 |
INTRODUCTION |
Midbrain dopaminergic neurons (DA) have been
implicated in several important functions in humans, including
movement, attention, learning, and reinforcement as well as in the
etiology of Parkinson's disease and various thought disorders, such as
schizophrenia, attention deficit disorders, and depression
(Carlson 1992
). Therefore it is essential to understand
how the firing pattern in these neurons is generated and modulated.
Whereas in vivo DA neurons can exhibit either single-spike or
burst-firing activity as well as spontaneous shifts between modes
(Freeman et al. 1985
; Grace and Bunney
1984
), in a slice preparation a regular, pacemaker-like firing
pattern usually is observed, presumably due to the loss of synaptic
afferents. In this study, we will focus on the intrinsic oscillatory
activity of DA neurons in a slice preparation.
In addition to the endogenous regular single-spike firing exhibited in
a slice preparation, burst firing has been induced by the application
of apamin (Ping and Shepard 1996
) or
N-methyl-D-aspartic acid (NMDA) (Johnson
et al. 1992
). The firing pattern in these neurons is
particularly significant because burst-firing in vivo is associated
with behaviorally relevant appetitive and sometimes novel stimuli and
possibly with movement (for a review, see Overton and Clark
1997
). This study will address the biophysical basis for the
calcium-dependent oscillations underlying regular single-spike firing
and apamin-induced burst firing.
Because midbrain DA neurons have an unusually depolarized firing
threshold (
33 ± 15 mV) (Grace and Onn 1989
), the
membrane must be depolarized repeatedly from rest (
60 mV) to this
threshold to fire repetitively. Indeed, such an underlying slow
oscillatory potential (SOP) has been observed when the sodium-mediated
spikes are blocked by tetrodotoxin (TTX) and is enhanced by the
application of the potassium channel blocker tetraethyl ammonium (TEA).
The depolarizing phase of the SOP has been variously termed the
pacemaker-like slow depolarization (PLSD) or simply the pacemaker-like
depolarization (PLD), which has been shown to be calcium-dependent.
The SOP sometimes can be converted into a square-wave oscillation by
the application of apamin (Nedergaard et al. 1993;
Ping and Shepard 1996
), revealing a crucial role for the
apamin-sensitive current IK,Ca,SK in the
repolarization of the SOP. Just as the SOP is the underlying
oscillation that sets the rhythm for pacemaker-like single-spike
firing, the square-wave oscillation sets the timing for burst firing;
in the absence of TTX, a burst of spikes is generated on the plateau of
the square-wave and quiescence during the trough (Gu et al.
1992
; Ping and Shepard 1996
; Shepard
1993
; Shepard and Bunney 1988
). In some cases,
apamin induced irregular firing instead of burst firing (Gu et
al. 1992
; Ping and Shepard 1996
), and in those
cases, the subsequent application of TTX and TEA revealed irregular
high-threshold calcium spiking (Ping and Shepard 1996
).
The generation of irregular firing patterns, however, is not addressed
in this study. Neither the SOP nor the square-wave should be confused
with the slow oscillation in membrane potential revealed by TTX in the
presence of NMDA, which also can underlie burst firing, appears to be
sodium rather than calcium dependent (Johnson et al.
1992
) and is also not addressed in this study.
The goal of this study was to test hypothesized mechanisms for the SOP
and the square wave. On the basis of data that suggest that
dihydropyridines abolish the SOP (see DISCUSSION), we
hypothesize that the SOP is generated by interplay between the L-type
calcium current ICa,L and the
calcium-activated IK,Ca,SK. We further
hypothesize that in the presence of apamin, the calcium-mediated
inactivation of ICa,L is responsible for
repolarization of the square wave. We also hypothesize that the
mechanisms driving these phenomena are located in or near the soma.
Hence we constructed a quantitative somatic model based on
voltage-clamp, morphological, and calcium-imaging data to confirm that
the currents described under voltage clamp are capable of generating
the electrical activity observed under current clamp. Minimum
expectations for model performance included production of SOPs similar
to those of Ping and Shepard (1996)
in the presence of
TTX and TEA, SOPs under conditions where either the calcium current
ICa,T or ICa,N
were blocked but not when ICa,L is blocked,
and square waves similar to those observed (Ping and Shepard
1996
) in the presence of apamin.
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MODEL DEVELOPMENT |
Our model of the DA neuron consists of a single compartment
Hodgkin-Huxley (HH)-type parallel conductance membrane model (Fig. 1A) and a lumped fluid
compartmental model (Fig. 1B). The HH equivalent circuit is
composed of a somatic membrane capacitance of 15.8 pF (Kang and
Kitai 1993b
), shunted by resistive ion-selective channels, as
well as ion pumps, a sodium-calcium exchanger, and other ionic
currents. The lumped fluid compartment model (Fig. 1B)
consists of an intracellular compartment containing constant concentrations of Na+ and
K+, and a calcium buffer (presumably calmodulin).
A material balance for Ca2+ describes the time
rate of change in cytosolic Ca2+ concentration.
The extracellular space is assumed to have a relatively large volume,
so that the ionic concentrations of Ca2+,
Na+, and K+ there are
assumed to be constant.

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Fig. 1.
Diagram of the dopaminergic (DA) neuron cell model. A:
electrical equivalent circuit for the membrane;
ICa is the combined calcium current
consisting of ICa,T,
ICa,N, ICa,L, and
ICa,HVA.
IK,tot is the combined voltage-dependent
potassium current consisting of IK and
IA.B: fluid compartmental
model, including intracellular and extracellular spaces.
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 |
MEMBRANE CURRENTS |
Under space-clamp conditions, the differential equation describing
the time-dependent changes in the membrane potential (V) is
|
(1)
|
where Cm is the whole cell
membrane capacitance. Model currents include a T-type calcium current
(ICa,T), an L-type calcium current
(ICa,L), an N-type calcium current
(ICa,N), a residual high-voltage
activated calcium current (ICa,HVA), a
delayed rectifier current (IK), a transient
outward current (IA), a small conductance calcium-dependent potassium current
(IK,Ca,SK), a hyperpolarization activated
current (Ih), a sodium-potassium pump
current (INa,K), a calcium pump
current (ICa,pump), and a
sodium-calcium exchanger (INa,Ca).
In the current descriptions that follow, the HH-type activation and
inactivation gating variables are solutions of the familiar first-order
differential equations described as
|
(2)
|
where
(V) is the steady-state value
of the general gating variable z at membrane voltage
V. We have characterized the steady-state gating variable
(V) by a sigmoidal, or Boltzman-type
relationship, and the time constant
z(V) by a Gaussian relationship.
Individual ionic membrane currents were characterized by fits to
published voltage clamp experiments where available. The temperature at
which the experiments were performed ranged between 30 and 35°C, thus
temperature adjustments (e.g., Q10)
were not needed during final integration of currents into the model. To obtain fits to the currents, membrane potential was held constant and
the differential equations characterizing the gating variables for each
current were integrated numerically and substituted into the
appropriate equation for ionic current. Examples of fits to the
component currents of the model are shown in Fig.
2.

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Fig. 2.
Model-generated approximations to voltage-clamp data for several
membrane currents. Steady-state activation and inactivation
characteristics of the currents were characterized by fits ( ) to
experimental data (- - -). A:
ICa,T, model was held at 88 mV and clamped
at various voltages from 68 mV through 54 mV in increments of 2 mV
following Kang and Kitai (1993b) . B:
IA, model was held at 90 mV and clamped to
various voltages from 60 to 75 mV in 15-mV increments after
experiments by Silva et al. (1990) . C:
Ih, model was held at 57 mV and clamped to
various potentials from 141 to 69 mV in 12-mV increments after
Mercuri et al. (1995). D:
IK, model was held at held at 40 mV and
clamped to 20 mV in 10-mV increments after experiments by Silva
et al. (1990) . E:
ICa,N, model was held at 84
mV and clamped to 58, 48, and 38 mV following Kang and
Kitai (1993b) . Steady-state activation and inactivation curves
for IA ( and
, respectively) and steady-state activation
curves for Ih ( ) and
IK ( ) are displayed in
F. Current parameters and equations are in Tables
1-3.
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Fits to voltage-clamp data are not the only criteria for formulating
ionic current descriptions, which may have to be modified to fit whole
cell transmembrane potential data. These additional adjustments are
justified considering that the voltage-clamp and the free-running SOP
and square-wave recordings were obtained from different cells and
in different laboratories and that the voltage-clamp experiments were
performed on a particular cell, and there is a considerable
variation in the waveshape of the ionic current response from cell to cell.
Calcium currents
ICa
The mathematical equations used in the model for the different
calcium currents are given in Table 1,
and the voltage dependence of the steady-state activation and
inactivation characteristics of these currents are shown graphically in
Fig. 3. The reversal potential for the
calcium currents has been set to a constant 50 mV by extrapolating
current-voltage curves from Kang and Kitai (1993b)
.

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Fig. 3.
Steady-state characteristics of calcium currents. d and
f represent the activation and inactivation gating
variables, respectively, for calcium currents. Steady-state activation
( T) and inactivation
( T) characteristics of
ICa,T were obtained from voltage-clamp
studies of Kang and Kitai (1993b) . Steady-state
activation of ICa,N
( N) was obtained by fitting
voltage-clamp data from the same source. Steady-state activation of
ICa,L ( L)
is more hyperpolarized than that of ICa,N.
Steady-state activation ( HVA) and
inactivation ( HVA) of
ICa,HVA, are adapted from Cardozo and
Bean (1995) and Kang and Kitai (1993b) ,
respectively.
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T-TYPE LVA CALCIUM CURRENT
ICA,T.
The T-type calcium current is based on data from Kang and Kitai
(1993b)
. The membrane potential was held at
88 mV and clamped at various voltages from
68 mV through
54 mV in increments of 2 mV.
Their experiments show a two-stage inactivation: a quick inactivation
to a small current followed by a slow inactivation that maintains the
small current for the duration of the voltage clamp (300 ms).
Characterization for the long secondary component is based on data from
neostriatal neurons (Hoehn et al. 1993
), whereas the
activation and fast inactivation component was characterized by fitting
data from Kang and Kitai (1993b)
. The voltage-clamp fits
to data are shown in Fig. 2A. The current description fits the quick activation and biphasic inactivation well at the expense of
matching the peak current values at some clamp potentials.
N-TYPE HVA CALCIUM CURRENT
ICA,N.
As modeled, ICa,N begins to activate
at
60 mV and continues to activate at
45 mV. The slow inactivation
that appears in voltage-clamp records (Kang and Kitai
1993b
) is probably calcium-mediated, because no
calcium-independent inactivation was detected in the presence of the
calcium buffer EGTA and the addition of the calcium buffer EGTA
increased the inactivation time constant. We were unable to find
specific data regarding the Ca2+-mediated
inactivation of ICa,N in DA neurons
and hence employed a simple Michaelis-Menten relationship to modify the
channel conductance. The parameters of the gating variable equations
associated with the equations for
ICa,N were determined generally by
fitting voltage-clamp data (Fig. 2E) from Kang and
Kitai (1993b)
in a cell with N-type channels which did not
exhibit calcium-dependent inactivation. The cell was held at
84 mV
and clamped to voltages of
58,
48, and
38 mV, respectively.
Because the characterization of the calcium-dependence of this current
was not based on voltage-clamp data, the parameters associated with the
steady-state calcium-dependent inactivation function
(fCa,N) were loosely set in
the fit to voltage-clamp data and determined finally in fits of the
complete model to SOP and square-wave data. The specific equations for
ICa,N are listed in Table 1.
L-TYPE HVA CALCIUM CURRENT
ICA,L.
On the basis of experiments in other cells (Fox et al.
1987a
,b
; Johnston and Wu 1995
), we have assumed
that the nifedipine-sensitive, long-lasting calcium current,
ICa,L, in DA neurons has activation characteristics that are similar to those of
ICa,N. The steady-state half-activation voltage (V0.5)
associated with ICa,L was
set to be more hyperpolarized compared with the
V0.5 value of the
ICa,N activation characteristic.
This is consistent with observations about the relationship between
dihydropyridine- and
-conotoxin-sensitive currents (Kasai and
Neher 1992
; Regan 1991
) (see Fig. 3).
Recent experiments have shown that calcium chelation with
bis-(o-aminophenoxy)-N,N,N',N'-tetraacetic acid
(BAPTA) prolongs the plateau of the square-wave oscillation, which is
blocked by nifedipine (Ping and Shepard 1997
). This is
consistent with the hypothesis that the L-type current, like
ICa,N, is inactivated by
[Ca2+]i. The inactivation
of ICa,L has been modeled as
calcium dependent (Johnston and Wu 1995
). A
Michaelis-Menten relationship,
(fCa,L in Fig.
4) has been used to characterize the
calcium-dependent inactivation of this current, and its parameters have
been adjusted to give reasonable fits to SOP and square-wave
oscillation data. The equations for
ICa,L can be found in Table 1.

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Fig. 4.
Steady-state calcium-dependent characteristics associated with various
ionic currents. A: summary of calcium-dependent ionic
currents. fCa,L,
fCa,N, and pK,Ca
are indicated. B: activation of
IK,Ca,SK by calcium during slow oscillation
potential (SOP). IK,Ca activation variable
(pK,Ca) is plotted against the internal
calcium concentration, [Ca2+]i. Extent of
[Ca2+]i excursion during SOP is indicated by
the shaded region. C: inactivation of
ICa,L and ICa,N
by the calcium-dependent fCa,L and
fCa,N, respectively, during square-wave
oscillations. Extent of [Ca2+]i excursion
during the square-wave oscillations is indicated by the shaded
region.
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COMBINED HVA CALCIUM CURRENT
ICA,HVA.
The remainder of the HVA current has been lumped together as
ICa,HVA. This includes the
-agatoxin-sensitive P-type current and the R-type current, which is
not selectively blocked by any known pharmacological agents. The
parameters used for voltage-activation (V0.5,act =
10.0 mV,
k = 10 mV) are similar to those obtained by
Cardozo and Bean (1995)
, which were
5.0 mV and 13.8, respectively. The parameters for voltage-inactivation are similar to
those obtained by Kang and Kitai (1993b)
,
(V0.5,inact =
48 mV,
k = 5.0 mV), which were
48 mV and 6.0, respectively.
CALCIUM-ACTIVATED POTASSIUM
CURRENT
IK,CA.
Dopaminergic neurons are known to contain at least two types of
calcium-activated potassium currents, the BK and SK currents (Shepard and Bunney 1988
; Silva et al.
1990
). The BK, or maxi type channel, has a high channel
conductance and voltage- and calcium-dependent activation and is known
to be blocked selectively by iberiotoxin and nonselectively blocked by
TEA. Specifically, Silva et al. have shown that
IK,Ca,BK is blocked by TEA in DA neurons
(Silva et al. 1990
). The smaller conductance SK channel has a purely calcium-dependent activation and obtains its voltage dependence indirectly via the voltage dependence of calcium entry mechanisms (e.g., calcium channels) (Hille 1992
). It is
blocked selectively by apamin and is insensitive to TEA. For the
purposes of simulation, we have assumed that apamin partially blocks
IK,Ca,SK, leaving it at ~20% of its
original strength. Further, we have assumed that
IK,Ca,SK is activated by intracellular
Ca2+ through the Boltzman-type function
pK,Ca with a half-activation calcium
concentration of KM,K,Ca given in Table
2 and seen in Fig. 4. The calcium
half-activation concentration value KM,K,Ca has been set to 190 nM. This is reasonable in view of experimental observations that the maximum calcium excursion of DA neurons during
SOP is ~200 nM (Callaway and Wilson 1997
). Because
IK,Ca,BK is inactivated by TEA application
as mentioned earlier, it cannot be essential for the slow underlying
oscillations (SOP). In the present modeling study, we have excluded
this active spiking mode of oscillation, and therefore we have not
included IK,Ca,BK in the model.
TRANSIENT OUTWARD CURRENT
IA.
The 4-aminopyridine (4-AP)-sensitive current,
IA, has been observed in DA neurons
and plays a role in the regulation of action-potential frequency by
slowing the recovery of membrane potential to baseline levels
(Silva et al. 1990
). The steady-state activation
(p) and inactivation (q) characteristics of
IA (Fig. 2F) were
determined by fitting published voltage-clamp data obtained from DA
neurons of the substantia nigra pars compacta (SNc) of the rat
(Silva et al. 1990
). The cell was held at
90 mV and
clamped to various voltages in the range
60 to 75 mV in 15-mV
increments. Equations describing this current are given in Table 2, and
Fig. 2B shows the model-generated fits to voltage-clamp data.
HYPERPOLARIZATION-ACTIVATED CATION
CURRENT
IH.
The hyperpolarization-activated cation current
(Ih) has been shown to be important
for pace-making activity in thalamic and cortical neurons
(McCormick and Pape 1990). In DA neurons, however, Ih has been shown to have a negligible
effect on spontaneous firing, resting membrane potential, and normal
resting conductance (Mercuri et al. 1995). Our
simulations confirm that Ih is not
essential for the generation of the SOPs or for setting the normal
resting input impedance of the model (see Table
3). However,
Ih is the major component of the input
impedance of the membrane at more hyperpolarized potentials (about
100 mV).
Ih activates with hyperpolarization beyond
approximately
60 mV and exhibits no inactivation. Its reversal
potential Eh is
35 ± 4 mV
(Mercuri et al. 1995), which lies between the reversal potentials for potassium and sodium currents (about
70 and 60 mV,
respectively). Ih therefore is modeled as a
mixed cation current predominately permeable to sodium and potassium
using a previously developed characterization (Demir et al.
1994
). Model-generated fits to voltage-clamp data from rat by
Mercuri et al. (1995) are shown in Fig.
2C. The cell was held at
57 mV and clamped to various potentials from
141 to
69 mV in 12-mV increments.
DELAYED-RECTIFIER CURRENT
IK.
The delayed-rectifier current, IK, was
characterized by fitting published voltage-clamp data from rat DA
neurons (Silva et al. 1990
) in which the cell was held
at
40 mV and clamped to 20 mV in 10-mV increments (Fig.
2D). The equations characterizing IK are given in Table 2. As mentioned
in the introduction, SOPs are observed after TTX application in the
presence IK and enhanced after its
blockade by TEA. The significance of this observation is that
IK is not necessary for SOP and,
unless otherwise noted, is not present in either SOP or square-wave oscillations.
BACKGROUND CURRENT
IB.
Although important, there is little quantitative experimental data upon
which to base a mathematical description of
IB. The linear background current is
modeled as a sum of three separate components: a sodium current
(IB,Na) a calcium current
(IB,Ca), and a potassium component
(IB,K) The equations for these linear leak conductances are given in Table 4.
The input impedance of the model was obtained by clamping the membrane
to
60 mV from a holding potential of
50 mV and observing the evoked
transient current. A
V of 10 mV was divided by the
resultant
I to obtain an input impedance of 365 M
,
which is within the experimentally reported range for these neurons
(Grace and Onn 1989
; Johnson and North
1992
; Yung et al. 1991
).
PUMP AND EXCHANGER CURRENTS
ICA,P,
INA,CA, AND
INA,K.
The equations (see Table 5) for the
sodium-calcium exchanger and the calcium pump are adapted from
Canavier et al. (1991)
and the weakly
voltage-dependent sodium-potassium pump from Lindblad (1996)
. The value for the half-activation calcium concentration for ICa,pump
(KM,CaP) was set to 0.50 µM, which is
within the range of experimental observations for other preparations
(Lichtman et al. 1981
; Michaelis et al.
1983).
Fluid compartment model
Calcium dynamics are an important part of DA neuron activity
because Ca2+-mediated processes are hypothesized
to be involved in the generation of the subthreshold oscillations, and,
as a result, the mechanisms of rhythmic firing. The material balance on
calcium depends on three processes: entry, extrusion, and buffering of
calcium. The differential equation representing the rate of change of
cytosolic Ca2+ is
|
(3)
|
where Voli is the cytosolic volume in
nanoliters, F is Faraday's constant in C/mol, and
[B]i is the concentration of the internal calcium buffer in millimolar with n binding sites
for Ca2+. nBi
gives the total concentration of binding sites for
Ca2+. The volume is calculated based on a
spherical soma of radius 15 µm for a total intracellular volume of
0.0141 nl.
The buffering of intracellular calcium has been modeled as binding to
an intracellular protein such as calmodulin. The formula for the buffer
(Robertson et al. 1981
) is given by
|
(4)
|
where OC is the buffer occupancy
or the fraction of sites that already are occupied by
Ca2+ and therefore unavailable for binding. All
binding sites are considered as a single population and assumed to be
independent. This assumption ignores the nonlinearities that occur as a
result of multiple cooperative binding sites.
 |
COMPUTATIONAL ASPECTS |
The complete system consists of 15 state variables: 3 differential
equations describing the time rate of change in membrane voltage
(V), internal calcium concentration
([Ca2+]i), and fractional
occupancy of the calcium buffer (OC)
and 12 other differential equations characterizing voltage-dependent gating variables. Equations 1, 3, and 4, along
with the tables for individual membrane currents (Tables 1, 2, 3, 4,
and 5); parameter values (Table 6); and
initial conditions (Table 7), contain all
the information necessary to carry out the simulations presented in
this paper. The units used in the model are time in milliseconds (ms),
voltage in millivolts (mV), concentration in millimoles/liter (mM),
current in nanoamperes (nA), conductance in microsiemens (µS),
capacitance in nanofarads (nF), volume in nanoliters (nl), and
temperature in degrees Kelvin (°K). All simulations were performed on
a Sun Ultra I by forward integration of the coupled system of
differential equations using an implicit fifth-order Runge-Kutta method
with variable step size designed for stiff systems of differential
equations (Hairer and Warner 1990
).
 |
RESULTS |
In addition to providing a quantitative test that the current
descriptions derived from voltage-clamp data are sufficient to
reproduce experimental data generated under current-clamp conditions, simulations of the dopaminergic neurons in this study enable us to gain
insight into the ionic mechanisms responsible for rhythmic firing as
well as to utilize this insight to make experimentally testable
predictions. Figure 5 summarizes the model results for the two oscillations of interest and compares model-generated and
experimentally observed behavior under different simulated experimental
conditions. Because TTX must be present to reveal slow sinusoidal
oscillations, INa has not been
modeled. TEA application is simulated by letting
K = 0. which results in slow
oscillatory potentials (SOPs, Fig. 5A). Simulation of the
application of apamin blocks most of the outward current
IK,Ca,SK and results in square-wave oscillations (Fig. 5B). In either case, the model provides
SOPs or square-wave oscillations that agree qualitatively with data from Ping and Shepard (1996)
.

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Fig. 5.
Model-generated and experimentally observed oscillations of DA neurons
under influence of blocking agents. A: model and data
comparison of the SOP. IK shut off to
simulate TEA application. B: model and data behavior
during oscillations produced by partial blockage of
IK,Ca,SK to simulate apamin application.
Data from rat scanned and digitized from Fig. 2C of Ping and
Shepard (1996) .
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Slow oscillatory potentials
Figure 6 shows the ionic currents
underlying the SOP. Figure 6A shows the SOP (same as Fig.
5A), whereas the intracellular calcium concentration is
shown in B. The major currents involved in SOPs are
ICa,L and
IK,Ca, whereas
ICa,N is small relative to ICa,L (Fig. 6C). Note that
ICa,T and
ICa,HVA (E) are not fully activated because the SOPs occur in a potential range that is outside
their range of activity. On the other hand, the calcium inactivation
characteristic of ICa,N (Fig.
4A) keeps the magnitude of this current below that of
ICa,L in the calcium range of
oscillation (nominally, 180-200 nM). The other model currents consist
of background and pump currents as seen in Fig. 6D as well
as the transient outward current, IA
(E). These currents are important in biasing the cell in the
appropriate range of membrane potential and cytosolic Ca2+ concentration. The
hyperpolarization-activated mixed cation current, Ih, can be eliminated with little
impact on the shape and frequency of the SOPs (not shown).

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Fig. 6.
Underlying currents of SOP. A: membrane potential during
SOP. B: [Ca2+]i oscillations
conform well to data from Callaway and Wilson (1997) .
C: ICa,L and
IK,Ca are the main ionic currents during the
SOP with minor contribution from ICa,N.
D: background currents IB,
sodium-dependent currents INa,K and
INaCa, and the calcium pump,
ICa,pump. E: remaining
calcium currents ICa,T and
ICa,HVA and the transient outward current
IA.
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Mechanisms underlying SOP
The mechanism underlying the SOP (Fig. 6A) can
be understood by considering the calcium- and voltage-dependent
processes in the model. Broadly, activation of
ICa,L depolarizes the membrane and
leads to increased
[Ca2+]i (Fig.
6C), which activates
IK,Ca,SK (Fig. 4B).
Activation of IK,Ca,SK, in turn,
hyperpolarizes the membrane, decreases
ICa,L, and reduces
[Ca2+]i.
Clearly, the calcium transient (Fig.
7C) lags behind the calcium
current ICa,L, whereas the potassium
current (IK,Ca,SK) is in phase. As
IK,Ca,SK increases it limits the
membrane depolarization due to ICa,L
and institutes a first stage of repolarization (slope L1 in Fig. 7A). The peak of
ICa,L is long lasting, and it is only after it begins to decline and concomitantly,
IK,Ca,SK peaks, that a second,
stronger phase of repolarization is brought about (slope
L2 in Fig. 7A).

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Fig. 7.
Important currents underlying sinusoidal oscillations. Membrane
potential during SOP (A) responds weakly (with slope of
line L1) at first to the activation of
IK,Ca (C).
ICa,L (B) remains active,
increasing [Ca2+]i (D).
Finally, ICa,L regeneratively deactivates,
hyperpolarizing the membrane (with slope of line
L2).
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The model-generated calcium transient during SOP conforms well to
experimental observations (Callaway and Wilson 1997
).
Calcium levels vary within the reported range of 160 and 220 nM, and
the calcium oscillations lag the voltage oscillations by ~90°.
Nedergaard et al. (1993) have shown that neither
Ni2+ nor
-conotoxin can block the SOP in
guinea pig SNc DA neurons, whereas nifedipine application alone is
effective. Figure 8 shows that blockade
of ICa,T,
ICa,N, and
ICa,L in the model for the purposes of
simulating Ni2+,
-conotoxin, and nifedipine
application, respectively, conforms to these experimental results. The
increase in amplitude after blockade of
ICa,T is due to the removal of a
depolarizing drive and reduction in calcium entry, which allows
ICa,L to remain active longer, raising
the peak potential.

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Fig. 8.
Simulation of Ni2+, -conotoxin, and nifedipine
application. A: control. Application of Ni2+
(B) and -conotoxin (C) produce small
changes in the character of the SOP, whereas nifedipine application
(D) eliminates the SOP.
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Square-wave oscillations
The component currents underlying the square-wave
oscillations are shown in Fig. 9 in the
same order as those for the sinusoidal case (Fig. 6). The major
depolarizing currents are again ICa,L and ICa,N; however, the outward
K+ current
IK,Ca,SK has been blocked. Our model
predicts that the mechanism for the square-wave oscillations need not
depend on the presence of a distinct hyperpolarization current in this
mode of oscillation. Without the activation of
IK,Ca,SK by calcium, internal calcium
concentration continues to rise and achieves a higher peak level than
in sinusoidal oscillations (Fig. 9B), and the frequency of
oscillation decreases. ICa,L is
inactivated more strongly at the elevated levels of
[Ca2+]i. As
ICa,L declines due to inactivation,
the remaining "residual" currents
(IRes, Fig.
10) sum to repolarize the membrane.

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Fig. 9.
Underlying currents of square-wave oscillations. A:
membrane potential during square-wave oscillations. B:
[Ca2+]i variation during square-wave
oscillations is greater than that seen during SOPs. C:
ICa,L is the main ionic current. The
residual IK,Ca,SK current and
ICa,N are minor contributers in this mode of
oscillation. D: calcium and sodium background currents,
IB,Ca and IB,Na.
E: remaining calcium currents
ICa,T and
ICa,HVA. F: sodium-dependent
pump currents, INa,K and
INa,Ca. G: transient outward
current (IA), potassium background current
(IB,K), and the calcium pump
(ICa,P).
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Fig. 10.
Time-expanded view of square-wave oscillations. A and
B: plotted as before and show the membrane potential and
cytosolic Ca2+ during square-wave oscillations.
C: ICa,L and the sum of the
remaining currents (IRes = Itot ICa,L).
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Functional tests
Certain functional tests reveal the robustness of the model and
provide explanations of experimental observations. It has been shown
that application of TTX alone will result in lower-amplitude and
higher-frequency sinusoidal oscillations than those resulting from the
coapplication of TTX and TEA (Nedergaard and Greenfield 1992; Kang and Kitai 1993a
). Figure
11B is a simulation of
membrane potential under TTX application alone. The resulting
oscillations in this case are due to interactions between
ICa,L and the two outward currents
IK, and
IK,Ca,SK. Recall from Fig. 7, that due to its calcium-dependent activation,
IK,Ca,SK reaches its maximum strength
~100 ms after membrane potential peaks. A voltage-dependent current
with relatively fast activation, on the other hand, would lead to a
hyperpolarizing drive that coincides with the peak voltage. This drive
leads to an earlier onset of hyperpolarization, leading to a lower
amplitude and a shorter period.

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Fig. 11.
Effect of TEA on the model. A: control SOP in the
presence of TTX and TEA. B: SOP with bath application of
TTX alone results in lower amplitude oscillations.
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The mechanisms of the square-wave oscillations also were tested via
model simulation and functional tests. Ping and Shepard (1997)
have shown that chelation of DA neurons during
square-wave oscillation by intracellular injection of the calcium
buffer BAPTA prolongs the plateau phase of the oscillation. Because
BAPTA is a calcium buffer, we assume that the effect of adding more
buffer to that already present in the cytosol is similar to increasing the concentration of the internal calcium buffer
[B]i. The value of [B]i
used nominally in the model (0.050 mM) was increased to 0.080 mM to
simulate calcium chelation. Figure 12
shows the model-generated response to this increase in
[B]i. At first, the results seem counterintuitive. An increase in buffer concentration should lead to a
decrease in the rate of change of
[Ca2+]i, resulting in a
smaller excursion in calcium concentration. However, on closer
inspection of the figure, we see that the increase in
[B]i also leads to a decrease in the rate of
ICa,L inactivation, which causes a
larger calcium influx and a greater excursion in [Ca2+]i.

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Fig. 12.
Increased buffering and calcium-mediated currents. A:
simulation of chelation by increasing the concentration of the internal
calcium buffer results in increased plateau duration as compared with
the control case. B: increased buffering delays the rise
in [Ca2+]i which allows
ICa,L (C) to remain active
longer. There is little change in the strength of the other
calcium-dependent currents, INa,Ca and
ICa,P.
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In addition to changes in buffer concentration, the calcium
availability can be changed by adjusting other calcium-dependent processes in the model. Two additional currents that have the potential
for altering calcium availability in the cytosol are the calcium pump,
ICa,pump, and the sodium-calcium
exchanger, INa,Ca. Increasing the
strength of either calcium extrusion mechanism should decrease
cytosolic Ca2+ availability and yield results
similar to increasing [B]i. In addition to
increasing calcium extrusion, calcium availability also can be
restricted. Because we have considered
ICa,N nonessential for SOPs, it can be
reduced without altering the self-excitability of the model. Figure
13 summarizes the results of these
adjustments. The maximal conductances for
INa,Ca,
ICa,pump, and
ICa,N
(KNa,Ca,
Ca,pump, and
Ca,N, respectively) were each
changed by 5% in these experiments. We found that
INa,Ca alone was able to affect duty
cycle, i.e., plateau duration could be changed without a significant
change in the period of the oscillation. Figure 13A shows
the control square-wave oscillations followed by three different schemes for increasing plateau duration by reducing cytosolic Ca2+. Of the three adjustments, only increasing
INa,Ca (Fig. 13D) has the
desired effect. Decreasing ICa,N
(B) and increasing ICa,pump (C) in fact have very little effect.

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Fig. 13.
Altering plateau duration. Control square-wave oscillations are shown
(A) followed by attempts at increasing plateau duration
by decreasing ICa,N (B),
increasing ICa,pump (C), and
increasing INa,Ca (D).
Lightly and darkly shaded regions divide the control period at the
minimum, and the dashed line marks the control period from that minimum
(P). As the lightly shaded regions show, only increasing
INa,Ca had the desired effect of increasing
plateau duration. Additionally, INa,Ca
adjustment allows for control of plateau duration without significant
change in oscillation frequency.
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DISCUSSION |
We have developed a semiquantitative model that can mimic the
behavior of DA neurons under a wide range of experimental conditions, including: SOPs in the presence of TTX and TEA; the changes in the
period and amplitude of sinusoidal oscillations in response to TEA
application; square-wave oscillations in the presence of TTX, TEA and
apamin; and the increase in the duration of the plateau phase of
square-wave oscillations observed in response to calcium chelation. The
key feature of the model is that it can be used to explore the putative
mechanisms underlying SOP and square-wave oscillations.
Somatic origin of calcium-dependent oscillations
This model is a representation of the soma only and therefore is a
first approximation of the full dynamics in a dopamine neuron, which
has a dendritic tree whose branches frequently extend >500 µm from
the soma. The data on which the current descriptions are drawn are
characteristic of the soma and proximal dendrites because the studies
on which the descriptions are based (Cardozo and Bean
1995
; Kang and Kitai 1993b
; Silva et al.
1990
) were conducted in acutely dissociated or thin (150-200
µm) coronal slice preparations. In both of these types of
preparations, the dendritic arbor can be truncated. In particular,
ICa,T may be localized preferentially on distal dendrites because only 11 of 15 cells in thin coronal slice
(Kang and Kitai 1993b
) and none in the dissociated
preparation (Cardozo and Bean 1995
) exhibited this current.
On the other hand, the presence of large calcium currents on the distal
dendrites has been demonstrated in thick (400 µM) coronal sections
(Dunia et al. 1996
). Nonetheless, the assumption that
the somatic currents and calcium dynamics drive the SOP is justified by
several experimental observations. First, during the SOP the calcium
transients in the soma are a substantial fraction of those observed in
the dendrites (Callaway and Wilson 1997
) despite the
large disparity in surface area to volume ratio. Hence there is a
significant amount of calcium entry into the soma. Second, the PLD can
be evoked by brief depolarizations of the soma, and its rate of
activation is strongly dependent on the membrane potential in the soma
(Grace and Onn 1989
). Finally, one study found that
sectioning the distal dendrites of DA neurons had no significant effect
on the pacemaker firing frequency (Nedergaard and Greenfield
1992). Assuming the firing frequency is strongly influenced by
the SOP, the distal dendrites do not contribute significantly to the
SOP as recorded at the soma.
Predictive value of model can aid in experimental design
Although the modeled voltage- and calcium-dependence of
ICa,L is consistent with available
data, the precise form of the voltage dependence and even the presence
of calcium inactivation of this current have not been directly
demonstrated in rat dopamine neurons. Hence the model description of
this current can be considered a prediction (see following text). In
addition, model predictions that
INa,Ca controls the duty cycle (Fig.
13D) could be tested if a specific
INa,Ca blocker could be applied to the bath.
Homogeneity of DA neurons within and across species
Homogeneity of midbrain DA neurons across species frequently is
assumed (Cardozo 1993
; Lacey et al. 1989
;
Sanghera and German 1984
; Tepper et al.
1987
; Yung et al. 1991
) because the PLD,
depolarized spike threshold, long-duration action potentials, inward
rectification, and other characteristics are observed across species.
It has been argued (Kotter and Feizelmeier 1998
) that
scaling alone does not ensure constant activity patterns between
species, but additional compensatory mechanisms (LeMasson et al.
1993
; Turrigiano et al. 1994
) likely are
involved. The model in the current study applies specifically to rat
mesencephalic neurons because it is based on data from rat
(Cardozo and Bean 1995
; Kang and Kitai
1993a
,b
; Ping and Shepard 1996
; Silva et
al. 1990
).
Within a species, these neurons also exhibit a degree of
electrophysiological homogeneity (Cardozo and Bean 1995
;
Johnson and North 1992
; Yung et al. 1991
)
and similar responses to stimuli (Schultz et al. 1995
),
although they are located in three adjacent regions: the ventral
tegmental area (VTA or A10), the substantia nigra pars compacta (A9),
and the retrorubral area (A8). Although we have treated the DA neurons
as a homogenous population, some have suggested that there are
subpopulations of DA neurons (Nedergaard and Greenfield
1992; Shepard and Bunney 1988
), and there is
evidence of some heterogeneity in their neurochemistry, pharmacology,
and electrophysiology (Roth and Elsworth 1995
). In
addition, the mix of calcium currents in these cells can vary with age
(DeFazio and Walsh 1995
).
Does the dihydropiridine-sensitive ICa,L
mediate the SOP in all cases?
We have assumed that dihydropyridines block the SOP, based on the
following data. Mercuri et al. (1994) reported
that 30 µM nifedipine blocked spontaneous pacemaker activity in rats,
and by inference, the SOP as well. This is consistent with findings in
rat that the calcium conductance mediating the SOP is dihydropyridine sensitive (Ping and Shepard 1997
) and with those of
Kang and Kitai (1993a
,b
), who found that the calcium
channel blocker Cd abolished the SOP in rat. These findings are also
consistent with the results of Nedergaard et al. (1993),
who reported that 0.5-20 µM nifedipine abolished the SOP in guinea
pig. On the other hand, Fujimura and Matsuda (1989)
reported that although the calcium channel blockers Cd2+
and Co2+ as well as Ca-free saline blocked the SOP in
guinea pig, 100 µM nifedipine did not. Furthermore Yung et al.
(1991)
reported that 500 µM Ni2+ abolished the
SOP in guinea pig, whereas Nedergaard et al. (1993) reported that 500 µM Ni2+ only slightly attenuated it.
Thus, in guinea pig at least, there are conflicting data. However, the
evidence argues for a dominant role for
ICa,L. For example, Kang and Kitai
hypothesized that ICa,N was responsible for
the PLD; however, they did not show that
-conotoxin abolishes the
SOP. Nedergaard et al. (1993) did perform this test in
guinea pig neurons and found that 1-10 µM
-conotoxin did not
block the SOP; this casts doubt on the importance of
ICa,N in generating these oscillations. The
data of Yung et al. (1991)
suggest a role for
ICa,T because Ni2+ selectively
blocks this current (but see Ellinor 1993
). Our
simulations have shown that it is very unlikely that
ICa,T, as characterized by the voltage-clamp
data of Kang and Kitai, contributes significantly in the voltage range
of the PLD. Similarly, although Kang and Kitai have shown that
ICa,N exists in the correct voltage range for the PLD, the simulations show that to generate both SOP and square-wave oscillations under the appropriate conditions, a calcium current with different activation characteristics from
ICa,N is necessary. We predict that
ICa,L, as we have characterized it, is this
missing current. The voltage range of activation of this current is
more negative than that typically associated with L-type calcium
currents; however, similar L-type Ca2+ channels have been
found in several locations, including guinea pig motoneurons
(Hsiao et al. 1998
), turtle motoneurons (Russo and Hounsgaard 1996
), and rat supraoptic neurons (Fisher
and Bourque 1996
). Further experimental work is required to
verify the validity of this prediction and determine the activation
characteristics of ICa,L in DA neurons.
It is widely accepted that a persistent, TTX-sensitive sodium current
also contributes to the SOP (Grace and Onn 1989
;
Nedergaard et al. 1993; Ping and Shepard
1996
). In the dopamine neurons of the retina, which exhibit
similar regular pacemaker activity in a slice preparation
(Feigenspan et al. 1998
), the sodium current, and not a
calcium current, is the dominant pacemaking current. Hence under
different circumstances, different currents may contribute to the SOP
in differing proportions, perhaps due to self-regulatory compensatory
mechanisms that dictate that dopamine cells are intrinsic pacemakers.
Comparison with previous models
Li et al. (1996)
developed a model that replicated
both NMDA- and apamin-induced burst firing but ignored the SOP and
regular pacemaker firing. Their model was conceptual in nature with
arbitrary parameters not based on the voltage-clamp data of the
particular currents in DA cells. The mechanism that they proposed for
NMDA-induced bursting is that same as that proposed by a later, more
physiologically based model by Canavier (1999)
, namely
the interaction of the NMDA-induced current
(INMDA) and the sodium pump.
Regenerative voltage activation of
INMDA is postulated to be responsible
for the depolarization during a burst, and sodium accumulation due to
entry via INMDA is hypothesized to
activate the electrogenic sodium pump, which is a net outward current,
until a regenerative hyperpolarization due to the voltage-dependent
closing of INMDA channels. Sodium is
removed during the hyperpolarizing phase, allowing the cycle to begin
again. On the other hand, the mechanism proposed by Li et al.
(1996)
for apamin-induced, calcium-dependent bursting assigns a
crucial role to ICa,T rather than
ICa,L, which is unlikely in view of
the more physiologically based simulations in the current study. In
addition to the models described above, Kotter and Feizelmeier
(1998)
also have modeled DA neurons, but it is difficult to
compare the models because that study did not discuss underlying ionic
mechanisms, but merely showed that different activity patterns, such as
single-spike firing and burst-firing, could arise from scaling
parameters to account for different neuronal sizes.
Whereas the model of Canavier (1999)
focused solely on
sodium dynamics and the current study focuses solely on calcium
dynamics, in reality these systems operate in tandem and interact with
each other. Further work is required to elucidate this interaction, as
apamin is known to facilitate NMDA-induced burst firing (Seutin et al. 1993
). As far as the relative contributions of these
types of burst-firing to the situation in vivo, it is well known that NMDA promotes burst firing in vivo (Overton and Clark
1992
) and that DA neurons receive excitatory amino acid input
(Carter 1982
; Robledo and Feger 1990
;
Scarnati et al. 1986
). To date, an endogenous substance
that modulates IK,Ca,SK in vivo has
not been identified. Some have argued that apamin-induced burst firing
resembles in vivo firing patterns more closely than burst firing
induced by NMDA (Overton and Clark 1997
), so a role for
IK,Ca,SK may yet be discovered in
addition to the well-established role for
INMDA.
Nullcline analysis was an invaluable tool in the development of
the model. Repeated, time-consuming simulations were avoided, and
considerable insight into the model was gained through nullcline analysis, which is a subset of the more general phase space analysis of
nonlinear systems (see Rinzel and Ermentrout 1998
).
A multidimensional system can be analyzed with two-dimensional
nullcline methods if it can be reduced to a two-dimensional system.
This is done by eliminating the kinetics of all but two state variables
and assuming that they can satisfactorily characterize the system. As
we have seen, the slow cyclic variation in
[Ca2+]i along with
membrane potential are essential contributors to oscillation in DA
neurons. Therefore we reduce the system to a two-dimensional one and
set all state variables to their steady-state values. We obtain the
nullclines for the reduced system by setting all state variables to
their steady state values and numerically solving the following
equations:
Rough tunings of the model were performed through analysis of the
nullclines of the reduced system. First, the slope of the potential
nullcline at the equilibrium point of the reduced system was a good
predictor of the stability of the complete system. Using this rule, we
were able to make large changes in the system equations while
maintaining the model in the correct range for oscillation. Also,
because [Ca2+]i varies
slowly relative to the potential, the trajectory in phase plane remains
more or less on the potential nullcline, jumping between stable
branches when an unstable region is encountered. This was very helpful
in estimating from the nullclines, the extent of
[Ca2+]i and V
excursion without running the full simulations.
The equations describing the dopaminergic neuron model are
contained within this appendix. Expressions for the transmembrane currents are given in Tables 1-4. The pump and exchanger currents are
given in Table 5. The gating variables in the model are n, dT,
fTf,
fTs,
dN,
dL,
dHVA,
fHVA, p,
q1, and
q2. The equations for
(V) and
z(V) can be found in Tables 1-3. A
listing of model parameter values is given in Table 6, and the initial
conditions needed to run the model in both modes of oscillation are
given in Table 7 with SOPs and square-wave oscillations in the
left and right columns, respectively.
This work was funded by a Biomedical Engineering Research grant
from the Whitaker Foundation and National Institute of Neurological Disorders and Stroke Grant NS-37963.
Address for reprint requests: C. Canavier, Dept. of Psychology,
University of New Orleans, New Orleans, LA 70148.
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.