1Laboratory of Neurophysiology, Biomedical Research Institute, Kaunas Medical Academy, 3000 Kaunas, Lithuania; and 2Department of Medical Physiology, The Panum Institute, Copenhagen University, Copenhagen DK-2200, Denmark
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ABSTRACT |
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Svirskis, Gytis,
Aron Gutman, and
Jørn Hounsgaard.
Electrotonic Structure of Motoneurons in the Spinal Cord of the
Turtle: Inferences for the Mechanisms of Bistability.
J. Neurophysiol. 85: 391-398, 2001.
Understanding how
voltage-regulated channels and synaptic membrane conductances
contribute to response properties of neurons requires reliable
knowledge of the electrotonic structure of dendritic trees. A novel
method based on weak DC field stimulation and the classical method
based on current injection were used to obtain two independent
estimates of the electrotonic structure of motoneurons in an in vitro
preparation of the turtle spinal cord. DC field stimulation was also
used to ensure that the passive membrane properties near the resting
membrane potential were homogeneous. In two cells, the difference in
electrotonic lengths estimated with the two methods in the same cell
was 6 and 9%. The majority of dendritic branches terminated at a
distance of 1 electrotonic unit from the recording site. The longest
branches reached 2. In the third cell, the difference was
36%, demonstrating the need to use both methods, field stimulation and
current injection, for reliable measurements of the electrotonical
structure. Models of the reconstructed cells endowed with
voltage-dependent conductances were used to explore generation
mechanisms for the experimentally observed hysteresis in input
current-voltage relation of bistable motoneurons. The results of
modeling suggest that only some dendrites need to possess L-type
calcium current to explain the hysteresis observed experimentally and
that dendritic branches with different electrotonical lengths can be
bistable. Independent bistable behavior in individual dendritic
branches can make motoneurons complex processing units.
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INTRODUCTION |
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This paper deals with the
well-known problem of defining the electrotonic image of a nerve cell.
Electrotonic measurements provide necessary parameters for complex
models of neurons and serve as a basis for understanding the functional
role of linear and voltage-dependent membrane properties. For this
reason, it is necessary to provide reliable estimates of the
electrotonical parameters. Electrotonic measurements have been
performed in morphologically reconstructed neurons of different types
(Clements and Redman 1989; Fleshman et al.
1988
; Major et al. 1994
; Rapp et al.
1994
; Thurbon et al. 1994
, 1998
). Unfortunately
with the methods used, it was not possible to check for homogeneity of
the passive membrane properties. Thus it was necessary either to assume
homogeneity or include a fourth unknown parameter to account for the
difference between somatic and dendritic membrane properties. However,
recordings with two electrodes in a single cell have suggested that
dendritic membrane properties can also be intrinsically heterogeneous
even near the resting potential (Stuart and Spruston
1998
).
Recently methods based on stimulation with weak electrical DC fields
were proposed for detecting membrane heterogeneity and injury shunts
and for estimating the electrotonic structure of neurons
(Svirskis et al. 1997a,b
). If the passive membrane
resistance of a neuron is homogeneous, then the transient in response
to an applied DC field has no characteristic shape peculiarities and
develops faster than the response to current injected through the
recording electrode (Svirskis et al. 1997b
). This
transient depends only on membrane time constant,
, and
electrotonical length constant,
(Svirskis et al.
1997a
), allowing for a more reliable estimation of the
electrotonical structure of neurons. Since the check for homogeneity of
the passive properties can be done during experiments, only cells
fulfilling these criteria were chosen for electrotonical measurements.
The combination of DC field method with the classical current injection
method can increase the reliability of electrotonical measurements significantly.
In the present study, the electrotonic parameters were found for three motoneurons in slices of the turtle spinal cord. Two independent estimates of the electrotonic structure were obtained for each cell from responses to weak DC field stimulation and current pulse injection. An acceptable difference between parameters estimated with different methods in two cells suggests that errors introduced by staining procedures and reconstruction were inessential.
Because numerous potential-dependent currents are present in the
neuronal dendrites, electrotonical structure alone can provide only a
basis for further exploration of how synaptic input is processed.
Models of reconstructed motoneurons were used to explore the generation
of nonlinear properties due to potential-dependent inward current
observed previously (Hounsgaard and Mintz 1988). It is
known that part of this inward current is generated in the dendrites of
motoneurons in turtles and cats (Hounsgaard and Kiehn 1993
; Lee and Heckman 1996
, 1998a
). In voltage
clamp, turtle and cat motoneurons show hysteresis in their input
current-voltage relation (I-V) (Lee and Heckman
1998b
; Schwindt and Crill 1980
; Svirskis
and Hounsgaard 1998
). The presence of the hysteresis during
very long voltage ramp stimulation (Lee and Heckman
1998b
) suggests that dendrites are bistable (Gutman
1984
; Jack et al. 1983
) as theoretical studies
had anticipated (Butrimas and Gutman 1978
, 1981
) from
early experimental findings (Schwindt and Crill 1977
,
1980
). Qualitatively, the hysteresis can be explained by the
relatively weak electrical coupling between soma and distal dendrites
that allows distal inward current to be activated even when the
proximal dendrite is relatively hyperpolarized. Here we explored how
strong and where the inward current should be to generate hysteresis as
observed experimentally.
Our models suggest that dendrites with different electrotonic length can be bistable and that only a fraction of the dendrites have to be nonlinear to generate the hysteresis observed experimentally. The results also show that a simple equation derived previously for nonlinear cables can be used to predict the behavior of more realistic models.
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METHODS |
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Methods for the experimental procedures
Transverse sections of the lumbar spinal cord were obtained from
turtles (Pseudemys scripta elegans) deeply anesthetized with pentobarbitone (100 mg/kg) (Hounsgaard et al. 1988). The
medium contained (in mM) 120 NaCl, 5 KCl, 15 NaHCO3, 20 glucose, 2 MgCl2, and 3 CaCl2.
6-Cyano-7-nitroquinoxaline-2,3-dione (CNQX; 40 µM; Tocris Cookson,
Bristol, UK) was applied to block excitatory synaptic potentials. For
experiments a section of the cord, 0.5-mm thick, was placed in the
recording chamber between two silver-chloride electrodes (see Fig.
2A in Svirskis et al. 1997b
) used to
establish an extracellular DC field.
For recordings, patch electrodes were pulled from borosilicate glass
tubes with an outer diameter of 1.5 mm and an inner diameter of 0.86 mm. Electrodes were filled with 125 mM potassium gluconate, 9 mM HEPES,
and 1% biocytine; pH was adjusted to 7.4 with KOH. During whole cell
recording, voltage transients were generated by injecting a current
pulse of 0.3-1.0 nA for 1 ms through the recording electrode or by
applying an extracellular current pulse of 1 µA between field
electrodes for 150 ms (Svirskis et al. 1997b). The
response to the DC field stimulation depends on the direction of the
field. Since the transmembrane potential induced by the electric field
is largest in direction of the applied field, the shape of the evoked
transient reflects electrotonic structure of the dendrites oriented
mainly in direction of the field. The DC field was applied in the
lateral direction because turtle spinal cord motoneurons have their
physically longest dendrites oriented laterally (Ruigrok et al.
1985
). To reduce noise, 256 sweeps were averaged on a HIOKI
digital oscilloscope (Hioki E. E. Corp., Nagano, Japan) and fed to
a computer for later analysis.
After all measurements were accomplished, the electrode was withdrawn from the cell, and the extracellular potential, induced by the same field step stimulus, was recorded, averaged and subtracted from the intracellular potential to get the transmembrane potential. The extracellular potential gradient was measured in nine points at the depth of the recording performed. Extracellular potential values were obtained in three rows each of three points with a 0.5-mm distance between them. The field strength was homogeneous in the range of 10-15% and was 3-4 mV/mm 50-100 µm below the surface of the tissue. Weak electric field stimulation assured that membrane potential changes were less than a few millivolts and did not activate potential-dependent conductances. Because the field strength used for calculations is a proportionality coefficient (see following text), it was allowed to change within the limits of measured strengths. For this reason, only the transient part of the response to the field was used for electrotonical measurements. The dimensions of the slice were measured to assess and correct for shrinkage due to histological procedures. The position of the electrode in relation to the borders of the slice was obtained.
Electrotonic estimates were obtained from three motoneurons recorded
with patch electrodes. All responses to the field step had no
characteristic shape peculiarities and decayed faster than the response
to current injection (Fig. 2), indicating homogeneity of passive
membrane resistance (Svirskis et al. 1997a,b
). The responses to current pulses of opposite sign were anti-symmetric, showing that membrane properties were linear in the range of the response amplitudes.
Staining and reconstruction
Standard procedures were applied to stain biocytin-injected
neurons (Horikawa and Armstrong 1988). In brief, slices
were kept in fixative overnight, then washed in phosphate buffer and
treated with H2O2. After
removal of hydroxy peroxide, slices were incubated with ABC complex and
1% Triton overnight. After wash, DAB treatment was used to visualize
the stained neurons. Nickel salt was used to get almost black colored
staining. To dehydrate slices, the incubation solution was gradually
changed to pure ethanol. Then slices were cleared in xylene and mounted
in Permount.
For reconstruction, semi-automatic Neurolucida software and hardware
were used with Zeiss microscope. Water-immersed objective allowed to
define the diameter with a precision of 0.3 µm in stained motoneurons. Soma was reconstructed as a part of the dendritic tree.
The slice contour was outlined to define the shrinkage, which was
1.3-1.6 times, and to set the field direction in relation to
reconstructed neuron. The position of the recording electrode was
estimated after the coordinates were corrected for the shrinkage. In
two motoneurons, this position hit the soma region, whereas in one, it
coincided with the proximal dendrite (Fig. 3). Diameters of the
dendritic branches were not corrected for the shrinkage, which could
cause either the reduction or enlargement of the diameters depending on
the intracellular contents of the dendrites (see Major et al.
1994) (see also DISCUSSION).
Calculations for electrotonic measurements
Homogeneous passive membrane properties were assured
experimentally by using DC field stimulation as described in the
preceding text. For the description of the linear response of the
neurons with homogeneous membrane properties, we therefore used a set of electrotonic parameters: membrane time constant, = RmCm, electrotonic length constant,
=
(Rma/Ri
)1/2,
and characteristic resistance (resistance of semi-infinite cable), R
=
(RmRi/a
)1/2,
where Rm is the specific membrane
resistance, Cm is the specific capacitance, and Ri is the specific
intracellular resistance. Here, a is the area of a
cross-section of the cable and
is the perimeter. The
apparent diameter, D, is a morphologically measured quantity,
~ D and a ~ D2 with constant proportionality coefficients
throughout the dendritic tree on a macroscopic scale
(Alaburda and Gutman 1996
). Thus
and
R
are constant over the same scale.
Because
~ D1/2 and
R
~ D
3/2, it is
necessary only to know values of parameters for the single particular
diameter to define the response to the current and DC field stimulation
of the complete dendritic tree. For this purpose, we used constants
1 and
R1
, which are defined as a
electrotonical length and characteristic resistance for a homogeneous
dendritic segment with an apparent diameter, D = 1 µm
(Svirskis et al. 1997a
). For any dendritic segment with
a diameter of D µm,
and
R
were found by multiplying
1 and
R1
by dimensionless number equal to
D1/2 and
D
3/2, respectively.
The electrotonic parameters, ,
1, and
R1
, were
estimated by comparing experimental and simulated transients after current pulse injection and stimulation with DC field. Calculations in
models of the reconstructed motoneurons were accomplished using Fourier
transformation. The results were checked by solving the system of
ordinary differential equations directly for the compartmental model of
reconstructed motoneurons using the method of Cranck-Nicholson (Press et al. 1992
). The same system of equations was
also solved as a matrix equation for the eigenvalues,
n, and eigencoefficients, Cn (Holmes et al. 1992
;
Perkel et al. 1981
), with Matlab (Fig. 1).
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Here we only outline the method using Fourier transformation
(Svirskis et al. 1997a). This gives us insight in the
dependence of transients on the parameters. For the current injection
the harmonic component of the response at the recording site
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(1) |
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(2) |
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(3) |
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(4) |
From Eqs. 1 and 2 and after inverse Fourier
transformation, it is evident that in the case of current injection the
response of transmembrane potential in any point of the neuron depends on 1,
, and linearly
on R1
; in the case of field
stimulation, the response depends only on
1 and
. Another
advantage of using the electrotonic set of parameters is that the
solution for the complete dendritic tree does not require specifying
the shape of the dendritic cross-section. Solution dependence on the
shape is hidden in the electrotonic parameters.
Calculations for nonlinear models
To analyze the mechanisms responsible for the experimentally
observed nonlinear properties of motoneurons, we added L-type calcium
channel conductance and potassium delayed rectifier conductance in the
membrane of models of the reconstructed neurons. Parameters and
equations for these conductances were similar to whose used previously
in models of bistable motoneurons (Booth et al. 1997). Calcium current Ica = Gca[m(V
Vca)
m
(
65
Vca)]; calcium channel activation
variable, m, was governed by the equation
dm/dt = (m
m)/
ca, where
m
= 1/{1 + exp[(V
L)/kL]}, Vca = 40 mV,
ca = 20 ms,
L =
35 mV, kL =
5 mV. The conductance parameter,
Gca, was used as a variable parameter. Since the activation kinetics of the K+ delayed
rectifier channel is very fast compared with the stimulus protocol used
(see following text), the activation was modeled as instantaneous. Thus
potassium current IK = GKn4
(V
VK), where
n = 1/{1 + exp[(V
K)/kK]},
VK =
90 mV,
K =
30 mV,
kK =
12 mV, and
GK
=3/Rm. Input I-Vs for the
reconstructed neurons were computed by simulating voltage clamp as
triangular ramps of the soma voltage. The speed of the ramp was made
0.005 mV/ms to get almost stationary input I-V. Calculations
were performed by directly solving the system of ordinary differential
equations for the compartmental models of reconstructed motoneurons.
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RESULTS |
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First of all, we illustrate that the responses to intracellular
current injection and field stimulation are independent, i.e., are not
linearly transformable. For this purpose, we calculated the eigenvalues
and eigencoefficients of the system of equations describing
reconstructed motoneurons in both cases of stimulation (see
METHODS). As seen in Fig. 1 the eigenvalues are the same for both types of responses except for the absence of the component with the membrane time constant in response to DC field stimulation (Svirskis et al. 1997b). However, the coefficients are
very different implying that transients after current injection and
field stimulation are independent. This shows that the two methods
provide independent measures of the electrotonic structure of neurons
and that they therefore also provide a mutual validation.
The transients in neurons with homogeneous passive properties depend on
the electrotonic parameters in a way that allows the parameters to be
estimated one by one. First, by using the response to injected current
pulses, the membrane time constant, , was estimated from
the exponentially decaying part of the transient (Fig.
2A). Values of
varied from 10 to 29 ms for the three motoneurons (see Table 1). Since
the transient after a current pulse is proportional to
R1
(see METHODS), the
value of this parameter was made equal to 1 M
when calculating the
response of the model of reconstructed motoneurons. Only
1 value was changed to achieve the
best correspondence between experimental and calculated transients. The
fitting was done by dividing the experimentally obtained transient by
the calculated transient; this gave the
R1
value for each point of the
transient (Fig. 2B, bottom). Because the
calculated transient should be proportional to the
R1
sought, large variance of
R1
values at the points of the
transient indicates a poor fit to the experimental data. Usually large
variance was due not to the scatter of the values but to the global
change with time in the transient. Thus the value of
1 was assumed to be the best
estimate if it produced minimal variance of
R1
without a global change in the
transient. The
1 values obtained in
this way were different for all three neurons and ranged from 550 to
850 µm (Table 1). The procedure described above simultaneously provided an estimate for the characteristic resistance,
R1
(see Table 1), which was
calculated as an average of R1
values for all points in the transient. To validate the procedure used,
we also calculated the average of squared difference between the
experimental and the theoretical transients. In this case the
calculated transients were multiplied by the estimated
R1
value. The estimates of
1 provided the minimal average of
the squared difference (Fig. 2B, inset). The
electrical parameters were then calculated assuming that the dendritic
cross-section is circular. The specific membrane capacitance,
Cm, ranged from 0.6 to 1.1 µF/cm2, specific membrane resistance,
Rm, had values from 14 to 26 k
· cm2, and cytoplasmic resistance,
Ri, ranged from 100 to 160
· cm (Table 1).
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To check the validity of measurements, we also estimated
1 by simulating the response of the
models of reconstructed motoneurons to the field stimulation. This
response depends only on
and
1 (see METHODS). Since
the membrane time constant,
, could be reliably estimated
from the response to current injection (Fig. 2A), we used it
in these simulations as a known parameter. We varied the value of
1 and found the estimate which
produced the minimum of the average of the squared difference between
experimental and calculated transients (Fig. 2C,
inset). In two cells the difference between estimates of
1 obtained by the two methods in
the same cell was 6 and 9%, showing that possible errors due to
histological procedures were inessential. In the third cell, the
difference was 36%, demonstrating the necessity to use current and
field stimulations to ensure the reliability of the measurements.
Knowing the electrotonical length constant,
1, defined for the diameter of 1 µm allows calculating electrotonical length constant for any
dendritic segment with a diameter of D µm as
=
1 · D1/2 (see METHODS).
Surprisingly, despite very different membrane time constants between
neurons the electrotonical structure of the dendritic trees did not
vary much. The majority of the branches terminated at the
electrotonical distance around 1
from the recording site
(Fig. 3) for all three neurons. However,
some branches reached as far as 2
(Fig. 3B).
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Since dendrites also possess voltage-dependent conductances, the
electrotonic structure provides just a starting point for the
exploration of synaptic integration. In motoneurons, dendritic inward
current mediated by L-type calcium channels is responsible for plateaus
and hysteresis in the input I-V (Hounsgaard and Kiehn 1993; Lee and Heckman 1998b
; Schwindt and
Crill 1980
; Svirskis and Hounsgaard 1998
). To
investigate the generation of hysteresis in input I-V,
reconstructed neurons were used to create nonlinear models with L-type
calcium and potassium delayed rectifier conductances (see
METHODS).
Input I-Vs were calculated using the complete model of
reconstructed motoneuron m2206 (Fig.
4A). To mimic experiments, a
slow triangular voltage-clamp ramp at the soma was used as a stimulus for these calculations. Hysteresis appeared in the I-V plot
when the conductance of calcium current,
Gca, was increased to
1.55/Rm. With the further increase of
calcium conductance, Gca = 2.95/Rm, the hysteresis became broader
and deeper (not shown). However, in experiments, the I-V
hysteresis observed is not very deep. Possibly, only some dendrites
have potential-dependent inward currents. As a check, we endowed only
the third dendrite of the motoneuron (Fig. 3) with the
voltage-dependent conductances while the other two dendrites were left
only with the passive leakage current. In this case, the simulated
I-V plot (Fig. 4A) became more similar to the
experimentally observed plot (Svirskis and Hounsgaard 1997,
1998
).
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The mechanisms of hysteresis could be understood from the distribution
of membrane potential in the dendritic tree (Fig. 4B) when
hysteresis is narrow, i.e., Gca = 1.55/Rm. When the soma voltage was
decreasing during the falling phase of the triangular ramp, in some
interval of the soma voltage the longest branches were more depolarized
than the cell body due to activation of the persistent inward current
and weak coupling to the soma (Fig. 4B). The depolarization
of the terminal dendrites was absent in the same interval of clamped
soma voltage when voltage ramp was rising because the inward current
was not yet activated. Thus this bistability of branches is reflected
as hysteresis in input I-V. The depolarization of the same
two branches also caused the last jumps in input I-V for the
larger value of Gca = 2.95/Rm (Fig. 4A), which
made hysteresis broader and induced dendritic bistability (Fig.
4B) in shorter branches. Thus in case of broad hysteresis
observed under experimental conditions the dendrites with different
electrotonic length could be bistable. As the number of bistable
branches increases, the variety of responses of motoneuron to synaptic
input also increases (Gutman 1991).
Interestingly, the value of calcium conductance,
Gca = 1.55/Rm, which induced hysteresis in
the model of reconstructed motoneuron could be estimated by using a
simple equation. For cables, a mathematical relation defines whether
hysteresis is present in the stationary input I-V: the
stationary negative slope-conductance,
SN, in membrane I-V (Fig.
4C) should be SN > 2/4RmLcable2,
where Lcable is cable electrotonic
length (Baginskas et al. 1999
; Gutman
1991
; Jack et al. 1983
). For the cable with the
electrotonical length 1
,
SN = 2.47. To achieve such a slope in
membrane I-V, the calcium conductance,
Gca, should be equal to
1.5/Rm, according to the equations in
METHODS. In agreement with this estimate only the dendritic
branches longer than 1
were bistable in the case of
Gca = 1.55/Rm (Fig. 4B). This
example shows that mathematical relations derived for the cable can be
used to get a qualitative estimate of the behavior of nonlinear,
branched dendrites. It also proves that hysteresis in the input
I-V is a very general phenomenon, which does not depend on
particular membrane channels generating the inward current in the dendrites.
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DISCUSSION |
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Electrotonic parameters were found for three spinal motoneurons in the turtle. The electrotonic structure was obtained by two independent methods after checking for homogeneity of the passive membrane properties. Comparison of electrotonic length constants obtained by two independent methods in the same cell suggests that anatomical procedures and reconstruction did not induce substantial errors in two out of three cells.
Electrotonic measurements are prone to errors. Tissue dimensions are
changed by the histochemical procedures employed for fixation and
staining. In our experiments, shrinkage in tissue dimensions was
1.3-1.6 times. While the lengths of dendrites were corrected, we have
chosen not to change the diameter values measured because the shrinkage
could cause either reduction or increase of the dendritic diameters
(see Major et al. 1994). The change in the diameters is
important for estimates of the electrical parameters
Rm,
Cm,
Ri. However the electrotonical
structure is less sensitive to global changes in diameters. For
example, increase of all diameters by r times causes the
reduction of the electrotonical length by the factor
r0.5 if
1 is kept constant. But this
results in faster transients, which require a decrease of
1 to fit the experimental
transient. We estimated
1 values
for all three neurons after diameters were corrected for shrinkage. The
reduction of
1 was 500/600 = 1/(1.44)0.5, 675/800 = 1/(1.4)0.5, and 500/550 = 1/(1.2)0.5 for the shrinkage of 1.6, 1.3, and 1.3 respectively in three cells. Thus the electrotonical structure would
not change significantly if diameters were corrected for shrinkage.
The other source of possible error is the morphological reconstruction.
We evaluated the electrotonical structure after inducing a significant
reduction in diameter of a compartment near the recording site. In this
case, the estimates of obtained by injection of current
pulses and by DC field stimulation differed several times. Since the
experimental estimates we obtained differ by only 6 and 9% in two
cells, we can be sure that the morphological reconstruction was without
major errors.
The only modest correspondence between the estimates of the electrotonic structure in cell m0207 could be explained by the heterogeneity in field strength, which equaled 15% and was the largest among the three cells, distortion of the response to the current impulse by pipette capacitance, and/or heterogeneous shrinkage. At this stage, we cannot distinguish between these explanations. Note, however, that for this cell the value of Cm is smaller and the value of Ri considerably larger than for the other two cells.
The calculated values of eigencoefficiences (Fig. 1) show a striking
difference between responses to the field stimulation and current
injection. The eigencoefficiences of the fast components are very small
in the response to DC field stimulation. Hence the decay rate of the
whole transient is a satisfactory reflection of the electrotonic
lengths of dendritic branches, which are oriented in the direction of
the field. This result validates the DC field stimulation as a method
for the estimation of the equivalent electrotonic length of the
dendritic tree (Svirskis et al. 1997a). Cell
m2206 is particularly good for illustration because the evolution
of the response evoked by the DC field depends mainly on two exponents (Fig. 1). The slowest time constant of the transient,
F = 5.6 ms, and the membrane time
constant,
= 29 ms, allow us to calculate an estimate of
the tip-to-tip electrotonic length L in the field direction.
According to the classical Rall equation: L =
/ (
/
F
1)1/2
1.5 and is much shorter than
L for the two other cells. The estimation fits quite well to
the electrotonical structure if we notice that cell m2206
has rather short laterally oriented dendrites and long
(Fig. 3A).
As shown here and elsewhere (Svirskis et al. 1997a,b
),
DC field stimulation offers several advantages for electrotonic
measurements. The essence of the method is that during weak DC field
stimulation the total current passing the membrane of a neuron equals
to zero (Svirskis et al. 1997b
)
there are no net
current sinks or sources. Although we used the extracellular current to
create a potential gradient for polarizing neurons, other methods could
be used as well. Recordings from the same neuron with two independent
electrodes have proved feasible (Stuart and Spruston
1998
; Stuart et al. 1993
). Electrotonic
measurements can be obtained with this configuration keeping the total
current flowing through the neuronal membrane zero by injecting current
of equal amplitude but opposite polarity through the two electrodes.
Also the check for homogeneity of the specific membrane resistance is
the same as with the field stimulation.
Neglecting the uncertainties in correcting diameters and assuming
circular cross-sections of dendritic branches, we have obtained electrical parameters for dendritic cables. The specific membrane resistance, Rm, was in the range from
14 to 26 k · cm2; specific membrane
capacitance was from 0.6 to 1.1 µF/cm2; and the
specific cytoplasmic resistance was from 100 to 160
· cm (see
Table 1). The values of these parameters are in the range observed in
other studies. Specific membrane capacitance was estimated to be from
less than 1 µF/cm2 (Major et al.
1994
) to more than 2 µF/cm2
(Rapp et al. 1994
; Thurbon et al. 1998
).
Estimates of other parameters varied even more. Specific membrane
resistance was estimated to be from tens of
k
· cm2 (Larkman et al.
1992
; Meyer et al. 1997
; Thurbon et al.
1994
, 1998
) to hundreds of k
· cm2
(Major et al. 1994
; Rapp et al. 1994
).
Specific cytoplasmic resistance had values from less than 100
· cm (Thurbon et al. 1994
, 1998
) to several
hundred
· cm (Larkman et al. 1992
; Major
et al. 1994
; Meyer et al. 1997
; Rapp et
al. 1994
). The huge variation of the estimates may be
attributed to the reasons outlined in the preceding text and/or to
biological differences between cell types. The electrotonic set of
parameters,
,
1, and
R1
, has several advantages over the
electrical set, Cm,
Rm, and
Ri. First, each parameter has its own
functional meaning for the response of dendritic cables. Second, the
estimates are less vulnerable to unverified assumptions, i.e., circular
cross-section of the dendrites. Third, the parameters can be estimated
one by one, which improves the reliability of estimates.
In motoneurons, voltage-dependent currents together with passive
conductance shape the response to synaptic input. Persistent dendritic
inward current mediated by L-type calcium channels is responsible for
generation of plateau potentials in current-clamp mode
(Hounsgaard and Kiehn 1993; Lee and Heckman 1996
,
1998a
) and for hysteresis in voltage-clamp mode (Lee and
Heckman 1998b
; Schwindt and Crill 1980
;
Svirskis and Hounsgaard 1997
, 1998
). In this study, we
used a model with L-type calcium current to show that dendrites with
different electrotonic length could be bistable in case of the
experimentally observed broad hysteresis in input I-V.
Although only three motoneurons were reconstructed in the present
study, previous morphological investigations (Ruigrok et al.
1984
, 1985
) showed that short terminal dendritic branches in
abundance is a general characteristic for turtle motoneurons.
In our model, membrane conductances were homogeneously distributed in the dendrites. This may not be true in real dendrites. Because the dendrites are not very long electrotonically, membrane potential changes in space are smooth (Fig. 4B) in our case of slow currents and slow somatic voltage-clamp ramps. Thus effects of any possible heterogeneities of potential dependent conductances would be smoothed over entire dendritic branches. In this case, negative slope in membrane I-V, which defines when dendritic bistability could occur, would represent an average of the membrane conductances over the dendritic branches.
As demonstrated here, even in branching dendrites, hysteresis in input
I-V depends only on electrotonic length of dendrites and the
maximal negative slope in membrane I-V (Fig. 4C).
Consequently the inferences made do not depend on the detailed nature
of persistent inward current and should be readily applicable to other
types of motoneurons with observed bistability (Hsiao et al.
1998; Lee and Heckman 1998b
; Rekling and
Feldman 1997
). Because of the independence of hysteresis on
particular membrane mechanisms, we did not include other channels, like
N-type calcium channel, calcium-sensitive potassium channel etc., in
the model, although these channels are known to be present in turtle
motoneurons. These currents could influence the temporal phenomenology,
but they would not change our qualitative findings regarding hysteresis
during very slow voltage ramps.
Very slow kinetic properties of inward currents could, however, have
profound influence. A slow depolarization induced facilitation of
inward current, possibly by changing voltage sensitivity of this
current, was observed in motoneurons (Bennett et al.
1998; Svirskis and Hounsgaard 1997
). An inward
current with shifting voltage sensitivity due to slow facilitation has
steep potential dependence and, accordingly, could increase the
negative slope in membrane I-V and cause hysteresis in input
I-V of electrotonically short cables (Baginskas et
al. 1999
). The slowness of inward current kinetics also
explains why jumps in hysteresis were not observed experimentally. Thus
it is very probable that dendritic branches with different electrotonic
lengths can be bistable in turtle motoneurons.
In turtle motoneurons the persistent inward current is also modulated
by neurotransmitters via metabotropic pathways (Svirskis and
Hounsgaard 1998). Facilitation of persistent inward current by
bath-applied agonist leads to bistability in current clamp and broad
hysteresis in voltage clamp (Svirskis and Hounsgaard 1998
). In contrast and in agreement with modeling results,
focal synaptic facilitation of dendritic inward current merely
increases excitability in current clamp mode and only induces narrow
hysteresis in voltage clamp (Delgado-Lezama et al. 1997
,
1999
).
In conclusion, potential-dependent inward current causing bistability
independently in numerous dendritic branches increases the richness of
synaptic processing and makes motoneurons complex processing units
(Gutman 1991).
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ACKNOWLEDGMENTS |
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We thank H. Markram for providing facilities for morphological reconstruction of neurons.
We acknowledge the support of the Lithuanian Department of Science and Education. The experimental work was supported by the Lundbeck Foundation, the Novo Foundation, and the Danish Medical Research Council.
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FOOTNOTES |
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Present address and address for reprint requests: G. Svirskis, Center for Neural Science, 4 Washington Place, Rm. 809, New York University, New York, NY 10003 (E-mail: gytis{at}cns.nyu.edu).
Received 7 March 2000; accepted in final form 12 September 2000.
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REFERENCES |
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