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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
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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).
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. 1995). 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. 1998) or
movement planning (Hollerman et al.
1998) 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.
1998), and the response modulation is often large enough to be
clearly seen in individual trial records
(Kawagoe et al. 1998). While
the reward-dependent activity may have cortical origins in part
(Wilson 1990), 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 1998). 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. 1991;
Mirenowicz and Schultz 1994;
Schultz et al. 1993). 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. 1998;
Kawagoe et al. 1998). This, in
conjunction with the demonstration of dopamine-mediated neuromodulation of
spiny neuron activity in vivo (Gonon
1997; Kiyatkin and Rebec
1996) and in vitro (Akaike et
al. 1987; Calabresi et al.
1987; Flores-Hernandez et al.
2000; Hernandez-Lopez et al.
1997; Surmeier et al.
1992a), 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. 2000). 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. 1997).
This state dependence arises from the coordinated modulation of ion channels
regulating these states (Flores-Hernandez
et al. 2000; Hernandez-Lopez
et al. 1997; Pacheco-Cano et
al. 1996; Surmeier et al.
1992a,
1995). 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 (1998)
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. 1991;
Schultz et al. 1993).
Correspondingly, Kawagoe and colleagues
(1998) suggested, and later
confirmed (Kawagoe et al.
1999), 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.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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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 1993;
Wilson and Kawaguchi 1996).
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
1995; Wilson and Kawaguchi
1996). The inward rectifying current, predominantly of the Kir2
type in spiny neurons (Mermelstein et al.
1998), 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 1995). 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. 1996).
The outward rectifying K+ current in spiny neurons includes slowly
and rapidly inactivating components
(Nisenbaum et al. 1996;
Surmeier et al. 1991,
1992b) 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 1995) 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. 1994;
Song and Surmeier 1996). 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. 1994). This subthreshold activation is attributable to the
expression of Cav1.3 L-type channels by medium spiny neurons
(Olson and Surmeier 2002).
This current is enhanced by D1 agonists in medium spiny neurons expressing D1
receptors (Surmeier et al.
1995,
1996), and this modulation is
critical to the increased excitability produced by D1 agonists at depolarized
membrane potentials (Cooper and White
2000; Hernandez-Lopez et al.
1997); it is therefore included in the model.
Our approach (Gruber and Houk
2000) 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 1978;
Rinzel and Ermentrout 1989), 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. 2002;
Wickens and Arbuthnott 1993),
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 (1998).
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 1970;
Selemon 1985) respond
phasically to sensory stimuli such as visual cue onset
(Colby et al. 1996;
Funahashi et al. 1990) and
exhibit context-dependent tonic activity
(Watanabe et al. 2002). This
input is excitatory (Kitai
1976) 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. 1996;
Surmeier et al. 1995). 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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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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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) |
i is the
maximum conductance and Li (Vm) is a
logistic function of the membrane potential
 | (4) |
). 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.
(1996). 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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Calcium currents are not well represented by a linear driving force model;
extremely low intracellular calcium concentrations result in a nonlinear
driving force (Hille 1992).
The Goldman-Hodgkin-Katz (GHK) equation accounts for this effect and is used
to model L-type Ca (Bargas et al.
1994)
 | (5) |
 | (6) |
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.
(1994). 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. 1994) (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 2002). The corresponding current is shown in
Fig. 2B.
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). 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.
1996). 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(Vm – Es).
The excitatory synaptic conductance is determined by the sum of the context
and target input signals
 | (7) |
is a
random variable included to simulate the noisy character of synaptic input.
The statistics of 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. 1997). 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
(Vm – Es) 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. 1997;
Pacheco-Cano et al. 1996;
Surmeier et al. 1995). 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. 1995;
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.
1996) and the predominantly low affinity of D1 receptors to the
ligand (Richfield et al.
1989). The potential importance of basal dopamine levels
(Grace 1991), 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 1997). Some
spontaneously active spiny neurons display an increased firing rate after
evoked dopamine transients elicited through stimulation of the medial
forebrain bundle (Gonon 1997).
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 (1990) 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
(1997) 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. 1993). 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 1997). 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) |
); 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. (1998). 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 |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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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 = 1. The equilibrium values of
Vm satisfy dVm/dt = 0; it
follows from Eq. 1 that such values are the solutions to
 | (9) |
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(Vm – Es), 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 (µ = 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 (UD) 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 1981;
Wilson and Kawaguchi
1996).
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 
gs 
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 gs 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 gs 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) |
, 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 
, and their
corresponding uncertainties . 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 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.
|
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) |
will
in this case depend on the ratio (µKir2/µL-Ca);
the precise location of the critical point in the Vm
– gs plane of
Fig. 4B will change
accordingly. If the ratio (µKir2/µL-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
(µKir2/µL-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
(1998).
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 DU
transition and from above by a UD 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 gs 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.
|
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 1996). 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 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.
|
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 Vm
– gs 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 1997;
Kiyatkin and Rebec 1996;
Williams and Millar 1990).
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
.
|
),
marked by a circle.
and
hyperpolarizes for
. These
processes become increasingly slow as the value ofgs
approaches 
.
Transient responses slow down and barely have time to develop as the value of
gs