INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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 R_min = 0.42, R_max = 7.96, Sf380/Sb380 = 10.98, fura k_D = 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
where Sb380/Sf380 is the ratio of fluorescence of bound and free fura-2 as used in Grynkiewicz et al. (1985), F/F is the change in fluorescence at 380 nm divided by the fluorescence measured immediately after the opening of the shutter, corrected for the autofluorescence as described in the following text. This formula is derived in the APPENDIX and was employed because it did not require a measurement of the maximal practically possible fluorescence change, which otherwise must be measured by loading the cell with calcium. It is also suitable for experiments in which the calcium concentration decreases below that present immediately after the shutter is opened. Fluorescence measurements were corrected for bleaching during the trial by measuring the bleaching that occurred when the cell was held hyperpolarized, filtering the resulting curve at 3 Hz, and subtracting the resulting curve from trials in which the cell was depolarized, or was allowed to oscillate. Autofluorescence correction was performed by subtraction of measured autofluorescence of a nearby region of the slice from the measured initial value of F.

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
The maximal conductance (_K or _Ca), half-activation voltage (V_HK or V_HCa) and the slope factor for activation (V_SK or V_SCa) were the only free parameters. Omission of the kinetics of activation of these conductances from the simulations did not alter the results due to the slow variation in membrane potential relative to the activation time constants in all cases. Neither of the voltage-dependent conductances inactivated. The calcium-dependent potassium conductance was represented as dependent on the fourth power of calcium concentration, to best represent the known characteristics of the SK channel (Köhler et al. 1996) and had only maximal conductance (_KCa) and half-activation calcium concentration (K_Ca) as free parameters
A small linear leak conductance, with no dependence on voltage or calcium, was included in all the simulations to keep input resistance bounded at membrane potentials below the activation range of the voltage-dependent conductances. The reversal potential for the leak current was set at 50 mV rather than at the potassium equilibrium potential (90 mV) to prevent the occurrence of a stable equilibrium at E_K. This was done to best represent the fact that dopaminergic neurons do not have such a stable equilibrium point and that the actual leak current in those cells (active at membrane potentials below those occurring associated with the spontaneous oscillation) is largely due to a hyperpolarization-activated cation current with a reversal potential positive to E_K (e.g., Mercuri et al. 1995).

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
where dl is the surface area of the membrane, (d/2)²l is the volume of the compartment, P_max is the maximum pump rate surface density [in µm/s as in Zador and Koch (1994)], K_P is the dissociation constant for the pump (in nM), and  represents a simplification of the effect of calcium buffering based on the assumptions that calcium interacts with the buffer more rapidly than it diffuses, and so can be thought of as instantaneous (Wagner and Keizer 1994; Zador and Koch 1994). It has no units, as it reflects the ratio of free to total calcium. For one fixed buffer B_S at a total concentration of [B_S]_T and with dissociation constant K_S
When the buffer is far from saturation,  becomes a constant. In all simulations, calcium buffering was represented using a constant ratio of free to total calcium.

Assuming that the pump is also not saturable [([Ca²⁺]_i/K_p) 1]
The assumption that the pump is not saturable reflects our uncertainty about which of the known calcium membrane pump mechanisms are responsible for extrusion and the low calcium concentrations observed in experiments (and expected from simulations). Slow calcium removal via the endomembrane system or mitochondria was not included in the simulations nor was calcium-dependent release from calcium stores. Although these are likely to play a role in dopaminergic neurons, including them would require estimates of their time courses, and none are available. It should be noted that addition of intracellular stores would reduce the effect of intracellular calcium diffusion kinetics as uptake and release of intracellular stores are distributed in the cytoplasm so do not produce a spatial gradient.

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
and
in which [B_M]_T is the total concentration of the mobile buffer, K_M is its dissociation constant, D_M is its diffusion constant (in µm²/s), D_Ca is the diffusion constant of calcium in the absence of buffers, and ₀ is the value of  calculated for a background calcium concentration [Ca²⁺]₀. The total amount of buffering is represented by , which goes from 0 to 1 with 0 meaning that all free calcium is immediately removed by buffering and 1 meaning no buffering. If all buffers were fixed, the diffusion constant of calcium would be reduced in proportion to the total buffering. The presence of some proportion of mobile buffer mitigates this, increasing the effective diffusion constant. In the experiments, the presence of fura-2 (a mobile calcium buffer) ensures that not all buffer will be fixed. For the simulations, D_app was treated as a parameter.

For the simulations presented here, the buffers and the calcium extrusion mechanism were represented as nonsaturable. For the pump, this implies [Ca²⁺]_i K_P, and for the buffers, it implies that [Ca²⁺] K_M and [Ca²⁺] K_S. The buffering value  thus becomes independent of calcium concentration, and represents the total concentration of buffer
The effects of buffers on diffusion were treated separately from buffering, to represent changes in the ratio of mobile to immobile buffers. The diffusion rate was represented using 40 discrete calcium shells following Sala and Hernández-Cruz (1990), with exchange determined by a variable diffusion coefficient. Although constraints on diffusion and buffering share common cellular substrates (e.g., Gabso et al. 1997), they thus were separated in the simulations to allow easy comparison of their effects.

The resulting equation for calcium at the interior surface of the membrane was

(1)

in which r is the radial distance from the center of the cylinder, D_app is the apparent diffusion constant for calcium, I_Ca is the calcium current,  is the ratio of free to total calcium, z is the valence of calcium, F is Faraday's constant, and d is the diameter of the compartment. The ratio of 4/d that appears in the calcium influx and efflux terms represent are the ratio of surface area to volume for a cylinder. Longitudinal diffusion of calcium was ignored. For interior regions of the compartment, only the diffusion term was used.

For voltage, the usual current balance equation was applied

(2)

in which I_KCa is the current through the calcium-dependent potassium conductance, I_K is the (TEA sensitive) potassium current, I_L is the leak current, as described in the preceding text, R_i is the longitudinal resistivity of the cytoplasm, and C is the membrane capacitance per unit surface area. The last term in the numerator represents the effect of longitudinal current flow.

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.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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.

View larger version (80K): [in this window] [in a new window]   Fig. 1. Calcium measurement and recording from dopaminergic neuron in the substantia nigra, pars compacta. A, top: IR-DIC image of the pars compacta, including the neuron from which measurements were taken (*). Bottom: same neuron, with the field translated to follow an in-focus dendrite, using epi-fluorescence imaging with excitation wavelength of 380 nm. Calcium measurements in B are taken using the correspondingly colored boxes shown in A. B: calcium concentration and membrane potential for the cell shown in A. The cell was held hyperpolarized to prevent spontaneous oscillation by passage of a small constant current (50 pA). One second after the shutter was opened, the current was removed for 10 s and the cell was allowed to oscillate. Note that in the control media, action potentials were present on most but not all cycles of the oscillation. In the presence of TTX and TEA, the oscillation continued to be evoked, and calcium oscillations were of similar amplitude. In all cases, oscillations slowed during the first few seconds after release of hyperpolarization. Dendritic calcium levels reached steady state more rapidly than somatic ones, and more distal dendrites reached steady state more rapidly than more proximal ones. Even very proximal dendrites (<10 µm from the soma) showed larger amplitude and more rapid accumulation and decay of calcium than did the somata.

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.

View larger version (36K): [in this window] [in a new window]   Fig. 2. Oscillation frequency dependence on diameter in a single compartment model of the dopaminergic neuronal oscillator. A: phase plane representation of the oscillatory mechanism. Calcium and voltage nullclines and trajectories for a 2- and a 10-µm compartment are shown. Diameter has no effect on the nullclines or their point of intersection, indicating that it does not alter the stability of the oscillation. Trajectories for the small and large compartment differ only slightly, and the difference is attributable to the time constant of the membrane (which affects the fast oscillation of the 2-µm compartment much more than that of the 10-µm compartment). B: calcium oscillation plotted as a function of time for the 2- and 10-µm compartments as shown in A. C: membrane potential oscillation for the 2- and 10-µm compartments shown in A.

Effect of limited diffusion on the single compartment model

Like changes in diameter, changes in the apparent diffusion constant for calcium (D_app) 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 D_app 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.

View larger version (55K): [in this window] [in a new window]   Fig. 3. Dependence of frequency on radial diffusion rate of free calcium in the single compartmental model. Data shown are for a 10-µm compartment. Diffusion coefficient range shown is 0-10% of the diffusion rate of calcium in sea water (600 µm2/s) (Hodgkin and Keynes 1957). Note that the diffusion coefficient affects frequency only for very low values (1% of the diffusion rate in sea water) and reduces the amplitude in addition to increasing the frequency of the oscillation.

View larger version (35K): [in this window] [in a new window]   Fig. 4. Even with very restricted radial diffusion, frequency of the oscillation varies substantially with diameter over the diameter range seen for neuronal somata and dendrites. A: natural frequency of oscillation vs. diameter for the single compartment model, with parameters as in Fig. 3, and 5 different values of the calcium diffusion coefficient. B: steady-state oscillation for 3 different diameters, showing similarity of amplitude and duty cycle.

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.

View larger version (44K): [in this window] [in a new window]   Fig. 5. Five-compartment model of the dopaminergic oscillator, with exponential tapering (tapering ratio 0.5). The model dendrite was hyperpolarized by a small constant current at the large end, which was released at time 0. Compartments were synchronized by longitudinal currents that enforced phase locking of the voltage in all compartments (bottom). Calcium concentration increased gradually toward steady-state values in the largest compartment and had a small modulation. In the smallest compartment, calcium modulation overshot steady state in the beginning and maintained modulation amplitudes that were inversely related to diameter. The mean calcium level was approximately the same for all compartments at steady state.

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 (D_app = 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.

View larger version (36K): [in this window] [in a new window]   Fig. 6. The coupled frequency is a weighted mean of the frequencies of the coupled compartments. Results of altering the coupling ratio in a 6-compartment model. The largest compartment had a diameter of 16 µm, and each subsequent compartment had a diameter that was related to its predecessor by the diameter ratio. Decreases in the diameter ratio increases the rate of tapering, and reduces the diameter of every compartment except the first. The natural period of each compartment (when uncoupled from the rest) is shown in the gray lines as a function of diameter ratio. The period of oscillation for the coupled model is shown in red. For the most gradual tapers (in which compartment diameters differed little), the coupled period was approximately the mean of the periods of the individual compartments. With increased tapering, the smaller compartments were overcome by the larger currents that could be generated by the larger ones, and the coupled system was increasingly dominated by the largest compartments.

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.

View larger version (19K): [in this window] [in a new window]   Fig. 7. Measurement of calcium transients and steady-state values in voltage-clamp experiments. Left: fura-2 image (380-nm excitation) of a dopaminergic neuron in the substantia nigra, pars compacta. The cell was identified by its position, morphology, and spontaneous electrical activity before voltage-clamp recordings were begun. Boxes indicate positions of integrated calcium measurements. Right: current and calcium traces for a step from 60 to 40 mV. After an initial large outward transient and its recovery, a slow outward current dominates the current trace. After the termination of the voltage step (and subsequent brief inward transient), the outward current decays slowly during several seconds. Calcium levels increase approximately exponentially at the start of the voltage step. Calcium accumulation is more rapid in the dendrite than in the soma, but the steady-state values for the two are similar. Calcium levels decay more slowly in the soma than the dendrite.

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).

View larger version (16K): [in this window] [in a new window]   Fig. 8. Steady-state calcium concentration measurements can be used to measure the voltage sensitivity of the calcium current. An example showing calcium transients (left) and steady-state calcium concentration (right) in the soma for 5 different voltage steps from 60 mV (), and the best fitting curve for the calcium nullcline based on the steady state values (). The equation for the calcium nullcline is that for the simplified one compartment model. Values for the 3 parameters, the scaling parameter a, the half-activation voltage for the calcium current VH, and the slope factor for activation VS are those obtained from the least-squares fit. The resting calcium concentration was determined by ratiometric measurement at steady state at 60 mV.

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.

View larger version (21K): [in this window] [in a new window]   Fig. 9. The dependence of the tail current on calcium concentration gives an estimate of the calcium sensitivity of the calcium-dependent potassium current. A: an example tail from a voltage step from 40 to 65 mV, plotted vs. somatic calcium concentration on the same trial. Total time represented by the decay curve is 20 s. B: result for simulations of the same experiment using a single-compartment model with various calcium diffusion coefficients. The sigmoid shape expected for the calcium dependence of the tail current is converted to an accelerating function with decreasing intracellular calcium mobility. Although a quantitative comparison would require certainty about the value of the half-activation concentration of calcium, results suggest that either the calcium dependence is less than expected for the SK channel or that mobility of somatic calcium is limited.

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.

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