Models of midbrain dopaminergic neurons

Dopaminergic (DA) neurons are defined as synthesizing and containing neurotransmitter and neurohormone dopamine. The neuron synaptically as well as dendritically releases dopamine (Bustos et al. 2004).

Midbrain DA neurons are positioned within three cell groups: ventral tegmental area (VTA), substantia nigra (SNc), and retrorubral area. There is no clear anatomical distinction or boundaries between the groups, but heterogeneities in biophysical properties of the DA cells both within and between these groups are documented (see e.g. Wolfart et al. 2001, Neuhoff et al. 2002). Nevertheless, the characteristics listed below are common for midbrain DA neurons.

## Contents

* 1 Basic biophysical characteristics of DA neurons
* 2 Firing patterns
* 2.1 Background firing
* 2.2 Burst firing
* 3 Models of firing activity of the DA cell
* 3.1 Background firing
* 3.2 Bursting evoked by bath application of NMDA
* 3.3 Apamin-induced bursting
* 3.4 In vivo bursting
* 4 Conclusions and open problems
* 5 References
* 6 See also

## [edit] Basic biophysical characteristics of DA neurons

* broad action potentials (>2 ms)
* a spontaneous pacemaker-like firing pattern (1-5 Hz)
* a “sag” in the membrane potential recorded during hyperpolarizing current pulses
* two distinct components of an action potential: an initial segment (IS) spike, and a somato-dendritic (SD) spike, which are displayed as separate peaks in extracelular recordings
* inhibition by dopamine or apomorphin (via D2 autoreceptors) and elimination of this effect by D2 receptor antagonists.

## [edit] Firing patterns

In isolation, the DA neurons display low-frequency regular tonic spiking. A less regular pattern of a similar average frequency is most commonly seen in vivo and has been suggested to maintain a constant background dopamine level in the target structures, such as striatum. Together, these two low-frequency patterns are called background firing. Additionally, DA neurons display burst firing, which is characterized by much higher firing rates attained for short time intervals. In vivo, these events have been observed to occur in response to reward-related stimuli, and it has been suggested that they serve to bind salient events with actions and movements in the immediate past to reinforce the repetition of those actions. For example, DA neurons fire a burst after the animal receives an unpredicted reward, but not a predicted one. Thus, they apparently fire in relation to reward prediction error (e.g. Schultz, 2002, but also see Redgrave and Gurney, 2006).

### [edit] Background firing

A slow (1-8 Hz) rhythmic single spiking pattern occurs spontaneously in vitro. This background firing is driven by a subthreshold oscillation called a slow oscillatory potentials (SOP), which persist after pharmacological blockade of spike-generating currents (Ping and Shepard 1996).

Calcium processes are essential for generating the SOP: persistent L-type calcium currents are responsible for depolarization and simultaneous accumulation of intracellular calcium, whereas SK-type calcium-activated potassium currents produce the repolarization phase of the oscillation. These two currents constitute a calcium-potassium mechanism for pacemaking in this neuron (see e.g. Harris et al. 1989). The oscillations are exhibited in both the soma and dendrites, suggesting that the pacemaking mechanism is distributed along the dendrites (Wilson and Callaway 2000). The accumulation and removal of intracellular calcium, which determines the period of the oscillation, has been shown to be slower in the soma than in the dendrites due to the difference in the surface area-to-volume ratio (Wilson and Callaway 2000). Nevertheless, the SOP occurs synchronously in the soma and dendrites.

### [edit] Burst firing

In contrast to other types of neurons, burst firing of DA cells cannot be elicited in vitro by somatic current injections. Moreover, the spiking frequency cannot be elevated in this way to the rates attained during bursts either. Instead, as increasing amounts of depolarizing current are injected, a steady firing rate of 10 Hz is the maximum rate achieved before spiking ceases (Richards et al 1997), apparently due to inactivation of spike-generating currents (depolarization block). Thus, unlike other neurons, the DA cell requires a mechanism designated for high-frequency spiking exhibited during bursts. This mechanism may or may not rely on a slow subthreshold wave that brings the cell closer to or further from the spike threshold.

Nevertheless, injection of a depolarizing somatic current can elicit a burst of the DA cell in vivo, and bursting also has been elicited in vitro pharmacologically. Bath application of an agonist of the NMDA synaptic receptor or a selective blocker of the SK current, apamin has been used to induce bursting. A burst was also evoked in vitro by stimulation of afferent fibers or iontophoresis of NMDA agonists differentially to dendrites of the DA cell. However, these methods result in bursts that are significantly different from each other (see e.g. Johnson and Wu 2004) and may be different from the in vivo bursts (see Overton and Clark 1997). The main properties of each type of bursting in the DA cell are described below.

In vivo bursting

Firing rates of ≥ 20 Hz can be achieved for irregularly distributed short time intervals in the DA neuron in vivo (Hyland et al. 2002). I.e., usually, the episodes of elevated firing rate don’t follow any apparent repetitive pattern, although rhythmic bursting may occasionally be observed (see e.g. Freeman et al. 1985). A burst in the DA neuron has been defined to start with an interspike interval shorter than 80 ms, and terminate with that greater than 160 ms (Grace and Bunney 1984). Within the burst, the spike amplitude progressively decreases, and the spike duration and interspike interval progressively increase (but see Hyland et al. 2002).

The high rates are thought to be driven by excitatory synaptic inputs to the DA neuron, the majority of which are glutamatergic. A number of studies indicate that bursts in vivo are dependent on activation of NMDA, but not AMPA/kainiate glutamate receptors (see e.g. Meltzer et al. 1997 for discussion). Additionally, blockade of GABAA synaptic currents has been shown to induce bursting in DA neurons, suggesting a disinhibition hypothesis of DA burst firing in vivo (Tepper et al. 1995).

The burst is critically dependent on calcium: along with a stimulatory influence of intracellular calcium, extracellular Ca2+ administration decreases the frequency of firing a burst (Grace and Bunney 1984). Injection of a calcium buffering agent (calcium chelator), which decreases the concentration of free cytosolic calcium, has been reported to abolish depolarization-induced bursting. Additionally, SK channel blockers induce burst firing in vivo (Ji and Shepard 2006, Waroux et al. 2005), similar to the action of SK blockers in vitro (see below).

The frequency of burst occurrence can be regulated differentially from the average firing rate. In particular, a 25-fold increase in the occurrence of a burst by iontophoresis of barium (presumably by blocking K+ currents) was accompanied by only a weak influence on the average firing rate (Grace and Bunney 1984). Cadmium has been shown to increase the occurrence of the bursts 10 times, but only moderately elevate the average firing rate (~ 50%). The burst induction by removal of GABAA inhibition has not been correlated with an increase in the firing frequency (Tepper et al. 1995).

In vitro bursting induced by bath application of NMDA

Repetitive bursting can be elicited in DA neurons via bath application of NMDA together with a hyperpolarizing somatic current injection (Johnson and Wu 2004). The regenerative nature of the NMDA current, which is displayed as a negative slope in its IV characteristics, has been shown to be critical for burst evocation.

This type of bursting is calcium-independent: bursts persist in Ca2+-free solution (Johnson et al. 1992). Bursting also persists upon chelation of intracellular Ca2+, in contrast to in vivo burst firing.

Bursting induced by bath application of NMDA is critically dependent on extracellular sodium. Activation of an electrogenic sodium pump, but not potassium currents, is involved in terminating the bursts and hyperpolarizing the cell between the bursts (Johnson et al. 1992). The depolarizing ramp initiating the burst is suggested to rely primarily on the sodium ion entry through the NMDA channel. The spike-generating fast sodium current is not implicated in burst generation because blockade of this current abolishes spiking, but does not block voltage oscillations underling bursts.

In vitro bursting induced by apamin

Repetitive bursting of the DA neuron can be observed during bath application of apamin (Shepard and Bunney 1991). Under apamin, however, DA cells are susceptible to depolarization block, and bursts are not easily elicited. Blockade of spike generation preserves voltage oscillations underling bursts (Ping and Shepard 1996), which are called square-wave oscillations.

Apamin-induced bursting is critically dependent on calcium: application of an L-type Ca2+ channel blocker has been shown to abolish the plateau potential oscillations, whereas a drug increasing the open probability for this channel induces burst firing, especially in the presence of apamin (Johnson and Wu 2004). Bursting elicited in such a way has been shown to be abolished in Ca2+-free medium. This stimulatory role of extracellular Ca2+ suggests a distinction between apamin-induced and in vivo bursting of DA neurons.

Bursting has been shown to be most robust in simultaneous bath application of NMDA and apamin. Properties of bursting in this case are similar to those for whole bath NMDA-induced burst firing, including its Na+, but not Ca2+ dependence (Johnson and Wu 2004). This suggests that apamin, by blocking the SK current, can facilitate two different bursting mechanisms, one of which is intrinsic and the other based on the NMDA receptor activation.

In vitro bursting induced by stimulation of afferent fibers or iontophoresis of glutamate

A burst has been observed after extracellular electrical stimulation (a train of short pulses) or iontophoresis of a glutamate receptor agonist to dendrites of the DA cell (Morikawa et al. 2003). Subsequent bath application of an NMDA receptor antagonist has been shown to inhibit the burst, suggesting involvement of NMDA receptors. By contrast to bursting elicited by the other two methods in vitro, here spikes within a burst are not superimposed on a depolarizing plateau. Moreover, hyperpolarization after the burst originates from a separate pathway. This pathway can be blocked or desensitized without affecting the burst.

## [edit] Models of firing activity of the DA cell

### [edit] Background firing

Figure 1: The Ca2+-K+ mechanism for subthreshold oscillations.

Two models that describe background firing were designed independently, but share several features (Figure 1). First, based on the persistence of the oscillations under blockade of the spiking currents, the problem was reduced to modeling of the underlying SOP. Second, all ionic concentrations, except for intracellular calcium, were assumed to be constant. Third, along with the influx via the calcium current and removal via a calcium pump, calcium balance included buffering. Different assumptions on calcium buffering and removal were used without affecting results significantly.

In the model by Amini et al. (1999), all ionic currents characterized in the DA neuron were included. A one-compartment representation of the neuron has been found sufficient to reproduce the subthreshold voltage oscillations (Figure 4A). Consistent with pharmacological data, the voltage dependence of the persistent calcium current responsible for oscillations has been identified as similar to the L-type found in other types of neurons. Currents other than the L-type and the SK-type, though not necessary for pacemaking, have been shown to bias the cell to the appropriate range of the membrane potential and Ca2+ concentration.

Figure 2: Oscillatory transient of calcium concentration at the somatic and different dendritic locations along with the somatic voltage (left: experiment, right: model). Reproduced from Wilson and Callaway (2000) with permission.

Wilson and Callaway (2000) introduced a multicompartmental representation of the neuron to study how the presence of the oscillatory mechanism in dendrites influences background firing. Confirming that a persistent calcium and the SK-type potassium currents are sufficient for pacemaking, the oscillations were reproduced by inclusion of these two currents only. The multicompartmental model has explained characteristics of an oscillatory transient observed in the DA cell after release from hyperpolarization, such as the increasing peaks of somatic calcium concentration and a gradual decrease in the oscillation frequency (Figure 2). In simulations, irregular low frequency oscillations have been shown to arise from asynchronous oscillations of neuronal compartments, suggesting a novel mechanism for irregular background firing. Thus, modeling suggests coupled oscillator representation of the neuron, where the soma and dendritic parts have different natural frequencies and may synchronize due to electrotonic coupling through cytoplasmic conductivity.

Mathematical analysis of this coupled oscillator model has shown that the transient dynamics corresponds to a slow drift along an attracting invariant subset in the phase space (Medvedev et al. 2003). The rate of calcium removal has been found to determine the duration of the transient. Robustness of this dynamics has been confirmed in a chain of FitzHugh-Nagumo oscillators that roughly mimics basic dynamical properties of the biophysical model (Medvedev and Kopell 2001).

### [edit] Bursting evoked by bath application of NMDA

Figure 3: The Na+ mechanism for bursting evoked by bath application of NMDA.

Several common features can be identified in different models of bursting induced by bath application of NMDA (Figure 3). First, multicompartmental representation of the neuron has been adopted in these models because experiments suggest that the neuron is not equipotential: dendrites of the DA neuron can sustain NMDA-induced oscillations even when the soma is voltage clamped. Second, the sodium-potassium exchanger and the NMDA current are assumed to be critical for the oscillations underlying bursts during bath application of NMDA. Assuming distal origin of the oscillations, these cellular mechanisms have been restricted to the dendrites. Third, the voltage sensitivity of the NMDA current has been assumed to be the same for all three ions conducted, and represented by a sigmoid voltage dependence in the assumption of instantaneous Mg2+ block.

Li et al. (1996) modeled the neuron as two compartments: the soma and a dendrite. Under simulated NMDA receptor activation, the NMDA inward current and the sodium-potassium exchanger outward current generated slow oscillations similar to that underlying NMDA-induced bursting. The timescale of the oscillations was determined by accumulation and removal of dendritic sodium. The regenerative nature of the NMDA current (i.e., the negative slope of its I-V characteristic) has been shown to be crucial for the mechanism of burst generation. Consistent with experimental observations, modeling has shown the persistence of slow oscillations in the clamping current when the soma is voltage clamped at -60mV.

Canavier (1999) and Komendantov et al. (2004) introduced a more morphologically realistic model, which allowed for electrical coupling among the compartments to be fitted to experimentally measured voltage gradients and spike attenuation. In accord with the earlier results on the role of NMDA current, bursting activity has been found to be favored by manipulations that enhance the region of negative slope in the whole-cell I-V characteristics and inhibited by those manipulations that dampen this region. In particular, activation of such ohmic (linear) currents as the AMPA or GABA eliminates bursting in the models. On the other hand, modeling predicts that excessive NMDA-induced depolarization also abolishes bursting and can be counteracted by a concurrent increase in the somatic GABAA current. The necessity of an applied hyperpolarizing current for this type of bursting has been explained as counteracting this excessive NMDA-induced depolarization. Additionally, simulated blockade of the SK current facilitated bursting, reproducing experiments with simultaneous bath application of NMDA and apamin.

Limitations

The models were focused on the voltage oscillations underlying bursts, and high-frequency firing was assumed to be a simple consequence of the depolarizing phase of the above oscillation. However, this is not the case for the DA neuron, in which action potential generation rapidly fails during depolarizing current injection.

The model without spikes (Canavier 1999) does not address this question at all, while the major limitation of the models with spikes (Li et al. 1996, Komendantov et al. 2004) is the hyperexcitability of the spike-producing mechanism. The ability to generate repetitive spiking due to the fast sodium and delayed rectifier potassium currents gives not only an abnormally elevated spiking frequency in the bursts, but also rebound bursting in the model by Lee et al. (1996) and other effects which have not been seen in the DA neuron.

### [edit] Apamin-induced bursting

Based on the persistence of the underlying plateau potential oscillation under blockade of spikes, the spike-producing currents have been excluded from models (Amini et al. 1999, Canavier et al. 2007). Both models had only one compartment based on the assumption that the plateau potential oscillations are generated near the soma. The same assumptions on ionic balance as in modeling of the SOP have been made. In the presence of apamin, Amini et al. (1999) suggests calcium-mediated inactivation of the calcium current as responsible for the repolarization of the plateau potential oscillation. By contrast, Canavier et al. (2007) includes a slow delayed rectifier (ether-a-go-go, ERG) current for such repolarization.

Figure 4: Subthreshod voltage oscillations: (A) control, (B) apamin application. Reproduced from Amini et al. (1999) with permission.

Both models reproduce the transition from the SOP to the plateau potential oscillation upon simulated blockade of the SK current (Fig.4B). However, Canavier et al. (2007) disproves the calcium-dependent mechanism of repolarization of the plateau potential oscillation hypothesized in Amini et al. (1999). Instead, the repolarization phase is shown to be independent of accumulation of cytosolic calcium and rely on the slowly activating voltage-dependent ERG current. Confirming the relevance of the modeled mechanism to the observed characteristics of the plateau potential oscillation, the depolarized plateau duration has been shown to increase in response to partial blockade of the ERG current.

Limitations

The models have not considered the interaction of the plateau potential oscillations with spike generation. Therefore, the limitations for modeling bursts induced by bath application of NMDA also apply here.

### [edit] In vivo bursting

Two independently designed models for in vivo bursting used substantially different assumptions. In Kuznetsov et al. (2006), no oscillatory mechanism causing the slow depolarizing wave that underlie repetitive bursting in DA cells in vitro was included. The subthreshold currents were designed to generate the SOP, and, therefore, the model included a simple version of the calcium-potassium mechanism for background firing similar to that in Wilson and Callaway (2000) (see assumptions above). To account for the difference in natural frequencies of the soma and dendrites, the model included two electrically coupled compartments. One compartment represented the soma. The other one stood for multiple distal dendrites, in the assumption that they are synchronous with each other.

Figure 5: Left: an interval of NMDA-induced high-frequency firing superimposed on the background low-frequency activity. Right: the current between the soma and dendrites indicating change in dominance at the onset of the high frequency. Reproduced from Kuznetsov et al. (2006) with permission.

The spike-producing fast sodium and delayed rectifier potassium currents were added in such a way that they alone were not able to generate repetitive firing. Spike-generating currents did not have a significant influence on pacemaking, firing one action potential on every cycle of the subthreshold oscillation. A depolarizing applied current elevated the firing frequency, but not above 10 Hz limit because it decreased the amplitude and eventually blocked the subthreshold oscillations. By contrast, NMDA current activation restricted to the dendrite elevated the spiking frequency to the levels comparable to that seen during NMDA-induced bursting in vivo (Figure 5 left). High-frequency firing has been shown to be dominated by dendrites, while the background firing is dominated by the soma(Figure 5 right). The mechanism of dendritic dominance is suggested to rely on a phenomenon of localization, where natural oscillations of one compartment are suppressed due to coupling with another.

In addition, either somatic or dendritic AMPA activation had an effect similar to that of somatic current injection rather than dendritic NMDA activation. The difference between AMPA and NMDA influence has been attributed to their distinct I-V characteristics. The difference has been explained via nullcline analysis in a single nonspiking compartment, where NMDA promotes while AMPA suppresses oscillations. In accord to the studies mentioned above, these results suggest distinct functional properties for apparently closely related synaptic inputs such as AMPA and NMDA.

Figure 6: Voltage traces in a model with realistic synaptic inputs. 1: Control, 2: Doubling AMPA, 3. Blocking SK. Reproduced from Canavier and Landry (2006) with permission.

The approach of Komendantov and Canavier (2002) and Canavier and Landry (2006) was based on the model for bursting evoked in vitro by bath application of NMDA (see assumptions above). In the latter article, in vivo-like synaptic inputs were introduced. Based on the evidence of preferential proximal effect of GABA in vivo, most of the GABAA current was introduced to the soma. By contrast, both the NMDA and AMPA currents were restricted to dendrites, reflecting a distal location of glutamatergic synapses. Simulated glutamatergic synaptic events obeyed a Poisson time distribution. The model with such realistic kinetics and statistics of glutamatergic synaptic inputs has been shown to closely replicate in vivo-like firing patterns (Figure 6). The work suggests that synaptic events along with intrinsic dynamics of the neuron control the firing pattern.

The same simulations have explained the variable degree of correlation between the average firing rate and the firing pattern observed in experiments. Increasing the short-lived AMPA EPSC has been shown to evoke additional spikes without regard to the background firing pattern, producing a comparable increase in the average spike frequency and the probability of burst occurrence (Figure 6 middle). By contrast, blocking the SK current allows a single depolarization to evoke more spikes, increasing the number of bursts rather than the average spike frequency (Figure 6 bottom). Thus, the rate and pattern of the neuron can be modulated either concurrently or differentially by the two proposed mechanisms.

In Komendantov and Canavier (2002), synaptic projections were omitted, but dopaminergic neurons were assumed to be coupled electrically by gap junctions. The authors have shown that electrical coupling between two model neurons can facilitate burst firing. This happens either when both neurons in isolation are tonically active at a high frequency due to a strong NMDA input, or when NMDA strongly affects only one of the model neurons.

Limitations

The model by Kuznetsov et al. (2006) does not address interactions among different bursting mechanisms of the DA neuron. In particular, repetitive spike generation was modeled critically dependent on the SK current. The inclusion of other currents known to contribute to the AHP is needed to verify if a model capable of fast firing in the absence of the SK current will otherwise fire only one action potential per each cycle of the SOP.

Hyperexcitability of the spike-generation mechanism in Komendantov and Canavier (2002) and Canavier and Landry (2006) is to be corrected, as well as in the models of bursting induced by bath application of NMDA. Additionally, in these models, calcium dynamics is restricted to the soma, while the pacemaking calcium-potassium mechanism is shown to be distributed along dendrites of the DA neuron.

Comparison of model assumptions Included cellular mechanisms  |  Ca2+ balance  |  Ca2+-K+ mechanism for SOP  |  Na+ balance  |  Spiking currents  |  NMDA input  |  AMPA input  |  GABA input  |  Realistic timeseries of synaptic stimulation

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Amini et al. 1999  |  yes, soma only  |  yes, soma only  |  no  |  no  |  no  |  no  |  no  |  no
Canavier 1999  |  no  |  no  |  yes, distributed  |  no  |  yes  |  yes  |  yes  |  no
Canavier and Landry 2006  |  yes, soma only  |  yes, soma only  |  yes, distributed  |  yes, hyperexcitable  |  yes  |  yes  |  yes  |  yes
Li et al. 1996  |  yes, soma only  |  no  |  yes, dendrite only  |  yes, hyperexcitable  |  yes  |  no  |  no  |  no
Komendantov et al. 2004  |  yes, soma only  |  yes, soma only  |  yes, distributed  |  yes, hyperexcitable  |  yes  |  no  |  yes  |  no
Kuznetsov et al. 2006  |  yes, distributed  |  yes, distributed  |  no  |  yes  |  yes  |  yes  |  no  |  no
Wilson and Callaway 2000  |  yes, distributed  |  yes, distributed  |  no  |  no  |  no  |  no  |  no  |  no

## [edit] Conclusions and open problems

The combination of experimental and modeling studies has not yet given a consistent picture of firing patterns and underlying electrophysiological properties of the neuron. First, most of the modeling studies don’t address the difference between the depolarizing wave that underlie bursting and a depolarizing current injection, which abolishes spiking instead. The only modeling article that differentiates these two cases (Kuznetsov et al. 2006), suggests that the spike-producing currents are incapable of repetitive firing and both high- and low-frequency spiking rely on additional afterhyperpolarization by the SK current. However, this mechanism cannot be applied to apamin-induced high-frequency spiking within bursts.

Second, several significantly different mechanisms for oscillations underlying bursts have been documented, and the contribution of these mechanisms to in vivo bursting remains controversial. Nevertheless, the latest modeling results suggest that in vivo bursts may be caused by weak or brief NMDA activation, insufficient to elicit rhythmic bursting. As such, in Kuznetsov et al. (2006), the high-frequency episode (Figure 5) directly reflects activation of the NMDA current, whereas repetitive bursting cannot be elicited since the corresponding mechanisms are not included. In Canavier and Landry (2006), the oscillatory mechanisms causing repetitive bursting are present, but activated only weakly when the appearance of in vivo bursting is best reproduced, as indicated e.g. by weak interburst hyperpolarizations (Figure 6). Therefore, a synaptic input rather than a slow repetitive depolarizing wave should trigger the burst in vivo. The synaptic trigger is consistent with the association of bursts with very specific behavioral circumstances in awake animals.

Both approaches to modeling in vivo bursting suggest that NMDA receptor activation plays the central role in burst evocation, but the proposed mechanisms of the NMDA influence are distinctly different. In Kuznetsov et al. (2006), high-frequency spiking within a burst can occur only as a result of a high-frequency subthreshold oscillation. In a similar way, background firing requires the SOP as a driving force. The dendritic NMDA current directly interacts with the calcium-potassium subthreshold oscillatory mechanism distributed along dendrites. It amplifies the high-frequency dendritic oscillations, and eventually allows them to dominate the slow somatic oscillation.

By contrast, in Canavier and Landry (2006), the role of the NMDA current is identified as opposing the SK current, and therefore reducing the amplitude and duration of the AHP. A resulting condition for grouping spikes into a burst is either low calcium concentration, or high level of NMDA activation, suggesting that both internal and external factors contribute to shaping the burst. Therefore, the extent of contribution of external inputs vs. internal dynamics is still to be determined.

Finally, the firing pattern of the DA neuron has been shown to depend on the neuron morphology (Kotter and Feizelmeier 1998). Assuming conservation of cellular mechanisms contributing to electrophysiological properties, the morphological difference of DA neurons from rats to primates has been shown to lead to different firing patterns. Thus, the electrophysiological properties of the DA neuron cannot be assumed to be the same in different species, and further investigation of possible compensatory mechanisms that provide the uniformity of firing patterns across species is essential.

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Internal references

* Peter Redgrave (2007) Basal ganglia. Scholarpedia, 2(6):1825.
* Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
* James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
* Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
* Eugene Roberts (2007) Gamma-aminobutyric acid. Scholarpedia, 2(10):3356.
* Peter Jonas and Gyorgy Buzsaki (2007) Neural inhibition. Scholarpedia, 2(9):3286.
* Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
* Wolfram Schultz (2007) Reward. Scholarpedia, 2(3):1652.
* Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.

## [edit] See also

Dopamine, Dopamine modulation, Midbrain