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The Journal of Neuroscience, October 1, 1999, 19(19):8271-8280
NMDA-Induced Dendritic Oscillations during a Soma Voltage Clamp
of Chick Spinal Neurons
L. E.
Moore1,
N.
Chub2,
J.
Tabak2, and
M.
O'Donovan2
1 Laboratoire de Neurobiologie des Reseaux
Sensorimoteurs, UPRESA-7060, Paris, France, and
2 Laboratory of Neural Control, National Institutes of
Health, Bethesda, Maryland 20892
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ABSTRACT |
An investigation of dendritic membrane properties was performed by
whole-cell patch measurements of the biophysical properties of intact
chick spinal neurons that are involved in rhythmogenesis. A whole-cell
voltage clamp of the somatic membrane was used to block NMDA-induced
voltage oscillations from the cell body, thus partially isolating the
intrinsic oscillatory properties of dendritic membranes from those of
the soma. An experimental approach was developed that takes into
account the complexity of the dendritic tree in an environment as
normal as possible, without the need for cell isolation or slice
preparations. A computational study of the experimentally determined
model showed that excitatory amino acid receptors on dendrites can
dynamically control the electrotonic length of the dendrites through
the activation of negative slope conductances. These experiments
demonstrate the presence of NMDA receptors on the dendrites and that
they induce intrinsic oscillations when the synaptic input from other
cells is significantly reduced.
Key words:
dendritic oscillations; chick spinal neurons; NMDA; electrotonic structure; frequency domain; impedance
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INTRODUCTION |
The determination of dendritic
membrane properties has been most successful using direct whole-cell
patch studies in both isolated and slice preparations (Major et al.,
1993a ; Spruston et al., 1993, 1994; Bekkers and Stevens, 1996;
Kavalali et al., 1997). Recently, measurements from single neurons
before and after isolation have introduced an innovative way to
estimate the "removed" dendritic conductances (Destexhe et
al., 1998). These techniques (Jackson, 1992; Surmeier et al., 1994;
Stuart and Spruston, 1998), in combination with a number of available
computational studies, have provided significant theoretical insight
into the role of dendritic processing for numerous neuronal behaviors
(Traub and Miles, 1991; Mainen et al., 1995; White et al., 1995; Koch
and Segev, 1998). As an extension of these methods to neurons in their functioning networks (O'Donovan et al., 1992; Roberts et al., 1995;
Marder and Calabrese, 1996; O'Donovan and Chub, 1997), we have
developed an approach that takes into account the complexity of the
dendritic tree in an environment as normal as possible, without the
need for cell isolation or slice preparations. Our approach is
fundamentally based on the pioneering work of Rall and his
collaborators (Segev et al., 1995) who developed much of the
mathematical formalism needed for the analysis of dendritic membrane
properties. Our method uses step voltage-clamp currents and small
signal frequency domain responses, both of which are elicited by the
same final command membrane potential.
Thus, both conventional large-step nonlinear responses as well as
small-signal linear perturbations are used to analyze dendritic conductances of intact neurons by explicitly taking into account the
currents originating from the unclamped cable processes attached to the
soma (Major, 1993; Major et al., 1993b). This approach allows an
accurate determination of electrotonic parameters, and under certain
conditions, the active ionic conductances of the dendrites can be
quantitatively described.
An assessment of dendritic properties is especially useful in
understanding rhythmogenesis in the embryonic chick spinal cord. Previous experiments (Sernagor et al., 1995; Chub and O'Donovan, 1998)
have suggested that excitatory amino acid conductances, which are
increased during rhythmic activity, are likely to be primarily
dendritic, whereas GABA/glycine conductances are predominantly somatic.
These conclusions were based on experiments in which transmitter
antagonists injected into the motor nucleus in the vicinity of the cell
bodies abolished presumed GABA-A responses but only partially depressed
excitatory amino acid responses (Sernagor et al., 1995).
Thus, to quantify the above issues, measured biophysical properties of
intact spinal neurons involved in rhythmogenesis of the chick spinal
cord were used to construct a quantitative somatic and dendritic model
of both the voltage-dependent ionic conductances and those induced by
NMDA activation. A whole-cell clamp of the somatic membrane was used to
block voltage oscillations from the cell body, thus partially isolating
the intrinsic oscillatory properties (Lampl and Yarom, 1997) of
dendritic membranes from those of the soma (Seutin et al., 1994). This
method was used to demonstrate that NMDA receptors are present on the
dendrites and show oscillatory behavior independent of the somatic
membrane receptors.
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MATERIALS AND METHODS |
Experimental procedures. The experiments were
performed on isolated spinal cords of embryonic day 10-11 White
Leghorn chicken embryos that had been maintained at 38°C. As
described previously (Chub and O'Donovan, 1998), the spinal cord was
prepared in cooled (10-15°C), oxygenated Tyrode's solution
containing (in mM): 139 NaCl, 12 glucose, 17 NaHC03, 2.9 KCl, 1 MgCl, and 3 CaCl2. Measurements were made at 28°C with
continuous perfusion. When indicated, 20 µM NMDA was
added to the perfusate. The whole-cell recording was performed with
electrode solutions containing (in mM): 130 K-gluconate, 10 NaCl, 10 HEPES, 1.1 EGTA, 1 MgCl2, 0.1 CaCl2, 1 Na2ATP, and 10 QX-314
(lidocaine, N-ethyl bromide quaternary salt). All
measurements reported here were performed in the continuous
voltage-clamp mode using an Axoclamp 2A (Axon Instruments, Foster City,
CA). All other procedures were identical to those described by Chub and O'Donovan (1998).
The whole-cell patch was made with an electrode under positive pressure
that was advanced through the ventral white matter of the chick spinal
cord until cell bodies were reached. Gigaseals were then made with a
slight negative pressure as described previously (Sernagor and
O'Donovan, 1991). Although this technique allows recordings from
neurons in a completely intact spinal cord, electrode resistances are
often increased over their value in the bathing solution before the
seal is made. Quantitative measurements require an accurate correction
of their effects (Wilson and Park, 1989). Most studies have used
bridge-balance techniques, for which there are numerous problems
because they are dependent on both the resistive and capacitative
properties of the electrode (Burke and Bruggencate, 1971; Jackson,
1992; Wright et al., 1996). In addition, electrode resistances are
known to change, which requires reassessment of the bridge balance
during the course of an experiment (Armstrong and Gilly, 1992). Thus,
unless some evaluation of the active electronic compensation filtering
of the signal is performed, it is impossible to determine the accuracy
of the measurement. The technique presented here provides a monitor of
the electrode properties during each voltage-clamp step.
Our technique consists of a whole-cell large-signal nonlinear
step-voltage clamp on which is superimposed a sum of low-amplitude sine
waves that evoke a linear response after all ionic conductance transient responses are complete (Moore and Christensen, 1985, 1987).
The analysis is thus a hybrid of linear (Moore et al., 1987, 1993;
D'Aguanno et al., 1989) and nonlinear methods in both the time and
frequency domains (Murphey et al., 1995, 1996). Figure 1A shows the current response to a hyperpolarizing
voltage command at the soma electrode, which consists of a step
followed by a sum of sines (white noise) superimposed on the step at
0.6 sec. The amplitude of the white noise voltage signal was maintained below 3-5 mV root mean square to obtain a linear response. The current
response shows an initial transient, attributable to both electrotonic
structure and the electrode (see Fig. 1A). With
depolarizing steps, this was followed by second ionic conductance
transient (see Fig. 2A) and finally a steady-state
response consisting of both DC and sine wave components (see Fig.
2A,B). The linear frequency domain
responses are ratios of the amplitudes of the voltage and current sine
waves and the phase shifts between them for each frequency (see Fig.
1B). This was performed with a 1024 point Fast
Fourier transform on the steady-state data collected from 0.8 to 1.6 sec that was low pass-filtered at 0.4 times the maximum frequency
measured (Moore et al., 1988). The transformed data are shown in the
figures as magnitude and phase relationships. The analysis is based on
data from 18 neurons, all of which show similar results.
Thus, the determination of dendritic properties consists of three
parts: (1) large- and small-stimulus voltage- and current-clamp experiments, (2) a parameter estimation that tests different neuronal models, and (3) a computational investigation of the nonlinear predictions of the experimentally determined models. This paper presents an analysis of whole-cell voltage-clamp data from real neurons
with dendritic trees and demonstrates that dendritic properties can be
quantitatively analyzed by this approach.
Computational models. Nonlinear compartmental models were
used to quantitatively fit soma voltage-clamp currents (Murphey et al.,
1995, 1996) to estimate (1) the electrotonic structure of the dendrites
and (2) their voltage-dependent conductances. Because neuronal
dendritic structures preclude a space clamp (Armstrong and Gilly,
1992), it is not possible to use the relatively simple analytical
solutions of voltage-clamp current responses traditionally used in a
Hodgkin-Huxley analysis (Hodgkin and Huxley, 1952). Therefore,
the complete nonlinear equations were solved numerically for all real
time parameter estimations and simulations of model responses. The
models used in the data analysis were required to describe all of the
time and frequency domain data from a particular neuron. All models
consisted of an electrotonic structure on which uniformly distributed
voltage-dependent ionic conductances were expressed. For normal
Tyrode's solution in which the sodium conductance was
pharmacologically blocked, a potassium conductance
(gk) was always used. In addition, most cells
required an inward calcium conductance. In the presence of NMDA, the
voltage-dependent conductances consisted of an inward NMDA-induced
channel (gnmda), the normal potassium conductance
(gk), and an additional slower potassium conductance
(gkslow) that appears only in
the presence of NMDA. The slow potassium channel is generally
attributed to a calcium-activated channel that is enhanced because of
the entry of calcium through the open NMDA channel.
As described above, at each voltage-clamp step a small-signal linear
response was evoked. These data were analyzed in the frequency domain
with theoretical functions that are the limiting linear response for
small perturbations of the complete nonlinear equations (see Figs.
1B, 2B). These linear functions do
have an analytical form for both the passive and voltage-dependent
properties. Previously described parameter estimation methods (Murphey
et al., 1995, 1996) were applied simultaneously to both the real-time nonlinear and the linearized data. The constraints imposed by fitting
two different types of data provide a more complete test of a
particular nonlinear model than an analysis based on one data type
(Bhalla and Bower, 1993).
The above considerations illustrate the need to analyze and model as
completely as possible the true experimental situation to obtain the
correct membrane kinetics that are superimposed on the passive
electrotonic structure. The only truly clamped potential is that of the
electrode. All other potentials and currents must be described by
solutions of the Hodgkin-Huxley-type (Hodgkin and Huxley, 1952),
nonlinear differential kinetic equations (Borg-Graham, 1991)
appropriate for the experimental condition being analyzed, and with due
attention given to the nature of cable equations used to model
dendritic processes (Rapp et al., 1994). Two methods were used to model
the equivalent dendritic cable: (1) a compartmental model consisting of
3-30 segments in series (D'Aguanno et al., 1989; Murphey et al.,
1995, 1996) and (2) a uniform frequency domain analytical model, also
with voltage-dependent conductances. This model is equivalent to an
infinite number of nonbranched compartments and thus has perfect
spatial resolution. In this paper we have shown that a minimal neuronal
model, consisting of a soma and one Rall-type equivalent cylinder (Rall
and Segev, 1985), quantitatively describes the soma voltage-clamp data
presented here. Although complicated branching models (Major and Evans, 1994; Major et al., 1994) could more explicitly represent the morphology, they were not required to quantitatively describe our data.
Thus, our ability to accurately fit the sensitive frequency domain data
near the resting potential with the analytical model of Equation 1
supports the hypothesis that the Rall equivalent cylinder is a
reasonable model for embryonic chick motoneurons. Other neuronal types
in the spinal cord might require more complicated models, such as two
or more equivalent cylinders with different electrotonic parameters, as
discussed by Major et al. (1994).
All analyses in the frequency domain were performed with the uniform
analytical model:
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(1)
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and
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(2)
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(3)
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where Yt is the total admittance with the
electrode, Ya is the admittance seen at the
soma, Ysoma(V,f) represents the
passive and active admittances of the soma compartment,
csoma is the soma capacitance,
gl is the passive soma conductance,
Vl is reversal potential for leakage
conductance, L is the electrotonic length, A is
the ratio of dendritic to soma area, ce is the
electrode capacitance, re is the electrode
resistance, j =   1, V is potential, and
f is the frequency in hertz (see below for the remaining definitions).
Parameter estimation. The finite difference method was used
to obtain the gradient of the real-time model response to apply nonlinear optimization directly to the dynamic response of the model.
The model equations were solved with the backward differentiation integration method (Byrne and Hindmarsh, 1975) or the NDSolve algorithm
of Mathematica (Wolfram, 1996). The gradient-type parameter estimation (Dennis et al., 1981) was used to fit the measured admittance spectrum and the solution of ordinary differential equations
to the real-time and frequency domain voltage-clamp data. In all cases
a simultaneous analysis of the temporal and frequency domain
measurements at different membrane potentials was performed. Iterative
gradient descent was used until the minima error that was reached
changed <0.1%. Local minimum were avoided by verifying that
changes in the relative weighting of the frequency and time domain data
did not move the fit to a different region in parameter space. Although
the final fits were performed by estimating all parameters over a range
of membrane potentials, the initial estimates were obtained by the
following sequential procedures. (1) By using one or two
hyperpolarizing records, seven passive
parameters csoma,
gl,
Vl, L, A,
ce, and re were
estimated over the measured frequency range. Because
Vl is not used in the frequency domain
(Equation 1), only six parameters are used for this condition. (2)
Next, Vl,
re, and ce were
fixed, and the remaining four electrotonic ones
(csoma,
gl, L, and A)
were estimated over one-half of the frequency range of procedure 1. (3)
Finally, the voltage-dependent conductance parameters were estimated as
in procedure 2, where gp is a generic
voltage-dependent ionic conductance with a reversal potential of
Vp and whose kinetics is governed by
the unitless variable x, which has a steady-state value of x . Thus, at the half-activation
(x = 1/2)
voltage, vx,
sx is the slope of
x, tx
is the time constant,  x, and
rx is the normalized slope of
x. In this paper
gp represents potassium
(gk)-, calcium (gca)-, and
NMDA-induced conductances, both inward
(gnmda) and outward
(gkslow) (Murphey et al.,
1995). All fits of active conductances involved a maximum of two
conductances, one inward gCa and one outward gK.
In each of the three steps, all of the time domain data, and the same
50 frequencies out of 400, logarithmically distributed from 0.5 to 250 Hz, were selected for fitting. Thus, low-frequency points were
emphasized by this selection, as indicated by the log plots. Procedure
3 was also used in estimating NMDA-induced conductances
(gnmda and
gkslow); however, all other passive
and active conductances were kept fixed from previous estimations. This
approach prevented mixing the effects of the normal voltage-dependent conductances with those induced by NMDA. Similar to classic
voltage-clamp experiments, a combination of voltage ranges, ion
substitutions, and the use of pharmacological blocking agents is
required to obtain a detailed description of multiple ionic
conductances. The advantages of incorporating frequency domain
measurements with real-time analysis are numerous: (1) electrotonic
structure is more easily taken into account, (2) the presence of
voltage-dependent conductances in the dendritic tree can be explicitly
evaluated, and (3) combinations of linear and nonlinear responses
constrain parameter estimation more than either one alone.
Electrode properties. Before discussing the active and
passive parameters, it is necessary to clarify the contribution of the
electrode (Armstrong and Gilly, 1992) to the data analysis. This was
achieved by simulating the model of Figure 1 with the fitted
parameters, but changing just the value of the electrode resistance
from its estimated high value of 36 M (solid lines) to a low one of
0.5 M (see Fig. 1A,B,
dashed lines). Under these latter conditions the
hyperpolarized current at 70 mV (see Fig. 1A,
dashed curve) is indicated as an electrotonic response
because the predicted transient is mainly caused by the dendritic cable structure of the model (Spruston and Johnston, 1992). Thus, the near
superposition of the two electrode model responses (dashed and solid
smooth lines) and data show that the slow component of the essentially
passive somatic current is relatively insensitive to the value of the
electrode resistance. Therefore, the passive dendritic cable of the
neuron contributes more significantly to the hyperpolarizing real-time
response than the electrode in series with the soma capacitance.
In contrast to the real time data, the frequency domain plot for the
70 mV in Figure 1B (dashed lines) shows
a dramatic change when the electrode resistance is lowered to 0.5. Similar high-frequency distortions were also seen with the depolarized
frequency domain records (data not shown). It is clear from this
simulation that changes in the electrode resistance and capacitance
that might occur during an experiment cannot be easily determined from
the current transients but are readily accounted for by the frequency domain functions that were measured in the steady-state at each voltage-clamp step.
Although the hyperpolarized time domain data are relatively insensitive
to a decrease in the electrode resistance, depolarized voltage-clamp
currents are markedly altered (see Fig. 2A,
dashed curves). The most obvious difference is the
steady-state current because the voltage drop across the electrode
affects the final membrane potential. The computed initial transients
of the low-resistance electrode model at 20 and 10 mV have a time
course similar to that of the measured hyperpolarized current, i.e.,
significantly faster than the observed slower response during a
depolarization. Clearly, the distortions throughout the depolarized
transients are caused by an interaction of the turning on kinetics of
the active conductances and the passive electrotonic behavior of the neuron plus the electrode (Armstrong and Gilly, 1992; Surmeier et al.,
1994; White et al., 1995). Unfortunately, such electrode and similar
electrotonic effects do not alter the general shape of the
current and could be mistaken for true conductance time courses and
therefore misinterpreted.
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RESULTS |
Somatic voltage clamp
The hybrid parameter estimation method was simultaneously applied
to hyperpolarized and depolarized potentials, i.e., data sets that were
both passive and active. The passive electrotonic parameters were
uniquely constrained by the hyperpolarized data (Borst and Haag, 1996),
and the voltage-dependent kinetic behavior was expressed through the
same electrotonic structure at depolarized potentials. Simultaneous
parameter estimation over a range of membrane potentials thus assured a
self-consistent model of both the passive electrotonic structure as
well as the voltage-dependent conductances. The hyperpolarized
voltage-clamp currents of Figure 1A were fitted with a
passive dendritic compartmental model (smooth curves) having an
electrotonic length of 0.5 and a dendritic to soma area ratio of 26. The corresponding frequency domain data of Figure 1B
were fitted simultaneously with the identical model in its analytical
form. Average passive electrotonic values for seven neurons (±SD) for
the electrotonic parameters were csoma = 12.6 (±16.3) pF, L = 0.38 (±0.17),
A = 16.4 (±8.2), gl = 5.6 (±14.7)
pS, re = 29.8 (±10.3) M,
ce = 6.3 (±1.8) pF.
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Figure 1.
Whole-cell soma voltage clamp. A,
Current response to a voltage-clamp command. The step voltage command
was applied for 0.2-0.6 sec, after which was added 1 sec of a small
signal deterministic white noise signal consisting of 400 sequential
sine waves with randomized phases. The step response from 0 to 0.6 sec
is data recorded from the neuron followed by a white noise curve that
is a schematic representation of the current response. The smooth
curves superimposed on the data represent model responses computed with
parameter values fitted simultaneously in the time and frequency domain
for a 20 mV hyperpolarizing voltage-clamp pulse from a holding level of
50 mV. The dashed lines are model calculations with
only a change in the electrode resistance,
re, from 36 to 0.5 M. More
complicated segmental models for the electrode properties gave a better
description of some electrodes; however, the simple lumped model
described above provided nearly identical membrane parameter estimates.
Therefore, the simpler electrode model was routinely used.
B, Fitted magnitude (Megaohms) and phase
(radians) functions for each voltage-clamp step are
superimposed on measured responses. The dashed lines at
70 mV represent a model simulation where the electrode resistance was
lowered to 0.5 M as in A. All measurements were
analyzed from 1.25 to 250 Hz. The parameter values for this neuron are
given in Figure 2 (Neuron #1).
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The depolarized soma voltage-clamp response to 20 mV in Figure
2 shows a modest outward current;
however, the frequency domain functions are dramatically modified in
both amplitude and phase compared with the passive hyperpolarized
condition (Fig. 1). Together, the time and frequency domain data
provide important information about the nature of the voltage-dependent
ionic conductances. The magnitude shows a pronounced increase and
resonance at a few hertz, and the phase decreases to a negative value
below 2 radians (Fig. 2B), i.e., considerably more
negative than 90° ( /2 radians). This result can only occur if
a negative slope conductance has been activated. Negative slope
conductances are generally associated with the activation of an
inwardly directed current carried by sodium or calcium ions. The phase
function of a conductance showing a positive I-V
slope can never be more negative than 90° (/2 radians).
Further depolarization to 10 mV shown in Figure 2B illustrates a shift of the resonance to higher frequencies, with an
accompanying sharp phase transition characteristic of highly resonant
responses. The step voltage-clamp current at 10 mV shows an increased
outward current suggesting the activation of a putative potassium
conductance. These data could be accounted for by an inactivating
positive conductance that showed a negative slope for its
I-V plot at 20 mV; however, this hypothesis
seems unlikely because no indication of a decrease in the outward
current during voltage-clamp steps was observed. It is plausible to
propose an inward calcium conductance to account for the negative
conductance because the sodium conductance was blocked by QX314 (Sigma,
St. Louis, MO). Thus, the neuronal model fits shown in Figure 2 consist of a putative outward potassium conductance and an inward calcium conductance.
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Figure 2.
Depolarizing soma voltage-clamp currents. The
entire data sets, both time (A) and frequency
(B) domains, were simultaneously fitted with a
neuronal model consisting of an isopotential soma to which were
attached the electrode resistance,
re, with its capacitance,
ce, plus a dendritic cable as
follows: (1) in the frequency domain, an analytical description with a
finite length, and (2) in the time domain, a three-compartment model
required for the solution of the nonlinear differential equations that
describe the voltage-dependent conductances (Murphey et al., 1995). The
superimposed voltage-clamp currents and model fits were elicited from a
holding potential of 50 mV followed by steps to the indicated
potentials. The parameter values for the model fits were
csoma = 4.9 pF, L = 0.45, A = 25.79, gl = 0.00012 µS, re = 36.5 M,
ce = 8.3 pF,
Vl = 47.9 mV, gk = 0.00014 µS, vn = 20.95 mV,
sn = 0.0688 mV 1,
tn = 0.259 sec,
rn = 0.2 mV1,
gca = 0.0003 µS, rs = 0, vs = 18 mV,
ss = 0.07 mV1,
ts = 10 µsec, Vca = 0, Vk = 95 mV (Neuron #1).
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Anomalous impedance increase with NMDA activation
The addition of 20 µM NMDA to the external solution
activated rhythmic activity similar to that seen in this preparation
either spontaneously or after a brief cathodal pulse to the spinal
cord. After some seconds the patterned behavior was replaced by tonic, disorganized activity that appeared to inhibit further synchronized responses. Application of TTX at this stage often led (four of five
neurons) to the appearance of intrinsic pacemaker activity similar to
that observed in other spinal cord preparations (Brodin et al., 1991;
Wallén et al., 1992; Scrymgeour-Wedderburn et al., 1997).
Voltage-clamp responses from such a preparation are shown in Figure
3A where a maintained
NMDA-induced inward current at the holding potential of 50 mV was
observed. A 5 mV step depolarizing pulse elicited a small increase in
the negative current; however, a larger 20 mV hyperpolarization
slightly reduced it. This type of response was never observed in normal
control solutions and was adequately modeled as an NMDA-induced inward
conductance in which a magnesium block of the NMDA receptor conferred
voltage-dependent properties (Moore and Buchanan, 1993). For this
limited polarization range, gkslow was
not needed to model the data, because its activation occurs at more
depolarized potentials. The model describes well the reduction of the
inward current with hyperpolarization caused by an enhancement of the
Mg block of the NMDA receptor.
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Figure 3.
NMDA-induced impedance increase. The soma
voltage-clamp responses of A show that the
hyperpolarizing current at 70 mV is less negative than the
depolarizing response at 45 mV elicited from a holding level of 50
mV. This reversal of the usual current response was modeled as a
voltage-dependent NMDA-induced current that turns off because of a
magnesium block at hyperpolarized potentials. B, This
conductance has a negative current-voltage relationship that
manifests itself in a small-signal frequency domain measurement as a
large negative phase at low frequencies. The model responses are
superimposed on the data for both time and frequency domain records.
The parameter values were csoma = 49 pF, Vl = 39 mV,
gl = 0.0002 µS,
re = 18 M,
ce = 7 pF, A = 4.1, L = 0.67, gk = 0.0028 µS,
vn = 2 mV,
sn = 0.02 mV1,
tn = 0.014 sec,
rn = 0.02,
gnmda =0.01 µS,
vm = 5 mV,
sm = 0.02 mV1,
tm = 0.0001 sec,
rm = 0, Vnmda = 0, Vk = 95, gca = 0, gkslow = 0.0. The electrode was modeled
with a series resistance, re, and a
capacitance to ground, ce. C,
The superposition of the 3 and 30 compartmental models are virtually
indistinguishable for the three voltage-clamp pulses. The dashed
lines represent the 30-compartmental model. The adequacy of the
spatial resolution of the compartmental model was evaluated by
computing the time domain responses in models having 30 dendritic
segments. The electrotonic parameters and distribution of active
voltage-dependent conductances of all models were identical. The
relationship between the compartmental and analytical models is given
by the following formulation in which x is a generalized
kinetic variable (Moore and Buchanan, 1993; Murphey et al., 1995):
Ii = Ip + Icore + Il,
Il = gl
(Vi Vl),
Ip =  p
gp x (Vi Vp), Icore = gcore (Vi
Vi+1), Icore = gcore (Vi
Vi+1) and dVi/dt = N Ii/A csoma, where
csoma is the capacitance of the soma,
Ii is the current in the ith
compartment, Icore is the current between
compartments, Vi is the membrane potential
in the ith compartment, gl
and Il represent a nonspecific leakage
conductance and current having Vl as a
reversal potential, L is the electrotonic length,
A is the ratio of dendritic to soma area,  = N/L is in units of the number of
compartments, N, and /gcore = L/(A*gl). D, The computed
potential profiles for the soma and most distal 30th compartment show
that the membrane potential is relatively uniform when the soma was
depolarized to 20 mV, which suggests that a smaller number of
compartments is adequate for this condition. The similarity of the
model behavior in A and C for 3 versus 30 compartments supports this hypothesis. This effect is most striking
because 20 mV is just the potential at which nearly zero net current
occurs, giving a ratio of the voltage to current of virtually infinity,
i.e., a limiting electrotonic length of zero and a perfect steady-state
space clamp. At hyperpolarized levels the conductances are essentially
turned off; thus even if there is a potential profile it does not lead
to different impedances for the different compartments as would be the
case with activation of voltage-dependent positive conductances (Neuron
#2).
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The corresponding frequency domain records of Figure 3B show
that the 70 mV response has a monotonically decreasing magnitude with
frequency that shows no corner frequency representing the membrane time
constant, indicating that it must be well below 1.25 Hz, the lowest
frequency measured. Such behavior is characteristic of a balance
between negative and positive conductances that leads to the anomalous
situation of virtually no change of the current in response to a change
in the membrane potential, as experimentally observed in Figure
3A. Under these conditions the DC impedance, or resistance,
approaches infinity, and the magnitude function versus frequency is
approximately linear on a log-log Bode plot. Thus, the impedance is
mainly capacitative, having a negative phase function at low
frequencies, often close to /2 radians or 90°; however, its
behavior with frequency is very sensitive to the electrotonic structure
of the cell. The frequency domain records for the 5 mV depolarization
(V = 45 mV) are not smooth because of evoked
oscillatory instabilities; nevertheless, the phase function appears to
be strongly negative, similar to that observed for the negative
conductance of Figure 2, seen at a much greater depolarization
(V = 20 mV). In normal Tyrode's solution no negative
conductance effect was observed over the limited potential range shown
in Figure 3. Average values of four neurons (±SD) for the peak
conductances were gk = 0.00024 (±0.00025) µS,
gca = 0.0005 (±0.0003) µS,
gnmda = 0.0027 (±0.0047) µS, and
gkslow = 0.0016 (±0.0019) µS.
Model predictions of experimental results
In the previous section parameter estimation methods were used to
obtain the best fits from the real time and frequency domain data. By
contrast, the following analysis consists of model predictions and
simulations at more depolarized membrane potentials based on the
voltage-clamp analysis presented above. These computational studies
consist of both current and somatic voltage-clamp predictions and
provide an interpretation of observed experimental results, where
NMDA-induced oscillatory effects are prominent.
Dynamic electrotonic length: dendritic potential profile
The data analysis showed that the NMDA-induced conductances, as
well as presumed intrinsic calcium conductances, lead to extremely large low-frequency impedances. To take into account the role of
steady-state active conductances on electrotonic behavior, it is useful
to define an effective electrotonic length as follows:
![L<SUB><UP>ef f</UP></SUB>=L*√(Mag[Y<SUB><UP>soma</UP></SUB>(V, f)]/gl),](/content/vol19/issue19/fulltext/8271/img005.gif)
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(4)
|
where
Mag[Ysoma(V,f)]
is the magnitude of Ysoma(V,f)
(see Eq. 2). Thus, Leff takes into
account the capacitance as well as the voltage-dependent conductances
and reverts to L at f = 0 for a passive
neuron. The consequence of a nearly infinite intrinsic resistance
(f = 0) is that
Ysoma(V,0) and therefore
Leff approach zero. Thus, at low
frequencies the neuron becomes isopotential, and all maintained
peripheral synaptic activity is virtually unattenuated at the soma. At
high frequencies Leff increases, and
significant attenuation will occur. To test these conclusions, model
simulations are presented to clarify the mechanisms that lead to these
unusual effects.
Figure 3C shows that the superimposed voltage-clamp currents
for models with 3 and 30 dendritic compartments are remarkably similar,
indicating that errors caused by spatial resolution of the
three-compartmental model of Figure 3A (smooth
curves) are minimal for these parameter values. The
hyperpolarizing potential profile of the last dendritic compartment
(Fig. 3D) compared with the soma shows a marked delay in the
development of a steady-state offset potential. This delay and offset
are related to the hyperpolarizing somatic current time course shown
for both the model and data, as would be expected of a simple passive
cable (Rall, 1960).
By contrast, the depolarization clamp to 20 mV shows the remarkable
result that the steady-state potentials of the soma and last dendritic
compartments are nearly the same, the latter having a damped overshoot
(Fig. 3D). The clamp control of the soma potential is
similar to the hyperpolarizing response, showing a slight delay attributable to the charging of the soma capacitance through the electrode. The computational result of virtually no steady-state current or potential drop between the soma and last dendritic compartment for a somatic step depolarization to 20 mV is a
manifestation of a steady-state negative conductance that increases the
effective membrane impedance and thereby reduces the effective
electrotonic length (Leff) of the
dendrite (Müller and Lux, 1993; Moore et al., 1994, 1995). The
dramatic nature of this computation is partly attributable to the lack
of a large potassium conductance in the model parameters, which would
counteract the negative conductance. Nevertheless, these calculations
show that this phenomenon provides a means to dynamically control the
space constant and significantly alter the effect of synaptic inputs
from peripheral dendritic regions (Buchanan et al., 1992; Stuart and
Sakmann, 1995).
These results demonstrate that the analysis of point voltage-clamp
experiments requires a consideration of at least three conditions: (1)
the electrotonic structure of the passive dendrite and electrode
properties, (2) an increase of the effective electrotonic length
(Leff) caused by the activation of
positive conductances, and (3) a decrease in the effective electrotonic
length (Leff) because of an
enhancement of the low-frequency impedance caused by the activation of
negative voltage-dependent conductances. In addition, these changes in
the electrotonic length are truly dynamic and must be quantitatively
accounted for if somatic voltage-clamp currents are described.
Dendritic membrane potential oscillations in somatic
voltage clamp
The instabilities inferred from Figure 3B for the 5 mV
cathodal step are fully developed by the larger depolarizations
illustrated in Figure
4A (solid
lines) where large oscillatory currents from the unclamped
dendritic regions provide a major contribution to the somatic
voltage-clamp currents. The difference in periods of the oscillations
observed between those at 30 and 10 mV is likely to be a
consequence of the voltage-dependent Mg block of the NMDA receptor.
Furthermore, the observation of current oscillations during a somatic
voltage clamp suggests that the dendritic membrane possesses NMDA
receptors. To support this conclusion, model simulations are shown
below to confirm that the soma is adequately voltage-clamped.
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Figure 4.
Dendritic oscillations during soma voltage clamp.
A, The oscillatory currents for somatic depolarizations
to 30 and 10 mV of A are superimposed with model
simulations using a dendrite with three compartments in which the
gnmda conductance was increased along with
the addition of a slow potassium conductance
(gkslow) to represent a
calcium-activated potassium conductance induced by the influx of
calcium during NMDA activation. The dashed model curves
are simulations of both the frequency and time domain data using the
parameter values of Figure 3 with the following replacements and
additions: gnmda= 0.03 µS,
sm = 0.033 mV1,
gkslow = 0.015 µS,
vq = 5.25,
sq = 0.065 mV1,
tq = 0.25 sec, and
rq = 0 (Neuron #2).
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These oscillatory responses are partially synchronized by the
voltage-clamp step and add a nonlinear component to the response during
the white noise measurement. At these depolarized levels, parameter
estimation could not be performed for the frequency domain data,
because the nonlinear uncontrolled responses prevented the achievement
of a true linear steady state. As such, this invalidates the
assumptions implicit in linear analysis; however, a coherence subtraction procedure was applied to these data to obtain a partial estimation of the frequency domain results. This consists of
subtracting current responses of two voltage-clamp steps to the same
potential, but with inverted white noise command stimuli. Thus, the
identical transient mean currents are cancelled, but the small-signal
white noise responses are added, i.e., the coherent responses are
eliminated and the small-signal data are averaged for two voltage-clamp
steps. Despite the removal of the mean oscillating response, it is
likely that potential dependent fluctuations are present in the
small-signal response and would not be accounted for in a linear
analysis. Nevertheless, this approach provides empirical estimates of
the magnitude and phase functions illustrated in Figure
4B. Although there are clear distortions, the results
are similar to that seen for smaller depolarizations, for example, a
clear negative phase at 30 mV exceeding /2 radians. The
analytical frequency domain model responses (Fig. 4B,
dashed lines) at somatic voltage-clamp steps to 30 and
10 mV show rather good agreement with the data. In addition, the
computational model gives resonant frequencies (Moore and Buchanan,
1993) related to the period observed in the real time nonlinear current response.
Because the somatic voltage currents were oscillatory, parameter
estimation was not easily performed in the time domain. Therefore, model simulations of the depolarized voltage-clamp responses of Figure
4A (dashed lines) were performed using
parameters estimated from the lower step depolarizations of Figure 3.
In addition, a slow potassium conductance,
gkslow, was added because of the increased levels of depolarization. Oscillatory responses in the dendrites during a somatic voltage clamp were found for
gnmda = 0.03 µS and
gkslow = 0.015 µS.
Membrane potential oscillations in current clamp
The oscillatory behavior is markedly different when the
experimental condition is changed from measuring somatic current at a
fixed soma potential to the current-clamp condition measuring somatic
potential at a constant current. This is not too surprising because the
voltage clamp clearly restricts the normal intrinsic oscillation by
holding the soma potential constant. The somatic current oscillations
observed during a voltage clamp are relatively sinusoidal (Fig.
4A); however, the soma membrane potential
oscillations observed under current clamp illustrated in Figure
5A are more complex. The shape
of the soma potential oscillations was markedly dependent on the
potential level similar to that observed in the lamprey spinal cord
(Wallén et al., 1992). During a hyperpolarization current of
0.02 nA, the rhythmic activity was irregular, having an oscillatory
plateau phase whose amplitude was ~20 mV and a duration of 0.2-0.3
sec. The plateau oscillations were of low amplitude and relatively high
frequency. When the holding current was removed, the membrane potential
remained at its plateau level of approximately 5 mV and showed only
the high-frequency oscillations. All oscillations were abolished by
hyperpolarizations greater than 50 mV in both current and voltage
clamp.
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Figure 5.
Somatic voltage oscillations during a constant
current. A, Voltage oscillations in motoneurons at two
different current levels. The current levels were 0 and 0.02 nA.
B, Model simulations with symmetrical polarizing
currents I = 0.01 and +0.01 nA show a change from
large plateau oscillations to lower amplitude high-frequency behavior,
as observed experimentally in A. To model the more
complex oscillatory behavior seen in current-clamp measurements,
gnmda was increased to 0.06 µS. All other
parameter values were identical to those of Figure 4. All simulations
of Figures 5 and 6 had 30 dendritic compartments.
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DISCUSSION |
The observation of oscillatory currents during NMDA activation of
whole-cell voltage-clamped neurons shows that the dendritic membrane is
well supplied with NMDA receptors. The analysis of time and frequency
domain data allowed the development of an electrotonic model of chick
motoneurons that can account for the current oscillations during a
voltage clamp. The voltage clamp controlled the intrinsic oscillations
near the resting potential. Under these conditions an initial estimate
of the NMDA-induced conductances was performed. At moderate
voltage-clamp depolarizations, oscillatory behavior was observed in the
current response that could be simulated by the estimated neuronal
model if the maximal NMDA-induced conductance was increased
approximately threefold.
A computational investigation of the model behavior can explain the
striking difference between the nature of the oscillatory behavior in
current versus voltage clamp. For the model to simulate the
constant-current plateau oscillations seen experimentally, the NMDA
conductance was increased another twofold, with all other parameters
the same as in the voltage-clamp simulations of Figure 4. Under these
conditions, a hyperpolarization of the model with 0.01 nA leads to a
high-amplitude oscillation with a distinct plateau phase (Fig.
5B). A depolarizing current of 0.01 nA abolishes the plateau
of the model and sustains a high-frequency oscillation, analogous to
that observed experimentally (Fig. 5A). The increase of
gnmda used in the successive modeling
steps was attributable to an inability to predict the maximal
conductance from data taken near the holding potential, where the
responses were stable and parameter estimation was more effective. This
illustrates the advantage of simulation studies to obtain parameter
values that are needed for a particular behavior and could then be used
as constraints for other experimental conditions.
In contrast to the current-clamp plateau oscillations (Fig. 5), the
simulated voltage-clamp currents (Fig.
6A,
Im) are relatively sinusoidal at a
soma depolarization to 30 mV, similar to that observed experimentally
(Fig. 4A). Thus, this model with two uniformly distributed potassium conductances, fast and slow, in combination with
a significant NMDA conductance, can reasonably well simulate the
experimental findings of complex plateau oscillations whose low-frequency component is reduced by a voltage clamp of the soma. Furthermore, these computations indicate that the oscillatory voltage-clamped somatic currents are a consequence of dendritic membrane potential (Fig. 6A,
Vm) oscillations (Seutin et al., 1994;
Li et al., 1996). This conclusion is dependent on an accurate determination of the electrotonic structure and electrode effects that
allow a rigorous assessment of the voltage-clamp control of the soma.
Previous simulations performed with only one potassium conductance
under constant current conditions (Moore and Buchanan, 1993; Murphey et
al., 1995) showed plateau oscillations without superimposed
high-frequency oscillations. Thus, the present computations suggest
that complex oscillatory patterns require at least three voltage-dependent conductances.
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Figure 6.
Voltage-clamp versus constant-current
oscillations. A, The computed somatic voltage-clamp
currents (Im) at 30 mV from a holding level of 50
mV, although increased, remain reasonably sinusoidal, similar to those
of Figure 4A at a lower
gnmda. The parameter values are identical to
those of Figure 5. B, The subsequent current clamp
(Im = 0 nA), computed with the initial conditions
at 50 mV of the voltage clamp, shows complex soma and dendritic
potential oscillations (Vm). In the bottom
panel, current-clamp simulations with a hyperpolarizing current
of 0.05 nA show an enhanced amplitude of the oscillatory plateau
phase.
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During a constant current, the potentials of the soma and peripheral
dendritic compartments show little difference between models with 3 and
30 dendritic segments, indicating that spatial resolution is adequate.
Under the conditions of strong intrinsic oscillations, whatever their
shape, each compartment is well synchronized such that there is
essentially no difference between the different locations on the cable.
This would not necessarily occur if there were a nonuniform
distribution of channel receptors. The 1-2 mV oscillation seen in the
soma (Fig. 6A) is a consequence of the imperfect soma
clamp attributable to an electrode resistance. Figure
6B shows constant current simulation where the soma
potential is unclamped such that at zero current all compartments
including the soma oscillate in synchrony. During a steady-state
hyperpolarization with 0.05 nA of current, the plateau amplitude is
enhanced and the potential profiles from different compartments are not
identical; however, the oscillations continue to be well synchronized
(Fig. 6B).
In conclusion, the finding that complex plateau membrane potential
oscillations observed in current clamp could not be completely abolished by a soma voltage clamp is a consequence of the impossibility of space clamping the dendritic tree. The experimental and
computational studies demonstrate the presence of intrinsic dendritic
membrane potential oscillations that are significantly different under current- versus voltage-clamp conditions. This behavior provides a
means to quantitatively distinguish between soma and dendritic NMDA
receptors and can provide an additional constraint in the selection of
neuronal models and their parameter values. Thus, parameter estimation,
using a combination of current- and voltage-clamp measurements in both
the time and frequency domains, allows a nearly unique estimation of
dendritic and soma conductances.
Finally, the experimental finding of oscillatory responses during a
voltage clamp of the soma definitively demonstrates that NMDA receptors
must be present on the dendrites and provides support for an earlier
hypothesis that the presence of excitatory amino acid receptors on
dendrites is important for rhythmic activity (Chub and O'Donovan,
1998). One important way in which they may act is to reduce the
electrotonic length of the dendrites during increased activity by
dynamically altering the relative impedance of the cable structures
based on the activation of negative conductances, which in turn
modifies synaptic efficacy. In addition, when the activation of NMDA
receptors is sufficiently large, intrinsic membrane oscillations are
established that are likely to stabilize rhythmic patterns.
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FOOTNOTES |
Received March 23, 1999; revised July 8, 1999; accepted July 20, 1999.
This work was supported in part by the Centre National de la Recherche
Scientifique (Paris, France).
Correspondence should be addressed to Dr. Lee E. Moore, Laboratoire de
Neurobiologie des Reseaux Sensorimoteurs, UPRESA-7060, Centre
National de la Recherche Scientifique, 45 rue des Saints-Peres, 75270 Paris Cedex 06, France.
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TOP
ABSTRACT
INTRODUCTION
