Alexey S. Kuznetsov, Nancy J. Kopell and Charles J. Wilson ...math.bu.edu/people/nk/Kuznetsov_Wilson_Kopell_2006.pdfTransient High-Frequency Firing in a Coupled-Oscillator Model of

95:932-947, 2006. First published Oct 5, 2005; doi:10.1152/jn.00691.2004 J NeurophysiolAlexey S. Kuznetsov, Nancy J. Kopell and Charles J. Wilson

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Transient High-Frequency Firing in a Coupled-Oscillator Modelof the Mesencephalic Dopaminergic Neuron

Alexey S. Kuznetsov,1 Nancy J. Kopell,1 and Charles J. Wilson21Center for BioDynamics and Mathematics Department, Boston University, Boston, Massachusetts; and 2Cajal Neuroscience Center andDepartment of Biology, University of Texas at San Antonio, San Antonio, Texas

Submitted 1 July 2005; accepted in final form 28 September 2005

Kuznetsov, Alexey S., Nancy J. Kopell, and Charles J. Wilson.Transient high-frequency firing in a coupled-oscillator model of themesencephalic dopaminergic neuron. J Neurophysiol 95: 932–947,2006. First published October 5, 2005; doi:10.1152/jn.00691.2004.Dopaminergic neurons of the midbrain fire spontaneously at rates�10/s and ordinarily will not exceed this range even when driven withsomatic current injection. When driven at higher rates, these cellsundergo spike failure through depolarization block. During spontane-ous bursting of dopaminergic neurons in vivo, bursts related to rewardexpectation in behaving animals, and bursts generated by dendriticapplication of N-methyl-D-aspartate (NMDA) agonists, transient firingattains rates well above this range. We suggest a way such high-frequency firing may occur in response to dendritic NMDA receptoractivation. We have extended the coupled oscillator model of thedopaminergic neuron, which represents the soma and dendrites aselectrically coupled compartments with different natural spiking fre-quencies, by addition of dendritic AMPA (voltage-independent) orNMDA (voltage-dependent) synaptic conductance. Both soma anddendrites contain a simplified version of the calcium-potassium mech-anism known to be the mechanism for slow spontaneous oscillationand background firing in dopaminergic cells. The compartments differonly in diameter, and this difference is responsible for the differencein natural frequencies. We show that because of its voltage depen-dence, NMDA receptor activation acts to amplify the effect on thesoma of the high-frequency oscillation of the dendrites, which isnormally too weak to exert a large influence on the overall oscillationfrequency of the neuron. During the high-frequency oscillations thatresult, sodium inactivation in the soma is removed rapidly after eachaction potential by the hyperpolarizing influence of the dendriticcalcium-dependent potassium current, preventing depolarizationblock of the spike mechanism, and allowing high-frequency spiking.

I N T R O D U C T I O N

Mesencephalic dopamine neurons play a key role in thefunction of the basal ganglia and parts of the neocortex.Dopamine released by the axons of these neurons has beenshown to modulate voltage-sensitive ion channels, affect therelease of other neurotransmitters, alter synaptic transmission,and modulate synaptic plasticity in target areas, primarily thedorsal striatum, nucleus accumbens, and prefrontal neocortex(Centonze et al. 2003; Nicola et al. 2000). Dopaminergicneurons are autonomously active (Fujimura and Matsuda 1989;Grace and Bunney 1983b, 1984a; Harris et al. 1989; Kita et al.1986; Lacey et al. 1987; Nedergaard and Greenfield 1992) andproduce a constant background rhythmic or irregular firingpattern on which bursts may be superimposed (Celada et al.

1999; Grace and Bunney 1984b; Hyland et al. 2002; Tepper etal. 1995). Dopaminergic neurons in behaving animals havebeen shown to fire distinct bursts of activity, with several-foldhigher rates within the burst than without, temporally locked tothe reward prediction error (e.g., Schultz 2002). That is, theyfire at the moment the animal receives an unpredicted rewardor is unexpectedly presented with the opportunity to begin abehavioral sequence known to end with a reward. The exis-tence of neurons that encode reward-prediction is required forone kind of learning process, called reinforcement learning,that increases the probability of re-occurrence of whateverbehavior preceded success (Dayan and Balline 2002; Schultz2002). Thus the shift of dopaminergic cells from backgroundfiring to burst and back is one of the few cellular events in thebasal ganglia for which a clear psychological meaning has beenproposed.

The mechanisms underlying the firing patterns of the dopa-mine neuron have been a subject of intensive study for severalyears. Periodic and irregular low-frequency single spiking andhigher frequency burst firing are all observed in anesthetizedand awake animals (Grace and Bunney 1984a; Hyland et al.2002; Overton and Clark 1997; Tepper et al. 1995). Burstsoccurring spontaneously in anesthetized animals appear similarto those seen during learning of operant tasks. In either case,firing within a burst can achieve rates of �20 Hz (Hyland et al.2002; Kiyatkin and Rebec 1998; Schultz 2002). These highrates of firing are thought to be driven by synaptic activitybecause spontaneous bursts are not seen in slices. Bursts areprobably not generated by giant long-lasting excitatorypostsynaptic potentials (EPSPs) because dopaminergic cellscannot be induced to fire at rates characteristic of bursts (above�10 Hz) by passage of somatic current pulses (Grace andBunney 1983a; Kita et al. 1986; Richards et al. 1997). Thisfailure to sustain high-frequency driven firing apparently arisesfrom a combination of powerful spike afterhyperpolarizationcurrents that follow spike generation and spike failure withprolonged depolarization (Grace and Bunney 1983a; Richardset al. 1997). High-frequency burst firing can be observed afterblockade of the calcium-dependent potassium current respon-sible for the postspike afterhyperpolarization (Johnson and Wu2004; Shepard and Bunney 1991). Under these conditions,however, dopamine cells are prone to depolarization blockespecially during passage of depolarizing currents, and burst-like episodes of high-frequency firing are not easily elicited.Another puzzling feature of burst firing in dopaminergic neu-

Present address and address for reprint requests and other correspondence:A. Kuznetsov, Dept. of Mathematical Sciences, IUPUI, Science Bldg., LD270,402 N. Blackford St., Indianapolis, IN 46202-3216 (E-mail: [email protected]).

The costs of publication of this article were defrayed in part by the paymentof page charges. The article must therefore be hereby marked “advertisement”in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

J Neurophysiol 95: 932–947, 2006.First published October 5, 2005; doi:10.1152/jn.00691.2004.

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rons is its dependence on N-methyl-D-aspartate (NMDA) re-ceptor activation. A number of studies have shown that bothspontaneous bursts in vivo (see review by Overton and Clark1997) and bursts evoked by stimulation in slices (Morikawa etal. 2003) are dependent on activation of NMDA receptors. Ifthe burst was simply due to a large and prolonged EPSP, whywould it matter whether it arose from AMPA or NMDAreceptors?

Our previous models, both abstract (Medvedev and Kopell2001), and biophysical (Medvedev et al. 2003; Wilson andCallaway 2000), have considered the firing elicited by thesomatic current pulses in slices. In this situation, a frequencysomewhat higher than the steady state (but much less than inspontaneous bursts) can be elicited by release of the neuronfrom hyperpolarization. The previous papers reproduced thatbehavior in models that described the neuron as a chain ofoscillators where each oscillator represented the soma or adendritic compartment. In the biophysical models, the essentialcurrents were a voltage-dependent calcium current and a cal-cium-dependent potassium current, and there was also extru-sion of calcium from the cell. In each compartment, the periodof oscillation was determined by the filling and emptying timesof the free calcium and thus depended on the ratio of surfacearea to volume ratio of the compartment. Hence the frequencyof the uncoupled compartments depended on the diameter ofthe compartment, with thinner compartments having a largernatural (uncoupled) frequency, and there was a large range inuncoupled frequencies. When the coupling was strong, thecoupled compartments displayed a slow periodic oscillation,like that seen in dopaminergic cells recorded in slices (Med-vedev et al. 2003; Wilson and Callaway 2000). If the strengthof coupling was not enough to synchronize the compartments,the model generates an irregular oscillation (Wilson and Cal-laway 2000), which resembles the most common firing patternseen in vivo (Tepper et al. 1995). In our work cited in thepreceding text, the models represented dynamics in a prepara-tion in which the spikes were absent because of the addition ofTTX to the bath.

In this paper, we extend our model to NMDA-evoked burstfiring of the dopaminergic neurons and suggest a mechanismby which these cells can burst in response to synaptic NMDAbut not AMPA activation. We also show why this would bemost effective for dendritic, rather than somatic, inputs. Al-though there are some similarities, we distinguish the transientburst firing that we study in this paper from the rhythmicbursting that can be elicited in the presence of apamin orbath-applied NMDA (Johnson and Wu 2004; Ping and Shepard1996) and for which different mechanisms have been proposed(Canavier 1999; Johnson et al. 1992; Li et al. 1996).

M E T H O D S

We start with a biophysical model similar to those in our previouswork (Medvedev et al. 2003; Wilson and Callaway 2000) but addAMPA or NMDA and spike-producing ionic currents. We consider aneuron in which there is a single large somatic compartment andmultiple small dendritic compartments all connected to the larger one.All compartments have the same oscillatory mechanism and spikingcurrents but differ in natural frequency of oscillation because of thedifference in their surface area to volume ratio.

Equations for the single compartment

We construct a spiking and a nonspiking version of each compart-ment in the dopaminergic cell model. Much of our work is performedusing the nonspiking compartment, which was originally presented torepresent the mechanisms active in the dopaminergic cell after block-ade of fast spike-related currents by TTX and TEA (Wilson andCallaway 2000). The nonspiking compartment consists of a noninac-tivating voltage-sensitive calcium current with half-activation at about�20 mV, a voltage-insensitive high-affinity calcium-dependent po-tassium current (half activated at 250 nM), a voltage-sensitive potas-sium current (half activation at �10 mV), and a small leak current thatlimits the input resistance at very hyperpolarized potentials when allthe other conductances can be zero. For simplicity, the voltage-sensitive conductances are treated as instantaneous. This has no effecton the result because the dynamics of voltage and calcium concen-tration are both slow relative to the activation of both of theseconductances. The equations for this compartment were

Cdv

dt� iapp � gCa�v�

4�ECa � v� � gK�v��EK � v� � gKCa��Ca�i��EK � v�

� gt�Et � v� � gNMDA�v��ENMDA � v� � gAMPA�EAMPA � v�

d�Ca�idt

�2�

��gCa�v�

zF�ECa � v� � PCa�Ca�i�

gKCa([Ca]i) � g�KCa�Ca�i

4

�Ca�i4 � k4

�c�v� � �0.0032�v � 50�/�e��v�50�/5 � 1�

�c�v� � 0.05e��v�55�/40

gCa�v� � g�Ca�c�v�/��c�v� � �c�v�

4

gNMDA�v� � g�NMDA/�1 � �Mg�10 e�v/12.5�gK�v� � g�K/�1 � e

��v�10�/7�

Here, v is voltage, iapp is applied current (as a current density, in�A/cm2), gCa(v) is the voltage-sensitive calcium conductance per unitsurface area (in ms/cm2), gKCa is the calcium-dependent potassiumconductance (in ms/cm2), gl is the leak conductance, gNMDA(v) is thevoltage-sensitive NMDA conductance density, and gAMPA is theAMPA conductance density. Potassium conductance was representedas a simple instantaneous sigmoidal function of voltage. The calciumconductance density was also a sigmoidal function of voltage. TheAMPA conductance density was voltage-independent, but the NMDAconductance showed voltage sensitivity as in Li et al. (1996). Themagnesium block of the NMDA receptor was treated as instantaneous.The reversal potentials and maximal conductances for each of thecurrents used in Figs. 1, 2, and 4–7 are given in Table 1.

Intracellular diffusion of calcium was not represented in the model.We have previously analyzed the effect of calcium diffusion in thismodel, and have shown that it does not qualitatively affect the results(Wilson and Callaway 2000). Calcium buffering was treated as

TABLE 1. Parameters for the non-spiking cell model, and thevalues used in Figs. 1, 2, and 7

Parameter Value Parameter Value

gLeak 0.05 mS/cm2 ECa 100 mV

g�Ca 0.2 mS/cm2 EK �90 mV

g�k 0.4 mS/cm2 Et �50 mV

g�KCa 0.3 mS/cm2 EAMPA 0 mV

[Mg] 1.4 mM ENMDA 0 mV� 0.05 PCa 2500 �m/s

933DOPAMINERGIC CELL BURSTING

J Neurophysiol • VOL 95 • FEBRUARY 2006 • www.jn.org

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instantaneous and unsaturable. Calcium was removed from the cell byan unsaturable pump with maximum rate density PCa, and the pumpwas treated as nonelectrogenic. Although these simplifications are notstrictly true, we verified in simulations that they did not alter theoverall outcome of the model, and they greatly simplified the analysis.Calcium entering through the NMDA channel was assumed to besmall compared with that entering via the calcium channel itself, andwas not included in the calcium equation, in which � is the ratio offree to total calcium, r is the radius of the compartment (in �m), z isthe valence of calcium, and F is Faraday’s constant.

Two-compartment model

All compartments were given a length of 1 �m. This has no effecton the solutions for the single compartment in which all currents areexpressed as densities anyway (membrane capacitance C was 1�F/cm2). For the coupled compartment model, absolute length is stillunimportant, but adjustments of the relative length of the dendritic andsomatic compartments effectively adjust the ratio of dendritic tosomatic surface area, and hence the strength of the dendritic contri-bution to the coupled oscillation. This occurs because the compart-ments are all treated as isopotential, and any increase in length onlyalters the total surface area of the compartment without affecting itssurface area to volume ratio. This is inaccurate of course becausedistribution of charge along long dendrites is slow enough to generatevoltage gradients along the dendrite that would complicate the result.Because we wanted to study the interaction between compartmentswithout these complications, we left the compartment lengths constantand adjusted the dendritic to somatic surface area ratio by adjustingthe number (nd) of identical synchronous dendritic compartmentsattached to the soma. Because we assumed all dendrites were syn-chronous, attaching nd isopotential dendritic compartments to thesoma is the same as attaching a single isopotential compartment withan nd-fold increase in surface area.

Compartments were coupled by voltage. There was no diffusion ofcalcium between compartments. The strength of the electrical cou-pling between the somatic and dendritic compartments is an importantissue for models of this type. Because the dendritic membrane isrepresented by a single compartment, the most realistic strength ofcoupling to use cannot be computed from the cytoplasmic resistivityas done in cable models (Li et al. 1996). We tried a variety of valuesfor the coupling parameter for the pair of compartments gc, but theresults shown here were obtained with values ranging from 0.25 to0.30 mS/�m. In a previous work, we considered the case in whichcoupling was very strong, and the entire cell was effectively isopo-tential (Medvedev et al. 2003), as likely is effectively the case duringspontaneous oscillations in the absence of spiking (Wilson and Cal-laway 2000). Distributed models of the dopaminergic neuron suggestthat under some conditions, substantial voltage differences exist alongthe dendritic membrane (Wilson and Callaway 2000), and these arelikely to be especially pronounced during action potential generation.Here we required that the coupling be moderately strong so thatsteady-state voltage gradients were small, but large voltage gradientswere possible during transients. This level of coupling approximatesthat seen in the most realistic distributed models currently available(Canavier et al. 1999; Wilson and Callaway 2000). The currentbalance equations governing each of the two compartments in thecoupled model were

Cdvddt

� �gion�vd � Vion� � gc rs2rdld�ld�ldrd2 � lsrs2� �vd � vs�C

dvsdt

� �gion�vs � Vion� � ndgc rd2rsls�lsrs2 � ldrd2 �vs � vd�in which vd and vs are the dendritic and somatic voltages, ld and ls aredendritic and somatic lengths (always 1 �m), and rd and rs are thedendritic and somatic radii.

Spiking model

To the preceding model, we added spike-producing sodium andpotassium conductances to each compartment. Specifically

�m�v� � �0.32�v � 31�/�e��v�31�/4 � 1�

�m�v� � �0.28�v � 4�/�ev�4�/5 � 1)

m��v� � �m�v�/��m�v� � �m�v��

�h�v� � 0.01e��v�47�/18

�h�v� � 1.25/�1 � e��v�24�/5�

dh

dl� �h�v��1 � h� � h�h�v�

gNa�v,h� � g�Nam��v�3h

�n�v� � �0.0032�v � 5�/�e��v�5�/10 � 1�

�h�v� � 0.05e��v�10�/16

dn

dt� �n�v��1 � n� � n�n�v�

gKS�n� � g�KSn4

in which h is the sodium inactivation variable, and n is the activationvariable for the delayed rectifier. The simplifications introduced bymaking the activation of voltage-dependent calcium and sodiumcurrents instantaneous did not affect the outcome because both ofthese are fast, and the relative speeds of the two currents was not veryimportant for the dopamine cell model, in which most calcium entryoccurred as a result of subthreshold oscillations rather than as aspike-triggered tail current. This was verified by comparing the resultsto an otherwise identical model with more accurate sodium andcalcium kinetics. The spike-triggered (delayed rectifier) potassiumcurrent was added to the spiking model without removing the smallerinstantaneous potassium current with a more negative activation thatwas used in the nonspiking model. The additional parameters used inthe spiking version of the model, were g�Na, which was set at 150mS/cm2 and g�KS, which was 4 mS/cm

2. Computer simulations wereperformed using XPPAUT (Ermentrout 2002) using the stiff methodand a time step of 0.1 ms.

R E S U L T S

Slow spontaneous oscillation in a single compartment

Our approach is based on the coupled oscillator model forthe dopaminergic neuron (Medvedev et al. 2003; Wilson andCallaway 2000). That reduced model did not address actionpotential generation but only the subthreshold oscillationsobserved after blockade of sodium currents by treatment of thecells with TTX. The fundamental features of this model areillustrated for a single compartment in Fig. 1. The depolarizingphase of the membrane potential oscillation occurred whenintracellular calcium concentration and calcium-dependent po-tassium current were low, the voltage-sensitive calcium con-ductance was activated, and calcium influx exceeded efflux.When the resulting increase in calcium concentration andincrease in calcium-dependent potassium conductanceachieved a sufficient level that calcium-dependent potassiumcurrent exceeded the calcium current, the model hyperpolar-ized regeneratively and stayed hyperpolarized until calciumlevels were reduced by the plasma membrane calcium pump.Calcium concentration lagged calcium current because of the

934 A. S. KUZNETSOV, N. J. KOPELL, AND C. J. WILSON


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time required to fill the volume of the compartment, andlikewise the time required for calcium removal was dependenton the volume. Both calcium influx and efflux depended on themembrane surface area, so the frequency of the oscillation wasrelated to the surface area to volume ratio of the compartment.Although not essential for the oscillation, the voltage-sensitivepotassium current was required to prevent large amplitudecalcium spikes in the model. This is consistent with theobservation of large-amplitude calcium spikes in real dopami-nergic cells after treatment with high doses of TEA (e.g., Graceand Onn 1989).

This highly reduced model for the dopaminergic neuronomits a number of known features of these cells. For example,there are a number of different calcium currents that have beenidentified in dopaminergic neurons (Durante et al. 2004; Ne-dergaard et al. 1993; Wolfart and Roeper 2002), and these haveall been subsumed into a single current in the model. Likewise,a variety of voltage-sensitive potassium currents are repre-sented by a single current, the current generated by the calciumpump is disregarded, and the dynamics of calcium diffusion arenot included. The kinetics of activation and deactivation of thecalcium and potassium conductances are fast compared withthe time course of changes in membrane potential, and thesecurrents are treated as instantaneous. We do not include avariety of ion channels known to be present in the dopaminer-gic cell but that play a modulatory role in the oscillation (e.g.,Neuhoff et al. 2002). Although these simplifications alter thesize and shape of the oscillations somewhat, the dependence ofthe oscillation frequency of any cell process on its diameter ismaintained (Wilson and Callaway 2000). The chief advantageof the reduced model is the ability to represent it in a two-dimensional phase plane as shown in Fig. 1. As pointed out

previously (Medvedev et al. 2003; Wilson and Callaway 2000),the positions of the nullclines in this simplified model, andhence the position of the equilibrium point at their intersectiondo not depend on diameter. For those sets of parameters atwhich the model exhibits membrane potential oscillations, thesurface area to volume ratio, and hence the diameter, primarilydetermines the period of oscillation. For very small diameters,(e.g., 1 �m as shown in Fig. 1), the waveform of oscillationdoes depend on diameter. But for a wide range of diametersthat yield low oscillation frequencies, the oscillation operatesin the relaxation mode in which calcium concentration changesmuch more slowly than the membrane potential, and thetrajectory remains close to the voltage nullcline except at therapid voltage changes that occur at the minimum and maxi-mum values of calcium concentration. Over this range ofdiameters, the amplitude of the oscillation does not depend ondiameter. Calcium concentration changes more rapidly in thesmall-diameter dendrites, but the voltage rate of change is notincreased. This increase in calcium rate of change reduces thedifference in time scales between voltage and calcium concen-tration, and the trajectory leaves the vicinity of the voltagenullcline. At these frequencies, the oscillation is smaller inamplitude, in both the voltage and calcium directions (Fig. 1).For very-small-diameter dendrites, the oscillation fails com-pletely, despite the absence of any change in the intersection ofthe nullclines, as the rates of change of voltage and calciumconcentration become comparable.

Oscillations in the coupled two-compartment system

Wilson and Callaway (2000) showed that the currents re-sponsible for this oscillation exist on both the soma and the

FIG. 1. Dependence of frequency and ampli-tude of oscillations on diameter in the isolatedcompartment. A: locations of nullclines does notdepend on diameter, but the frequency is deter-mined largely by the surface area to volumeratio, which decreases with diameter. At verylow diameters, the phase trajectory deviates fromthe usual relaxation oscillator pattern because atvery high frequencies, the membrane time con-stant limits the rate at which the voltage canchange. B: relationship between frequency anddiameter for the single compartment dopaminecell model. Note that for compartments of diam-eter comparable to the soma, the resting fre-quency is near 2 Hz, and for a 1-�m dendrite, itis maximal near 14 Hz. C: amplitude of theoscillation decreases dramatically for fine den-drite-sized compartments, and the oscillationfails near 1 �m. D and E: voltage and calciumconcentration oscilations respectively for the 10-and 1-�m-diam compartment



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dendrites of dopamine neurons. These neurons possess asparsely branched dendritic tree consisting mostly of longsegments of dendrite on the order of 1 �m in diameter,connected to a soma of diameter 15–25 �m by some shortlarge-diameter primary dendrites (Juraska et al. 1977; Tepperet al. 1987; Wilson and Callaway 2000). It was proposed thatthe different regions of the neuron should tend to oscillate atdifferent frequencies and that the resulting oscillation wouldoccur at a frequency higher than the natural (uncoupled)frequency of the soma but lower than that of the dendrites.Wilson and Callaway (2000) further showed that under restingconditions, the dendrites and soma oscillate at a frequencycorresponding approximately to the somatic natural frequency.This is expected if the various parts of the cell are stronglycoupled by voltage (i.e., if the cytoplasmic resistance betweensoma and dendrites is low) (Medvedev and Kopell 2001;Medvedev et al. 2003) and if the amplitude of dendriticoscillations is lower than that of the soma.

The effect of dendritic diameter on the peak-to-peak ampli-tude and frequency of the coupled system is shown in Fig. 2.Reducing the diameter of one compartment relative to the otherwill increase the frequency of the pair, but this effect is smalland does not continue over the entire range of diameters. Theability of a high-frequency small compartment to influence

the coupled pair is limited by the smaller current generated bythe reduced surface area of a smaller compartment. Thus as thesmaller compartment is reduced in diameter below about halfthe diameter of the larger one, the influence of the dendrite isreduced and frequency decreases and finally becomes equal tothat of the large compartment alone. This result can be alteredby increasing the dendritic surface area, by either making thedendrites longer or increasing the number of dendrites. Increas-ing dendritic length is complicated by issues of charge transferalong the longitudinal resistance of the dendrite, so in Fig. 2 weillustrate the effect by altering the number of dendritic com-partments. As the number of small dendritic compartmentsattached to the single large somatic compartment is increased,the maximum oscillation frequency increases, and the dendriticdiameter that is optimal for generating a high-frequency oscil-lation decreases. However, the amplitude of the oscillation alsodecreases with decreasing diameter and increasing frequencyof oscillation, as shown in Fig. 1. For the value of membranecapacitance and leak conductance used in all figures presentedhere (1 �F/cm2 and 0.05 mS/cm2), the passive time constantwas 20 ms, which is close to that observed in dopaminergicneurons, and of course this was independent of diameter.Altering the time constant by adjusting the leak conductancecould not overcome the upper limit on the frequency of thecoupled system (�10 Hz) because increases in leak conduc-tance decreased the amplitude of the oscillation and decreasesin the leak conductance lowered its frequency. Like dopami-nergic neurons oscillating spontaneously in slices, this two-compartment model is restricted to a frequency range between0 and �10 Hz. In this mode of firing, dopaminergic cells fireone action potential on every cycle of the oscillation, andcalcium levels during the course of the oscillation are not muchaffected by the presence or absence of action potentials (Wil-son and Callaway 2000). The pattern of firing is determinedentirely by the subthreshold oscillation.

Addition of action potential currents

In the absence of biophysical data on the action potentialcurrents of dopaminergic cells, the choices concerning theproperties of spiking currents to be added were based onknown features of the overall spiking activity of dopaminecells. If the spiking currents were adjusted so that the cell couldfire repetitively at high rates in response to powerful depolar-izations like those seen after depolarization block of the slowoscillation, then the model would exhibit high-frequency firingindependently of the slow oscillation, whenever high values ofcurrent were applied. This outcome was incompatible with theobservation that dopaminergic neurons do not fire at high ratesin response to injected current, so spiking mechanisms of thisclass were excluded. One way to limit the maximum rate ofevoked spiking to that seen in dopaminergic cells was toinclude powerful long-lasting voltage-sensitive potassium cur-rents triggered by action potentials. This was ruled out on thebasis of reports that most single-spike afterhyperpolarizationby dopaminergic cells is sensitive to apamin, a blocker of thecalcium-dependent potassium current responsible for the sub-threshold oscillation (Nedergaard 2004; Ping and Shepard1996; Shepard and Bunney 1991).

FIG. 2. Effect of diameter of a set of small compartments (D) coupled to asingle larger 1 of constant diameter (S, diameter � 20 �m). All dendrites areassumed to be identical and synchronized and so are represented by a singlecompartment. The influence of the dendritic compartment on the somatic oneis increased with addition of extra dendritic compartments. When the numberof dendrites is low, the frequency and amplitude of the oscillation of thecoupled system is only slightly affected by decreases in the diameter of thedendrite (number of dendrites � 2). The overall frequency and amplitude areapproximately that for the larger compartment. With larger numbers ofdendrites, the coupled system can achieve higher frequencies, ultimatelyachieving the high natural frequency and the low-amplitude characteristic ofthe fine dendritic compartment (as in Fig. 1)



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For the spiking compartment models presented here, spikefrequency during the slow oscillations and during currentinjection was limited by voltage-dependent sodium channelinactivation. There is ample indirect evidence of failure ofsomatic action potentials because of depolarization block. Forexample, the initial segment-somatodendritic (IS-SD) delay forspontaneous action potentials is increased, and finally thesomatodendritic spike is blocked, by small constant somaticdepolarizations (Grace 1983a,b). To maintain the simplicity ofthe model, this was implemented by adjusting the voltagesensitivity of activation of the potassium conductance, and thetime constant and voltage sensitivity of inactivation of thespike-generating sodium channels. Most importantly, the volt-age sensitivity of the potassium channel that repolarized theaction potential was set so that it deactivated at a more positivevoltage than required for removal of sodium inactivation (half-activation at about �20 mV). This ensured that the spikerepolarizing current generated by an action potential was notby itself sufficient to remove sodium channel inactivationgenerated because of that action potential. Thus the actioncurrents would not by themselves support repetitive firing inresponse to constant current. The voltage sensitivity of inacti-vation was adjusted so that 90% of sodium channels wereavailable at �45 mV and about half of sodium channels wereinactivated at �35 mV. Sodium channel activation was instan-taneous and the channel was half activated at �10 mV. Thisresulted in a small and narrow window conductance (0.025%of maximum) centered at �35 mV. Because sodium channeldensity was set high (usually 150 mS/cm2), this windowconductance produced a noninactivating inward current thatcontributed to the slow oscillation. These choices were notmeant to represent accurately the biophysical properties of the

dopaminergic cell but to recreate its most fundamental firingproperties with the minimal number of additional variables.

When spiking compartments of two different sizes wereconnected, the net oscillation was slightly higher frequencythan that observed for the same compartments without sodiumaction potentials. The interaction between two spiking com-partments was more complex than that for the nonspikingmodel, but it was basically similar. An example is shown inFig. 3, in which a compartment representing 10 synchronousspiking dendritic segments of 1 �m diameter are connected toa 20-�m-diam soma. Spiking is nearly synchronous (Fig. 3A),but the membrane potential in the soma and dendrites aredifferent throughout most of the cycle, and there are substantialcurrents flowing between compartments (Fig. 3B). Action po-tentials begin in the somatic compartment, but spike repolar-ization is more rapid in the dendrite because of the largecalcium concentration transient, which produces a large rapidpotassium current that is not present in the somatic compart-ment. A substantial proportion of the somatic repolarizationthat occurs after a single action potential in this model arisesfrom the dendritic compartment. This can be seen in thelongitudinal current that passes between the two compart-ments, shown in in Fig. 3B. During the ramp-like depolariza-tion that leads to the next action potential, the dendrite isprevented from oscillating by hyperpolarizing current from thesoma, which is seen as negative in Fig. 3B. This can also beseen in the phase plane in Fig. 3D by the fact that the dendrite’strajectory follows the calcium nullcline during this portion ofthe cycle. The difference in membrane potential between thetwo compartments, and the inter-compartmental currents thatresult, cause each compartment to distort the phase plane of theother. In Fig. 3D, the voltage nullcline is shown for a point at

FIG. 3. Oscillations of a 2-compartmentspiking model, with number of dendrites �10, dendritic diameter of 1 �m, and somaticdiameter of 20 �m. A: voltage oscillationsin the soma and dendrites. Note the oscil-lation frequency is only slightly faster thanthe nonspiking model (see Fig. 1). Theoscillation frequency is primarily deter-mined by the somatic compartment. B: cur-rent passing between the soma and dendriteduring the oscillation. Upward indicatescurrent flow from soma to dendrite, down-ward deflections means that the dendrite ismore depolarized than the soma. Note thelarge component of spike afterhyperpolar-ization current contributed by the dendrite,but the soma is depolarizing the dendritethrough most of the slow ramp period be-tween spikes. C: calcium concentration inthe soma and dendrite. Note large calciumtransients in the dendrite that are responsi-ble for the fast hypolarizing influence of thedendrites. D: phase plane representation ofthe dendritic and somatic voltage and cal-cium trajectories. The blue line is the volt-age nullcline for both compartments, calcu-lated at their isopotential point preceedingeach spike. The green line is their sharedcalcium nullcline. During the ramp pre-ceeding spiking, the dendrite is effectivelypassive, and its trajectory follows the cal-cium nullcline.



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which the soma and dendrites are at the same voltage. Duringthe oscillatory cycle, the voltage nullcline of each compartmentshifts over a wide range, reflecting the effect of longitudinalcurrent flowing between compartments.

Applying current to the somatic compartment of this modelcannot drive the model to fire at rates faster than that observedfor the nonspiking model. This limit exists because repolariza-tion of the somatic membrane relies on calcium-dependentpotassium current that arises from the calcium-driven oscilla-tion. To clarify why injected current cannot drive the cell tohigher frequencies, effects of current and synaptic input to thenonspiking model will be described first, and the spiking modelwill be revisited afterward.

Current injection or synaptic activation of thesingle compartment

Injection of somatic current in the dopaminergic neuroncannot drive the cell at rates much above 10 Hz, even tran-siently (e.g., Richards et al. 1997). Figure 4 illustrates theeffect of current application on the calcium-driven oscillationin a single nonspiking compartment. Injecting current increasesthe frequency of oscillation, but the amplitude of the oscillationis also decreased, and the oscillation rapidly fails. This failureoccurs because of a block of the calcium oscillation that occurswhen the calcium-dependent potassium current can no longer

exceed that of the combined calcium current and injectedcurrent. Block occurs when the intersection of the voltage andcalcium nullclines becomes a stable equilibrium for the cell.Stability cannot be assessed simply from examination of theslope of the voltage nullcline at the intersection because, forsmall diameters, the oscillation deviates from the relaxationmode of operation as described earlier. This makes block of theoscillation diameter-dependent. In Fig. 4, for example, the1-�m-diam dendrite has ceased oscillating with injected currentsabove �4 �m/cm2, whereas the 20-�m one continues to oscillateat higher current injections, albeit with small amplitude.

Synaptic activation of the dopaminergic neuron’s dendritescan generate a burst of firing at rates that exceed the 10-Hzlimit seen for the spontaneous oscillation. This response isdependent on activation of NMDA receptors and can also beevoked by dendritic application of NMDA (Morikawa et al.2003). A comparison of the effects of NMDA and of AMPA onthe calcium oscillation in the single-compartment model isshown in Fig. 5. AMPA receptor activation has an effectsimilar to current injection in that it raises the voltage nullcline.But because of its positive reversal potential, the effects ofAMPA activation are most pronounced in the hyperpolarizingphase of the oscillation, for which the driving force for AMPA-mediated current is greater. This causes the oscillation toundergo a powerful reduction in amplitude, and then to fail inthe soma-sized compartment at frequencies �10 Hz. For asmaller compartment, the oscillation frequency can achievemuch higher frequencies, but the amplitude is very small at allfrequencies.

Because of its voltage dependence, the NMDA receptorundergoes voltage-dependent block and unblock on each cycleof the oscillation. While NMDA activation is slow, the voltage-dependent block of the receptor is rapid (Mayer and Westbrook1985). Because it is mostly blocked during the hyperpolarizingphase of the oscillation, activation of the NMDA receptor hasmuch less effect on the phase plane and voltage trajectory inthe hyperpolarizing phase but primarily influences the depo-larizing phase of the oscillation. This voltage sensitivity can beseen in the voltage nullcline. Thus NMDA receptor activationincreases the amplitude of the oscillation for all diametercompartments. Increasing NMDA receptor activation first in-creases frequency moderately and then at higher levels de-creases frequency. The increased rate of change of voltage inboth the hyperpolarizing and depolarizing directions producedby the extra voltage-dependent current through the NMDAchannel causes the change in the voltage nullcline, allowing theoscillation to survive in finer dendritic processes (compare withFig. 1B). At very high levels of NMDA conductance, oscilla-tion frequency decreases, and larger calcium-dependent potas-sium currents become required to repolarize the cell in oppo-sition to the combined depolarizing influence of inward cal-cium current and NMDA current. With still higher NMDAconductance, the cell would exhibit slow plateau potentialsdominated by the NMDA current (not shown). In Fig. 5, allcurrent carried by the NMDA receptor is sodium. The oscilla-tion frequency would be increased somewhat if a portion of thecurrent were carried by calcium, as in natural NMDA recep-tors, because it would increase the rate of calcium filling of thecompartment.

FIG. 4. Effects of applied current on the single oscillatory (nonspiking)compartment model. A: applied current acts on the voltage nullcline (black),raising it but also reducing the amplitude of the oscillation both in voltage andcalcium. The calcium nullcline (gray) is not changed. B: effect on frequency ismuch greater for smaller diameter processes. For a 20-�m soma, stability ofthe oscillation is lost by depolarization block at a maximum frequency of �12Hz. For a 1-�m dendrite, the frequency range can extend as high as 50 Hz andwould go higher except that stability of the oscillation is lost at lower currentlevels (at �4.7 �A/cm2). The amplitude of the oscillation for the smalldendrite becomes increasingly small at frequencies 25 Hz, reducing itspotential for influencing the soma in the coupled model.



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Two compartments, with synaptic activation restrictedto the dendrite

The effect of AMPA receptor activation restricted to a set of10 1-�m dendritic compartments attached to a single 20-�msoma is shown in Fig. 6. In the single-compartment case, wesaw that AMPA conductances 0.03 mS/cm2 drove the iso-lated dendritic compartment to depolarization block of thecalcium oscillation. In the coupled model, the fast oscillation ofthe dendrite is not visible in the membrane potential of eithercompartment. However, over the range of AMPA conduc-tances that allowed dendritic oscillation in the isolated dendrite(0–0.03 mS/cm2), the amplitude of the dendritic oscillation isat its greatest. The amplitude of the oscillation in the dendritedecreases steeply over this range as it did in the isolatedcompartment (Fig. 6A). The frequency of the coupled systemincreases slightly over a wide range of AMPA conductances(Fig. 6A). At all levels of AMPA conductance, the somaticfrequency dominates the coupled system (as in Fig. 6B). Theeffect of the slow somatic membrane potential on the dendriteis illustrated by drawing the phase plane for the dendriticcompartment, as shown in Fig. 6C. Because of voltage cou-pling from the relatively slow somatic oscillation, it is accurateto represent the voltage nullclines of the fast compartment as afamily of curves, one for every voltage visited by the somaduring its oscillation. Three such nullclines are shown in Fig.6C, one when the soma is at its most positive voltage (�20mV), one at the soma’s negative extreme (�70 mV), and one

near the center of the somatic membrane potential trajectory(�50 mV). At all times during the cycle, the dendrite’soscillation is blocked, and the dendrite’s trajectory stays closeto the calcium nullcline. When the soma is hyperpolarizedbetween spikes, its hyperpolarization is strong enough to drivethe dendrite to hyperpolarization block, despite AMPA activa-tion, and when the dendrite is released by somatic depolariza-tion, it is driven to depolarization block of calcium oscillationsby AMPA receptors. As a result, the dendrites passively followthe somatic oscillation. This was typical for AMPA activationor current injection into the dendrites for a wide range ofparameters.

NMDA activation of the dendritic compartment in the samecoupled configuration produced a very different result, shownin Fig. 7. Three different kinds of outcomes were observed,depending on the strength of NMDA activation. At low levelsof NMDA input to the dendrites (region labeled 1 in Fig. 7, Aand B), there was little change in the amplitude or frequency ofthe oscillation. In the uncoupled dendrite, this is the range ofNMDA activations associated with the rapid increase in am-plitude and frequency of dendritic oscillations. In that range,the slow oscillation at the soma continued to dominate thecoupled system, and the oscillations resembled those seenduring spontaneous firing. At higher levels of NMDA (region2 in Fig. 7, A and B), the slow oscillation continued, but duringthe slow depolarizing phase of the oscillation, the dendrite andsoma expressed a high-frequency oscillation (Fig. 7C). The

FIG. 5. The effects of AMPA and N-methyl-D-aspartate (NMDA) activation on the propertiesof the single-compartment model. A: AMPAactivation is similar to current injection, raisingand distorting the voltage nullcline (black). Thecalcium nullcline (gray) is not changed. High-frequency oscillations cannot be evoked in largediameter compartments by AMPA receptor acti-vation. Small-diameter dendrites can oscillate athigh-frequency, but those oscillations have verylow amplitudes. At high levels of AMPA acti-vation (near 0.07 mS/cm2 in this example), os-cillations cease, as they do for injected current.B: NMDA activation has a very different effecton the oscillation due to its voltage sensitivity.This difference is especially evident from theeffect on the voltage nullcline. As a result,NMDA receptor activation increases the ampli-tude of oscillations, and has a moderate andbiphasic effect on the frequency. NMDA doesnot produce depolarization block in the activa-tion range illustrated.



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higher-frequency oscillation increased in frequency and de-creased in amplitude toward the end of the burst. The ampli-tude relationship between dendrite and soma were reversed forthe two rhythms, the somatic amplitude being larger for theslow oscillation and the dendritic amplitude larger for the fastone. The amplitude of the faster oscillation grew with increasesin NMDA activation, but its mean frequency was greatest atthe lowest levels of activation (Fig. 7B). Such mixed fast-slowoscillation was observed over the range of NMDA activationslabeled 2 in Fig. 7, A and B. Mixed oscillations in the coupledsystem were seen at levels of NMDA activation that began toproduce large-amplitude oscillations in the isolated dendriticcompartment (Fig. 5B). With higher NMDA activations (re-gion 3 in Fig. 7), the slow oscillation was lost, and the dendriteand soma both oscillated at high frequency with the dendriticamplitude exceeding the somatic one (Fig. 7D). The oscillationfrequency in this range decreased with additional NMDA inputbut remained higher than ever seen for an isolated compart-ment with NMDA activation and was comparable to that seenduring NMDA-evoked bursts of activity in dopaminergic cellsin slices and reward-related firing in vivo.

The mechanism of the mixed oscillations is illustrated inFig. 7E. Again, the slow changes in voltage imposed on thedendrite by the soma are indicated by drawing nullclines for

the dendrite at three different somatic voltages, the extremaand the midpoint of the somatic oscillation. At the mosthyperpolarized part of the somatic oscillation, the dendriticoscillation is stopped, the dendrite exhibits a stable hyperpo-larized fixed point, and the dendritic trajectory approaches thatfixed point. As the soma depolarizes, the dendritic voltagenullcline shifts up, and the dendritic compartment becomesunstable and oscillates. With additional upward shift of thevoltage nullcline, the dendritic oscillations become lower am-plitude and higher frequency. During the brief somatic finaldepolarization, the dendrite is driven briefly to depolarizationblock and then makes a final small cycle of its fast oscillationbefore the dendritic oscillation is abolished by the abruptsomatic hyperpolarization. This pattern characterized all theoscillations seen in region 2 of Fig. 7, A and B.

The mechanism of the fast sustained oscillations seen inregion 3 and in Fig. 7D is shown in Fig. 7F. This pattern ofoscillation is dominated by the dendrite, the oscillations ofwhich are more rapid than the somatic oscillatory mechanismcan follow. Effectively, the dendrite acts as a constant load onthe soma approximately equal to the average voltage of thedendrite. The somatic nullclines that would apply if the den-dritic voltage were fixed at the average of their oscillation areshown superimposed on the somatic trajectory in Fig. 7F. The

FIG. 6. AMPA activation applied to the dendrite of the 2-compartment model. A: amplitude and frequency of the membrane potential oscillation in the somaand dendrite with increasing AMPA activation. The dendrite undergoes rapid reduction in oscillation amplitude and subsequent voltage block of its ownoscillatory mechanism over a narrow low range of conductances (�0.1 mS/cm2). At higher levels of activation, the dendrite is effectively passive, and providesonly a tonic current injection to the dendrite on the remainder. B: voltage waveform of the oscillation with 0.25 mS/cm2 AMPA activation. The dendriticwaveform mostly mimics the somatic 1. C: phase plane for the dendrite is shown for 3 different points during the oscillation. As the soma oscillates, the currentapplied to the by the soma distorts the phase space of the dendrite on a time scale slower than the dendrite’s natural time scale. Thus the voltage of the somacan be treated as a slowly changing parameter. The trajectory of the dendrite stays close to the calcium nullcline, while tracking the changing stable equilibiumimposed by the soma.



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net effect of the dendritic oscillation is a hyperpolarization ofthe soma that blocks the somatic oscillation. The oscillation ofthe dendrite is passively conducted to the soma and is respon-sible for the small changes in voltage seen in Fig. 7F. Becausethe somatic calcium concentration cannot follow the highfrequency of the driving oscillation from the dendrite, somaticcalcium stays at a point determined by the average voltagenullcline and the somatic voltage follows the dendritic oscil-lation but with reduced amplitude and a phase shift caused bythe somatic capacitance. The soma maintains a constant lowcalcium concentration and very little calcium-dependentpotassium current. The soma exerts a small constant depo-larizing influence on the dendrite, and it is the combinationof somatic depolarizing current and NMDA receptor acti-vation that drives the dendrite to high-amplitude, high-frequency oscillations. This phenomenon, in which coupledoscillators exhibit widely different amplitude, is known aslocalization and has been studied in a variety of contexts(e.g., Kuske and Erneux 1997; Rotstein et al. 2003). In thedopaminergic cell model, as well as in others (Rotstein et al.2003), the results depend strongly on the strength andsymmetry of the coupling.

With increases in the coupling parameter, or with reductionof the difference in diameters of the soma and dendriticcompartments, application of low NMDA conductances to thedendrite could stabilize both soma and dendrite, causing re-

gions 1 and 2 in Fig. 7 to be separated by a region of stableresting potential (not shown). This occurred because eachcompartment shifted the voltage nullcline of the other to aposition with a stable equilibrium. The interaction betweencoupled oscillatory compartments depends on the strength ofcoupling between them, the sensitivity of the oscillation toshifts in the voltage nullcline (how far each compartment isfrom hyperpolarization block), and the asymmetry of the ef-fective coupling.

The results shown in Fig. 7 suggests that brief periods ofNMDA activation in the dendrite might be capable of creatingbrief bursts of oscillatory activity in the dendrites and somata,which could trigger bursts of action potentials. The criticalissue is whether the coupled system could shift from pattern 1to pattern 3 firing and back rapidly. Such rapid shifts do occurwith very little aftereffect, as demonstrated in Fig. 8, for a pairof spiking compartments as in Fig. 3. In Fig. 8A, a 500-msapplication of NMDA to the dendrites produces a period ofhigh-frequency oscillation superimposed on the slow sponta-nous regular spiking. The dendrite and soma both fire actionpotentials during both the slow and fast oscillations, but theroles of the two compartments are different in the two patterns.Note that there is minimal buildup of somatic calcium duringthe burst despite the high-frequency oscillations, as in thephase plane of Fig. 7F, and after the burst, the cell returns to itsresting oscillation with very little delay. Because individual

FIG. 7. Effect of dendritic NMDA activation on the 2-compartment model (number of dendrites � 10, soma diameter � 20 �m, dendrite diameter � 1 �m).A: increasing NMDA receptor activation produces 3 qualitatively different patterns of oscillation. At low levels (region 1), amplitude of the dendritic oscillationincreases slightly and frequency is almost unaffected. With small additional increases in NMDA activation, the cell exhibits low-amplitude bursting in thedendrite, which is superimposed on the slow, large-amplitude oscillation (region 2). The amplitudes and frequencies of both the slow and fast oscillation areshown in A and B for this pattern. The fast oscillation is larger in the dendrite, and the slow 1 in the soma. At higher levels of NMDA activation (region 3), theslow oscillation disappears, and the fast oscillation becomes constant (not bursting) and remains larger in the dendrite. In this region, increases in NMDAactivation produce an increase in amplitude and a decrease in the frequency of the oscillation. The frequency of oscillation in region 2 is between 20 and 40 Hz.C: voltage waveform for the soma (black) and dendrite (red) for oscillations in region 2. D: voltage waveforms for the soma (black) and dendrite (red) foroscillations in region 3. Note change in time scale. E: phase plane trajectory for the dendrite during bursting in region 2. Voltage nullclines are drawn for variousphases of the oscillation as indicated by the somatic membrane potential. Note that the dendrite is passive during the hyperpolarizing phase of the slow oscillation,but its oscillation is released from soma-imposed hyperpolarization block during the bursting phase of the cycle. F: phase plane trajectory for the soma duringthe oscillation in region 3. Note that the dendritic oscillation blocks the soma because the average dendritic voltage is hyperpolarized enough to drive the somato hyperpolarization block through most of the cycle. Somatic depolarization occurs rapidly enough to respond during the release from hyperpolarization imposedby the dendrite, but somatic calcium cannot change concentration fast enough to respond to the rapid oscillation rate of the dendrite, and so the somatic calciumlevel remains low and nearly constant.



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action potentials do not generate large calcium concentrationchanges in dopaminergic neurons, the rapid action potentialsdo not result in a large calcium transient, and the cell resumesits slow oscillation after only a brief period following the burst.

The mechanism responsible for the burst is similar to that forthe nonspiking two-compartment system in that it relys onoscillation generated by subthreshold currents. During sponta-neous firing, the somatic natural frequency of oscillation dom-inates, and action potentials are generated initially in the somaand subsequently in the dendrites. The dendrite is held inhyperpolarization block throughout much of the cycle, but itsoscillation is released only briefly at the time of the actionpotential. Calcium-dependent potassium current in the dendriteprovides much of the early afterhyperpolarization current at thesoma. During application of NMDA, the roles of the dendriteand soma are reversed, as can be seen from the longitudinalcurrent flowing between compartments (Fig. 8, C and D). Thefast calcium oscillation of the dendrite is increased by NMDAactivation. During the fast oscillation, action potentials areinitiated in the dendrite and propagate to the soma. Althoughthe average effect of the dendrite on the soma is hyperpolar-izing (as it was for the nonspiking model), some calciumaccumulation does occur in the soma due to calcium currentactivation during action potential generation. It is critical thatthe powerful calcium-dependent potassium current in the den-drite produces the hyperpolarization of the soma required toremove sodium inactivation and to enable a subsequent action

potential (Fig. 8D). The dendrite then immediately depolarizesagain and initiates another action potential. Because the oscil-lation is primarily generated on the faster time scale of thedendrite, its onset and offset are rapid and have little influenceon the somatic calcium concentration.

D I S C U S S I O N

The dopaminergic neurons of the substantia nigra (andventral tegmental area) are among the few cell types whoseactivity has been seen to correspond to clearly defined psycho-logical states in animals (Dayan and Balleine 2002; Hyland etal. 2002; Schultz 2002), exhibiting brief episodes of high-frequency firing, or brief pauses, corresponding to changes inreward expectation. These changes in firing rate are superim-posed on a �10/s background-firing rate, which is essential ifthe neuron is to be able to signal reward expectation error inboth directions (Tobler et al. 2003). Although this finding isexciting, it is puzzling because it seems to endow the dopami-nergic neuron with the ability to use sensory inputs and pastexperience to predict reward. How can the dopaminergic cellgain access to this information? Possibly, the dopaminergiccell only relays this signal, which may be generated elsewhere.Or perhaps only components of the information required arepresent in any one class of the various inputs to the dopami-nergic cell, and the overall signal must be synthesized by theneuron from those components. The search for the afferents

FIG. 8. NMDA-induced bursting in the 2-compartment (10x 1 �m dendrites, 20 �m soma) spiking model used in Fig. 3. NMDA (0.4 mS/cm2) is applied for500 ms starting at 600 ms. A: The background firing rate is slow, and is dominated by the soma. During the application of NMDA, firing rapidly increases to�20 Hz. B: During the rapid firing, somatic calcium increases slowly with each action potential contributing much less calcium increase than that seen duringspontaneous firing. Dendritic calcium transients remain large throughout high-frequency firing. C: Current passing between soma and dendrite during the firingin A and B. The mechanism of rapid firing is a switch from soma-dominated oscillation to dendritic oscillation. During the slow oscillation, somaticdepolarization leads dendritic at the threshold of firing, and action potentials originate in the soma. This is indicated by the large amplitude fast positive currentpulse at the beginning of each spontaneous oscillation. The dendrite is primarily responsible for the rapid repolarization following the action potential. Duringthe burst, dendritic depolarization leads during spiike initiation and the dendrite fires first, as indicated by the large downward transients. D: Expanded view ofcurrents associated with spontaneous (left) and bursting (right) spikes. Time of onset of spikes is indicated by red dashed lines. Parameters as in Table 1, exceptg�ca � 0.15 ms/cm

2, gc � 30 nS/cm2.



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that inform the firing of dopaminergic cells may be guided bya knowledge of the cellular-level properties of the dopaminer-gic neuron that determine its response to synaptic inputs. Thusthe cellular mechanisms of firing of the dopaminergic cell havebeen the subject of intense study during the past 20 yr.

Bursting requires a synaptic trigger

Starting with the pioneering in vivo intracellular recordingsof Grace and Bunney (1984a, 1984b), the rhythmic, back-ground firing of the dopaminergic neuron has been recognizedas arising from an autonomous pacemaker mechanism. Subse-quent studies have identified the specific ion channels respon-sible for pacemaking in this neuron (Fujimura and Matsuda1989; Grace and Onn 1989; Harris et al. 1989; Kang and Kitai1993a,b; Kita et al. 1986; Mercuri et al. 1994; Nedergaard et al.1993; Shepard and Bunney 1991; Wolfart and Roeper 2002;Wolfart et al. 2001; Yung et al. 1991). Building on theidentification of ion channels responsible for that mechanism,there have been several computational models that have exam-ined the interactions among ion channels that give rise to thesubthreshold mechanisms responsible for pacemaking (Aminiet al. 1999; Li et al. 1996; Wilson and Callaway 2000). Themechanisms responsible for the irregular pattern of firing oftenseen in vivo (Hyland et al. 2002; Tepper et al. 1995; Wilson etal. 1977) are less clear. Presumably, the irregular firing patternrequires synaptic mechanisms because it is not seen in slices.However, this does not rule out constituitive properties of thedopaminergic cell that might generate irregularities in firing inthe presence of synaptic influences (Canavier et al. 2004;Rodriguez et al. 2003).

Irregular and rhythmic firing are not discrete states of thedopaminergic neuron. Some indication of the single spikepacemaker responsible for rhythmic firing is apparent in thefiring patterns of cells firing irregularly (Rodriguez et al. 2003;Wilson et al. 1977). By convention, any pattern that includesbursts in vivo is called a bursting pattern, but bursts in dopa-minergic neurons are actually singular events superimposed onthe rhythmic or irregular single spiking pattern (Grace andBunney 1984a; Hyland 2002; Tepper et al. 1995). In awakeanimals, bursts are associated with very specific behavioralcircumstances (Hyland 2002; Schultz 2002) and so must betriggered by, or at least tightly controlled by, synaptic inputs.Spontaneous bursting in dopaminergic neurons in anesthetizedanimals does occur, and even in this preparation, bursts notgenerated rhythmically but are isolated events that are super-imposed on a single-spiking pattern. Spontaneous bursts inanesthetized animals are a curiosity in any case because thebehavioral conditions for burst generation are ostensibly ab-sent. Perhaps spontaneous bursting reflects random occur-rences of the same circuit events required for behaviorallysignificant bursting in awake animals. It is possible that spon-taneous bursts arise from an entirely different mechanism.Given their relation to behavior, bursts are likely to depend onsynaptic input, but it is not certain the extent to which synapticinput serves only to trigger a burst, or to guide and shape thebursts (but see Paladini and Tepper 1999).

Alternative mechanisms of burst firing

In principle, the explanation for synaptically driven burstingneed not be complex. For those cells having the capacity to fire

much faster than their spontaneous rate, all that is needed is asuprathreshold depolarization of the correct duration. Synapticexcitation is a likely source of such depolarizations, anddopaminergic neurons receive excitatory glutamatergic andnicotinic cholinergic input from a number of sources (Kitai etal. 1999; Overton and Clark 1997). But in dopaminergicneurons, the precondition for this mechanism is not met.Depolarizing pulses applied to these cells via intracellularelectrodes have consistently failed to drive firing that resem-bles natural bursts either in pattern or in rate (Grace andBunney 1983a; Kang and Kitai 1993a; Kita et al. 1986;Richards et al. 1997). Firing in response to modest currentpulses that allow repetitive firing never achieves the high ratesobserved in natural bursts. When driven with currents in excessof those required to get above the maximum spontaneous rate(4–10 Hz), firing of dopaminergic cells fails rapidly, appar-ently due to depolarization block of the spiking currents (Rich-ards et al. 1997). Application of excitatory neurotransmitters,such as nicotinic cholinergic agonists (Grillner and Mercuri2002; Sorenson et al. 1998), or AMPA agonists (Chergui et al.1993; Johnson et al. 1992), likewise can produce increases infiring rate but do not drive the cells to fire at the 10 Hz ratescommonly seen during in vivo bursts.

Because the firing rate of the dopaminergic neuron is limitedby the prominent afterhyperpolarization that follows each spikeduring rhythmic spontaneous spiking, one class of possiblealternative mechanisms could arise from suppression of thecalcium dependent potassium current that is responsible for theafterhyperpolarization (Ping and Shepard 1996; Shepard andBunney 1991). Spiking in dopaminergic neurons treated withapamin (to block this current) is fragile, however, and cantrigger large plateau depolarizations that rapidly lead to spikefailure. Sustained spiking can occur only in a narrow range ofpotentials in which the cell is depolarized sufficiently to main-tain sustained firing but does not undergo depolarization block(Johnson and Wu 2004; Shepard and Bunney 1991). Identifi-cation of a synaptic mechanism with an apamin-like effect onthe AHP sufficient to generate bursts has not been successful,although SK-induced current has been shown to be reducedsomewhat by NMDA activation and by muscarinic cholinergicagonists (Kitai et al. 1999; Paul et al. 2003).

The calcium channels underlying subthreshold membranepotential oscillations are mostly responsible for calcium-de-pendent K� current in dopaminergic cells. The presence orabsence of an action potential superimposed on these oscilla-tions does not have a large effect on calcium influx or on theamplitude or duration of the hyperpolarizing phase of thesubthreshold oscillation, which is primarily due to SK channels(Kang and Futami 1999; Wilson and Callaway 2000). Theabsence of a big contribution from fast action potentials isapparently due to the low threshold of voltage-sensitive cal-cium channels responsible for the AHP (Kang and Kitai 1993a;Wilson and Callaway 2000; Wolfert and Roeper 2002). It isprobably also the reason the firing pattern that results of SKblockade does not greatly resemble the bursting seen in vivobut consists primarily of an alteration of the subthresholdoscillation, characterized by long-duration plateau potentialsthat trigger a few spikes at the beginning before the spike failsbecause of depolarization block of the spike.



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NMDA receptor activation is specifically associatedwith bursting

A number of studies suggest that NMDA receptor activationmay play a crucial role in generating bursts in dopaminergiccells in vivo. Dopaminergic neurons express NMDA receptorson their somata and dendrites (Albers et al. 1999; Lin andLipski 2001; Paquet et al. 1997). Application of NMDAantagonists to dopaminergic cells in vivo regularizes theirfiring and abolishes spontaneous bursting, which effect is notshared by non-NMDA glutamate antagonists (Chergui et al.1993; Overton and Clark 1993). Bursting can also be evokedby stimulation of cortical regions with direct or indirect syn-aptic connections with dopaminergic neurons, and that burstingis also suppressed by NMDA antagonists (Tong et al. 1996). Itis especially interesting that in these experiments the effectiveantagonists did not block the excitatory response to stimulationcompletely, but only the bursts. Bursts can also be evoked invivo by stimulation of other structures contributing glutama-tergic inputs to the substantia nigra, particularly the peduncu-lopontine nucleus (Lokwan et al. 1999) and the subthalamicnucleus (Smith and Grace 1992).

NMDA receptor activation can also produce burst firing ofdopaminergic neurons in slices. Bath application of NMDAand NMDA antagonists has been studied the most and has beenone of the principal models for bursting in dopaminergic cells(Amini et al. 1999; Canavier 1999; Johnson and Wu 2004;Johnson et al. 1992; Li et al. 1996). Bath application of NMDAto slices can induce rhythmic plateau potentials in dopaminer-gic neurons expecially if combined with blockade of SKchannels or intracellular calcium chelation (Johnson and Wu2004). The mechanism of these plateau potentials is apparentlyconnected to dendritic sodium disposition and has been thesubject of extensive examination, both experimentally andtheoretically (Canavier 1999; Johnson et al. 1992; Li et al.1996; Shen and Johnson 1998). Even in the absence of apaminor other treatment that could reduce the AHP, about one-thirdof dopaminergic neurons in slices exhibit spontaneous burstingon treatment with 10 mM NMDA (Johnson and Wu 2004). Aswith apamin treatment, bath application of NMDA agonistsproduces rhythmically occurring plateau potentials that do notresemble bursts as seen in vivo. However, this treatment canevoke high-frequency firing (in excess of the 10-Hz limit forbackground and driven firing). Perhaps the in vivo burst iscaused by a brief period of NMDA activation, insufficient toproduce rhythmic plateau potentials but long enough to evokea single burst.

In support of this view, bursting can be elicited by directiontophoretic application of glutamate or NMDA to dendritesof dopaminergic cells or stimulation of glutamatergic afferentsto the dendrites of dopaminergic neuron (Morikawa et al.2003). Bursts of firing at 10/s could be evoked by synapticstimulation or applied aspartate and were shown to be depen-dent on activation of NMDA receptors. Bursts happened in theabsence of any plateau potential and resembled natural burstsin appearance. They were followed by a long-duration pause infiring, which was blocked by antagonists of metabotropicglutamate receptor, which also inhibited release of calciumfrom intracellular stores. These results suggest that NMDAreceptor activation may be the principal mechanism of burstingin dopaminergic neurons and may occur without generation of

plateau potentials. They also indicate that the pause in firingfollowing bursts may arise from a separate cellular mechanismfrom the burst, explaining the absence of pauses followingbursts seen in behaving animals (e.g., Schultz 2002).

How does NMDA receptor activation promote bursting?

Previous models of the dopaminergic neuron have focusedeither on the sodium-based mechanism of NMDA-evokedrhythmic plateau potentials (Canavier 1999; Komendantov andCanavier 2001; Li et al. 1996) or on the calcium-mediated slowoscillation responsible for the background firing (Amini et al.1999; Wilson and Callaway 2000). In the models of NMDA-induced rhythmic bursting, the main purpose was to explain theplateau potential not the high-frequency firing. High-frequencyfiring in those models was a simple consequence of thatdepolarization, and depolarizing current injections at the somawould have the same effect. Thus while these models doexplain the rhythmic plateau potentials seen during bath appli-cation of NMDA, they do not account for high-frequency firingduring natural bursts (or during NMDA-induced plateau po-tentials) given that in nature the dopaminergic neuron’s actionpotential fails during prolonged depolarizations. For the samereason, they cannot account for the NMDA bursts seen in theMorikawa et al. (2003) study or for the dissociation betweenbursting and the postburst hyperpolarizations and pauses seenin that study. The model presented here does not address thesodium mechanism responsible for plateau potentials but ratherthe interaction between NMDA receptor activation and thecalcium mechanism responsible for the slow rhythmic back-ground firing. We have shown that the special status of NMDAin driving high-frequency firing is a natural consequence of thecoupled oscillator model of the dopaminergic neurons. In thismodel, burst firing evoked specifically by dendritic NMDAreceptor activation arises, not from a special set of ion chan-nels, but by the same mechanism responsible for the slowsingle spiking oscillation. The change that is responsible forthe burst is not a shift in the ion channels responsible forshaping the firing of dopaminergic neurons but the transientdominance of dendritic oscillation over that of the soma.

In the coupled oscillator model, the entire somatodendriticsurface of the dopaminergic neuron possesses the mechanismresponsible for rhythmic single spiking. The essential compo-nents of the mechanism are a low-threshold but mostly nonin-activating calcium current, plasma membrane calcium pump,and the SK-type calcium-dependent potassium current. Othermechanisms, including intracellular calcium stores and buffers,a persistent sodium current, transient voltage-sensitive potas-sium currents, and hyperpolarization-activated cation current,all participate and adjust the oscillatory period of this mecha-nism, but the main determinant of the period of oscillation isthe geometry of the neuron. Because most of the time duringeach cycle of the oscillation is occupied by the process ofcalcium filling of the cytoplasm or calcium removal, thesurface area to volume ratio of each part of the cell determinesthe natural frequency of the oscillation at that part. Voltagecoupling is strong enough (part of the time) to enforce a singleoscillatory frequency across the somatodendritic membranedespite the wide variation of natural frequencies represented.During rhythmic single spiking, as seen in slices and often invivo, the slow somatic membrane dominates the much faster



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oscillation of the dendrites. This happens because, despite thelarge dendritic surface area, the fast oscillations of the dendriticmembrane have a low amplitude and thus provide little currentwith which to counteract the large amplitude slow oscillationsof the somatic compartment. Dendritic NMDA receptor acti-vation, because of its voltage dependence, acts as a currentamplifier for the dendritic oscillation. The large surface area ofthe dendrites can, when amplified in this way, escape controlby the soma and express their high-frequency oscillation alongwith that of the slower one. With higher levels of NMDAactivation, the dendrites may dominate the soma and suppressthe slow oscillation completely. The processes responsible forthis are already present in a very simple and primitive repre-sentation of the dopaminergic neuron.

The mechanism of bursting relies on this switch betweensomatic and dendritic subthreshold oscillations but also on thepeculiar spiking behavior of dopaminergic neurons. The pro-pensity of dopaminergic cells to depolarization block of spik-ing was pointed out by Grace and Bunney (1983a, 1984a) andhas been noted by many others. It is visible in the low safetyfactor for antidromic invasion of the soma and dendrites andfor the decomposition of action potentials during spontaneousand driven bursting (Grace and Bunney 1983b). On the otherhand, the dopaminergic neuron also exhibits a high reliabilityof dendritic backpropagation of action potentials when they doinvade the somatodendritic membrane, and dendrites are capa-ble of generating action potentials (Hausser et al. 1995; Ne-dergaard and Hounsgaard 1996). In the model presented here,the origination point of action potentials shifted from thesomatic to dendritic compartments during bursts, but both thesomatic and dendritic membrane either fired or failed together.The crucial ingredient for the action potential to follow thehigh-frequency subthreshold oscillation generated by the den-drite was fast repolarization of the cell by the subthreshold(calcium-dependent potassium current) mechanism. In ourmodel, the dopaminergic cell’s spike mechanism is fragile, notbecause it has unusual sodium currents, but because the spike-generated AHP currents are not sufficient to repolarize the celland remove sodium channel inactivation. During slow singlespiking, the cell fires single spikes per cycle of the subthresh-old oscillation because the fast sodium channel retains substan-tial inactivation after a single spike to produce an elevatedthreshold and prevent subsequent spiking. This sodium inacti-vation is removed only on the hyperpolarizing phase of thesubthreshold oscillation (by the calcium-dependent K current).Thus high-frequency spiking can occur only when the cell isdriven by a high-frequency subthreshold oscillation. This ex-plains the fragility of the spiking mechansm after poisoningwith apamin and the failure of the cell to fire rapidly inresponse to constant depolarizing currents (including thosearising from AMPA and nicotinic ACh receptors). The crucialrole of NMDA receptor activation comes from its ability toamplify a high-frequency dendritic oscillation that includes apowerful spike-independent repolarizing current after eachaction potential.

Afferent control of bursting in dopaminergic cells

Our results suggest that the synaptic inputs responsible forburst generation in dopaminergic neurons may act primarily toalter the intrinsic membrane properties responsible for sub-

threshold oscillation. This mode of synaptic action is verydifferent from the usual formulation in which synaptic currentsdeposit charge on the membrane that changes the membranepotential through redistribution on the cell membrane. In themechanism we envision, there is no need for a synapticpotential, or a graded potential that decays over time and space,and no question of whether a synapse will bring the cell closeror farther from spike threshold. Instead, the voltage-dependentsynaptic current from NMDA receptors is added to the balanceof currents that make up the subthreshold oscillator, increasingthe contribution of the stimulated dendrite to a discrete transi-tion between alternative intrinsic firing modes. There is still theopportunity for spatial summation among synapses. In ourmodel, all dendritic compartments are simultaneously treatedwith NMDA. In a real neuron, only a subset of the dendritesmay receive synaptic input at any one time. The generation ofa burst will require some threshold number of dendrites acti-vated, comparable to the parameter nd in our model.

An intriguing feature of the mechanism we have proposed isits dependence on cell shape. The cause of the wide range ofnatural frequencies in our model is the discrepancy in sizebetween the somatic and dendritic compartments. Without this,and without the differential distribution of synapses on the finedendrites and soma, burst generation could not occur in thecoupled oscillator model. In this formulation, small distaldendrites are not a less effective site for excitatory synapses inthe control of firing, but on the contrary, the smallest dendritesare most advantageous for synaptic generation of bursts.

G R A N T S

This work was supported by National Institute of Neurological Disordersand Stroke Grant NS-047085 to C. J. Wilson and National Science FoundationGrant DMS-0109427 to N. Kopell.

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