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J Neurophysiol 90: 1095-1114, 2003. First published March 20, 2003; doi:10.1152/jn.00618.2002
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Modulation of Striatal Single Units by Expected Reward: A Spiny Neuron Model Displaying Dopamine-Induced Bistability

Aaron J. Gruber1,2, Sara A. Solla2,3, D. James Surmeier2 and James C. Houk1,2

1 Department of Biomedical Engineering, Northwestern University Medical School, Chicago, Illinois 60611; 2 Department of Physiology, Northwestern University Medical School, Chicago, Illinois 60611; 3 Department of Physics and Astronomy, Northwestern University Medical School, Chicago, Illinois 60611

Submitted 31 July 2002; accepted in final form 15 March 2003


    ABSTRACT
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 DISCLOSURES
 REFERENCES
 
Single-unit activity in the neostriatum of awake monkeys shows a marked dependence on expected reward. Responses to visual cues differ when animals expect primary reinforcements, such as juice rewards, in comparison to secondary reinforcements, such as tones. The mechanism of this reward-dependent modulation has not been established experimentally. To assess the hypothesis that direct neuromodulatory effects of dopamine on spiny neurons can account for this modulation, we develop a computational model based on simplified representations of key ionic currents and their modulation by D1 dopamine receptor activation. This minimal model can be analyzed in detail. We find that D1-mediated increases of inward rectifying potassium and L-type calcium currents cause a bifurcation: the native up/down state behavior of the spiny neuron model becomes truly bistable, which modulates the peak firing rate and the duration of the up state and introduces a dependence of the response on the past state history. These generic consequences of dopamine neuromodulation through bistability can account for both reward-dependent enhancement and suppression of spiny neuron single-unit responses to visual cues. We validate the model by simulating responses to visual targets in a memory-guided saccade task; our results are in close agreement with the main features of the experimental data. Our model provides a conceptual framework for understanding the functional significance of the short-term neuromodulatory actions of dopamine on signal processing in the striatum.


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 DISCLOSURES
 REFERENCES
 
The classic notion of the basal ganglia as being involved in purely motor processing has expanded over the years to include sensory and cognitive functions (Brown et al. 1997Go). This view is substantiated by single-unit recordings in the neostriatum, an input structure of the basal ganglia, that reveal a wide range of neural activity associated with sensory stimuli, with motor planning as well as execution, and with working memory and other cognitive functions (Schultz et al. 1995Go). A surprising new finding is that much of this activity shows a motivational component. For instance, neostriatal activity related to visual stimuli (Kawagoe et al. 1998Go) or movement planning (Hollerman et al. 1998Go) is dependent on the expected reinforcement a behavior will elicit. Task-related activity can be enhanced or suppressed when a reward is anticipated for correct performance. The effect of reward expectation is significant. Most task-related units show a dependence on the type of reinforcement (Hollerman et al. 1998Go), and the response modulation is often large enough to be clearly seen in individual trial records (Kawagoe et al. 1998Go). While the reward-dependent activity may have cortical origins in part (Wilson 1990Go), the neostriatum receives a prominent neuromodulatory signal with properties that seem quite appropriate for mediating reward dependence.

Substantia nigra pars compacta (SNpc) neurons, which send a dopaminergic input to striatal spiny neurons, discharge in a reward-dependent manner (Schultz 1998Go). These dopamine neurons respond not only to the delivery of unexpected rewards but also to sensory cues that reliably precede the delivery of expected rewards (Ljungberg et al. 1991Go; Mirenowicz and Schultz 1994Go; Schultz et al. 1993Go). The data also suggest that the responses of dopamine neurons occur at approximately the same time as striatal spiny neuron responses to visual targets (Hollerman et al. 1998Go; Kawagoe et al. 1998Go). This, in conjunction with the demonstration of dopamine-mediated neuromodulation of spiny neuron activity in vivo (Gonon 1997Go; Kiyatkin and Rebec 1996Go) and in vitro (Akaike et al. 1987Go; Calabresi et al. 1987Go; Flores-Hernandez et al. 2000Go; Hernandez-Lopez et al. 1997Go; Surmeier et al. 1992aGo), suggests that the release of dopamine in the neostriatum could be responsible for the reward dependence of neostriatal neurons. Nevertheless, the functional effects of dopamine on the electrophysiological properties of spiny neurons remain to be fully elucidated.

Dopamine can alter the responsiveness of medium spiny neurons through the modulation of synaptic efficacy or through the modulation of voltage-dependent ionic currents that govern the response to synaptic inputs (Nicola et al. 2000Go). The model presented here focuses exclusively on the latter modulatory effects of dopamine, which should accompany burst activity of SNpc neurons. These effects cannot be viewed as simply excitatory or inhibitory. For example, activation of the D1 type dopamine receptors alone can either enhance or suppress responses of spiny neurons depending on the prior state of the neuron (Hernandez-Lopez et al. 1997Go). This state dependence arises from the coordinated modulation of ion channels regulating these states (Flores-Hernandez et al. 2000Go; Hernandez-Lopez et al. 1997Go; Pacheco-Cano et al. 1996Go; Surmeier et al. 1992aGo, 1995Go). Here, we use a computational approach to assess the hypothesis that the modulation of two channel types resulting from the activation of D1 receptors is sufficient to explain both enhanced and suppressed single-unit responses of medium spiny neurons to reward-predicting stimuli.

Our goal is to construct a minimal biophysically grounded model of spiny neurons whose simplicity allows us to perform a detailed analysis of D1 receptor-mediated modulation of the model response properties and to extract from this analysis qualitative features that explain the reward dependence of neostriatal single-unit responses. We validate the model by simulating responses to visual targets in the memory-guided saccade task described by Kawagoe and colleagues (1998Go) and by comparing our results to the main features of their experimental data. In any given block of trials, these investigators selectively rewarded saccades made to only one of four potential targets. This allowed them to compare the response of a specific unit to a given target in rewarded as opposed to unrewarded cases. For many cells, there was a substantial reward-dependent difference. The majority of these neurons showed a reward-related enhancement of the intensity and duration of discharge, and a smaller number exhibited a reward-related depression. Dopamine neurons in the SNpc are known to have a selective response to the presentation of visual targets that precede reward in a learned task (Ljungberg et al. 1991Go; Schultz et al. 1993Go). Correspondingly, Kawagoe and colleagues (1998Go) suggested, and later confirmed (Kawagoe et al. 1999Go), that the presentation of the target in the rewarded trials serves as the conditioned stimulus that elicits SNpc discharge, which should then release dopamine in the striatum. They speculated that D1 receptor activation might explain enhanced responses, whereas D2 receptor activation might explain depressed responses. The model presented here confirms that realistic biophysical assumptions about the neuromodulatory effects of dopamine acting through D1 receptors account well for the reward-dependent enhancement of striatal unit discharge. Furthermore, due to the emergence of bistable responsiveness, D1 effects also account well for the depressed responses. Bistability constitutes a qualitative change in response characteristics, and its emergence in spiny neurons could be a very important consequence of dopamine neuromodulation.


    METHODS
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 DISCLOSURES
 REFERENCES
 
Model description

As stated in the INTRODUCTION, our goal is to construct a biophysically grounded membrane model that includes sufficient detail to reproduce the qualitative effects of D1 receptor activation on the characteristic up/down state behavior of spiny neurons, while remaining simple enough for detailed analysis and for inclusion in future network simulations. A detailed investigation of the static and dynamic properties of the membrane model in high and low dopamine conditions reveals generic features of reward dependence. A specific simulation illustrates the modulation of the response to target presentation for comparison to reward-dependent single-unit activity in the visually guided saccade task.

The membrane properties of the model spiny neuron result from a biophysically grounded representation of a minimal set of currents needed to reproduce the characteristic up/down state behavior of spiny neurons. These cells exhibit pseudo two-state behavior in vivo; they spend most of their time either in a hyperpolarized "down" state around – 85 mV or in a depolarized "up" state around –55 mV (Wilson 1993Go; Wilson and Kawaguchi 1996Go). This bimodal character of the response to cortical input has been attributed to a combination of inward rectifying and outward rectifying potassium currents (Nisenbaum and Wilson 1995Go; Wilson and Kawaguchi 1996Go). The inward rectifying current, predominantly of the Kir2 type in spiny neurons (Mermelstein et al. 1998Go), contributes a small outward current at hyperpolarized membrane potentials above the K+ equilibrium potential, thus providing resistance to depolarization and stabilizing the down state. This current accounts for most of the ionic current near the resting potential and is blocked on depolarization (Nisenbaum and Wilson 1995Go). The enhancement of Kir2 currents by D1 agonists is thought to be an important component of the suppressive effects of D1 activation at subthreshold potentials (Pacheco-Cano et al. 1996Go). The outward rectifying K+ current in spiny neurons includes slowly and rapidly inactivating components (Nisenbaum et al. 1996Go; Surmeier et al. 1991Go, 1992bGo) that are attributable to different channel types. The rapidly inactivating K+ components inactivate after the transition to the up state; it is the slowly inactivating component that influences the membrane potential for the remaining duration of the up-state episode. This current becomes activated at subthreshold potentials and opposes the depolarizing influences of excitatory synaptic and inward ionic currents; it is the balance between these inputs that determines the membrane potential of the up state. The two K+ currents included in our model, Kir2 and Ksi (si, slowly inactivating), have been shown (Nisenbaum and Wilson 1995Go) to account for the characteristic nonlinear voltage dependence of the outward current measured in spiny neurons. We recognize that the si K+ current is likely to arise from at least two channel types, but for the sake of simplicity we have treated it as a single conductance. This combined outward current acts in opposition to inward ionic and synaptic currents to regulate membrane potential in the up/down states.

The other major ionic mechanism included in the model provides an inward, depolarizing drive. L-type calcium currents are found in all medium spiny neurons (Bargas et al. 1994Go; Song and Surmeier 1996Go). In contrast to a number of other cell types, L-type currents in medium spiny neurons begin to activate at subthreshold membrane potentials, thus modulating the voltage range of the up state (Bargas et al. 1994Go). This subthreshold activation is attributable to the expression of Cav1.3 L-type channels by medium spiny neurons (Olson and Surmeier 2002Go). This current is enhanced by D1 agonists in medium spiny neurons expressing D1 receptors (Surmeier et al. 1995Go, 1996Go), and this modulation is critical to the increased excitability produced by D1 agonists at depolarized membrane potentials (Cooper and White 2000Go; Hernandez-Lopez et al. 1997Go); it is therefore included in the model.

Our approach (Gruber and Houk 2000Go) is to design a model that provides a consistent description of membrane properties in the 100 to 1,000 ms time range. This is the characteristic range of duration for up- and down-state episodes; it also spans the time course of short-term modulatory effects of dopamine. To provide a reliable description of the dynamical properties of spiny neurons in this intermediate time range, the model is constructed according to the principle of "separation of time scales" (Bender and Orszag 1978Go; Rinzel and Ermentrout 1989Go), a successful and fundamental technique in the study of dynamical systems. Processes that operate in the 100 to 1,000 ms range are modeled as accurately as possible. Processes that activate on a much shorter time scale are assumed to have instantaneously achieved their steady-state values. Similarly, processes that inactivate on such short time scales are not included. The time variation of processes that occur over longer time scales, such as slow inactivation, is neglected. The model therefore cannot provide a good description of rapid events such as the generation of action potentials or the precise time course of transitions between up and down states. This approach excludes many currents that contribute to the control of the membrane potential. The addition of such currents would improve the ability of the model to provide quantitative descriptions of short-term phenomena to relate to dynamical biophysical data accounted for in other models (Kitano et al. 2002Go; Wickens and Arbuthnott 1993Go), but it would not improve the usefulness of our model for determining if the enhancement of Kir2 and L-type Ca currents is sufficient to account for the reward dependence of single-unit activity. A reliable answer to this question follows from the type of detailed analysis that can only be performed on a simple model such as the one proposed here.

In addition to a detailed analysis of the generic features of the membrane model in high and low dopamine conditions, we provide a simulation of responses to target presentation in the memory-guided saccade task used by Kawagoe and colleagues (1998Go). This simulation demonstrates one specific instance of the generic properties of the model for particular values of input magnitudes and duration, chosen to illustrate that response modulation accounts for the qualitative features of reward-dependent single-unit responses. The input parameters are chosen so as to be consistent with experimental data from various sources but do not impact the generic properties of the membrane model.

The components of the model used for the simulation of the saccade task are shown schematically in Fig. 1. The model spiny neuron (components inside the dashed box) receives two types of input: excitatory input from cortex and modulatory input from SNpc. Neurons in cortical regions that provide input to spiny neurons (Kemp 1970Go; Selemon 1985Go) respond phasically to sensory stimuli such as visual cue onset (Colby et al. 1996Go; Funahashi et al. 1990Go) and exhibit context-dependent tonic activity (Watanabe et al. 2002Go). This input is excitatory (Kitai 1976Go) and is modeled here through the increase of a depolarizing current conductance gs. The model also incorporates short-term modulatory actions of dopamine release resulting from the phasic activation of SNpc neurons triggered by the detection of reward-conditioned stimuli. The short-term effects of elevated dopamine concentration on the membrane conductances of spiny neurons, represented by the neuromodulatory factor µ in Fig. 1, is not direct but is mediated through D1 receptor activation (Pacheco-Cano et al. 1996Go; Surmeier et al. 1995Go). The specification of the magnitude and time course of µ is based on an attempt to extract a coherent description from a variety of ambiguous and at times controversial biophysical and single-unit data. In this attempt, we relied more heavily on data from experiments that explore the time scales relevant to the saccade task and to processes that are likely to take place in behaving animals.



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FIG. 1. Schematic diagram of the two input pathways to the model spiny neuron (components inside the dashed box): cortico- and nigro-striatal. The cortical input is composed of a tonic context-related component sc and a phasic target-related component st that is activated by the target onset. It provides excitatory input to the spiny neuron by increasing a depolarizing current conductance gs in the membrane model. The dopamine input from the substantia nigra activates if a reward is anticipated as a result of target onset and acts through a second-messenger system to control the neuromodulatory factor µ. The cortical and nigral inputs alter the membrane model properties to influence the membrane potential Vm. The output of the model is a spike train r generated as a deterministic function of Vm. A time delay {Delta}t is associated with signal conduction through the visual pathways.

 

The output of the spiny neuron model in Fig. 1 is a firing rate r that is needed only to construct rasters for comparison with single-unit data. For this purpose, we obtain r from an interspike interval chosen to be a deterministic nonlinear function of the membrane potential.

Model formulation

The membrane of a spiny neuron is modeled here as a single compartment with active ion currents. A first-order differential equation relates the temporal change in membrane potential (Vm) to the membrane currents (Ii)

(1)

The right-hand side of the equation includes active ionic, leakage, and synaptic currents; µ is the neuromodulatory factor.

We use a standard formulation to model the ionic currents based on parameter data obtained from the biophysical literature. The biophysical characterization of ionic currents is often done in conditions that deviate from the in vivo environment so as to facilitate data collection. Experimental conditions typically involve blocking agents, ion substitutions, and altered extracellular ionic concentrations to distinguish currents of interest. These techniques can modify the ionic current profiles away from their in vivo manifestations. Adjustments were therefore made to the parameters reported in the literature to compensate for the specific experimental conditions used in characterizing the currents. This procedure led to model currents that more closely match in vivo realizations. The parameters used in our model are listed in Table 1. The compensatory adjustments are described as specific parameters are introduced in the following description of the corresponding model currents. Eq. 1 is integrated numerically using a fifth-order Runge-Kutta method with a 0.5-ms time step and an error tolerance of 0.1 mV/ms to determine the dynamical evolution of Vm as the inputs to the model are varied.


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TABLE 1. Model parameters

 

All currents except for L-type Ca are modeled by the product of a conductance and a linear driving force

(2)

The reversal potential of ion species i is indicated as Ei; these parameters are set at biologically plausible values. The reversal potential for potassium EK is used for all potassium currents and the leak current. The factor gi represents the conductance for ionic current type i. The leakage conductance is constant. The conductances for Kir2 and Ksi are voltage dependent

(3)
where i is the maximum conductance and Li (Vm) is a logistic function of the membrane potential

(4)
L varies smoothly between 0 and 1. The half activation parameter Vh determines the value of Vm at which L = 1/2, while Vc controls the slope of the curve. The parameters that define the conductance function for Kir2 are derived from experiments by Mermelstein and colleges (1998Go). These experiments were performed at EK = –50 mV. The half activation parameter for Kir2 (VhKir2) derived from their data is shifted by –40 mV for consistency with the value of EK = –90 mV used in our model. Parameters for Ksi follow from a least-squares fit to data in Nisenbaum et al. (1996Go). The maximum conductance for Ksi derived from data in their paper is increased by 40% to compensate for the effect of the potassium channel blocker used in their experiment. The relative conductance profiles of Kir2 and Ksi are shown in Fig. 2A. Note that only the tail of the Kir2 conductance function is operational in the normal physiological range for Vm. The resulting currents are shown in Fig. 2B.



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FIG. 2. Voltage dependence of relative ion conductances and permeability (A) and the resulting ionic currents (B) in high (µ = 1.4; —) and low (µ = 1; · · ·) dopamine conditions. The high-dopamine condition increases Kir2 and L-type Ca currents by 40%. Note that the rightward tail of the Kir2 gating function in A extends beyond the K+ reversal potential (–90 mV), which causes the small inactivating outward Kir2 current seen in B.

 

Calcium currents are not well represented by a linear driving force model; extremely low intracellular calcium concentrations result in a nonlinear driving force (Hille 1992Go). The Goldman-Hodgkin-Katz (GHK) equation accounts for this effect and is used to model L-type Ca (Bargas et al. 1994Go)

(5)
where z is the valence of the Ca2+ ions, F is Faraday's constant, T is the temperature, R is the gas constant, and [Ca]i and [Ca]o refer to intracellular and extracellular concentration of Ca2+, respectively. The membrane permeability is voltage dependent

(6)
where L-Ca is the maximum permeability and LL-Ca(Vm) is a logistic function of the membrane potential (see Eq. 4). Parameters for L-type Ca are obtained from Bargas et al. (1994Go). The parameter values derived from these experiments are adjusted to compensate for their use of Ba2+ in place of Ca2+ as the charge carrier; the amplitude of the peak current is seven times larger and the corresponding value of the potential exhibits a depolarized shift when measured at external concentrations of 10 mM Ba2+ as compared with 2 mM Ca2+ (Bargas et al. 1994Go) (see Fig. 4). The reported value of PL-Ca is accordingly reduced by a factor of 7 and VhL-Ca is shifted by –24 mV to represent the properties of the L-type Ca current at 1–2 mM Ca2+, which more accurately represents the in vivo environment. The resulting relative permeability for L-type Ca (Fig. 2A) starts to rise around –50 mV, within the physiological range for L-type Ca current activation as predicted for Cav1.3 (L-type) Ca2+ channels in medium spiny neurons (Olson and Surmeier 2002Go). The corresponding current is shown in Fig. 2B.



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FIG. 4. A: the operational curve for µ = 1 (dotted curve) is a single valued monotonically increasing function. The operational curve for µ = 1.4 (solid curve) consists of 3 branches: 2 stable branches (black solid curves) and an unstable branch (solid gray curve). "D-> U" and "U-> D" mark the edges of the bistable region and indicate the transition thresholds to the depolarized and hyperpolarized states, respectively. Small dots along the horizontal axis identify the values of gs used in Fig. 3A. B: operational curves for µ = 1.0 (black), 1.1 (yellow), 1.2 (green), 1.3 (purple), and 1.4 (red). Note the increasing width of the bistable region in gs for increasing µ. All lines intersect at the critical point (), marked by a circle.

 



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FIG. 3. A: intersections between the net ionic current (thick curves) and the negative of the synaptic current (thin lines) are fixed points where dVm/dt = 0. The 5 thin lines in A correspond to gs = 7, 9.74, 12, 14.17, and 17 µS/cm2. The slope of the net ionic and synaptic currents at an intersection determine the stability of corresponding fixed point. In low dopamine conditions (A, dotted curve; µ =1), there is 1 stable fixed point for any value of gs. The arrowhead indicates the stable solution for gs = 12 µS/cm2. In high dopamine conditions (A, solid curve; µ =1.4), there is 1 stable fixed point for gs < 9.74 µS/cm2 or gs > 14.17 µS/cm2. In the intermediate regime, there are 3 fixed points for each value of gs. The circles show the solutions for gs = 12 µS/cm2; the outer solutions (filled circles) are stable and the intermediate solution (open circle) is unstable. Current-voltage curves for gs = 6, 12, 18, and 24 µS/cm2 at µ = 1 (B) and µ = 1.4 (C) illustrate the nonlinear effect of gs on Vm. The curved arrows indicate the direction of increasing gs. Note that although a region of negative slope conductance exists for small values of gs at µ =1 in B, no bistability is present in the operational curve at µ = 1 due to this nonlinear relationship.

 
The model includes two types of excitatory input signals: context sc and visual target st (see Fig. 1), which represent the collective input from many cortical neurons. The context signal is the sum of all tonic inputs representing persistent neural activity related to the environment and task expectation (Watanabe et al. 2002Go). The visual target signal is the sum of all phasic inputs representing a simplified version of the responses of cortical neurons to a visual cue (Colby et al. 1996Go). These two types of input are combined to obtain the synaptic current, which is modeled as the product of a conductance and a linear driving force, Is = gs(VmEs). The excitatory synaptic conductance is determined by the sum of the context and target input signals

(7)
where wc and wt are the synaptic efficacies that transform input signals into their corresponding synaptic conductances. Inhibition is not included in this model. The factor {zeta} is a random variable included to simulate the noisy character of synaptic input. The statistics of {zeta} are chosen so that the fluctuations of Vm in the up state have a similar amplitude and power spectrum as the high-frequency (>10 Hz) membrane potential fluctuations seen in intracellular recordings of spiny neurons (Stern et al. 1997Go). The synaptic input used here is more biologically relevant than the constant injected current term frequently used in computational models. An injected current independent of the membrane potential would be represented as a horizontal line in Fig. 3A. The absolute value of the synaptic driving force (VmEs) used in our model decreases linearly as the membrane potential is depolarized, and becomes zero at the reversal potential Es = 0 (thin lines in Fig. 3A). The net conductance gs in Eq. 7 determines the slope of the synaptic current lines in Fig. 3A.

Dopamine modulates the properties of ion currents through the activation of dopamine receptors. Agonists for the D1 type receptor enhance the Kir2 and L-type Ca currents observed in spiny neurons (Hernandez-Lopez et al. 1997Go; Pacheco-Cano et al. 1996Go; Surmeier et al. 1995Go). This effect is modeled by the neuromodulatory factor µ, which scales Kir2 and L-type Ca currents (see Eq. 1). An upper bound at µ = 1.4 is derived from physiological experiments on the modulation of Kir2 and L-type Ca currents when D1 receptors or their effector mechanisms are activated (Surmeier et al. 1995Go; Surmeier, unpublished observations). The lower bound at µ = 1.0 corresponds to low dopamine levels; this is the experimental condition in which the ion currents have been characterized. The assumption of µ = 1.0 neglects any effects of background dopamine, a reasonable approach given the low basal concentration of dopamine in the striatum (Herrera-Marschitz et al. 1996Go) and the predominantly low affinity of D1 receptors to the ligand (Richfield et al. 1989Go). The potential importance of basal dopamine levels (Grace 1991Go), the effect of which on ion current properties is not well characterized, is likely to manifest itself in relation to predominantly high-affinity D2 or D5 receptors not incorporated in our model.

The time course over which µ varies between the upper and lower bounds is controlled by transmitter diffusion, the rate of receptor activation, and the kinetics of the intracellular cascade that ultimately leads to the modulation of ion currents. There is insufficient data to accurately model the time course of these processes to specify µ(t) and thus describe how µ changes with time after dopamine release. To minimize the dependence of our results on an explicit form for µ(t), we first perform a detailed analysis of those generic properties of the modulation that are independent of the dynamics of µ. It is only when we come to the simulation of the saccade task that we need to chose an explicit form for µ(t) to test if the interaction between the dynamics of the inputs and the dynamics of the membrane model leads to reward-dependent responses similar to those observed in the single-unit data.

To approximate µ(t), we rely on experiments in which dopaminergic neurons are stimulated in a manner that mimics the naturally occurring bursts in response to conditioned visual stimuli (Gonon 1997Go). Some spontaneously active spiny neurons display an increased firing rate after evoked dopamine transients elicited through stimulation of the medial forebrain bundle (Gonon 1997Go). The enhancement of activity begins with a latency of 200 ms after the initiation of the stimulation and trails off up to 1,000 ms later. This latency reflects both the lag due to second messengers and the subsequent dynamical response of the membrane potential. Earlier experiments by Williams and Millar (1990Go) used a high stimulation frequency (50 Hz), delivering a minimum of 25 pulses, as opposed to the 1–4 pulses delivered at 15 Hz in the Gonon (1997Go) experiments. This more intense stimulation of the medial forebrain bundle produced responses that lasted tens of seconds, which we are presuming not to reflect naturally occurring bursts in response to conditioned visual stimuli.

The form of µ(t) we choose here is a fast exponential rise toward a maximum value beginning with a delay of 80 ms after the onset of SNpc activity, followed by a slower exponential decay to a baseline level beginning with a delay of 600 ms. The response of SNpc neurons to a visual cue follows the onset of the stimulus by ~100 ms (Schultz et al. 1993Go). This delay is also included in the model; the µ transient thus begins 180 ms after the onset of the visual cue. The relatively brief modulatory effects considered here are to be distinguished from longer-lasting effects observed in experiments that employ direct application of dopamine for long periods or at high concentrations (Umemiya and Raymond 1997Go). These long-term effects, which could be either synaptic or modulatory in nature, are not explored in the present model.

In the model described by Eq. 1, the membrane potential Vm is a state variable, the value of which depends on two inputs: the synaptic conductance gs and the neuromodulatory factor µ. Although Eq. 1 is linear in both µ and gs, the currents at the right-hand side of the equation exhibit a significantly nonlinear dependence on Vm. Both the equilibrium and the dynamical dependences of Vm on gs and µ are therefore nonlinear. These nonlinearities play an essential role in determining the response properties of the membrane model.

The output of the model neuron is expressed as a spike train r, which is used to construct rasters and histograms so as to allow for comparison to the neurophysiological data on single-unit response properties in the saccade task. The spike train is chosen to be a deterministic function of membrane potential and time

(8)
where Vf is the firing threshold for generating spikes, t is the time in milliseconds, and tp is the time at which the previous action potential occurred. In this formulation, no spike occurs if the membrane potential is below threshold or if the time (ttp) elapsed since the last spike is less than the instantaneous interspike interval calculated as a deterministic function of Vm. When Vm is below the firing threshold, tp is set to a small value so as to allow a spike to occur as soon as the firing threshold is exceeded. However, the interspike interval is never smaller than 20 ms even if Vm fluctuates rapidly above and below the firing threshold. Vf is set at –58 mV, consistent with experimental data (Wilson and Kawaguchi 1996Go); this value is slightly more depolarized than the normal up-state potential of spiny neurons. Values for Vh and Vc are chosen so that the excitatory input corresponding to target presentation results in a firing rate of 50 spikes/s in unrewarded trials, as seen in Fig. 1 of Kawagoe et al. (1998Go). The values of these parameters do not influence the qualitative properties of the model; they only determine the magnitude of the response in high relative to low dopamine conditions.


    RESULTS
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 DISCLOSURES
 REFERENCES
 
In the ensuing section, we present a detailed analysis of the spiny neuron membrane model introduced in METHODS. This minimal model displays complex nonlinear dynamics. We begin by identifying the fixed points that characterize the stationary properties of the model; we then show that dopamine-induced bifurcations of these fixed points lead to bistability. The dynamical responses of the model first to cortical inputs and then to neuromodulatory inputs at fixed levels of cortical activity are analyzed. Finally, responses to combined cortical and neuromodulatory inputs are considered. This analysis begins with a discussion of generic features of the model's response properties, which are independent of the spiking mechanism and of the neuromodulatory dynamics. We then illustrate these features through a simulation of the saccade task that incorporates specific assumptions about the dynamics of both cortical and neuromodulatory inputs as well as the spiking mechanism. The resulting predictions for the characteristics of reward-dependent single-unit discharge are presented in the final part of this section. In the DISCUSSION, we will relate the results presented here to existing experimental data.

Fixed points and dopamine-induced bifurcations

The dynamical evolution of the membrane potential Vm is controlled by the sum of ionic and synaptic currents that appear in the right-hand side of Eq. 1. The stationary solutions to Eq. 1 correspond to equilibrium values of the membrane potential Vm consistent with specific values of the dopamine-controlled neuromodulatory factor µ and the synaptic conductance gs. Fluctuations of this conductance around its mean value gs = gc + gt are ignored until later by setting {zeta} = 1. The equilibrium values of Vm satisfy dVm/dt = 0; it follows from Eq. 1 that such values are the solutions to

(9)
These solutions are called fixed points to emphasize that such values of Vm, once attained, will not change with time. The stationary values of Vm for given values of gs and µ remain fixed through a cancellation between the net outward ionic current and the inward synaptic current. In such a configuration, the membrane potential cannot change unless the system is perturbed.

A useful visual tool for the identification of fixed points and the analysis of their stability properties is shown in Fig. 3A. Although synaptic and ionic currents play a mathematically equivalent role in determining the equilibrium values of Vm, we choose to differentiate them so as to illustrate the separate dependence of the stationary properties of the model on the two parameters that determine them: µ and gs. The sum of the ionic currents is plotted in Fig. 3A as a function of the membrane potential Vm for both low (µ =1, dotted curve) and high (µ = 1.4, dark solid curve) dopamine levels. The shape of this relationship depends markedly on the value of µ. Note that the net ionic current is outward whenever Vm is more depolarized than the resting potential at Vm = –89.99 mV, which is very close to the reversal potential for potassium at EK = –90mV. Synaptic current is represented in Fig. 3A by a family of straight lines that pivot around the reversal potential at Es = 0. The increasing steepness of the lines corresponds to increasing values of gs. Synaptic current is inward: Is is negative, whereas the plot shows –Is = gs(VmEs), which is positive. This separate consideration of the synaptic currents is particularly useful for the analysis of our model; as opposed to in vitro scenarios where the magnitude of injected currents is a control variable, we address the case of cortical inputs that control instead the synaptic conductance gs.

Intersections between a curve representing the net ionic current and one of the straight lines representing the negative of the synaptic current determine the stationary values of the membrane potential, the fixed points of the model. These solutions can be followed as a function of gs for fixed µ by varying the slope of the straight line. At low-dopamine levels (µ = 1), there is only one such intersection for each value of gs; the arrowhead in Fig. 3A shows this solution for gs = 12 µS/cm2. The corresponding stationary membrane potential is a single-valued function of the synaptic conductance. At high-dopamine levels (µ = 1.4), the membrane potential is a single-valued function of the synaptic conductance for either gs < 9.74 µS/cm2 or gs > 14.17 µS/cm2. In contrast, there are three fixed point solutions to Eq. 9 for each value of gs in the intermediate regime 9.74 µS/cm2 < gs < 14.17 µS/cm2; the circles in Fig. 3A show these solutions for gs = 12 µS/cm2. This switch from single solutions to multiple solutions results in a qualitative change in the dynamical properties of the membrane model, as will be discussed in detail in the following text.

It is important to analyze the stability of fixed point solutions against perturbations. A perturbation to a stable fixed point results in a dynamical convergence back to the fixed point; a perturbation to an unstable fixed point results in a dynamical divergence away from it. For the fixed point solutions of Eq. 1, stability is determined by the competition between ionic and synaptic currents in the vicinity of the fixed point. The relevant comparison is between the slope of the net ionic current and the slope of the negative of the synaptic current at the intersection point; the latter is always negative and given by –gs. If the slope of the net ionic current at the fixed point is greater than –gs, the ionic current dominates to the right of the solution while the synaptic current dominates to the left. The total current is positive (outward) to the right of the solution, which will decrease Vm toward the fixed point; the total current is negative (inward) to the left of the solution, which will increase Vm toward the fixed point. This scenario corresponds to a stable fixed point: a perturbative depolarization (hyperpolarization) is followed by a dynamical hyperpolarization (depolarization), and the stationary state is restored. This condition is met by all fixed point solutions of Eq. 1 for µ = 1 (arrowhead in Fig. 3A) and by the outer solutions (filled circles in Fig. 3A) for µ = 1.4. This condition will always be satisfied when the slope of the net ionic current at the intersection is positive; it will also be satisfied if the slope of the net ionic current at the intersection is negative but smaller in absolute value than gs (i.e., the net ionic current is less steep than the negative of the synaptic current). The intermediate solution for µ = 1.4 (open circle in Fig. 3A) illustrates the opposite scenario: the slope of the net ionic current at the fixed point is less than the slope of the negative of the synaptic current. In this case, the slope of the net ionic current at the intersection is negative but larger in absolute value than gs (i.e., the net ionic current is steeper than the negative of the synaptic current). The synaptic current dominates to the right of the solution while the ionic current dominates to the left. The total current is negative (inward) to the right of the solution, which will increase Vm further away from the fixed point; the total current is positive (outward) to the left of the solution, which will decrease Vm further away from the fixed point. A perturbative depolarization (hyperpolarization) is enhanced by a dynamical depolarization (hyperpolarization); the fixed point is unstable. Note that the boundary between stability and instability occurs where the slope of the ionic current at the fixed point is negative and equal to gs in absolute value. For the curves shown in Fig. 3A, this condition is met for gs = 9.74 µS/cm2 and gs = 14.17 µS/cm2.

An alternative and perhaps simpler approach to the stability analysis follows directly from Eq. 1, which can be written as –Cm dVm/dt = I(Vm, gs, µ). The dynamical evolution of Vm is controlled by the sum I(Vm, gs, µ) of all the currents; this total current is shown as a function of Vm for various values of gs in Fig. 3, B (µ = 1) and C = 1.4). Fixed point solutions to Eq. 1 correspond to I(Vm, gs, µ) = 0 (dashed horizontal lines in Fig. 3, B and C). The stability of these solutions is controlled by the sign of the slope of the corresponding curve as it passes through I(Vm, gs, µ) = 0. A positive slope implies that dVm/dt is negative (outward net current) to the right of the fixed point and positive (inward net current) to its left; this scenario results in negative feedback and corresponds to a stable solution. Conversely, a negative slope implies that dVm/dt is positive (inward net current) to the right of the fixed point and negative (outward net current) to its left; this scenario results in positive feedback and corresponds to an unstable solution. Curves for µ = 1 (Fig. 3B) exhibit a single stable fixed point for each value of gs. Curves for µ = 1.4 (Fig. 3C) exhibit a single stable fixed point for low and high values of gs but three fixed points for intermediate values of gs. As in Fig. 3A, the outer fixed points (filled circles) are stable, while the one in the middle (open circle) is unstable.

It is worth remarking on an interesting feature displayed by the I-V curves shown in Fig. 3, B and C: a region of negative slope conductance is a necessary but not a sufficient condition for instability. The I-V curves are N-shaped at µ = 1 for low values of gs, and yet no instability is observed at µ = 1; increases in gs do not merely shift the I-V curve vertically downward (as voltage-independent increases in the magnitude of an injected current would) but also affect the shape of the curves so as to gradually shrink away the range of values of Vm for which negative slope conductance is observed. The occurrence of instability at µ = 1.4 is due to a persistence of this region of negative slope conductance. Both the IKir2 and IL-Ca currents, whose conductances are enhanced by dopamine, contribute to the positive feedback associated with the existence of an unstable fixed point at voltages intermediate to those characteristic of the down and up states and thus provide a mechanism for the persistence of negative slope conductance.

Operational curves and emergence of bistability

A full characterization of the membrane model is provided by the relationship between the state variable Vm and the two input variables: the synaptic conductance gs and the dopamine controlled neuromodulatory factor µ (Fig. 1). Treating the dependence on µ as a parameter, we can plot Vm as a function of gs for different values of µ and use these operational curves to explore the consequences of neuromodulation. For low values of µ, Vm is a smooth, monotonic function of gs. The dotted curve for µ = 1 in Fig. 4A exhibits a steep but smooth transition from hyperpolarized values of Vm corresponding to the down state to depolarized values of Vm corresponding to the up state. The curve for µ = 1.4 in Fig. 4A is qualitatively different; it consists of a lower branch (dark curve) corresponding to a stable hyperpolarized down state, an upper branch (dark curve) corresponding to a stable depolarized up state, and an intermediate unstable branch (gray curve) connecting these two. The resulting bistability for intermediate values of gs, due to the enhancement of the Kir2 and L-type Ca ionic currents, has a drastic effect on the response properties of the model in high dopamine conditions. This qualitative change in the dynamical properties of the membrane model as µ increases from 1 to 1.4 is the signature of a bifurcation.

Consider a quasistatic experiment in which µ is fixed at 1.4 and the synaptic conductance changes slowly so that the membrane potential is allowed to reach its corresponding equilibrium value. As gs increases, a hyperpolarized down state evolves following the lower dark solid curve in Fig. 4A. When gs reaches 14.17 µS/cm2, the synaptic current starts to overcome the mostly Kir2 hyperpolarizing current, and Vm depolarizes abruptly until it reaches the up state, which is stabilized by the hyperpolarizing Ksi current. This jump in Vm is a discontinuous change in state, the down state to up state transition (D->U) in Fig. 4A. If gs is increased further, the depolarized up state follows the upper dark solid curve in Fig. 4A, with a small amount of additional depolarization. If gs is now decreased, the depolarized up state will follow the upper dark solid curve in Fig. 4A in the downward direction. It is the enhanced effect of the inward L-type Ca current that counteracts the hyperpolarizing effect of the Ksi current and stabilizes the up state until gs reaches 9.74 µS/cm2. At this point, the net hyperpolarizing ionic current starts to overtake the synaptic current, and Vm hyperpolarizes abruptly until it reaches the down state. This jump in Vm is the up to down state transition (U->D) in Fig. 4A. Throughout the intermediate range 9.74 µS/cm2 < gs <14.17 µS/cm2, Vm will reach either of its two stable values, depending on the previous state; this memory of prior state is called hysteresis.

The emergence of bistability in high dopamine conditions, characterized by the appearance of sharp and distinct state transitions, results in a prominent hysteresis effect. The state of the model, as described by the value of the membrane potential, depends not only on the current values of µ and gs but also on the particular trajectories followed by µ and/or gs to reach their current values. The appearance of bistability at high-dopamine levels gives additional meaning to the notion of a down state and an up state, as in this case there is a well-defined gap between the two stable branches (dark solid curves in Fig. 4A) that characterize the membrane potential. There is a maximal value of Vm for the lower branch; this is the most depolarized potential attainable in the down state. The minimal value of Vm in the upper branch is the most hyperpolarized potential attainable in the up state. Intermediate values of Vm correspond to the unstable branch (solid gray curve) in Fig. 4A. The model cannot sustain membrane potentials in this range without an external driving force such as could be provided through voltage clamp. In contrast to this sharp separation, the transition between down and up states in low dopamine conditions (dotted curve in Fig. 4A) is smooth with no clear separation between them. We will nevertheless refer to hyperpolarized potentials as the down state and depolarized potentials as the up state for consistency with the terminology conventionally used in the description of spiny neuron electrophysiology (Wilson and Groves 1981Go; Wilson and Kawaguchi 1996Go).

Bistability in high dopamine conditions arises in this model through a saddle-node bifurcation with increasing µ. To investigate the bifurcation, it is useful to consider a family of operational curves for subsequent values of µ, as shown in Fig. 4B for µ = 1.0, 1.1, 1.2, 1.3, and 1.4. Curves for µ = 1.0 and µ = 1.1 follow a single stable solution for which Vm is a smooth, monotonically increasing function of gs. The curve for µ = 1.2 displays an unstable branch for –71.4 mV < Vm < –65.4 mV; this instability at hyperpolarized potentials is due primarily due to the enhancement of the Kir2 current. The resulting hysteresis loop is extremely narrow: it corresponds to a change of {Delta}gs {cong} 0.07 µS/cm2 in synaptic conductance. The "double S" shape of the curve for µ = 1.3 reflects the existence of two unstable branches separated by an additional intermediate stable branch; this type of operational curve results in two distinct ranges of unstable values for Vm. The associated hysteresis loop is still narrow: it corresponds to a change of {Delta}gs {cong} 0.54 µS/cm2 in synaptic conductance. These narrow hysteresis loops are to be contrasted with the one observed for µ = 1.4, characterized by a change of {Delta}gs {cong} 4.43 µS/cm2 in synaptic conductance. It is at this higher value of µ, of relevance to our model, that bistability is present for a significantly wide range of synaptic inputs and thus plays an important role in determining the dynamical properties of the membrane model.

A remarkable feature of Fig. 4B is that curves for all values of µ intersect at a unique point, at which mV and . The existence of this critical point is due to a cancellation between the Kir2 and the L-type Ca currents for this particular value of Vm, which arises as a solution to the equation

(10)
When this condition is satisfied, solutions to Eq. 9 become independent of µ; for , a change in µ does not result in a corresponding change in the equilibrium value of the membrane potential.

The location of this critical point follows from the model formulation of the Kir2 and the L-type Ca currents; the value of thus depends on the values of the parameters needed to characterize these two currents. A first-order sensitivity analysis allows us to quantify the expected variation in due to fluctuations in these parameter values. The results of this analysis are reported in Table 2. The first three columns in this table list the parameters, their values {Omega}, and their corresponding uncertainties {Delta}{Omega}. The derivatives listed in the fourth column are evaluated at the fixed point; they provide a mechanism for transforming parameter uncertainties into uncertainties in . The product of the derivatives in column four with the corresponding values of {Delta}{Omega} in column three result in the uncertainties listed in column five. Note that the location of the critical point at mV, a slightly more depolarized membrane potential than the firing threshold at Vf = –58 mV, is well established within ±2.5 mV.


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TABLE 2. Results of first-order sensitivity analysis

 

It is one of our model's simplifying assumptions that an increase in dopamine results in identical modulation of the maximal conductance for the Kir2 current and maximal permeability for the L-type Ca current. If we allow for the possibility that D1 dopamine receptor activation might result in unequal time courses for the modulation of the amplitude of the Kir2 and L-type Ca currents, the cancellation between these two currents will still result in a critical point. The value of would in this case no longer arise as a solution to Eq. 10, but as a solution to the equation

(11)
where µKir2 is the conductance gain factor for the Kir2 current and µL-Ca is the permeability gain factor for the L-type Ca current. The resulting value of will in this case depend on the ratio (µKir2L-Ca); the precise location of the critical point in the Vm gs plane of Fig. 4B will change accordingly. If the ratio Kir2L-Ca) is itself a function of time, the model will exhibit a dynamically generated critical line, a line of critical points that includes the critical point at mV for (µKir2L-Ca) = 1.

The existence of a critical point is an interesting aspect of our model. It introduces a slowdown effect that affects the dynamical response of the membrane potential to both cortical and neuromodulatory input. Although the presence of a critical point is not necessary for bistability in high dopamine conditions, its existence provides a simple explanatory mechanism for a dual response to dopamine which can either enhance or depress the response of the membrane model. We discuss this effect in detail later in this section, as we use our model to interpret the results by Kawagoe and colleagues (1998Go).

We conclude our discussion of dopamine-induced bistability by demonstrating the robustness of this effect. Consider the ranges of unstable values of Vm associated with the existence of unstable branches in the corresponding operational curves as shown in Fig. 4B for various values of µ. These unstable intervals, bounded from below by a D->U transition and from above by a U->D transition, are shown as a function of µ in Fig. 5A. Note the bifurcation at µ = 1.14, due primarily to the Kir2 current, followed by a second bifurcation at µ = 1.26, due primarily to the L-type Ca current; these two lobes coalesce at µ = 1.37. The "double-S"-shaped operational curve for µ = 1.3 in Fig. 4B is representative of this regime, in which two unstable branches are separated by a third intermediate stable branch. In spite of their intrinsic interest, the dynamical properties of the system in this regime are not especially relevant to our analysis, because they appear only for 1.26 < µ < 1.37 and manifest themselves only over a very narrow range {Delta}gs {cong} 0.54 µS/cm2 of synaptic conductance. It is the wide interval of unstable values for Vm found for µ > 1.37 that is especially relevant to the dopamine modulated dynamical responses of the membrane model to synaptic input.



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FIG. 5. Transition thresholds mark the extreme depolarized and hyperpolarized edges of the unstable regions. Green curves mark transitions from a hyperpolarized to a depolarized state (D-> U), and red curves mark transitions from a depolarized to a hyperpolarized state (U-> D). A: as µ increases, a bifurcation due mostly to the Kir2 current occurs at µ = 1.14, followed by a 2nd bifurcation due mostly to the L-type Ca current at µ = 1.26. These two coalesce at µ = 1.37 into 1 larger unstable region. The interaction of baseline Kir2 conductance and enhanced L-type Ca permeability implemented through the permeability gain factor µL-Ca (B) or the interaction of baseline L-type Ca permeability and enhanced Kir2 conductance implemented through the conductance gain factor µKir2 (C) suffice to elicit membrane potential instability.

 

Because the effects of D1 type dopamine receptor activation on these currents are unlikely to be strictly identical, we consider as extreme cases the possibility that only one of these two currents is affected by dopamine. As shown in Fig. 5B, the interplay between an enhanced L-type Ca current and the baseline Kir2 current suffices to account for most of the bifurcation diagram. The contribution of Kir2 enhancement, shown in Fig. 5C, is as expected restricted to hyperpolarized potentials. If both currents are simultaneously enhanced by a common factor µ, the wider unstable region in Fig. 5A is recovered. The analysis of Fig. 5 demonstrates that the existence of dopamine-induced bistability is a robust property of the model that does not rely on the simplifying assumption that dopamine release results in an identical enhancement of the L-type Ca and the Kir2 currents.

Dynamical responses to cortical and neuromodulatory inputs

We now investigate the dynamical evolution of the membrane potential Vm. Changes in Vm due to changes in the synaptic conductance gs and the dopamine enhancement factor µ follow from the integration of Eq. 1.

We first consider the response of the membrane model to cortical inputs not associated with reward; µ remains constant at the low dopamine level (µ = 1). We monitor changes in Vm in response to stepwise increases and decreases in gs. It is under similar conditions that cortically driven transitions between the down state and the up state have been observed (Wilson and Kawaguchi 1996Go). The model displays such state transitions; the corresponding time constants exhibit strong dependence on the proximity of the baseline and target values of gs to the critical point at .

The dependence of Vm on gs follows from Eq. 1 for µ = 1. The parameter {zeta} in Eq. 7 is allowed to be a random variable so as to simulate the noisy character of synaptic input; the spike-generating model is not included. From a hyperpolarized baseline value of Vm = –88.1 mV for gs = 3 µS/cm2 (Fig. 6B) and from a slightly more depolarized baseline value of Vm = –78.7 mV for gs = 10 µS/cm2 (Fig. 6C), gs is increased stepwise to values uniformly spaced between 10 and 22.5 µS/cm2. These instantaneous increases in gs are followed by slower increases in the membrane potential Vm as it moves toward its equilibrium value. After 400 ms, the value of gs is instantaneously returned to its baseline value, and Vm decays back toward its original value. We show in Fig. 6A two of the corresponding trajectories in the Vm gs plane. One trajectory (squares) describes the evolution of the system from a baseline value of gs = 3 µS/cm2 to an increased value of gs = 12.5 µS/cm2 and back; a second trajectory (diamonds) describes the evolution of the system from a baseline value of gs = 10 µS/cm2 to an increased value of gs = 17.5 µS/cm2 and back.



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FIG. 6. Dynamic responses of Vm to step changes of gs at µ = 1. In these experiments, gs is increased from a baseline level to a step level of 10, 12.5, 15, 17.5, 20, and 22.5 µS/cm2 for 400 ms and then is returned to baseline. A: sample trajectories corresponding to 1 experiment in B (squares) and 1 in C (diamonds). The open circle identifies the critical point. B: in these experiments, gs is increased in steps from a baseline of 3 µS/cm2. For the 2nd trajectory from the bottom, gs is increased to 12.5 µS/cm2. The instantaneous increase in gs at t = 0 corresponds to the horizontal line that originates at the filled square in A; Vm reacts to this change by depolarizing along the vertical dashed line toward the open square. The close proximity to the critical point slows down the depolarization process. A subsequent step decrease to the baseline level leads to hyperpolarization along the dashed vertical line toward the filled square. This process is fast, as are the other hyperpolarizing processes shown in B, because they occur away from the critical point. The characteristic rate for the depolarizing processes depends on the proximity of the corresponding trajectories to the critical point. C: in these experiments, gs is increased in steps from a baseline of 10 µS/cm2. For the 3rd trajectory from the bottom, gs is increased to 17.5 µS/cm2. The corresponding trajectory originates at the filled diamond in A and depolarizes to the open diamond. The characteristic rate for the depolarizing processes depends on the proximity of the corresponding trajectories to the critical point. The hyperpolarizing processes are slow because all vertical trajectories terminating at the filled diamond are close to the critical point.

 

Changes in synaptic conductances and the resulting membrane potential traces are shown in Fig. 6, B and C. Rapid fluctuations in Vm are due to the inclusion of synaptic noise. Some of these traces exhibit a noticeable slowdown during depolarization. This critical slowing down is a generic consequence of the existence of a critical point. The effect is particularly noticeable whenever trajectories in the Vmgs plane, such as those shown in Fig. 6A, pass near the critical point. Depolarizing trajectories triggered by increases in the synaptic conductance up to a value of gs = 12.5 or 15 µS/cm2 come close to the critical point at , and the dynamical convergence of Vm to its new equilibrium value is slow (see the corresponding traces in Fig. 6, B and C). Depolarizing trajectories triggered by increases in gs to values further removed from do not exhibit this slowdown effect. All hyperpolarizing trajectories returning to a baseline of gs = 10 µS/cm2 pass much closer to the critical point than those returning to a baseline of gs = 3 µS/cm2 (see Fig. 6A). It is this proximity to the critical point that explains the slowdown in the hyperpolarizing Vm traces in Fig. 6C not observed in the hyperpolarizing Vm traces in Fig. 6B.

We now consider the dynamical response of the membrane model to changes in dopamine level; the cortical input gs is kept constant while the dopamine-controlled neuromodulatory factor µ varies with time. These conditions mimic those of experiments that monitor the modulation of tonic striatal activity due to the application of dopaminergic agents or due to the electrical stimulation of dopamine fibers (Gonon 1997Go; Kiyatkin and Rebec 1996Go; Williams and Millar 1990Go). This set of numerical experiments displays dramatic dynamical slowdown for gs close to (Fig. 7). The results also reveal a novel effect: increased dopamine levels can result in either depolarization or hyperpolarization depending on whether gs does or does not exceed .



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FIG. 7. Dynamic responses of Vm to changes in µ at fixed gs; the values of gs are close to the critical . A: in these experiments, µ is increased in a step from µ = 1 to 1.4. The cell membrane depolarizes for and hyperpolarizes for . These processes become increasingly slow as the value ofgs approaches . B: in these experiments, µ is increased in a step from µ = 1 to 1.4 for 200 ms and allowed to decay to its baseline level with a time constant of 70 ms. The transient response is depolarizing (hyperpolarizing) for . Transient responses slow down and barely have time to develop as the value of gs