1Cajal Neuroscience Center, Division of Life Sciences, University of Texas at San Antonio, San Antonio, Texas 78249; and 2Department of Anatomy and Neurobiology, University of Tennessee, Memphis, Tennessee 39163
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ABSTRACT |
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Wilson, C. J. and
J. C. Callaway.
Coupled Oscillator Model of the Dopaminergic Neuron of the
Substantia Nigra.
J. Neurophysiol. 83: 3084-3100, 2000.
Calcium imaging using fura-2 and whole cell recording
revealed the effective location of the oscillator mechanism on
dopaminergic neurons of the substantia nigra, pars compacta, in slices
from rats aged 15-20 days. As previously reported, dopaminergic
neurons fired in a slow rhythmic single spiking pattern. The underlying membrane potential oscillation survived blockade of sodium currents with TTX and was enhanced by blockade of voltage-sensitive potassium currents with TEA. Calcium levels increased during the subthreshold depolarizing phase of the membrane potential oscillation and peaked at
the onset of the hyperpolarizing phase as expected if the pacemaker potential were due to a low-threshold calcium current and the hyperpolarizing phase to calcium-dependent potassium current. Calcium
oscillations were synchronous in the dendrites and soma and were
greater in the dendrites than in the soma. Average calcium levels in
the dendrites overshot steady-state levels and decayed over the course
of seconds after the oscillation was resumed after having been halted
by hyperpolarizing currents. Average calcium levels in the soma
increased slowly, taking many cycles to achieve steady state. Voltage
clamp with calcium imaging revealed the voltage dependence of the
somatic calcium current without the artifacts of incomplete spatial
voltage control. This showed that the calcium current had little or no
inactivation and was half-maximal at 40 to
30 mV. The time constant
of calcium removal was measured by the return of calcium to resting
levels and depended on diameter. The calcium sensitivity of the
calcium-dependent potassium current was estimated by plotting the slow
tail current against calcium concentration during the decay of calcium
to resting levels at
60 mV. A single compartment model of the
dopaminergic neuron consisting of a noninactivating low-threshold
calcium current, a calcium-dependent potassium current, and a small
leak current reproduced most features of the membrane potential
oscillations. The same currents much more accurately reproduced the
calcium transients when distributed uniformly along a tapering cable in a multicompartment model. This model represented the dopaminergic neuron as a set of electrically coupled oscillators with different natural frequencies. Each frequency was determined by the surface area
to volume ratio of the compartment. This model could account for
additional features of the dopaminergic neurons seen in slices, such as
slow adaptation of oscillation frequency and may produce irregular
firing under different coupling conditions.
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INTRODUCTION |
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The mechanisms responsible for generating the
firing patterns of dopaminergic neurons in the mammalian brain stem
have been the subject of intense study in recent years. Their tonic
background firing is believed to be important in maintaining ambient
dopamine levels in the neostriatum necessary for the proper functioning of that structure (Grace 1991a; Romo and
Schultz 1990
), and their phasic firing during learning is
believed to play a critical role in motivation (Ljungberg et al.
1992
; Schultz et al. 1997
; Wise and
Rompre 1989
). Neurophysiological studies have demonstrated that
the cells are spontaneously active in vivo and in vitro and do not
depend on excitatory synaptic input to maintain their spontaneous activity (Fujimura and Matsuda 1989
; Grace and
Bunney 1983a
,b
, 1984a
; Harris et al. 1989
;
Kita et al.,1986
; Lacey et al. 1987
; Nedergaard and Greenfield 1992
). In vivo, three distinct
firing patterns have been observed. These are: a rhythmic single
spiking pattern, characterized by highly periodic low-frequency (<10
Hz) firing, an irregular firing pattern in the same low frequency range, and bursts of firing, usually <10 spikes, at higher frequency than, and superimposed on, the background regular or irregular firing
(Grace and Bunney 1984a
,b
; Tepper et al.
1995
; Wilson et al. 1977
).
The mechanism of the slow rhythmic single spiking pattern that occurs
spontaneously in slices (Grace and Onn 1989;
Harris et al. 1989
; Kang and Kitai
1993a
,b
; Lacey et al. 1989
; Yung et al.
1991
) and in dissociated substantia nigra cells
(Hainsworth et al. 1991
; Silva et al.
1990
) has been studied in detail. The slow membrane potential
oscillation is reduced in amplitude and in frequency after treatment
with TTX, indicating that action potentials, and perhaps also a
persistent sodium current, are important participants in the
oscillation but not essential for its occurrence (Fujimura and
Matsuda 1989
; Grace 1991b
; Grace and Onn
1989
; Harris et al. 1989
; Kita et al.
1986
; Nedergaard and Greenfield 1992
;
Yung et al. 1991
). The oscillation is abolished in
calcium-free media or after treatment with cadmium or cobalt, indicating that calcium currents are essential for the pacemaker (Grace and Onn 1989
; Harris et al. 1989
;
Kita et al. 1986
; Nedergaard and Greenfield
1992
; Yung et al. 1991
). The slow frequency of the oscillation (and the long time course of the inward current), the
relatively depolarized voltage range over which it occurs, and its
insensitivity to nickel suggests that rapidly inactivating low-threshold (t-type) calcium currents are not responsible for the
pacemaker (Harris et al. 1989
; Kang and Kitai
1993a
; Nedergaard et al. 1993
) [although note
that it was blocked by 500 µM nickel by Yung et al.
(1991)
]. The oscillation is blocked by known antagonists of
high-voltage-activated (HVA) calcium currents such as nifedipine (Mercuri et al. 1994
; Nedergaard et al.
1993
), even though the pacemaker current seems to be engaged at
relatively low voltages (
50 to
40 mV). The hyperpolarization phase
of the oscillation is blocked by apamin, and the oscillation is
sensitive to intracellular calcium and calcium buffers (Grace
and Bunney 1984b
; Ping and Shepard 1996
;
Shepard and Bunney 1988
, 1991
). These observations have
led to general acceptance of the view that the oscillation is primarily
due to inward calcium current and calcium-dependent potassium current,
with amplification by voltage-sensitive sodium current.
The facts that the necessary currents are present on the soma, that the
somata of dopaminergic neurons show the slow oscillation in isolation
(Hainsworth et al. 1991; Silva et al.
1990
), and the strong oscillations in slices, where distal
dendrites are often cutoff (Nedergaard and Greenfield
1992
), have suggested that the slow oscillator may be located
primarily on the soma. No experiments have ruled out the presence of
calcium channels or calcium-dependent potassium currents on the
dendrites, and there is some evidence for dendritic calcium currents
(Nedergaard et al. 1988
), but it is widely believed that
the dendrites are dominated by other ionic mechanisms that are thought
to be necessary for the generation of the irregular and burst firing
(Canavier 1999
; Johnson et al. 1992
;
Li et al. 1996
). This has led to the most widely
accepted model of the dopaminergic neuron, with the slow oscillation
arising proximally, while synaptic and burst generation mechanisms are
located in the dendrites (Amini et al. 1999
;
Canavier 1999
; Li et al. 1996
). That
model has been instantiated in computer simulations and shown to
produce a variety of firing patterns seen experimentally in
dopaminergic neurons.
We have tested this view of the ionic mechanism of the oscillation of dopaminergic neurons and its location on the neuron using calcium imaging to visualize the location of calcium influx. Our results confirm the basic mechanism of the oscillation but indicate that it is located on the dendrites as well as the soma. We propose an alternative model based on electrically coupled oscillators that better accounts for the rhythmic single spiking firing pattern seen in the dopaminergic cell and that also could incorporate the irregular and burst firing patterns without resort to widely different dendritic and somatic ionic mechanisms.
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METHODS |
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Experimental methods
Slices were prepared from the brains of Sprague-Dawley rats
ranging from 15 to 20 days of age. The rats were anesthetized deeply
with a 5:1 mixture of ketamine and xylazine, their brains were removed,
and the midbrain was sliced in the coronal plane at a thickness of 300 µm. Slices were maintained in a mixture of (in mM) 124 NaCl, 2.5 KCl,
2.0 CaCl2, 2.0 MgCl2, 1.25 NaH2PO4, 26 NaHCO2, and 10 D-glucose (bubbled
with 95% O2-5% CO2, pH
7.4). Slices were stored at room temperature prior to recording, but all the recordings were obtained at 32°C as it was found that the
oscillations were much more robust at temperatures approximating that
in vivo. Slices were viewed with an Olympus BX50WI upright microscope
equipped for DIC optics using a ×40 (0.8 NA) objective, under IR
illumination (780 ± 30 nm) using the same CCD camera used for Ca
imaging (see following text). Micropipettes had resistances of 6-8
M and were filled with a solution containing (in mM) 135 K-Gluconate, 5 KCl, 4 NaCl, 10 HEPES, 1 Na-ATP, 1 Mg-ATP, 0.3 Na-GTP,
and 0.05-0.2 fura-2 (Na salt) and 0.25% biocytin (pH 7.4). Current-clamp recordings were made using a Neurodata IR283 active bridge amplifier, and voltage-clamp recordings employed an Axon Instruments Axopatch 200B amplifier. Electrical and optical data were
collected synchronously using a single computer. Electrical records
were digitized at 16-bit resolution at 10 kHz, and corrected for a
10-mV liquid junction potential. Optical recordings were obtained using
a Photometrics EEV37 cooled CCD camera in frame transfer mode. Frame
rates of 20-50 per second were used, depending on the size of the
field of view. Fluorescence values were converted to calcium
concentration using a modification of the method described by
Lev-Ram et al. (1992)
. Single ratiometric measurements
were taken while the membrane potential was held hyperpolarized to prevent oscillations (in current clamp) or while fixing the membrane potential at
55 or
60 mV. These were converted to calcium
concentration in the usual manner (Grynkiewicz et al.
1985
) using a value for the fura-2/calcium dissociation
constant and the maximal and minimal fluorescences of fura-2 in our
electrode filling solution. These values were measured using
commercially available materials (Molecular Probes), and were
Rmin = 0.42, Rmax = 7.96, Sf380/Sb380 = 10.98, fura kD = 266 nM. Each trial began with a 1-s
segment of data gathered at this same membrane potential. Subsequent
changes in fluorescence at 380 nM then were converted to calcium
concentrations using the
formula
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After the experiment, slices were fixed by immersion in 4%
formaldehyde in 0.1 M phosphate buffer, treated with Avidin-biotin complex and stained with diaminobenzidine (DAB) as a whole
mount using the method of Horikawa and Armstrong (1988).
Stained neurons were visualized using an Olympus ×40 water-immersion
long working distance lens (NA, 1.2; WD, 0.2 mm), and in some cases
reconstructed using a computer reconstruction system developed within
the laboratory.
Modeling
The simulations represented a minimal model of the oscillation
mechanism, based on a simplification of the somatic compartment used by
Amini et al. (1999) but including calcium diffusion
kinetics. Conductances were represented as simple functions of voltage
or calcium concentration. For the voltage-dependent potassium and calcium conductance, a Boltzman function was employed
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The calcium current was generated using a reversal potential of +100 mV. In preliminary tests, this produced results indistinguishable from those obtained using the constant field equation, and so this simplification was considered valid.
Calcium removal was treated as a single process with a constant rate
and dependent only on calcium concentration. Calcium buffering and
diffusion were treated separately. For a single pump
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Assuming that the pump is also not saturable
[([Ca2+]i/Kp)
1]
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The presence of calcium buffer and the value of are expected to
have effects on the rate at which calcium diffuses within the
cytoplasm. Wagner and Keizer (1994)
have shown that the
more realistic (and more complicated) case of one mobile and one fixed buffer can be approximated by a similar equation to that for a single
fixed buffer, but adjusting the diffusion constant of calcium to
reflect the effect of the mobile buffer. In that case they show that
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For the simulations presented here, the buffers and the calcium
extrusion mechanism were represented as nonsaturable. For the pump,
this implies [Ca2+]i
KP, and for the buffers, it implies that
[Ca2+]
KM and
[Ca2+]
KS.
The buffering value
thus becomes independent of calcium concentration, and represents the total concentration of buffer
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The resulting equation for calcium at the interior surface of the
membrane was
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(1) |
For voltage, the usual current balance equation was applied
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(2) |
Computer simulations were generated using xpp (Bard Ermentrout, Univ. Pittsburgh) for the one- and five-compartment models, and Saber (Analogy, Beaverton OR) for the larger dendritic and full anatomic simulations. In both cases, integration was performed using the second-order gear method with a 1-ms time resolution, minimum time step of 1 ns, and a maximum step of 10 ms. Calcium concentrations for purposes of comparison with experimental data were calculated by averaging the concentration in each shell, weighted by its volume. Model description files for both the Saber and the xpp simulations are available from the authors. In both cases, compartments were represented as 40 shells, with diffusion between shells controlled by the apparent diffusion constant which was a parameter. The ratio of free to total buffer was treated as a separate parameter, as were the maximal conductance densities for the voltage-dependent calcium, voltage-dependent potassium, leak, and calcium-dependent potassium current. The maximal pump rate and diameters of compartments were likewise controlled by parameters. Typical parameters used in the simulations are given in Table 1.
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RESULTS |
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Time course of calcium accumulation after onset of rhythmic oscillation
As has been reported previously (Callaway and Wilson
1997), calcium concentration changes associated with
spontaneous membrane potential oscillations were synchronous in the
soma and proximal dendrites. Calcium levels increased during the steep
part of the ramp phase of the pacemaker potential immediately preceding
action potential generation and reached a peak immediately after the single action potential that terminated the depolarizing phase of the
oscillation. Calcium levels declined through the recovery from
afterhyperpolarization. These features of the oscillation of
intracellular calcium are seen in the example in Fig.
1. Fluctuations of calcium concentration
were smaller in the soma than in dendrites. This is unlikely to be the
result of differences in the voltage achieved in the soma and dendrites
as the difference was apparent even with the most proximal portion of
the dendrite, within 10 µm from the soma (Fig. 1, A and
B, purple boxes and lines). When oscillations were allowed
to resume after a period of sustained hyperpolarization (and associated
reduction in mean calcium concentration), the mean somatic calcium
concentration recovered more slowly than the dendritic concentration
(Fig. 1B). After the oscillation achieved steady state, the
cyclic fluctuations in calcium concentration continued to be larger in
the dendrites than in the soma, but the mean calcium concentrations
became approximately equal. Although the action potential was clearly
responsible for part of the calcium influx during each cycle of the
oscillation, cycles in which the action potential did not occur still
showed a subthreshold voltage transient and a calcium influx. That the
action potential contributed to, but was not necessary for, the calcium
influx was confirmed by blockade of action potential
Na+ currents using TTX (Fig. 1B).
After treatment of slices with TTX (1 µM), action potentials were no
longer evoked, but membrane potential oscillations persisted. These
were slower but otherwise similar in waveform and were increased in
amplitude if voltage-sensitive potassium currents were blocked by
treatment of the slices in TEA (2-20 mM). For the oscillations
occurring in the absence of action potentials, it was especially clear
that dendritic calcium transients exceeded those of the soma and that
the dendrites achieved steady-state oscillation more quickly than did
the soma during resumption of oscillations after a period of
hyperpolarization. Calcium concentrations achieved during the
subthreshold oscillations in the absence of action potentials with TTX
achieved approximately the same mean and transient levels as seen in
control solutions. Mean calcium concentrations in the dendrites
overshot the steady-state mean in the start of released oscillation;
the somatic mean increased gradually to steady state. These
observations with current-clamp recording were repeated with 20 neurons
in control and TTX or TTX and TEA solutions, with results as shown in
Fig. 1. Calcium signals were observed in dendrites as far as 200 µm
from the soma with no apparent decrease in the amplitude of the
transients.
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Single-compartment model of the oscillation
The observations of changes in calcium concentration in the soma of the dopaminergic neuron as described in the preceding text are consistent with the conclusions of neurophysiological studies of these cells as described in the INTRODUCTION. Those studies predict that a large proportion of the charge carried by the pacemaker current is calcium conducted by low-voltage-activated (but not rapidly inactivating) calcium channels. Thus during the last phase of the pacemaker potential preceding an action potential (or during the most depolarized phase of the subthreshold oscillation), there should be a large influx of calcium into the dopaminergic neuron. That calcium influx should achieve somatic calcium levels that can invoke a strong calcium-dependent potassium current that terminates the depolarizing phase, generates the hyperpolarized phase, and gradually decays due to removal of calcium through calcium pumps and perhaps sequestration into intracellular stores.
The dynamics of systems of this sort have been studied extensively and
are well known (e.g., Rinzel 1987). Calcium
concentration changes at a rate determined primarily by the calcium
current, the degree to which calcium is buffered, and the rate at which it is pumped out of the cell. Net calcium efflux and influx rates must
be similar, and the peak influx must exceed the efflux to maintain
oscillations of voltage and calcium concentration. As expected, the
most important feature for maintaining the oscillation was the time
constant of calcium efflux. If it was too slow, calcium concentration
would build up and a low-input resistance equilibrium would be achieved
with a strong calcium-dependent potassium current at a relatively
hyperpolarized membrane potential and small constant calcium current.
If calcium efflux was too rapid, a more depolarized equilibrium
associated with a relatively strong constant calcium current and a low
steady calcium level would be seen. For intermediate rates of efflux,
calcium currents would transiently become high, but calcium
concentration (being related to the integral of the current) would
build up slowly, giving a prolonged pacemaker current. When calcium
concentration reached a level at which the calcium-dependent potassium
current exceeded the calcium current, the cell would hyperpolarize
rapidly, and the intracellular calcium concentration would gradually be
reduced by efflux. If buffering was high, the depolarizing phase would
be prolonged (increasing the duty cycle), whereas if the efflux were
slowed, the recovery phase of the oscillation would be longer. This
dependence on parameters was best illustrated by examining their
effects on the nullclines for calcium and voltage as drawn in the
V/[Ca2+] phaseplane (e.g., Fig.
2A). In addition to the
important roles played by buffering and efflux (both of which act on
the calcium but not the voltage nullcline), the diameter of the
compartment had a key effect on the oscillation of the single
compartment model. At steady state there is no spatial gradient for
calcium so the diffusional term in Eq. 1 vanishes and the
nullcline includes terms only for calcium influx and efflux (which must
be equal), and diameter can be removed from both terms. Diameter also
does not appear in the voltage nullcline, but scales the rate of change of calcium when the system is not on the calcium nullcline. Both influx
and efflux are faster for a small diameter compartment because the
volume changes faster with diameter than does surface area (see
expression for calcium nullcline in the preceding text). Thus for
oscillations at rates much slower than the membrane time constant, the
diameter acts simply as a time scale for the oscillation with smaller
compartments oscillating more rapidly and fast compartments more slowly
but not affecting the amplitude or duty cycle of the oscillation. This
influence of diameter is illustrated in Fig. 2.
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Effect of limited diffusion on the single compartment model
Like changes in diameter, changes in the apparent diffusion
constant for calcium (Dapp) did not
alter the calcium nullcline but acted to change the rate at which
calcium concentration approached its equilibrium. Its influence was
seen in the steady-state oscillation and especially in the transients
caused by simulated current injections. We simulated the experimental
protocol in which oscillations of the model were prevented by
hyperpolarization and then released to allow the oscillation to resume.
When Dapp was large, so that calcium
was effectively well-mixed, the single-compartment model failed to
reproduce the transient calcium concentration changes seen in
dopaminergic neuron. During the first cycle of the model cell's
oscillation, calcium concentration rose to the peak value seen for all
subsequent cycles. The model's oscillation attained its steady-state
limit cycle over the course of a single cycle, and as a result, the
first cycle of the oscillation had an especially long depolarizing
phase. This occurred in the one-compartment model because the
calcium-dependent potassium current was not engaged until calcium
concentration was sufficiently high. Thus after a long-term decrease in
calcium concentration, a long depolarization was required to allow
calcium concentrations to raise to the level required to engage the
calcium-dependent potassium current. The well-mixed single-compartment
version of the model could not duplicate the gradual rise in calcium
and slowing of the oscillation seen in vivo because of the dependence
of repolarization on the calcium-dependent K current and its absolute
dependence on calcium concentration. A large reduction in the apparent
calcium diffusion constant, as occurs in cells containing nondiffusible
calcium buffers, produced more realistic results because calcium could
not diffuse rapidly away from the interior surface of the membrane, and
so the concentration there reached levels required to activate the
potassium current even though the average calcium concentration was
still low. The gradual increase in average calcium over many cycles at
the beginning of the transient reflected the time course of calcium
redistribution within the cell. To represent this in the model, the
total amount of buffering was kept constant, but the diffusion
coefficient of calcium in the cell was varied. For example, if
buffering was set to 1:100 (1% of entering calcium remains free), the
diffusion coefficient of calcium could be adjusted to 6 µm2/s (1% of the diffusion rate in saline)
(Hodgkin and Keynes 1957) to represent all buffers being
nondiffusible, to 100% of the diffusion rate in saline to represent
all buffers being as diffusible as calcium itself, or various values
between to represent combinations of mobile and diffusible buffers.
Decreasing calcium diffusion increased the natural frequency of
the oscillation and had its largest effect on the largest diameter
compartments. This is expected because the oscillation depends on the
calcium concentration at the surface shell, which is higher and changes
faster than the average calcium within the compartment. This effect is
seen in Figs. 3 and
4, for buffering set to 1:1,000 and a
wide range of calcium diffusion coefficients. For a 10-µm-diam
cylindrical compartment, the apparent diffusion rate of calcium had no
appreciable effect on natural frequency as it was reduced from 600 to
10 µm2/s. Further changes in apparent diffusion
constant had a dramatic effect, with extremely low diffusion rates (<1
µm2/s) increasing the natural frequency by a
factor of 4 (Fig. 3). It should be noted that apparent diffusion
coefficient of 0.6 µm2/s corresponds
approximately to 100% immobile buffer (when buffering is 1:1,000) and
so is the minimal amount for the level of buffering used in that
figure. Thus most of the effect of restricted calcium for the 10-µm
compartment occurs when
99% of the calcium buffers are immobile.
Restricted calcium diffusion had much larger effects on the natural
frequency of larger compartments and smaller effects when the diameter
was smaller. The effect of calcium diffusion rate on the relationship
between diameter and natural frequency are shown in Fig. 4. Low
apparent calcium diffusion constants had relatively little effect on
compartments <2 µm in diameter (comparable with dendrites of
dopaminergic neurons, see following text) but did affect the
frequencies of processes 5 µm (comparable with the proximal
dendrites and somata of dopaminergic neurons). The effect of diameter
on natural frequency continued to be manifest, although reduced in
size, even with calcium diffusion rates <1% of the calcium diffusion
rate in saline. With diffusion as low as 5 µm2/cm, there was still a substantial decrease
in natural frequency with diameter (Fig. 4), and the decrease occurred
over the same range of diameters (1-20 µm) found over the
somatodendritic region of dopaminergic neurons.
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Although the diffusion-limited single-compartment model could reproduce the gradual buildup in integrated calcium concentration seen in the somata of dopaminergic neurons, a similar mechanism could not account for the rapid increase and overshoot in the dendrites. Additionally, the presence of large calcium signals in the dendrites suggested that calcium channels in the dendrite may contribute directly to the oscillatory mechanism. Because the natural frequency of oscillation was dependent on compartment diameter but the stability of oscillation was relatively insensitive to diameter (as indicated by diameter independence of the nullclines), the soma and dendrites of various diameters have different natural frequencies of oscillation and must compete for control of the oscillatory process. To examine this competition, we constructed a small multicompartment model of a dopaminergic dendrite.
Coupled oscillator model
A minimal model of the dopaminergic neuron dendrite was
constructed of five compartments identical to the one used in the single-compartment model but varying in diameter. Because dopaminergic neurons exhibit a rapid initial decrease in diameter, an exponential taper was employed. Thus each compartment was smaller than the preceding one by a constant ratio. This arrangement is shown
diagrammatically in Fig. 5. Because each
successive compartments was of smaller diameter, it had a higher
natural frequency than its predecessor. Strong voltage coupling between
the compartments was present for all values of intracellular
resistivity tried (100-1,000 -cm) and enforced a common oscillation
frequency that was a compromise among the natural frequencies of the
components. The coupled frequency was always intermediate between the
largest and smallest compartment. Thus the largest compartment was
forced to oscillate at a higher frequency than it would if uncoupled
from the dendrite, and the smallest compartments were slowed. Because
in each cycle the smaller compartments were kept depolarized longer
than required to achieve the normal peak calcium concentration and also
kept hyperpolarized longer than they required to clear the calcium,
their peak and trough calcium levels were exaggerated beyond that
obtained for the smaller compartments when measured alone. The large
diameter compartments showed the opposite effect with smaller than
normal calcium transients at steady state.
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During oscillatory recovery from hyperpolarization, the
five-compartment model reproduced the gradual buildup of average free calcium in the soma and also the overshoot of average calcium in the
smaller processes (compare Figs. 1 and 5). This effect did not rely on
restriction of calcium diffusion but was seen for diffusion
coefficients ranging from 0.6 to 600 µm2/s
(Dapp = 20 µm2/s in Fig. 5). In the coupled-oscillator
model, there is also a gradual decrease in the oscillation frequency
during the transition period after release from hyperpolarization. The
gradual decrease in average calcium concentration in the dendrite is
presumably due to the decreased oscillation frequency, similar to that
in models of spike-induced calcium influx (Wang 1998).
Like those, this decrease is associated with the rise in average
calcium in the largest compartments and increased ability of the
oscillation there to influence the compromise frequency of the coupled compartments.
The compromise frequency obtained in the multicompartment dendritic
model was determined by both the variation in natural frequencies and
the ability of each compartment to influence its neighbors. The
smallest compartments, with the highest frequencies, also had the
smallest surface area and so generated less current with which to
influence the system. This is illustrated in the simulations shown in
Fig. 6, which shows the natural
frequencies for each of six equal-length (100 µm) compartments in a
simulated dendrite with exponential tapering, and the corresponding
steady-state frequency (at the end of a 60-s simulation) observed for
the coupled dendrite. The rate of tapering was varied from extremely
rapid (diameter ratio near 0) to no tapering (diameter ratio of 1). In
all cases, the diameter of the largest compartment was set to 16 µm
(to approximate the soma) and had a natural frequency near 0.26 Hz
(period 3.8 s). The natural frequencies of the other
compartments increased as the diameter ratio approached zero. For
slowly tapering dendrites, the frequency of the coupled compartments
was approximately the average of the frequencies of the various
compartments. As tapering increased, the smaller compartments came to
be less capable of influencing the larger ones. For extremely rapidly
tapering dendrites (diameter ratio <0.2), the largest compartments
came to dominate the compromise frequency by way of their current
sourcing ability.
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Experimental measurement of steady-state calcium levels during voltage clamp
The coupled oscillator model presented above was based on the
difference in rate of calcium disposition in the dendrites and soma due
strictly to their sizes. These modeling results suggested that there
should be differences in the time course of calcium accumulation and
disposal in the soma and proximal dendrites under constant voltage
conditions (which cannot be guaranteed in current-clamp recordings
unless there are electrodes in each part of the neuron). Because the
voltage dependence of the calcium current is the only voltage
dependency in the calcium nullcline, the steady-state calcium
concentration obtained in voltage-clamp data provide a measure of the
calcium current from calcium-imaging experiments with constant voltage.
We compared somatic and proximal dendritic calcium transients in
voltage-clamp experiments to determine the voltage sensitivity of
calcium influx, the time course of calcium accumulation and decay, and
the sensitivity of the calcium-dependent potassium conductance. The
basic design of these experiments is illustrated in Fig.
7. Whole cell recordings were obtained
under visual control as before, but after confirming the basic
physiological features of the dopaminergic cells (slow rhythmic
oscillation, long action potential waveform and strong sag in response
to hyperpolarizing currents in current clamp), the neurons were held at
60 or
65 mV using a voltage-clamp amplifier. This voltage range was
selected to be near the minimum current point for the cells so that
voltage-clamp error would be minimized and the dendritic tree would be
nearly isopotential. Calcium imaging revealed no calcium fluctuations in the dendrites at the holding potential. Long (3-16 s) voltage pulses to more depolarizing potentials (
50 to +10 mV) were applied, and the accumulation and decay of calcium was monitored using calibrated single wavelength measurements over the duration of the
pulses and for 10-16 s after their termination. Current was monitored,
primarily to assess the voltage error due to access resistance. Series
resistance was never more than 16 M
. Currents were <500 pA at all
times, and the resulting voltage error was never more than 10 mV.
Voltage errors >5 mV were corrected off-line. In five cells, this
experiment was performed in the presence of TTX, but there was no
consistent difference between the outcome in the presence or absence of
TTX and so the data were pooled.
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Voltage-clamp experiments were completed on 24 dopaminergic neurons, identified by their morphological features and their physiological features in current clamp. The average resting calcium concentration was measured ratiometrically at the holding potential in all of these cells, as described in METHODS and ranged from 20 to 430 nM [median = 130 nM, mean = 167 ± 141 (SD) nM].
In response to voltage pulses to 50 mV or more depolarized, all
neurons showed an increase in calcium concentration that approached a
steady-state value within a few seconds. There was no sign of the sag
in the calcium concentration that occurs with inactivation of the
calcium current (Gorman and Thomas 1978
). The rate of
calcium accumulation and decay was always greater in the dendrites than
in the soma, but the final steady-state value of calcium concentration
was similar or identical for all measurable (proximal) parts of the
cell. The steady-state calcium concentration achieved in this way is
especially useful because it is insensitive to calcium diffusion
kinetics (which are not known quantitatively for dopaminergic neurons).
Because the net calcium flux across the membrane is zero at steady
state, there is no calcium gradient within the neuron and no net radial
calcium diffusion. Comparison of the soma and the proximal dendrites
showed that these approach similar or identical values at steady state, so that longitudinal diffusion also cannot complicate the outcome. These features are all illustrated in the example in Fig. 7. The calcium concentration achieved at steady state is an experimental measurement of the calcium nullcline as presented in Fig. 2 for the
single-compartment model. As the calcium nullcline depends only on a
scale factor (which is a composite of the buffering ratio, the maximal
pump rate and the maximal calcium current), the resting calcium
concentration (at the start of the voltage step), the half-activation
voltage of the calcium current, and the slope factor for activation of
the current, these last two can be extracted from a fit of the
theoretical nullcline to the experimental values of calcium
concentration. The equation for the calcium nullcline and an example
experimental fit from five steps are shown in Fig.
8. Experimental fit of the nullcline was obtained from 15 neurons. These experiments revealed a relatively rapid
rundown of calcium currents over the first ~30 min of recording, and
so only a few points (4-8) could be obtained reliably from each cell.
Resting calcium concentration was obtained from a ratiometric measurement at the holding potential. For the sample of 15 neurons, the
half-activation voltage ranged from
29.8 to
42.6 mV [mean =
39.9 ± 5.3 (SD) mV]. The slope factor obtained in this way ranged
from 2.2 to 10.0 mV (mean: 5.2 ± 1.9 mV).
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In the well-mixed compartment model in which calcium diffuses rapidly, the accumulation and decay of calcium in the soma and dendrites should be exponential, depending only on the pump rate and the local surface area to volume ratio. Deviations from exponential accumulation and decay may be expected if calcium is released from intracellular stores in response to calcium influx, if calcium channels undergo some voltage- or calcium-dependent inactivation, or if calcium diffusion is slow due to the effect of fixed cytoplasmic buffers. Without assuming anything about the presence of these complications, we noted that calcium accumulation and decay could be approximately fit with single exponential functions simply for purposes of comparing the somatic with the dendritic rates of accumulation and decay. In most cells, there was little gained by adding a second exponential process to either the accumulation or the decay of calcium, and there was no particular pattern to the cases in which there was substantial deviation from a single exponential process. For the mean of 17 cells analyzed in this way, the somatic time constant of calcium decay was 3.6 ± 1.3 (SD) s, and the dendritic decay time constant averaged 51 ± 10% (mean ± SD) of the somatic one. When a similar approach was taken to comparing the time course of calcium accumulation, the dendritic calcium accumulation was faster than that in the soma by approximately the same proportion (1.8 ± 0.9 s). In these comparisons, dendritic or somatic calcium accumulation was on average faster than the corresponding time course of decay. The average somatic onset time constant was 57% of the average offset time constant, and the dendritic onset time constant was 47% of the dendritic offset time constant. The asymmetry between dendrite and soma was expected from the difference in their surface area to volume ratios, but the difference between accumulation and decay rates was not. One possible source of this is a partial inactivation of the calcium current. To test for this, we compared the time constant of calcium accumulation to the amplitude of the voltage step across the entire sample. There was no correlation between accumulation time constant and either the size of the voltage step (r = 0.22, df = 1,83, P > 0.1) or the steady-state calcium concentration measured at the end of the step (r = 0.14, df = 1,83, P > 0.1). These data argue against inactivation as the primary cause of the asymmetry in onset and offset time constants.
After the end of the voltage step, a powerful but brief inward tail
current was always seen, followed by a long-lasting outward current
that decayed with a time course comparable with the decay of calcium.
If calcium diffusion was fast relative to removal, it would be possible
to plot the calcium concentration against the size of the tail current
and recover the relationship between calcium concentration and
calcium-dependent potassium current. That is, because the
calcium-dependent potassium current changes rapidly in response to
changes in calcium concentration, the experiment consists of a gradual
decrease in calcium concentration and a measurement of
calcium-dependent potassium at each calcium concentration. In this
case, the tail current versus [Ca2+] curve
should be sigmoid, reflecting the sigmoid relationship between calcium
concentration and calcium-dependent potassium current. A sigmoid curve
of this kind, the result of a computer simulation, is as shown in Fig.
9B, for an apparent calcium
diffusion constant of 600 µm2/s. The
half-activation concentration of the current is the corresponding dissociation constant of the calcium-dependent potassium channel, and
the slope at that point is determined by the cooperativity (4 in these
simulations). If the diffusion of calcium is limited by fixed buffers,
the calcium-dependent potassium current will appear to be less
sensitive to calcium and will lose its sigmoid shape as shown in the
simulations in Fig. 9B. This occurs because the average
calcium concentration is not a good reflection of calcium concentration
at the interior surface of the membrane. At the beginning of the
calcium relaxation curve (after the step back to 60 mV), calcium
concentration is homogeneous throughout the cell, but as calcium is
removed from the cell, this creates a depletion layer near the membrane
that causes the calcium-dependent potassium current to decrease more
rapidly than expected from the average calcium measured in the
experiment. This results in an apparent shift in the calcium dependence
of the tail current in the positive direction and a distortion of the
sigmoidal shape of the curve as seen in computer simulations (Fig.
9B). Simulations of an alternative explanation, based on the
idea that most of the potassium current might be located in the
dendrites and that space clamp of the dendrites was not good, did not
produce a large distortion of the curve and could not match the
experimental data. This produced distortions of the curve only in the
beginning of the transient (not shown). Dependency of the tail currents
on calcium was studied in 18 cells. In all cases, the curves had the
shape shown in Fig. 9A. All showed a calcium dependency in the 200- to 500-nM range as expected for apamin-sensitive SK channels, but in no case did the curve appear sigmoid as expected for the well-mixed model. This suggested that calcium diffusion in dopamine neurons may be very restricted but did not allow a quantitative estimate of the diffusion coefficient or the half-activation
concentration of the calcium-dependent potassium current.
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Coupling was less reliable with a realistic dopamine neuron geometry
Although the five-compartment model of the dopaminergic dendrite
qualitatively reproduced most of the effects seen in of the calcium-imaging experiments, they did not provide an accurate representation of the peculiar morphological features of the
dopaminergic neuron. To determine whether the coupled-oscillator model
may be valid for dopaminergic neurons given the values of
calcium-conductance voltage sensitivity, calcium decay rate, and
potassium-conductance calcium sensitivity obtained in the preceding
text, we used these values in an anatomically realistic model of the
dopaminergic neuron. A representative neuron from the sample was
reconstructed using a computer-aided light microscopic neuron drawing
program and converted to a Saber input file using compartments with the parameters set from the voltage-clamp data. The morphology of the
reconstructed neuron is shown in Fig.
10. The calcium conductance had a half
activation voltage of 35 mV and a slope factor of 7 mV. The
calcium-dependent potassium conductance had a half-activation calcium
concentration of 180 nM. The ratio of bound to free calcium was 1,000, but the calcium diffusion rate was ignored to speed computation. The
calcium pump was set so that the somatic calcium cleared with an
effective time constant of ~10 s, and the maximal calcium conductance
was adjusted to achieve a peak somatic calcium concentration of 1,000 nM. The maximal calcium-dependent potassium conductance was set to
obtain a tail current of 400 pA at a somatic calcium concentration of
1,000 nM when measured at
60 mV immediately after the end of a 30-s
voltage-clamp step to
20 mV as in the experiments in the preceding
text. An example of the resulting spontaneous activity from an
morphologically accurate simulation of this kind is shown in Fig.
11. Steady-state (40 s after release from hyperpolarization) oscillations from selected points on the neuron
in are shown in Fig. 11B. The spontaneous oscillation rate of the electrically coupled model also is compared with the natural frequencies of the individual compartments Fig. 11. As in the
five-compartment model, robust oscillation was observed at a frequency
substantially greater than the natural frequency for the soma but less
than that of the finest dendrites. On release from hyperpolarization, the simulation reproduced the major features seen in dopaminergic neurons in slices. The somatic oscillation (compartment 1)
was lower amplitude than the dendrites, and showed the gradual average increase. The proximal dendrite (compartment 264) also
showed a gradual but more rapid rise to steady-state values, whereas a
distal dendrite (compartment 863) showed an initial
overshoot and gradual decrease in average calcium concentration to
steady-state values.
|
|
The global coupling and synchrony described in the preceding text was
resistant to disruption by increases in cytoplasmic resistivity. For
example, increasing Ri from 100 to
1,000 -cm did not disrupt voltage-enforced synchrony of the
oscillation in the model as described in the preceding text. However,
the coupling was disrupted easily by changes that weakened the currents responsible for the oscillation, making it marginal in the finest dendrites. In the example shown in Fig.
12, the maximum calcium current was
reduced by 33%, from 0.15 to 0.1 mS/cm2. At this calcium current
density, all parts of the neuron continued to oscillate and could
sustain steady-state oscillations in the absence of coupling, although
the amplitude of the oscillations was reduced and the frequency was
increased (compare uncoupled oscillations in Figs. 11 and 12). When the
realistic neuron model was coupled, the long dendrite with
compartment 620 at its tip failed to synchronize fully with
the rest of the neuron. This is illustrated in Fig. 12, in which the
oscillation in compartment 620 is not fully entrained with
the soma and proximal dendrites. This kind of partial synchronization was seen under circumstances that reduced the amount of current generated by the oscillation and weakened the ability of compartments to influence their neighbors. Phase locking of calcium signals was
poorer than the voltage coupling in such cases. As in the case shown in
Fig. 12, the result as seen at the soma was a less regular oscillation.
In extreme cases, the somatic voltage waveform did not repeat over the
period of the simulation (although no more than 50 s of
oscillation was ever simulated).
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DISCUSSION |
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Mechanism of the oscillation in dopaminergic neurons
Our results employing calcium imaging in dopaminergic neurons
confirm the mechanism of oscillation in dopaminergic neurons already
established by neurophysiological studies in slices. The phase
relationship between calcium influx in the soma and dendrites of the
dopaminergic neurons is as expected if calcium was the major charge
carrier for the pacemaker current as was concluded previously from the
insensitivity of the oscillation to TTX and its sensitivity to
extracellular calcium and calcium blockers (Grace and Onn
1989; Harris et al. 1989
; Kang and Kitai
1993a
; Nedergaard and Greenfield 1992
;
Yung et al. 1991
). The decreased amplitude of the
pacemaker potential in the absence of TTX has been interpreted as
evidence that a persistent sodium current participates in the
oscillation. Our results are also consistent with that conclusion but
indicate that the sodium current acts in phase with, and amplifies, the
inward current produced by calcium currents activated in the
subthreshold range. The calcium current responsible for the pacemaker
potential in dopaminergic neurons is unusual in that it has a low
voltage threshold but does not readily inactivate. When evidence
suggesting this first was presented by Kang and Kitai
(1993b)
, there were few examples of such low-threshold noninactivating calcium currents. In recent years, examples of calcium
currents with similar characteristics have become more commonplace. Our
voltage-clamp measurements of the voltage sensitivity of the current
produced on the basis of steady-state calcium concentrations are in
reasonable agreement with the measurements of Kang and Kitai
(1993b)
made using conventional voltage-clamp techniques. Unlike conventional voltage-clamp measurements, our technique has the
advantage of not requiring voltage control beyond the soma. The other
key claim regarding the calcium current responsible for the pacemaker
potential was that it is largely noninactivating. The argument for this
was based on the continuation of oscillations at voltages more
depolarized than those normally required for inactivation of T-type
calcium channels (Kang and Kitai 1993a
). We have
confirmed this conclusion directly using the combination of voltage
clamp and calcium imaging. Although the difference in time course of
calcium accumulation and decay may suggest a small amount of rapid
inactivation of calcium current, the absence of a significant sag in
intracellular free calcium over tens of seconds at depolarized
potentials shows that there is a large noninactivating component to the
low-threshold current. This is consistent with recent reports that the
calcium pacemaker potential is sensitive to blockade by antagonists of
dihydropyridine sensitive calcium channels, which often show little
voltage-sensitive inactivation (Mercuri et al. 1994
;
Nedergaard et al. 1993
).
Studies of the localization of the S-type calcium-dependent potassium
channel have shown that the substantia nigra, pars compacta, is rich in
mRNA encoding this molecule (Köhler et al.
1996). This confirms reports that membrane potential
oscillations of dopaminergic neurons are extremely sensitive to
concentrations of apamin known to be specific for this channel
(Shepard and Bunney 1988
, 1991
; Shepard and Stump
1999
). Our results show that elevated calcium concentrations
give rise to an outward tail current with a calcium dependence close to
that of the SK channel. Membrane potential oscillations seen in
dopaminergic neurons in slices require only these two currents (a
low-threshold noninactivating calcium current and the SK current), and
a mechanism for extruding calcium from the cell active at nanomolar
concentrations. Calcium-dependent potassium channels spatially
associated with the calcium channel and therefore primarily sensitive
to calcium flux rather than average calcium concentration would be much
less suited for the generation of slow oscillations.
The time course of the oscillation depends on the rate of buildup of free calcium in the neuron during the depolarizing phase and the rate at which calcium is pumped out during the hyperpolarization. The period of the oscillation is determined by the sum of these time courses and the duty cycle by their ratio. During the calcium entry phase, the input resistance of the neuron and the strength of internal buffers are of major importance. For realistic values of input resistance, the amount of calcium current required to depolarize a neuron enough to create a regenerative response generates a large influx of calcium. In the absence of buffers, calcium concentration would increase dramatically and calcium-dependent potassium currents would turn on before the depolarization was strong enough to generate a regenerative electrical response. Strong calcium buffering is required to make a suitable delay between activation of the calcium current and activation of the calcium-dependent potassium current so that oscillations can ensue. The rate of calcium efflux is also dependent on buffering and on calcium diffusion rate within the cell. To achieve realistic oscillations in our model neuron, we employed very heavy buffering, with the ratios of free to bound calcium ranging from 1:100 to 1:2,000 (1:1,000 was used in all the illustrations). This requirement for strong buffering also was supported by our observation that dopaminergic neurons were relatively insensitive to fura-2 concentration in the electrode. We used fura-2 in concentrations from 50 to 200 µM with no apparent change in oscillation strength or frequency. Because of its reliance on SK channels, the oscillatory mechanism in dopaminergic neurons studied here has a large dependence on the surface area to volume ratio with calcium influx and efflux proportional to surface area but dilution (and thus the time course of calcium concentration change) approximately proportional to volume. The volume dependence is modified due to restricted calcium diffusion, but on the scale of the processes of the dopaminergic neuron, there is still a strong relationship expected between the natural frequency of a compartment and its surface area to volume ratio. This expectation is confirmed by our experimental voltage-clamp observations of 1) a much more rapid accumulation and decay of intracellular calcium in the dendrites (even the most proximal dendrites), quantitatively consistent with that effect of diameter and 2) the observation that the steady-state value of calcium concentration was nearly the same for the soma and dendrites over a wide range of voltages. These transient time-course differences between soma and proximal dendrites and between proximal and distal dendrites might be attributed to a gradient in calcium currents; but if so, they must be exactly matched by a similar gradient in the rate of calcium disposition if they are to account for our data. Given the ease with which the volume to surface area effect matches the experimental observations, it is unnecessary to postulate a discrete change in density of calcium channels and transmembrane calcium pumps over the course of 5-10 µm at the somato-dendritic boundary and a subsequent gradient of those same mechanisms down the dendrites. However, it should be noted that the effects of the coupled oscillator model on the overall function of the cell are largely independent of the mechanism underlying the natural frequency gradient, and it could well be applied to neurons employing very difficult oscillator mechanisms.
Most of our experiments and all of our modeling were performed under
conditions that prevented the generation of sodium action potentials.
Although the oscillatory mechanism of dopaminergic neurons does not
require action potentials or TTX-sensitive sodium currents, it is
strongly affected by them. The oscillations observed in neurons
permitted to spike are faster than those in TTX (e.g., Fig. 1). Also,
whereas most calcium entry in dopaminergic neurons is associated with
the pacemaker potential, oscillatory cycles in which no spike occurs
are associated with somewhat smaller calcium peaks, suggesting that
some additional calcium entry is associated with fast action
potentials. That this must be the case is indicated by the fact that
dopaminergic neurons almost never fire more than one action potential
on each cycle of the oscillation The oscillatory cycle is terminated by
the action potential, which is always followed by a profound
afterhyperpolarization. After blockade of SK channels with apamin,
dopaminergic neurons do exhibit bursts, indicating that the
oscillation-resetting effect of action potentials is caused primarily
by calcium-dependent potassium currents (Ping and Shepard
1996). Thus the increased oscillatory rate observed when the
action potential mechanism is intact can be accounted for by the early
termination of the calcium accumulation phase of the oscillation and
accompanying decrease in duty cycle.
Coupled oscillator model
Dissociated dopaminergic somata have been reported to continue
rhythmic single spiking, suggesting that the oscillatory mechanism in
dopaminergic neurons is present in the soma (Hainsworth et al.
1991). Although models of dopaminergic neurons often have been
based on the hypothesis that the ionic mechanisms of the soma and
dendrites are different, our results indicated that, at least for the
first 100-300 µm from the soma, the oscillatory mechanism is not
detectably different in the dendrites and cell body. In the coupled
oscillator model proposed here, the proximal dendrites play a primary
role in setting the frequency of oscillation. This accounts for the
fact that oscillations are not strongly dependent on the amount of
trimming of distal dendrites that occurs in slices. In fact, distal
dendrites tend to destabilize the oscillation due to their remote
location and their high natural frequency. Their loss may contribute to
the highly rhythmic nature of the oscillation seen in slices and
dissociated cells.
The presence of the calcium-dependent oscillation in the dendrites has substantial implications for the operation of the neuron. Because the natural frequency of oscillation is dependent on the surface area to volume ratio, the various regions of the dendritic tree tend to oscillate at different frequencies. This conclusion is based on the differences in calcium accumulation and decay seen in the somata and dendrites in the voltage-clamp experiments as well as on the results of the model. The synchrony of calcium-concentration oscillations seen in current-clamp recordings from the same cells is a direct demonstration of the effectiveness of voltage coupling among the soma and dendrites, at least in cells in tissue slices. Thus the fundamental features of the coupled-oscillator model are demonstrated by the evidence presented in this paper.
The presence of multiple oscillators with different natural frequencies has a number of implications for the generation of firing patterns and for synaptic integration in dopaminergic neurons. They are all based on the availability of multiple frequencies of oscillation in the neuron and the necessity of a compromise frequency obtained via electrical coupling. In the most simple case, the compromise is simply an average of all the oscillators, perhaps weighted according to their current strength and location. Even the simple example used in this experiment (the oscillation is halted by passage of hyperpolarizing current and then allowed to resume) shows a slow settling time for the oscillation, during which frequency and amplitude slowly approach the steady state. These changes suggests that the balance between the soma and dendrites may be dynamically adjustable. In the simple case, the oscillations began at a faster frequency than that achieved at steady state because calcium concentration was slow to rise in the soma. During the several seconds that were required for somatic calcium to become high enough to influence the oscillation, dendritic oscillations of higher frequency were more effective than they would otherwise have been. The effective control of the oscillation thus starts in the distal dendrites and moves proximally as calcium builds up in the more proximal regions. As the oscillation slows, the mean calcium level in the distal dendrites is lowered. At one point along the dendrites, the natural frequency and the amplitude of calcium transients matches that of the oscillation frequency of the coupled system. This position moves closer to the soma over the first few seconds and is stationary at steady state. Our simulations suggest that it does not reach the soma at steady state, and the dendrites continue to force the cell to oscillate more rapidly than it would if the soma were isolated. Other alterations of the balance between the soma and dendrites might alter the firing rate of the neuron by shifting the effective center of oscillation proximally (slowing the cell) or distally (increasing the frequency).
Possible mechanism of irregular firing
The effective center of oscillation described is a valid concept
for the case of complete synchronization of voltage throughout the
neuron. This seems to be the case for dopaminergic neurons in slices of
the substantia nigra as described here, but the simulations of Fig. 12
suggest that it need not always be so. When the inward current driving
the oscillation is reduced so that less current is available to perform
the coupling among nearby cell regions with different natural
frequencies, some distal regions of the cell may pass in and out of
synchronization with the rest of the cell. The occurrence of
dendritically propagated sodium action potentials (Hausser et
al. 1995; Nedergaard and Hounsgaard 1996
) are
likely to play a large role in synchronizing the neuron, but between
spikes the various parts of the cell may go out of synchrony. The
results shown in Fig. 12 show that when this happens, the
out-of-synchrony distal region may interact with the proximal part in
complex ways, varying the amplitude of the oscillation over time and
causing the proximal regions to skip a cycle or more in an irregular
pattern. In its extreme form, this will produce an irregular
oscillation pattern at the soma that can result in an irregular single
spiking firing pattern in the dopaminergic neuron. A single-spiking
irregular pattern of firing is the most common firing pattern seen for
dopaminergic neurons in vivo (Tepper et al. 1995
). It
generally is assumed that this pattern represents the moment to moment
changes in an irregular pattern of synaptic input to the cell. The
present results suggest that the generation of an irregular single
spiking firing pattern could arise simply from a weakened coupling of
the parts of the dopaminergic neuron, perhaps due to a decrease in the
strength of calcium currents (e.g., due to modulation). In the model
presented in Fig. 12, each compartment must generate enough current to
sustain its own oscillation and in addition supply current for
longitudinal interactions with its neighbors that maintain synchrony.
If the natural frequency difference is great (i.e., the diameter
changes rapidly), as occurs at branch points and at the origins of
dendrites at the soma, the longitudinal currents required to maintain
voltage coupling may be substantial. These currents also vary during
the oscillatory cycle because of the opening and closing of ion
channels in each compartment throughout the cycle. When the current
density is insufficient, coupling may become marginal, as shown in Fig. 12.
Bursting in dopaminergic neurons could arise from similar transient changes in coupling between the soma and dendrites but over a time course of several cycles. For example, if the dendritic oscillation could temporarily control firing before somatic calcium levels could respond, the cell might fire at the high rates natural for the dendrites for a few cycles, gradually slowing as the effect of the more proximal dendrites is expressed.
Implications for synaptic integration in dopaminergic neurons
One of the unexplained features of the dopaminergic neuron is the difference between its firing pattern in vivo and in vitro. This also usually is attributed to the loss of synaptic input that is supposed to deregularize the cell by way of its own lack of a repeating pattern. Noisy synaptic input on an oscillatory membrane acts in a somewhat different way than on a cell with a stable membrane potential, and this generally not taken into account. For example, during the low-input resistance portion of the oscillatory cycle (the time when calcium-dependent potassium current is high), conventional fast synaptic input is likely to be relatively ineffective at disrupting the ongoing pattern. During the depolarizing phase, an oscillatory single compartment would be sensitive to conductance changes that alter voltage and so prevent or enhance the regenerative approach to firing threshold. Thus synaptic input could produce a phase advance or lag at the point of action. In the coupled-oscillator model, the local compartment receives powerful longitudinal currents maintaining synchrony along the dendrites. For synaptic input to produce a change in the oscillation at one compartment, it must overcome these currents that will oppose it. If coupling is relatively weak, phase disturbances may propagate along the dendrites and synaptic inputs may temporarily uncouple regions of the dendritic tree producing the irregularities in firing described in the preceding text. If coupling is strong, as it appears to be in slices and under some conditions in vivo, synaptic input must produce a phase shift over the entire neuron or not at all. Under these conditions, synaptic integration may function in ways very different from those seen in cells with stable resting potentials or cells in which synaptic input is localized on a nonoscillatory region of an otherwise oscillatory neuron.
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APPENDIX |
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The calculation of calcium concentration follows from
Grynkiewicz et al. (1985)
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(A1) |
By definition of the dissociation constant of fura-2
(kD)
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(A2) |
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(A3) |
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ACKNOWLEDGMENTS |
---|
We thank Miriam Chong for participating in the beginning of this project and Dr. Nancy Kopell for helpful discussions and suggestions.
This work was funded by National Institute of Neurological Disorders and Stroke Grant NS-36843.
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FOOTNOTES |
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Present address and address for reprint requests: C. J. Wilson, Div. of Life Sciences, Univ. Texas at San Antonio, 6900 N. Loop 1604, San Antonio, TX 78249.
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 30 August 1999; accepted in final form 18 January 2000.
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REFERENCES |
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