or current response, respectively, as well as a corresponding linear frequency domain point impedance or admittance function. Finally, the neuronal model obtained from the analysis developed in the following text was required to be consistent with the four types of data.
Space clamp issues
A spatially uniform voltage clamp of a cell is clearly the
preferred method for analyzing voltage-dependent membrane properties. Unfortunately, real neurons cannot be space clamped because of their
dendritic structure (Major et al. 1993, 1994
;
Rall and Segev 1985
;Spruston et al.
1993
). A perfect somatic voltage clamp records somatic currents
that originate from the cell body and unclamped dendritic regions.
Alternatively, the current clamp injects a constant current into the
soma that distributes itself between the soma and the dendritic tree.
The transient potential responses for these two types of experiments is
clearly different but can be modeled exactly by the appropriate cable
equations. A comparison of such a model in both current and voltage
clamp (Clements and Redman 1989
; Müller and
Lux 1993
) provides a good test of the accuracy of the model,
especially if voltage-dependent conductances are activated. If the
neuronal model with its dendritic tree is incorrect, it will not be
able to describe both the voltage- and current-clamp data because the
dendritic potential responses for the two modes are so different. On
the other hand, a good description of both clamp modes provides strong
evidence that the cable model can provide an adequate description of
the total neuron.
Figure 2 illustrates the essence of our experimental approach where a voltage clamp of the soma leads to potential responses in the dendritic compartments that resemble the measured somatic current. For comparison, the active membrane current generated by the somatic compartment is shown to be significantly smaller than the dendritic effect. By contrast, the response of the same neuronal model to a constant current shows a nearly synchronous response for the somatic and dendritic compartments. It can be seen from these simulations that the similarity between the somatic voltage-clamp currents and the dendritic membrane potential transients indicates that the somatic voltage-clamp current contains significant information about dendritic membrane behavior. This analysis is dependent on an accurate assessment of the electrotonic structure that is obtained readily in the frequency domain.
|
Kinetic models
The model formulations are identical to those of previous paper
(Saint-Mleux and Moore 2000) In this paper
gp, the maximum value of a
voltage-dependent conductance, represents potassium (gK,
gK2, or
gKCs), calcium
(gCa), and sodium
(gNa) conductances, where the
unitless, voltage-dependent variable, x, becomes
n or q, s, and m, respectively (see
APPENDIX of companion paper). The conductance,
gK, is the normal fast potassium
conductance, gK2, is a slow potassium
conductance possibly calcium ion dependent, and
gKCs, is a potassium-cesium conductance
having a reversal potential, VKCs, where
potassium ions carry inward currents and predominantly cesium ions
carry outward currents. The activation variables, m and
s, were assumed to be at their steady-state values because
the sampling frequency was too slow to measure the activation time
constants. The term, h, was only used with the sodium
conductance to described inactivation,
gNa*m*h* (V
VNa).
The use of the full nonlinear differential equations for parameter
estimation is far more difficult than curve fitting voltage-clamp transients of single compartments; however, it is absolutely essential because of the dendritic structures. We have used a modification of the
Hodgkin-Huxley (Hodgkin and Huxley 1952) kinetic model (see Borg-Graham 1991
; Murphey et al.
1995
) that does not use a power model for the voltage-dependent
variables. This choice was made for two reasons, a dendritic structure
masks the delay of the voltage-clamp transients needed to determine the
power function and the general form of the linearized equations is
unaffected by the power. A 500-Hz band was chosen to obtain the best
compromise between resolution and sampling frequency that was also
sensitive to the activation of the voltage-dependent conductances. Our
goal is to obtain the minimal model that will fit all the data, namely the nonlinear and linear responses.
The estimation methods, as previously described (Murphey et al.
1995, 1996
), were applied initially to the passive data for an
estimation of the electrode and electrotonic parameters
(Saint-Mleux and Moore 2000
). Using fixed passive
parameters, the data taken at depolarized membrane potentials were
fitted simultaneously in the frequency and time domains with the linear
analytic model and a three-dendritic nonlinear compartmental
model, respectively. The adequacy of both the frequency and time domain
fits was evaluated using a model with 30 dendritic compartments
(Bush and Sejnowski 1993
; Saint-Mleux and Moore
2000
). If necessary, 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).
It should be emphasized that consistent fits of both the constant
current- and voltage-clamp data are essential to achieve the best fits
of the minimal models used here (Bhalla and Bower 1993).
Multiple nonunique fits were possible using either voltage or current
clamp alone; however, only one set of parameters could be found if all
four data types were used. Only three of the data sets actually were
needed because the frequency domain results for the voltage and current
clamp were required to be consistent as a criterion for adequate instrumentation.
![]() |
RESULTS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The data and figures presented in this section are the result of
an extensive analysis of voltage-clamp data near the resting potential
to accurately determine electrotonic parameters and both voltage- and
current-clamp experiments over a wider range of membrane potentials. As
pointed out in the companion paper (Saint-Mleux and Moore
2000), this requires a determination of the adequacy of
different dendritic models with regard to numbers of compartments and
steady-state errors that can occur due to the potential profile along
the dendritic tree. The quantitative analysis in this paper was done on
the neurons of the previous paper, which are identified by the same
names in Tables 2-4 (see Tables 4 and 5 of preceding paper). Data over
a range of membrane potentials were collected with and without using
TTX during internal perfusion of potassium gluconate or cesium sulfate.
The basic strategy was to analyze the properties of individual neurons
in a sequential manner such that fitted parameters were fixed before proceeding to the next procedure, namely the passive electrotonic structure [hyperpolarized neurons with TTX present (see Saint-Mleux and Moore 2000
)], potassium conductances in the presence of TTX, calcium and cesium conductances with internal cesium ions in the presence of TTX, and finally sodium and calcium conductances that were
determined at subthreshold depolarizations with and without internal
potassium ions. All procedures except for the analysis of the sodium
conductance were done in the presence of TTX. Internal cesium ions were
used to block the potassium conductances to more effectively measure
inward sodium and calcium currents. This approach limits the number of
estimated parameters for a given experimental condition as well as
providing a partial separation of the ionic conductances by both
pharmacological and voltage modulation of the individual levels of
activation. The result of this analysis provides an experimentally
determined model that nevertheless is dependent on some theoretical
assumptions. As pointed out by others, such "parameter estimation"
should be done with care on as few variables as possible (Dale
and Kuenzi 1997
; Tabak and Moore 1998
).
Voltage-clamp I-V curves
In the companion paper (Saint-Mleux and Moore
2000), it was found that neurons with larger dendritic areas
and shorter electrotonic lengths (L) show more accommodation
(Type B) than those with smaller areas and larger values of
L (Type A). It was also shown that Type B neurons have a
highly rectifying current-voltage (I-V) curve. Although the
values of the potassium conductance kinetic parameters were similar,
the mean slope of the activation curve versus potential
(sn) was greater in Type B than Type A
neurons. This type of voltage dependence tends to cause a more abrupt
increase in the current near the half-activation potential and
correspondingly leads to a sharp transition in an I-V curve.
In addition to the effect of outward potassium currents on the
rectification of the I-V curve, both sodium and calcium ions
carry steady-state inward currents that counterbalance or override the
outward currents.
The voltage clamp is necessary to observe inward currents, which in turn leads to an I-V curve that has a negative slope, usually referred to as a negative conductance. Therefore I-V curves were measured under voltage-clamp conditions because negative slopes cannot be observed in a constant current experiment. Figure 3 illustrates that voltage clamp I-V curves for Type A and B neurons are in dramatic contrast, such that the slope of some Type B cells is zero or possibly negative at potentials that show the maximum rectification. Figure 3B superimposes the fitted model with its parameters and data for a Type B neuron (97J30A) showing that a lower minimum (negative) slope was found in voltage versus current clamp. These results support the hypothesis that steady-state negative conductances are contributing to the I-V curve of these neurons (see following text).
|
TTX-sensitive negative conductance (gNa)
The measurement of sodium and calcium inward currents is best done if the internal potassium ions are replaced by cesium. Figure 4 illustrates the time required to exchange by passive diffusion the normal potassium concentration with cesium ions by observing the consequential changes in the falling phase of the action potential. The decrease in the potassium conductance leads to multiple action potentials even for short-duration pulses. Finally, after nearly complete removal of K+, the neuron remains depolarized unless a hyperpolarizing current is injected.
|
Although inward currents are easily observed in voltage-clamped cesium perfused neurons, their analysis is especially difficult in dendritic structures because of the instabilities that occur in step-clamp measurements. In general, it is not possible to control the sodium action potentials in a somatic voltage clamp; however, subthreshold TTX-sensitive responses can be used to estimate kinetic parameters. As discussed in the preceding text, current-clamp data provided further constraints on the parameter values, which then led to a set of self-consistent parameters for the sodium conductance. In addition, simulations were carried out to explore the sensitivity of the voltage-dependent parameters on action potential behavior (see following text).
Figure 5, A and B,
illustrates inward currents associated with marked negative phase
functions at low frequencies for 55,
50, and
30 mV, despite an
uncontrolled spike at
30 mV (see Fig. 5A, *). The inward
currents and low frequency negative phases at
50 and
55 mV were
blocked by 1 µM TTX (Fig. 5, C and D) but not
at
30 mV. The absence of a TTX-insensitive low-frequency negative
phase function at
50 mV (Fig. 5D), and its presence at
-30 mV is consistent with a calcium conductance that has a threshold
higher than the sodium conductance.
|
The analysis of the sodium conductance was done after the presumed calcium (see following text) and potassium conductance parameters had been determined and fixed. Tables 1 and 3 show the estimated parameters for the activation and inactivation of the sodium conductance of the Type A neurons. The activation variables of both gNa and gCa were assumed to be in a steady state because the activation time constants are too rapid for our sampling interval. The sodium conductance of Type B neurons was generally not measured because of the difficulties of obtaining controlled responses from a somatic voltage clamp of neurons that had a relatively large electrotonic area, A (see Fig. 5A for one neuron).
|
TTX-insensitive inward conductance (gCa)
In Type A neurons, inward currents such as calcium currents
(ICa), are overwhelmed by the
potassium conductance and difficult to measure. However, replacement of
intracellular potassium with cesium ions shows a TTX-insensitive inward
conductance in both types. Figure 5, C and D,
illustrates that these data are well described by one noninactivating
putative calcium conductance (gCa).
The voltage-clamp current at 30 mV is less positive than those
observed at the smaller voltage clamp steps to
55 and
50 mV. Thus
an inward current is likely to have been activated as shown by the
negative phase function in Fig. 5D (
30 mV). A negative phase function at low frequencies is an extremely sensitive indication of an inward current and can be observed in the presence of a net
outward current. This occurs because the negative phase is a
manifestation of a negative slope conductance that may or may not be
associated with a net inward current.
The mean half-activation potential of the Type A putative calcium
conductance was 21 mV (Table 2), which
is consistent with a high-threshold calcium conductance (Gu and
Spitzer 1993
; Spitzer 1991
). This conductance
was subject to rundown; however, its kinetic properties did not change
with time. A small outward current was observed in most of the cesium
perfused neurons and probably represents an imperfect selectivity of
the potassium channel (gKCs). The
parameters of this conductance, given in Tables 2 and
3 as
gKCs, show a relative low
half-activation potential and most likely represent cesium and
potassium ions passing through a slow potassium channel. A net inward
TTX-insensitive current is shown for a Type B neuron in Fig.
6A. The frequency domain curves show marked negative phase functions (Fig. 6B) that
have been fitted with a noninactivating calcium conductance and
gKCs. The calcium conductance kinetic
parameters for Type B neurons given in Table 2 show a half-activation
potential of
31 mV, which is more negative than found for Type A
neurons.
|
|
|
|
Constant current kinetics
We have emphasized the voltage-clamp data in the preceding analysis because the observation of net inward currents provides convincing evidence of both sodium and calcium conductances. Nevertheless the final analysis of all aspects of these measurements always includes a confirmation of current-clamp data. In some instances, the current-clamp responses are more sensitive indicators of underlying conductances than the voltage clamp. One such case is illustrated by Fig. 7 in which regenerative oscillatory responses were observed. This neuron in TTX was fitted with a model containing one inward (gCa) and two outward conductances (gK, gK2). The inward putative calcium conductance (gCa) was responsible for the inflection on the rising phase, and a slow outward conductance (gK2) caused the oscillatory response seen at the initial depolarizations. The damped overshooting constant current responses at the more depolarized levels were described by a fast potassium conductance (gK). It was virtually impossible to fit these responses with only one potassium conductance. The gK2 was not required for the Type A neurons; however, this may be due to dominance of the normal delayed rectifier compared with other conductances (see following text).
|
Interestingly, the parameters of gK2 from Type B
and gKCs from Type A neurons are quite
similar; however, the gCas of the two
types are quite different (Tables 3 and 4). The more negative half-activation potential, vs = 31
mV, for Type B neurons suggests a relatively low-threshold calcium
conductance (Gu and Spitzer 1993
; Sun and Dale
1998b
). The determination of a low-threshold, presumed calcium
conductance for Type B neurons in K gluconate neurons implies that the
potassium conductances were less activated at lower depolarizations
than Type A neurons. This is consistent with a more shallow activation
curve (sn = 0.04) for Type A than the
steeper slope, sn = 0.09, of the fast
gK activation curve of Type B neurons,
despite similar half-activation potentials (vn =
10 to
15 mV) for both types
(see Table 3 of preceding paper). The additional Type B potassium
conductance, gK2, is two orders of
magnitude smaller than the fast gK and
has an activation threshold, vq =
39
mV, that is significantly more negative (see Table 2) than found for
the gK of Type A or B neurons
(Saint-Mleux and Moore 2000
).
Steady-state potential profile
The current-clamp data were well fitted by the three-dendritic
compartmental model for the initial depolarizations; however, discrepancies in the frequency domain occur at the more depolarized levels (Fig. 7, B and C). The magnitude of the
impedance of a 30-compartmental model was significantly greater (Fig.
7D) than given by the analytic model having the same
parameter values (Fig. 7C). At the resting potential, the
two versions of the same model, compartmental and analytic, give nearly
identical results. This result is an example of a steady-state
dendritic potential profile that leads to a decreased activation of the
potassium conductance in the peripheral regions of the dendrite. When
an analytic model is used to describe the frequency domain, it is
assumed that the voltage-dependent conductances are activated
identically in all compartments, thus leading to a lower impedance
magnitude (greater activation of gK) than actually measured.
It is rather remarkable that the parameters, which give a good fit only
at potentials more negative than 30 mV with the analytic model,
provide an excellent prediction at all potentials using a
30-compartmental dendritic model (Fig. 7D).
Relationship between electrotonic structure and firing patterns
The passive conductance of the dendritic cable (see Eq.
A14 of Saint Mleux and Moore 2000),
gdendrite, is given by the expression, (A/L)*gsoma*tanh
L, where gsoma is the
reciprocal of the passive soma resistance, A is the
dendritic to soma area ratio, and L is the electrotonic
length. Thus the dendritic structure is composed of three variables
that were shown to form a cluster of parameter values that allowed the
definition of two groups of neurons, Type A and Type B, presumably with
different cable morphologies (Saint-Mleux and Moore
2000
). Because the Type B neuron shows relative high values of
A and low L values, the ratio,
A/L, provides a single parameter to distinguish
Type A and B neurons. Figure 8
illustrates at least two clusters of neurons in a three-dimensional
plot of gdendrite, A, and
A/L. One group (
and
), identified as Type B has A/L > 24 and
gdendrite < 500 µS, and the other,
Type A (
and
), consists of the remaining neurons that also have
a broad range of dendritic area ratios that tend to cluster near 4. The advantage of the coordinates of Fig. 8 is ability to separate the
neurons by one plane, which was not the case for plots with gsoma, A, and L
as axes. The finding that A/L rather than
gdendrite is more discriminatory
suggests that there is a greater difference in the structural
parameters, A and L, compared with other passive membrane properties. In Fig. 8,
and
represent instances where the firing pattern of the neuron was measured, which in all cases showed a correspondence between repetitive firing for Type A and significant accommodation for Type B.
|
A computational analysis (Bush and Sejnowski 1993;
Mainen et al. 1995
) was done using the parameters
associated with each type of neuron to compare the predicted and
observed action potentials. Type I repetitive firing patterns were
obtained when simulations were done with the parameter values of Type A
neurons. In these computations, the mean passive and active parameters
of Type A neurons were used with the exception that the maximum fast
potassium conductance was increased. The slow potassium conductance in
this model was assumed to be identical with
gKCs, which is similar to the
gK2 parameters of Type B neurons (see Table
2 and legend of Fig. 9). The need for an
increase in the fast potassium conductance probably represents an
underestimate of our maximum conductance in these experiments because
they were done at relative low membrane depolarizations to characterize the cells near their resting potential. Simulations of Type II behavior
were also done with the measured Type B parameters as given in the
legend of Fig. 10.
|
|
The Type A constant current simulations (Fig. 9, A-C) show
increased numbers and frequencies of action potentials with current intensity as was observed for Type I neurons (Saint Mleux and Moore 2000). A clear rectification was obtained for the model with Type A parameters (Fig. 9D) that corresponds well with
that observed for the Type I neurons (Fig. 3A). The action
potential frequency of the Type A model also could be increased by
decreasing the slow potassium conductance (Fig. 9E) with
minimal changes in the I-V curve.
Figure 10 illustrates that a neuronal model derived from the averaged
data of Type B neurons (Tables 1 and 2) gives a single action potential
response to 0.02-nA maintained constant current if the
gNa is one-half that of Type A neurons
and the half-activation potential of the sodium conductance,
vm, is 28 mV (Fig. 10A). This single response can be converted to a rapidly adapting train of
action potentials by slightly shifting the
vm by 3 to
25 mV, as illustrated in
Fig. 10B. Under these conditions, reducing the current to
0.01 nA (Fig. 10C) elicits two action potentials after which
the potential decays nearly to the resting value despite the
maintenance of the constant current. This behavior is similar to that
observed in Type II neurons (Saint Mleux and Moore
2000
). The correspondence of the Type B model with the Type II
neurons suggests that neurons with relatively large dendritic membrane areas show less sustained activity. The repetitive firing of Type A
models is conserved for a range of gNa
half-activation potentials (vm)
between
25 to
30 mV.
An important aspect of the fast potassium conductance parameters of
Type B neurons is the steepness of the activation curve (high value of
sn), which has a half-activation value
of 15 mV. Thus although the peak conductance is high, it is not
significantly activated at the plateau potential of this model and the
action potential does not repolarize if the current is maintained.
Computed Type B current-voltage (I-V) curves determined with
a voltage clamp show a negative slope whereas the constant current
I-V plots are flat between
50 and
30 mV (Fig.
10D). The I-V curves (Fig. 10D) were
computed with the FindRoot procedure of Mathematica for both constant
current- and voltage-clamp conditions. The two values of
vm were the same as used in the action
potential simulations. Thus there is apparently a correlation between
Type II(B) neurons showing one action potential, a relatively negative
vm and a marked rectification in the
I-V plot.
Although the variation among the accommodating Type II neurons may be
partly due to electrotonic differences, a small shift of the sodium
conductance activation is sufficient to produce the observed Type II
behavior in the Type B model. The rectification of the I-V
plot is enhanced by the sodium and calcium conductances that are both
responsible for the negative slope under voltage clamp and flattening
under constant current. An increased negative slope is seen with the
simulation using vm = 28 mV due to
the activation of a negative conductance at a more negative potential than occurs for vm =
25 mV. A
negative slope in an I-V plot cannot be observed under
constant current conditions because a steady-state potential cannot be
maintained in such an unstable region. The corresponding I-V
plots for current- or voltage-clamp conditions of the Type A model are
essentially identical because no unstable negative slopes were observed
under voltage clamp.
Current voltage curves measured under soma voltage clamp illustrate the
marked differences in rectification of Type I and II neurons (Fig.
3A). Current voltage curves for Type I neurons appear
identical for voltage (Fig. 3A) and current clamp
(Saint Mleux and Moore 2000); however, a small increase
in the rectification during a voltage clamp could be observed in Type
II neurons, as illustrated in Fig. 3B. Thus the Type A and B
model simulations are consistent with the experimentally observed
I-V curves for the Type I and II neurons.
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
In addition to the use of frequency domain functions for an
analysis of electrotonic structure, such functions also provide a
complete description of the dynamic impedance of neurons at subthreshold membrane potentials. The linear neuronal models determined from the measurements in this paper include all of the active conductances (Surkis et al. 1998) and can be used to
predict small signal synaptic events. The measurements themselves are
single point impedance functions (data) that can predict responses to small currents at somatic locations. Current inputs at peripheral regions also can be computed from the derived models with two point
transfer functions (model). The main limitation of this approach for
synaptic events is the requirement of a quasi steady state condition;
however this is often the case for incoming synaptic phenomena.
Anomalous impedance increase during activation of negative conductances
In addition to the negative phase functions, an activation of the
sodium and/or calcium conductances causes increased impedance magnitudes at low frequencies. Figures 5 and 6 show impedance functions
that have maximal low-frequency magnitudes at membrane potentials of
55 to
50 mV for a cesium perfused neuron and
23 mV in the
presence of TTX. The former represents the effects of both
gNa and
gCa in a Type B neuron and the later
is due to a presumed gCa in a Type B
neuron. These effects have been enhanced by the lack of a significant
potassium conductance; however, this can be expected to occur normally
and play a role in the integration of synaptic events. Such increased
impedances decrease effective electrotonic lengths (Moore et al.
1999
) at relatively low frequencies and provide a way to
control somatic responses to dendritic synaptic inputs (Buchanan
et al. 1992
; Moore et al. 1994
, 1995
;
Stuart and Sakmann 1995
). This phenomenon occurs because
of a dynamic decrease in the effective electrotonic length that can be
caused by the activation of any negative conductance.
The enhanced negative conductance seen during cesium perfusion may be
caused partially by some inward potassium current passing through an
A-current channel. A cesium-insensitive potassium A current has been
demonstrated in mammalian central neurons (Sanchez et al.
1998), which is inward at potentials more negative than
35
mV. We have not been able to fit the cesium data with
VKCs =
35 mV; however, good fits are
possible with VKCs =
50 to
90 mV.
In all cases an inward calcium current was required to describe the
data. We occasionally have observed strongly inactivating A-type
currents in cesium-perfused neurons that show a marked negative slope
conductance (unpublished data). These currents were always outward and
much larger than the inward calcium currents.
Comparison of embryonic and larval neurons
Ionic conductances of spinal neurons from the Xenopus
preparation have been quantitatively analyzed in isolation at stages 37/38 (Dale 1995a; Harris et al. 1988
;
Kuenzi and Dale 1998
) and in culture or reduced
preparations at earlier stages (Desarmenien et al. 1993
;
Lockery and Spitzer 1988
). The mathematical descriptions in these investigations were single-compartment models of the soma
because dendritic structures are minimal at the earlier stages. The
neurons investigated in this paper represent later developmental stages, 42-47, and have been measured within an essentially intact spinal cord. Although the theoretical and experimental conditions of
these different preparations are not identical, some comparison of the
derived experimental parameters is possible. In addition to cell
isolation, an additional important difference between our experiments
and those of Dale and his coworkers (Dale 1995a
; Sun and Dale 1998a
,b
) is the external calcium
concentration. The 10 mM calcium levels used by Dale
(1995a)
are likely to shift all activation curves to more
depolarized values than would be observed at 1 mM calcium ions used in
our experiments. This effect also would diminish the differences that
appear to exist between early and late stages of development that are
indicated in the following text. A further point is that high calcium
levels act like a hyperpolarization that tends to enhance the delay in
the turning on of voltage-dependent conductances. This delay was
originally modeled by Hodgkin and Huxley (1952)
by the
power on the voltage-dependent variable. Because such a power function
causes a difference between the half-activation potential of the
conductance and the voltage-dependent variable
(x
), we generally have made our
comparisons with the conductance rather than
x
.
The current-voltage curves of the embryo (Dale 1995a)
suggest that the fast and slow potassium slope conductances have
reached their maximum values by 0 mV and are half-activated between 0 and
20 mV. Our analysis at later stages gave similar results for the
half-activation potentials (vn =
10
to
15 mV) of the fast potassium conductance, gK,
(Saint Mleux and Moore 2000
) and a more negative
vn of
39 mV for
gK2, the slow potassium conductance. Pharmacological
studies on isolated cells at stage 37/38 (Kuenzi and Dale
1998
) have shown that dendrotoxins significantly block the slow
potassium currents and correspondingly enhance repetitive firing. This
result was in contrast to the reduction by catechol of the fast
potassium current, which had relatively minor effects on sustained
action potentials; however, isolated neurons from stage 42 show firing
properties that are partially dependent on a fast-activating
Ca2+-dependent K+ current
(Sun and Dale 1988a
). These measurements suggest that a
fast calcium-dependent potassium conductance replaces part of the fast
and slow embryonic potassium conductances (Wall and Dale 1993
,
1994
) without a change in the overall magnitude of the total potassium current (Sun and Dale 1998a
). Such
developmental changes appear to occur by downregulation of the normal
K+ conductances, possibly both fast and slow.
This phenomenon could account for the lack of a significant slow
gK2 in our voltage-clamp measurements of
Type A neurons (Saint Mleux and Moore 2000
). It is thus
likely that a significant fraction of the potassium currents from
intact neurons at stages 42-47 may be due to a fast
Ca2+-activated K+ current.
Furthermore our Ca2+-activated potassium currents
in 1 mM Ca2+ may be low because it has been shown
that reducing the Ca2+ from 10 to 2 mM reduces
the current by 40% (Sun and Dale 1998a
).
The maximum calcium currents for isolated embryonic and larval cells
occurred ~10 mV (Dale 1995a; Sun and Dale
1998b
), which provides an upper limit for the half-activation
of the conductance. The lower limit would be
10 mV, the potential for
the maximum negative slope conductance (Dale 1995a
).
Thus the half-activation potential of
18 mV for Type A intact larval
neurons is more negative than estimated for stage 37/38. The
half-activation potential, vm
0, found at stage 37/38 for the sodium conductance (Dale 1995a
) is also less negative than
22 mV obtained from Type A larval neurons (Table 1).
Although there are different half-activation potentials for embryo and
larval neurons, the potential at the peak values of the time constants
are usually very similar. These comparisons are partly model dependent
and thus the effect of development on membrane properties needs to be
explored more thoroughly. Nevertheless the finding that the
half-activation potentials of some ionic conductances observed at later
larval stages are at more negative membrane potentials than found for
stage 37/38 is consistent with the findings of O'Dowd et al. (1988) on
cultured Xenopus neurons at different stages of development
(Spitzer 1981
). The voltages for half-activation of the
steady state variables for the conductances of young versus mature
neurons (Lockery and Spitzer 1988
) were as follows:
potassium: 5 and
10 mV; sodium:
16 and
12 mV; and calcium: 4 and
2 mV, respectively. Although, the sodium conductance showed a 4-mV
shift in the opposite direction, the calcium-activated potassium
conductance showed a difference similar to the delayed rectifier
(Lockery and Spitzer 1988
). Thus more mature neurons seem to have lower activation potentials for their voltage-dependent potassium conductances and consequently are likely to possess a wider
repertoire of responses such as bursting and repetitive activity.
Our reconstructed action potentials for Type A and B neurons suggest
that electrotonic structure is associated strongly with the neuronal
firing properties. Previous simulations of embryonic neurons
(Dale 1995b) have shown that repetitive behavior is
principally dependent on the slow potassium conductance. Although our
slow potassium conductance for larval neurons is relatively small, Fig.
9E shows that reducing it to zero increases the frequency of
simulated action potentials in a manner similar to that found by
Dale's single compartment model of embryonic neurons (Dale 1995b
). Decreasing the value of L had little effect
on the firing behavior; however, increasing the dendritic area,
A, reduced repetitive firing. Despite this dependence on
A, simulations with only the somatic compartments of both
Type A and B neurons maintains the appropriate firing pattern. The main
difference in the voltage-dependent conductances in our Type A and B
neurons is in the parameter, sn, the
slope of the activation curve. Interchanging sn values converts a Type
B model to a repetitive firing neuron and changes the repetitive
response of the Type A model to one with plateau oscillations. Thus our
model simulations show that repetitive firing is strongly
dependent on voltage-dependent potassium conductances and slightly
modified by changes in electrotonic structure. Nevertheless the firing
patterns are clearly correlated with different electrotonic structures.
Finally, the analysis of these experiments provides a quantitative description 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 and experimental conditions. Because computational efficiency is an important aspect of neuronal modeling, it is useful to use a minimal number for both data analysis and simulation of network behavior. The criteria developed in this paper provide a means to determine the number of compartments needed for any electrotonic structure with its associated voltage-dependent conductances. The greatest discrepancies occurred in prediction of the voltage-clamp currents. The more nonlinear constant current responses tend to equalize their compartmental potentials due to the presence of active conductances. The use of analytic models avoids compartmental issues; however, errors occur with these models when a large dendritic potential profile leads to an inhomogeneous activation of ionic conductances. Fortunately reasonably good initial estimates of steady-state conductances can be obtained with analytic formulations; however, at large depolarizations the potential profile effect must be taken into account.
In summary, our analysis strategy has addressed the electrode and
electrotonic structure by linear analysis and the active properties by a nonlinear analysis of the voltage-dependent
behavior expressed through the electrotonic structure. The assumption
that the voltage-dependent conductances for sodium, calcium, and
potassium ions are distributed uniformly was adequate for the analysis. This method could be extended further to address the spatial
distribution of the receptors in more detail (Murphey et al.
1995). The reconstructed action potentials based on all the
data reproduced the behavior of two broad distributions of interneurons
characterized by their degree of accommodation. Although previous
analyses of embryonic neurons suggest that the control of repetitive
firing appears to be principally due to a slow potassium current
(Dale and Kuenzi 1997
), it is clear that electrotonic
structure is also an important aspect of this behavior in larval
neurons. The equivalence between Types I and II firing patterns and
Types A and B neurons shows that this behavior in larval neurons is
associated with both the electrotonic structure as well as the specific
voltage-dependent conductances. It remains to be determined just how
these neurons interact in a network to produce the complicated
locomotor patterns characteristic of Xenopus larvae.
![]() |
ACKNOWLEDGMENTS |
---|
This work was supported in part by the Centre National de la Recherche Scientifique, France.
![]() |
FOOTNOTES |
---|
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-Péres, 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.
Received 7 July 1999; accepted in final form 28 October 1999.
![]() |
REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|