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The Journal of Neuroscience, April 1, 1999, 19(7):2807-2822
A Neurocomputational Theory of the Dopaminergic Modulation of
Working Memory Functions
Daniel
Durstewitz,
Marian
Kelc, and
Onur
Güntürkün
Arbeitseinheit Biopsychologie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany
 |
ABSTRACT |
The dopaminergic modulation of neural activity in the prefrontal
cortex (PFC) is essential for working memory. Delay-activity in the PFC
in working memory tasks persists even if interfering stimuli intervene
between the presentation of the sample and the target stimulus. Here,
the hypothesis is put forward that the functional role of dopamine in
working memory processing is to stabilize active neural representations
in the PFC network and thereby to protect goal-related delay-activity
against interfering stimuli. To test this hypothesis, we examined the
reported dopamine-induced changes in several biophysical properties of
PFC neurons to determine whether they could fulfill this
function. An attractor network model consisting of model neurons was
devised in which the empirically observed effects of dopamine on
synaptic and voltage-gated membrane conductances could be represented
in a biophysically realistic manner. In the model, the dopamine-induced
enhancement of the persistent Na+ and reduction of
the slowly inactivating K+ current increased firing
of the delay-active neurons, thereby increasing inhibitory feedback and
thus reducing activity of the "background" neurons. Furthermore,
the dopamine-induced reduction of EPSP sizes and a dendritic
Ca2+ current diminished the impact of intervening
stimuli on current network activity. In this manner, dopaminergic
effects indeed acted to stabilize current delay-activity. Working
memory deficits observed after supranormal D1-receptor stimulation
could also be explained within this framework. Thus, the model offers a
mechanistic explanation for the behavioral deficits observed after
blockade or after supranormal stimulation of dopamine receptors in the PFC and, in addition, makes some specific empirical predictions.
Key words:
dopamine; prefrontal cortex; working memory; D1 receptor; theory; neurocomputation; delayed matching-to-sample
 |
INTRODUCTION |
The prefrontal cortex (PFC) plays a
major role in working memory (Passingham, 1975
; Fuster, 1989
;
Goldman-Rakic, 1990
, 1995
; Petrides, 1995
; Kesner et al., 1996
). In a
series of recent experiments, Miller and colleagues (Miller et al.,
1991
, 1993
, 1996
; Miller and Desimone, 1994
) extended the classical
delayed matching-to-sample (DMS) paradigm, which is often used to
assess working memory functions, by introducing intervening stimuli
between the presentation of the sample stimulus and the matching
stimulus, after which a response had to be given. While monkeys were
performing this task, these authors recorded from PFC neurons. As shown
previously (for review, see Fuster, 1989
; Goldman-Rakic, 1990
), Miller
et al. (1996)
found that many neurons exhibited stimulus-selective
delay-activity. In addition, in contrast to delay-active neurons in the
temporal (Miller et al., 1993
, 1996
; Miller and Desimone, 1994
) and
posterior parietal (Di Pellegrino and Wise, 1993
; Constantinidis and
Steinmetz, 1996
) lobe, delay-activity of most PFC neurons persisted
even when intervening stimuli were presented. Thus, PFC neurons or networks seem to be equipped with a mechanism that enables them to hold
active neural representations of goal-related information and to
protect this goal-related delay activity against interfering stimuli.
However, the nature of this neural mechanism is largely unknown.
We speculated that dopamine (DA) might play a crucial role in such a
mechanism. Dopaminergic midbrain neurons become active (Schultz et al.,
1993
) and DA levels in the PFC significantly increase (Watanabe et al.,
1997
) during working memory performance. Prefrontal DA depletion
(Brozoski et al., 1979
; Simon et al., 1980
) or blockade of D1 receptors
in the PFC (Sawaguchi and Goldman-Rakic, 1991
, 1994
; Seamans et al.,
1998
) causes severe working memory deficits, whereas D1 agonists might
enhance delay task performance [Arnsten et al. (1994)
; Müller et
al. (1998)
; but see Zahrt et al. (1997)
]. Moreover, DA strongly
modulates the electrical activity of PFC neurons in vivo and
in vitro by multiple D1- and D2-receptor-mediated presynaptic and postsynaptic mechanisms (Bernardi et al., 1982
; Ferron
et al., 1984
; Mantz et al., 1988
; Sesack and Bunney, 1989
; Sawaguchi et
al., 1990a
,b
; Godbout et al., 1991
; Williams and Goldman-Rakic, 1995
;
Yang and Seamans, 1996
). DA enhances a persistent Na+ and reduces a slowly inactivating
K+ and a dendritic HVA Ca2+
current in rat PFC pyramidal neurons via D1 stimulation in
vitro (Yang and Seamans, 1996
; Gorelova and Yang, 1997
; Shi et
al., 1997
), it reduces glutamatergic synaptic inputs (Pralong and
Jones, 1993
; Law-Tho et al., 1994
), and it enhances activity and
spontaneous transmitter release of GABAergic neurons in the PFC
(Penit-Soria et al., 1987
; Rétaux et al., 1991
; Pirot et al.,
1992
; Yang et al., 1997
). We asked whether DA by modulating these
biophysical properties of PFC neurons could act to stabilize and
protect goal-related delay-activity in PFC networks. To answer this
question, we constructed a network of leaky-integrator model neurons
that allowed for a biophysically realistic implementation of the
cellular effects of DA. Network simulations were performed, and the
stability of representations was evaluated while DA-dependent neural
and synaptic parameters were varied.
 |
MATERIALS AND METHODS |
Single neuron model. The goal of the present model
was to explain the stability of delay-activity in the PFC and the
dopaminergic modulation of this activity. We tried to keep the model as
simple as possible within the constraints imposed by this goal. For
example, because only average spike rates but not single spike codes
are considered in the literature dealing with delay-activity during working memory and the dopaminergic modulation of this activity, spike
rates were directly derived from membrane potential fluctuations in the
model neurons instead of implementing an explicit spike-generating mechanism. On the other hand, we attempted to put sufficient detail into the equations to realistically describe the impact of DA on the
state and behavior of the model neurons. For the purposes of the
present study, we extended a network model extensively studied by Amit
and colleagues (Amit et al., 1994
; Amit and Brunel, 1995
), who also
showed that this model could reproduce very well various aspects of the
electrophysiological behavior of neocortical neurons recorded in
vivo.
The excitatory model neurons used in the present network study were
intended to represent deep-layer PFC pyramidal cells. Layer V/VI
pyramidal cells are the ones most densely innervated by dopaminergic
fibers in the rat PFC (Berger et al., 1988
, 1991
; Joyce et al., 1993
)
and constitute the major portion of neurons with sustained delay
activity (Fuster, 1973
). To account for the electrotonically separated
proximal (soma, basal, apical oblique) and distal (apical tuft)
dendrites of these neurons, the model neurons were chosen to consist of
a distal and a proximal "dendritic" compartment, connected by a
coupling resistance, as depicted in Figure 1.
The proximal and distal compartments were described by simple
"leaky-integrator" differential equations. The equation describing the proximal compartment contained additional nonlinear terms that
represented the contribution of a persistent Na+
(VNaP) and a slowly inactivating
K+ (VKS) current to
the membrane potential:
|
(1a)
|
|
(1b)
|
where Vp,i is the proximal and
Vd,i the distal membrane potential of unit
i,
p the proximal and
d the
distal membrane time constant,
pd a coupling strength
between the proximal and the distal compartment (which may be
interpreted as a length constant of the model neuron),
exc and
inh the general excitatory and inhibitory synaptic efficiency, wij the specific
excitatory synaptic coupling strength (weight) from neuron j
to neuron i, fexc and finh the (instantaneous) firing rates of the
excitatory neurons and the inhibitory unit, respectively, and
Iaff represents an afferent input arising from
other association or higher sensory areas (see below) (all variables in
arbitrary units).
Because two of the major cellular effects of DA (see Implementation of
the dopaminergic modulation) are on the persistent Na+ (INaP) (Alzheimer
et al., 1993
; Brown et al., 1994
; Yang and Seamans, 1996
) and slowly
inactivating K+ current
(IKS) (Huguenard and Prince, 1991
; Spain
et al., 1991
; Yang and Seamans, 1996
), a simplified biophysical
description of these currents was implemented in the model. Both
currents activate on depolarization and were represented by the product of a maximum current (INaP,max and
IKS,max, respectively) with a
voltage-dependent (steady-state) activation gate, given by a sigmoid
(Boltzmann) function (the slow inactivation process of these currents
was omitted for simplicity). The change in membrane voltage induced by
these currents relates to these currents by the constant passive
membrane resistance R, which was set to 1.0 for simplicity
(but could be set to any other value provided that the maximum currents
are scaled accordingly):
|
(2)
|
In the present paper, we were not seeking precise quantitative
matches with empirical data. Rather our goal was to demonstrate some
important functional principles that work regardless of the detailed
activation kinetics (which are fast compared with the membrane time
constants) or the exact parameter settings of the gating functions. For
the present line of argument, the key point is that there are inward
and outward currents in the model neurons that increase with membrane
potential and could be differentially modulated by DA.
The spike frequency output fexc of an excitatory
neuron was set to zero below a certain threshold
exc and
depended logarithmically on membrane potential above this threshold (as
described by Amit and Brunel, 1995
):
|
(3)
|
The population of GABAergic interneurons was lumped into a
single inhibitory unit (Amit et al., 1994
; Amit and Brunel, 1995
). For
this unit, membrane potential and spike frequency output were given by
the following equations:
|
(4)
|
where Vinh is the membrane potential,
inh the time constant,
inh the firing
threshold of the inhibitory unit, and NS the (mean) number of units coding for a stimulus (see below), introduced into the equation as a scaling constant.
A parameter configuration defining a "standard model" at baseline
DA activity is summarized in Table 1
(column
base). Parameters regulating the maximum
persistent Na+ and slowly inactivating
K+ current, the maximum synaptic current, and the
current flow between the two compartments were derived from simulations
carried out with a detailed compartmental model of PFC pyramidal
neurons (Durstewitz and Seamans, 1997
; Durstewitz, 1998
). We determined
from this pyramidal cell model, which could faithfully reproduce
somatic and dendritic recordings from real rat PFC neurons, the size of the total (subthreshold) INaP, the
(subthreshold) axial current from the dendrites to the soma, and the
mean total synaptic currents (AMPA + NMDA, or
GABAA) under simulated "in vivo"
conditions. We then used these values to adjust the relative
contributions of these currents to membrane potential in the simple
leaky-integrator neuron used here.
Network model. Figure 1
depicts the structure of the network model. The network consisted of a
PFC layer, a motor output unit, and a DA unit. The PFC layer consisted
of N = 100 excitatory "layer V/VI pyramidal
neurons," arranged in a 10 × 10 square, and an inhibitory
feedback unit, representing a population of GABAergic cells (Amit and
Brunel, 1995
). Within the PFC layer, each pyramidal cell made
excitatory synaptic contacts only on the proximal compartments of other
pyramidal neurons. In contrast, higher sensory afferent inputs
(Iaff in Eq. 1b) were simulated by charging the
distal model compartments. The inhibitory feedback unit received input
from all pyramidal neurons and projected back onto the proximal
compartments of all of those neurons. This pattern of connections was
chosen according to neuroanatomical data. Lübke et al. (1996)
and
Markram et al. (1997)
reported that deep-layer neocortical pyramidal
cells made synaptic contacts on other deep-layer pyramidal cells
preferentially within the proximal dendritic tree, and Kritzer and
Goldman-Rakic (1995)
and Levitt et al. (1993)
showed that axonal
arborizations of PFC deep-layer pyramidal neurons extending laterally
across column boundaries mainly stayed within the same deep layers. In contrast, Mitchell and Cauller (1997)
showed that afferent fibers from
other cortical association and higher sensory areas terminate in the
upper PFC layers I-II, which seems to be a common pattern of
association fiber connections in the neocortex (Jones, 1984
; Cauller,
1995
; Cauller et al., 1998
). Finally, axons from Chandelier and
basket GABAergic interneurons primarily terminate on the initial axonal
segment, soma, and proximal dendrites (i.e., the proximal compartment) of deep-layer pyramidal cells (Douglas and Martin, 1990
).
However, it should be emphasized that all of the results presented here
except the ones regarding dopaminergic modulation of dendritic
Ca2+ currents (see Results) do not depend on these
anatomical assumptions.

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|
Figure 1.
Structure of the PFC network model. Within the PFC
network, excitatory neurons (representing deep-layer pyramidal cells)
are interconnected by excitatory synapses on their proximal
compartments (prox) while receiving afferent input
from other cortices on their distal compartments (dis). An
inhibitory unit (INH) provides feedback inhibition. A
"striatal" motor unit (MOTOR) receives excitatory input
from all PFC "pyramidal cells" and inhibits the "mesencephalic"
DA unit, which modulates parameters of PFC neurons and synapses in an
activity-dependent manner.
|
|
In the Miller et al. (1996)
task, which will be used here for the
simulations, the monkeys gave their responses by pressing a lever.
Thus, suprathreshold activity of the motor output unit in the present
simulations was meant to indicate such a lever press. The motor output
and the DA unit consisted of just one compartment as described by
Equation 1a, without the terms for voltage-gated and axial current flow
and without inhibitory feedback. Parameters for these units are given
in Table 1. The motor unit received (nonreciprocal) connections from
all PFC pyramidal neurons, with a fixed total weight
exc × w = 1/(10 × NS) = 0.005, where NS = 20 is the number of neurons
participating in the representation of a given stimulus (see below).
This connection in the model network corresponds to the unidirectional
glutamatergic fiber connections from PFC layer V pyramidal cells to
striatal neurons (Selemon and Goldman-Rakic, 1985
; Goldman-Rakic, 1995
;
Heimer et al., 1995
).
The DA unit received excitatory input from all PFC pyramidal neurons,
with a fixed total weight
exc × w = 1/(10 × NS) = 0.005, and inhibitory
input from the motor unit with fixed weight w = 1.0 (
mot =
5.0). Glutamatergic projections from the PFC to
the substantia nigra (SN) have been demonstrated by glutamate uptake and ablation studies (Carter, 1982
; Kornhuber et al., 1984
). In addition, it has been shown that electrical and chemical stimulation of
the PFC induces burst firing in the ventral tegmental area (VTA) and SN
dopaminergic neurons (Murase et al., 1993
; Tong et al., 1996a
,b
), which
resembles natural burst events as elicited by behaviorally significant
stimuli (Schultz and Romo, 1990
; Ljungberg et al., 1992
; Schultz et
al., 1993
). Burst firing in turn (as opposed to a single spike mode)
leads to increased DA release at dopaminergic terminals in the
forebrain (Gonon, 1988
). Thus, PFC activity can stimulate DA release
via direct or indirect (Tong et al., 1996a
) excitatory fiber
connections to the VTA/SN (Karreman and Moghaddam, 1996
), justifying
the assumptions made here.
The inhibitory signal from the motor unit to the DA unit represented
the GABAergic input that SN and VTA dopaminergic neurons receive via
the striatonigral pathway (Heimer et al., 1995
). In an operant feeding
task carried out by Nishino et al. (1991)
, VTA neurons increased their
firing rate initially while monkeys pressed a lever for food reward but
became inhibited thereafter during ingestion, i.e. when the goal has
been achieved (see also Richardson and Gratton, 1998
). Goldman-Rakic et
al. (1992a)
speculated that in an oculomotor delayed-response task, a
command signal that originates in the PFC activates neurons in the
striatum, which in turn inhibit the SN (Hikosaka and Wurtz, 1983
),
which finally results in disinhibition of the medial dorsal thalamic nucleus and superior colliculus. Thus, PFC neurons could directly excite dopaminergic midbrain neurons and stimulate DA release, or could
inhibit them indirectly by initiating a motor output that results in
goal achievement.
The kind of implementation outlined above was chosen to demonstrate the
functioning of a completely self-regulating system. However, for the
central ideas pursued in this paper, it does not matter whether the PFC
itself activates the DA unit, whether it is activated by some other
brain structure (e.g. the amygdala; Goldstein et al., 1996
), or whether
increased DA release in the PFC during working memory is regulated at
the terminal level (Blaha et al., 1997
), as long as it is provided that
prefrontal DA levels rise at the onset of the working memory task [for
a discussion of related points see Schultz (1992)
]. Evidence for this
comes (1) from the in vivo studies by Schultz et al. (1993)
,
who demonstrated that VTA/SN neurons increase their firing rate with
the presentation of the first, instructing stimulus in a spatial
delayed response task, and (2) from Watanabe et al. (1997)
, who
observed increased DA levels especially in the dorsolateral PFC of
monkeys during performance of a delayed alternation but not during a
sensory-guided control task.
Synaptic connections within the PFC layer and neural
representation of stimuli. A number of different patterns,
representing different environmental stimuli, were stored in the
synaptic weight matrix of the network by connecting the neurons coding
for a particular stimulus by high synaptic weights. For simplicity,
patterns were represented by binary vectors S = {si}, and were stored in the PFC network according to a
Willshaw matrix (Amit and Brunel, 1995
):
|
(5)
|
For all simulations, the weights were set to w = 1/(NS
1). All stimuli were represented by
binary patterns with fixed coding level c = NS/N = 0.2, where
N = 100 is the total number of excitatory neurons in
the PFC layer, and NS = 20 as defined above [if
different patterns are to be represented by varying numbers
NS of units, NS in the
equations above should be set to the average number of units
participating in a pattern (see Amit and Brunel, 1995
)]. For the
purpose of illustrating general network performance, patterns forming
integer numbers ("0," "1," ...) were used as easily
recognizable stimuli. All input patterns that were used during a
simulation run were stored a priori in the synaptic weight matrix [for
mechanisms of online learning, see Amit and Brunel (1995)
]. In the
simulations demonstrating general network performance, there were
always 10 patterns stored in the weight matrix. However, different
numbers of patterns, different coding levels, or the assumption of
small, non-zero, randomly initialized weights (<10% of w
as defined above) between all network units all yield the same basic
results (using parameters as given in Table 1). For different coding
levels, only the weights have to be adjusted according to
NS as described above.
Stimuli (input patterns) were presented to the net by directly charging
the distal compartments of the PFC layer pyramidal neurons (i.e.,
clamping Iaff in Eq. 1b). It is generally
assumed that the increased firing rate of PFC neurons during the delay periods of working memory tasks represents the active holding of
stimuli (Fuster, 1989
; Goldman-Rakic, 1990
, 1995
; Funahashi and Kubota,
1994
). Likewise, in the present network model, the active holding, or
active short-term representation, of stimuli was reflected in the
elevated firing rates of the neurons participating in the neural
representation of the respective stimulus (Amit, 1995
; Amit et al.,
1997
). Thus, different stimuli activated different representations
encoded in the synaptic weight matrix of the network, which were then
eventually maintained by recurrent excitation and thus transformed
into active working memory representations. Amit and Brunel (1995)
expressed conditions under which increased firing rates will be
maintained in the absence of external input in associative networks of
the type used here.
Implementation of the dopaminergic modulation. In the PFC,
pyramidal cells seem to be the preferential targets of dopaminergic afferents and the major cell population expressing the
D1-receptor-related phosphoprotein DARPP-32 (Goldman-Rakic et al.,
1989
; Berger et al., 1990
; Smiley and Goldman-Rakic, 1993
). However,
dopaminergic inputs to smooth stellate and thus probably GABAergic
interneurons and D1-like immunoreactivity in these neurons have also
been observed (Smiley and Goldman-Rakic, 1993
; Muly et al., 1998
). In
the rat PFC, layers V-VI receive the densest dopaminergic input, and
in monkeys, in addition, the superficial (supragranular) cell layers (Berger et al., 1988
, 1991
; Goldman-Rakic et al., 1992b
; Lewis et al.,
1992
). For simplicity we have assumed that biophysical parameters of
all pyramidal neurons in these layers and thus in the model
net are affected by DA [which seems also to be reasonably justified
according to DARPP-32-positive cell countings in association areas of
the avian "neocortex" (Durstewitz et al., 1998
)].
DA has a variety of different effects on intrinsic membrane currents
and synaptic parameters of pyramidal and GABAergic neurons in the PFC.
We will focus here on the effects that are best documented. Furthermore, because the effects of DA on neural activity and single
ion channels exhibit considerable substrate specificity [e.g., compare
Cepeda et al. (1995)
with Yang and Seamans (1996)
], we will use in the
present study only effects confirmed in rat PFC neurons. Dopaminergic
actions considered in the present study were the following.
First, DA has been shown to enhance a persistent Na+
current, probably by shifting the activation kinetics of this current into the hyperpolarized direction and by prolonging its inactivation time constant (Yang and Seamans, 1996
; Gorelova and Yang, 1997
). This
effect might underlie the DA-induced depolarization of PFC neurons
observed in vitro and in vivo (Bernardi et al.,
1982
; Yang and Seamans, 1996
; Shi et al., 1997
) and the dramatic
increase in spike frequency (Yang and Seamans, 1996
; Shi et al., 1997
). In the model, this effect was implemented by shifting the activation curve of the persistent Na+ current (i.e., parameter
NaP) toward less positive potentials in a
DA-dependent fashion (see below).
Recently, Gulledge and Jaffe (1998)
reported that DA reduces
the spike frequency of PFC pyramidal cells in vitro, in
conflict with the findings cited above. The fact that Gulledge and
Jaffe (1998)
did not yet unravel the ionic mechanisms underlying this effect makes an assessment of its possible role in working memory processes very difficult. More importantly, these authors found that
the depressive effect of DA is D2-mediated, whereas D1 receptor agonists and antagonists had no effect. In contrast, working memory performance (Arnsten et al., 1994
; Sawaguchi and Goldman-Rakic, 1994
;
Zahrt et al., 1997
; Müller et al., 1998
; Seamans et al., 1998
)
and delay activity in vivo (Sawaguchi et al., 1990b
;
Williams and Goldman-Rakic, 1995
) are mainly susceptible to D1 but not D2 receptor stimulation or blockade, as were the DA-dependent ionic
mechanisms unraveled in the studies by Yang and Seamans (1996)
and
Gorelova and Yang (1997)
. In accordance with these findings, the
D2-mediated depressive effect on spike frequency might only occur
initially, shortly after bath application of DA in vitro
(starting from a zero DA concentration), until the D1-mediated effects
take over (J. Seamans, unpublished observations), and hence might not
show up in an in vivo situation in which there is a constant
baseline level of DA in the PFC (Abercrombie et al., 1989
; Moore et
al., 1998
). Finally, D1 receptors are also much higher in density than
D2 receptors in the PFC (Goldman-Rakic et al., 1992b
; Joyce et al.,
1993
). Thus, the D2-mediated effects described by Gulledge and Jaffe
(1998)
might not play a prominent role in working memory processes, and
we focused here on the D1-mediated effects elucidated by Yang and
Seamans (1996)
and Gorelova and Yang (1997)
.
Second, DA has been shown to reduce a slowly inactivating
K+ conductance (IKS)
in PFC pyramidal cells (Yang and Seamans, 1996
). This effect might
contribute to the depolarization and increased firing rate of PFC
neurons under DA action. In the model, DA reduced IKS,max (see Eq. 2).
Third, DA reduces the amplitude and half-width of isolated dendritic
Ca2+ spikes generated by a dendritic high
voltage-activated (HVA) Ca2+ current (Yang and
Seamans, 1996
; Formenti et al., 1998
). In vitro (Seamans et
al., 1997
) and compartmental modeling data (Bernander et al., 1994
;
Durstewitz and Seamans, 1997
) have demonstrated that dendritic HVA
Ca2+ currents can amplify EPSPs induced in the
distal dendrites of deep layer PFC neurons on their way to the soma. In
addition, Schiller et al. (1997)
have shown that synaptic inputs to the distal dendrites activate local HVA Ca2+ currents,
causing increased responses at the soma. Hence, dendritic HVA
Ca2+ currents affect the ability of distal EPSPs to
effectively depolarize the soma, and DA might reduce this ability. In
addition, the data of Yang and Seamans (1996)
made it likely that the
DA-affected Ca2+ current resides primarily in the
distal dendrites and might thus be of the N-type [more directly shown
by Surmeier et al. (1995)
and Formenti et al. (1998)
], which reaches a
local maximum in the distal dendrites of pyramidal cells (Westenbroek
et al., 1992
; Yuste et al., 1994
). Thus, DA might specifically diminish
distal EPSPs, or might at least attenuate them more strongly than
proximal EPSPs [as first suggested by Yang and Seamans (1996)
]. This
has in fact been shown by Yang et al. (1996)
and Law-Tho et al. (1994)
. The effect that DA induces via reduction of a dendritic
Ca2+ conductance could be interpreted as an increase
in the electrotonic distance between the distal and the proximal
dendritic region, which causes a reduction in current flow from distal
to proximal. Thus, in the model neurons, DA increased the electrotonic
distance between the distal and the proximal compartment by reducing
the coupling strength (
pd). However, in addition
to this simple way of representing in the model dopaminergic effects on
dendritic Ca2+ conductances, we also examined
whether a more explicit representation of the DA-induced reduction of a
dendritic Ca2+ current (see Results) basically
yields the same results.
Fourth, in the PFC, as in other cortical areas (Pralong and Jones,
1993
), DA depresses the AMPA as well as the NMDA component of EPSPs via
D1 receptors (Law-Tho et al., 1994
), thus reducing the amplitude of
EPSPs evoked by layer I stimulation by up to 50%. Cepeda et al. (1992)
also found that DA strongly suppresses glutamate-induced responses but
reported enhanced responses to NMDA. (They did not, however, record
from PFC neurons, and their slices were obtained from human brains that
might have undergone pathological changes, so that the applicability of
their results to the healthy PFC might have to be interpreted with some
caution.) The suppressing effect of DA on glutamate-induced responses
and EPSPs was implemented in the model by reducing the general
excitatory synaptic efficiency
exc.
Fifth, DA has been reported to enhance spontaneous activity of
GABAergic interneurons and to increase IPSP size (Penit-Soria et al.,
1987
; Rétaux et al., 1991
; Pirot et al., 1992
; Yang et al.,
1997
). This effect, in addition to the DA-induced reduction of EPSPs,
might be responsible for the inhibition of neural activity in the PFC
observed in vivo in anesthetized rats after local DA application or VTA stimulation (Ferron et al., 1984
; Godbout et al.,
1991
; Pirot et al., 1992
, 1996
) (note that these studies were performed
in rats anesthetized by ketamine, i.e., with NMDA currents blocked and
outside a behavioral context). However, it should be noted that a
reduction of evoked GABAergic activity and a reduction of
IPSP sizes by DA in the PFC have also been reported (Rétaux et
al., 1991
; Law-Tho et al., 1994
). We will show here what the effect of
varying the general inhibitory synaptic efficiency
inh
on network behavior might be.
All of the DA-modulated network parameters were chosen to depend
linearly on the deviation of the DA unit output from some baseline
firing frequency. That is, with the DA unit firing at baseline, all
network parameters had the values given in Table 1, and these values
were varied according to the deviation
fDA of
the DA unit output from some arbitrary baseline:
|
(6)
|
where
denotes some network parameter, and
base and
shift are given for each
DA-modulated parameter in Table 1. The deviation of the DA unit output
from the baseline depended sigmoidally on the membrane potential of the
DA unit, with a constant offset:
|
(7)
|
where VDA is the membrane potential of
the DA unit, and
DA is a parameter normally set to 1.0 except for "pathological" conditions of DA hypoactivity (
DA
condition, see below) or DA hyperactivity (++DA condition, see below).
During the simulation runs demonstrating general network performance,
the DA-induced parameter changes were roughly adjusted according to
estimations derived from in vitro data (Law-Tho et al.,
1994
; Yang and Seamans, 1996
) and to values obtained with simulations
with a detailed compartmental model of PFC pyramidal cells (Durstewitz,
1998
).
Testing the stability of working memory representations. To
demonstrate the general performance of the PFC model network, the DMS
task with intervening stimuli as used by Miller et al. (1993
, 1996
) and
Miller and Desimone et al. (1994)
was used here. An arbitrary sample
pattern was first presented to the network by clamping distal afferent
inputs (Iaff in Eq. 1b), followed by an
arbitrary sequence of partly overlapping "intervening stimuli" (e.g., 4, 0, 0, 2, 4). The network had to respond, i.e., to activate the motor output unit, when the initially presented sample pattern appeared the second time (target condition). A new trial could then be
started. Note that also a delayed non-matching-to-sample (DNMS) task could easily be implemented in the network by assuming that
the initial sample pattern activates a representation of the target
pattern (TP), i.e., evokes a representation of the goal state [as
suggested, e.g., by the electrophysiological data of Quintana et al.
(1988)
]. This essentially would internally transform the DNMS problem
into a DMS problem. It is also important to note that in the Miller et
al. (1993
, 1996
) task, stimuli were not trial unique, i.e.,
intervening stimuli appeared as sample/target stimuli on other trials
and were all known in advance to the animals, raising the opportunity
for considerable interference between trials and stimuli and justifying
the a priori storage of stimuli in the present network.
The central hypothesis of the present paper was that DA stabilizes
goal-related delay activity and protects it against interfering stimulation. To investigate the effect of the DA-induced parameter changes on stability of goal-related neural representations more systematically, the afferent input (Iaff in Eq. 1b) needed to disrupt the ongoing network activity (i.e., the current
neural representation) was taken as an index for stability. The higher
the afferent input needed to establish a new activity pattern (i.e., a
new attractor state) in the PFC network, the more stable is the
currently active neural representation. We termed this current
Iaff,crit in the following. In these
simulations, a single (DA-dependent) network parameter was
systematically varied while for each run a sample pattern was
established in the isolated PFC network (no motor and DA unit), and the
dependence of Iaff,crit needed to override this
pattern on the value of the respective parameter was determined. The
afferent input was presented for a time long enough for the activity of
the stimulated neurons to reach their approximate maximum.
Computational techniques. The simulation software was
programmed in C++, and all simulations were run on a 200 MHz Pentium PC
using LINUX, a UNIX version for PCs, as operating system. The whole
system of differential equations was simultaneously solved by a
fourth-order Runge-Kutta method or the semi-implicit extrapolation method as described in Press et al. (1992)
, yielding the same results
for the error criterion used here.
 |
RESULTS |
General network performance
The central hypothesis put forward here was that the function of
DA during working memory processing is to stabilize active neural
representations, i.e., to maintain goal-related neural activity in the
PFC even in the presence of interfering afferent stimulation. The
simulations presented in Figure 2
demonstrate that (1) the DA-induced changes of biophysical parameters,
taken all together, are appropriate for fulfilling this function, and (2) the prefrontal system via known anatomical connections could dynamically regulate its DA level to achieve the proposed function.

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Figure 2.
Activity of the PFC network under
conditions of normal (+DA) and reduced ( DA) DA
unit output during two successive trials of a DMS task with intervening
stimuli. A, Mean proximal membrane potential (top
parts) of all PFC net units belonging to the pattern "4,"
which is the TP in the first trial (TP1), and pattern
"0," which is the TP in the second trial (TP2).
Gray bars indicate the time intervals during which stimuli
were presented. Three intervening stimuli (IS1-IS3 and
IS4-IS6, respectively) were presented between the first
(sample) and the second (matching) presentation of a TP. Note that
IS1 = IS2 = TP2 and
IS4 = IS5 = TP1.
Bottom parts give the gray level-coded membrane
potential of all 100 PFC network units at discrete time points during
presentation of the stimuli in the first trial. Lighter gray
levels indicate higher activity. With normal DA output, TPs stay
stable during a whole trial (+DA), whereas intervening
patterns override currently active patterns under conditions of reduced
DA unit output ( DA). Iaff = 0.55. B, Membrane potential of the motor unit under the
+DA and the DA conditions. C, Mean
firing frenquency of the inhibitory unit under the +DA and
the DA condition.
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Figure 2A compares the activity of the pyramidal
units of the PFC network under two different conditions of simulated
dopaminergic activity during a simulated DMS task with intervening
stimuli as outlined in Materials and Methods. In the first condition
(+DA), the intact network as given by the configuration in Table 1 was used. In the second condition (
DA), the output of the DA unit (or, in
other terms, the DA level in the PFC) was strongly diminished by
reducing the scaling factor
DA in Equation 7 to 0.3. Another interpretation of this manipulation might be a partial blockade of DA receptors in the PFC.
Figure 2A shows the mean membrane potential of all
model neurons participating in the representation of a pattern that was the target in the first trial (TP1), and of all neurons participating in the representation of a pattern that was the target in the second
trial (TP2). The first target served as an intervening stimulus (IS)
during the second trial (i.e., IS4 = IS5 = TP1), and the
second target as an intervening stimulus during the first trial (i.e.,
IS1 = IS2 = TP2). All patterns were presented by injecting
Iaff = 0.55. Membrane potential instead of mean
frequency output fexc was chosen for
visualization to depict also the subthreshold activity of the units not
participating in the representation of the actual TP, and to allow
comparison of this activity with that of the TP units. Because spike
frequency in the model units relates logarithmically (i.e., strictly
monotonically) to membrane potential above threshold, essentially the
same picture would have been obtained for mean frequency of the TP
units. Furthermore, for discrete time points during the first trial
when the TP or one of the intervening stimuli was presented, the gray
level-coded activity of all 100 network units is depicted.
As can be seen from Figure 2A, in the +DA network,
activity related to the TP stayed stable during the whole trial even at times where interfering stimuli were presented to the network. Note
that the mean activity of the delay-active units temporally increases
during presentation of each of the intervening stimuli and reaches a
maximum with the second presentation of the TP. This same basic pattern
has been observed in vivo for single PFC neurons as well as
for the mean frequency of a sample of delay-active neurons (Miller et
al., 1996
, their Figs. 4, 5), indicating that the model net could
reproduce very well the basic electrophysiological pattern of
delay-active neurons in the monkey PFC during DMS tasks with
intervening stimuli. The increased firing rate during the second
presentation of the target, which was termed "match enhancement" by
Miller et al. (1996)
and was observed in the majority of monkey PFC
neurons with significant delay activity, triggered a suprathreshold motor output (Fig. 2B). Thus, the motor unit received
suprathreshold activation only when a goal-related match occurred. [In
this sense the motor unit encoded a "pure match" as it has been
observed for some neurons by Miller et al. (1996)
]. As described by
Miller et al. (1996)
, simple repetition of a stimulus (IS1 = IS2,
IS4 = IS5 in Fig. 2A) or intervening
presentation of a stimulus that was the target in other trials
(IS1 = TP2, IS4 = TP1 in Fig. 2A) did not
disrupt delay activity. However, after the motor response had been
delivered, the strong inhibition of the DA unit enabled the transition
to and active storage of a new TP.
For the
DA condition, the situation was quite different. The first
intervening pattern wiped out the target-related delay activity,
preventing any significant match enhancement effect and thus disabling
correct motor output (Fig. 2B). Thus, in the model
network the DA-induced parameter changes were a necessary prerequisite
for the protection of goal-related delay activity against interfering
afferent stimulation.
Figure 2B illustrates that a motor response during
the DMS simulation depicted in Figure 2A is delivered
only in the +DA model when a match between sample and target stimulus
occurred. Figure 2C shows that the activity of the
inhibitory unit closely mimics that of the excitatory model neurons.
Furthermore, the activity of the inhibitory unit is higher in the +DA
condition than in the
DA condition because of the increased firing
frequency of the excitatory units.
Having demonstrated that DA, in principle, could subserve the proposed
function, we will investigate in the next sections in detail which of
the DA-modulated biophysical mechanisms could lead to stabilization of
goal-related neural representations in working memory.
Contribution of the persistent Na+ current to
stabilization of delay activity
The DA-induced shift of the INaP activation
curve (i.e., parameter
NaP) toward less positive
potentials makes the currently active representation more stable in
terms of the afferent input needed to disrupt the current delay
activity (see Materials and Methods) (Fig.
3). This generally holds for different
steepnesses of the INaP activation function
(data not shown), for different degrees of overlap between the target
and the intervening pattern (Fig. 3A), and for different
degrees of "connectivity" (Fig. 3B).

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Figure 3.
Effects of DA-induced variations in
INaP parameters on the stability of TPs.
Stability is measured in terms of the afferent input needed to disrupt
current neural representations (Iaff,crit).
Dashed vertical lines indicate the range within which the
respective parameters or variables during full network simulations
varied. All other network parameters had the baseline values given in
Table 1. OVL, Overlap; CON, connectivity.
A, Reduction of the INaP activation
threshold ( NaP) increases the stability of the
active neural representation at different overlaps (CON = 25%). B, Reduction of the INaP
activation threshold ( NaP) increases the
stability of the active neural representation at different
connectivities (OVL = 15%). C, Dependence
of the stability of an active representation on the amplitude of a
constant (i.e., non-voltage-dependent) Na+ current
at different connectivities (OVL = 15%). D,
Dependence of the stability of an active representation on the
effective INaP amplitude
(INaP,eff), compared for the constant
INaP (labeled INaP,cons),
for INaP,max variation of the voltage-dependent
INaP (labeled INaP,max),
and for NaP variation of the voltage-dependent
INaP (labeled NaP).
OVL = 15%; CON = 25%.
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The term "overlap" (OVL) denotes the number of pyramidal
units shared by the target stimulus and the intervening stimulus representation. As is evident from Figure 3A, increasing the
overlap of the intervening stimulus with the TP reduces the afferent
input needed to override the currently active pattern. However, no
interaction of overlap with the INaP amplitude
or activation threshold is apparent. Dashed vertical lines in Figure 3
illustrate in which range
NaP varies within the full
network simulation as depicted in Figure 2.
By connectivity (CON) we mean the number of inputs
that units of the intervening pattern receive from the TP units, and
vice versa. For the calculation of this index, all connections
including units participating in the representation of both
patterns (i.e., the overlapping units) were excluded. Thus, the
connectivity as defined here depends solely on the other patterns
stored in the network. In the extreme case, if there is a very large
number of patterns stored in the network that overlap with the TP and the intervening pattern, each unit of the intervening pattern might
receive inputs from all pattern units (and vice versa). Thus, the number of inputs may vary between 0 and
NS, and CON was expressed as percentage
of the maximal number of possible inputs NS.
Furthermore, the connectivity between the target and the intervening
pattern was always fully symmetrical. If the connectivity is too high,
too many network units may become simultaneously active so that the TP
breaks down because the stable membrane potential is temporarily pushed
below the excitatory threshold by the increased inhibitory feedback
(note from Equation 10 in the Appendix that for
inh > 1 the inhibitory feedback increases more steeply with rising mean
frequency fexc than the excitatory feedback).
This in general was the case for CON
70% in the present network
(depending on the specific settings of the parameters). Thus, only
values of CON <70% were used in the present simulations. In general,
also for CON <70%, increasing the connectivity diminishes stability
(as evident in Fig. 3B) because the non-TP units receive more excitation whereas nothing changes for the TP units as long as
fexc = 0 for the non-TP units.
The DA-induced shift in the INaP activation
function enhances INaP in the normal potential
range of the excitatory neurons, and thus provides an additional source
of current. Figure 3C demonstrates that this additional
current alone, within a wide range of reasonable INaP amplitudes, could be sufficient for
stabilizing current delay activity. That is, neglecting the voltage
dependency of INaP and representing the
dopaminergic modulation of INaP just by adding a
constant INaP,cons to all excitatory
neurons suffices to make the representation more stable in the PFC
network. This, again, holds for various degrees of overlap between the
target and the intervening pattern (data not shown), and for various
connectivities (Fig. 3C). As shown more analytically in the
Appendix, an additional excitatory current pushes the stable frequency
of the delay-active neurons to higher values. Concurrently, the
difference in membrane potential between the TP and the non-TP units
increases (see Appendix). As indicated by simulation results (data not
shown), within the range of INaP,cons amplitudes
examined here, the rise in the firing rate of the TP units via
increasing the inhibitory feedback causes a decrease in the membrane
potential of the non-TP units. Thus, higher afferent inputs are needed
to drive these units above threshold; that is, it becomes more
difficult for intervening stimuli to override the currently active
neural representation. However, if INaP,cons
becomes too big, there is again a downward trend in stability (evident
from Fig. 3C as a slight downward bend in the curve for the
highest connectivity). This occurs when
INaP,cons becomes high enough to push also the
non-TP units closer to threshold (despite the increased inhibition),
especially in concert with high connectivities. Nevertheless, within a
reasonable range of INaP amplitudes, the pure
enhancement of INaP attributable to DA action
could be a major determinant of the stabilizing effect.
The voltage dependency of INaP as implemented in
the present model neurons makes an additional contribution to the
stability of currently active representations. This is shown in Figure
3D, where the afferent input needed to disrupt the actual
pattern is compared for the constant INaP and
the voltage-dependent INaP as implemented here.
To compare these two conditions, the respective INaP,cons values were set to exactly the same
values that the voltage-dependent INaP of the
delay-active neurons had at the time when the afferent input was
injected (termed INaP,eff here), while
INaP,max was varied (for the constant
INaP, INaP,eff = INaP,cons = INaP,max; for the voltage-dependent
INaP, INaP,eff
INaP,max). Furthermore, shifting the
INaP activation threshold
NaP
toward less positive potentials essentially has the same effect as
increasing INaP,max, as illustrated in Figure
3D. Only for very high effective INaP
amplitudes, shifting the threshold becomes less efficient than
increasing INaP,max. Note also that for the
threshold variation simulations, INaP,eff is
limited by INaP,max = 0.09.
The additional stabilizing effect of the voltage-dependent
INaP compared with a constant current
INaP,cons is attributable to the fact that
INaP increases with membrane potential and does so steeper for higher potentials (unless
Na becomes very
low). As a consequence, if
NaP is reduced, there is a
bigger increase in INaP in the TP units that
reside at a higher membrane potential than in the non-TP units. Thus,
reducing
NaP not only increases an excitatory membrane
current but in addition enlarges the difference in
INaP activation between the TP and the non-TP
units. In Figure 3B, the points where the trend in stability
reverses with decreasing
NaP values indicate the points
of maximum difference in INaP activation between
the TP and the non-TP units. Below these points, the non-TP units gain
more additional current than the TP units with decreasing
NaP values. In general, whether a decrease in
NaP results in an enlargement or a reduction of the
difference in INaP activation between the TP and
the non-TP units depends on whether the membrane potential of the TP or
the non-TP units resides in a region of greater steepness of the
sigmoid INaP activation function. In real
neocortical pyramidal cells, this should always be the case for the
membrane potential of the TP units because the point of maximum slope
conductance (dI/dV) of the
Na+ currents lies well above firing threshold
(Cummins et al., 1994
; Fleidervish et al., 1996
).
The persistent Na+ current has been ascribed to a
major role in synaptic amplification (Schwindt and Crill, 1995
, 1996
;
Stuart and Sakmann, 1995
). Basically, this view is fully compatible
with the account given above. The model neurons are driven by
excitatory synaptic inputs that activate INaP by
pushing the membrane potential to higher levels. The amplification
these synaptic inputs gain by activating INaP
increases at higher membrane potential levels, so that the TP units
profit more from a shift of
NaP. However, one might
argue that this is attributable to the fact that
INaP was placed only into the proximal
compartment (see Eq. 1), where the recurrent synapses terminate.
Although this model assumption is completely justified by the results
of Stuart and Sakmann (1995)
and immunocytochemical data on the
distribution of Na+ channels by Westenbroek et al.
(1989)
, it might not be well supported by the data of Schwindt and
Crill (1995)
and Mittmann et al. (1997)
, which suggest an amplification
of distal EPSPs by INaP all along the apical
dendrite. Hence, to rule out any possible objection one might base on a
homogeneous dendritic distribution of
INaP, the simulations shown in Figure
3A,B were rerun with model neurons where
INaP was inserted also into the distal
compartment (with the same properties as in the proximal compartment).
The results were essentially the same, and the stabilizing effect was
not less in magnitude than the one shown in Figure 3A,B.
Contribution of the slowly inactivating K+
current to stabilization of delay activity
Because the slowly inactivating K+ current like
the persistent Na+ current increases with membrane
potential but acts in the opposite direction, reduction of its
amplitude (IKS,max) as induced by DA
mirrors the effect that is produced by increasing the amplitude of
INaP. TP representations become more stable at
various overlaps with the intervening stimulus (Fig.
4A) and various degrees
of connectivity (Fig. 4B) with decreasing values of
IKS,max. Just diminishing a source of (constant)
negative current has a similar effect as an increase in
INaP,cons. In addition, reducing
IKS,max also results in a proportionally bigger
decrease of the effective IKS in the TP than in
the non-TP units because of the fact that IKS
increases with depolarization (see Eq. 2). Another way to put this is
that IKS at higher membrane potentials more
strongly withholds further depolarization induced by excitatory
synaptic input, so that the TP units are affected to a bigger degree
than the non-TP units by a partial removal of this current. As
it was the case with the persistent Na+ current,
this stabilizing effect is generally robust with respect to other
parameter settings, like that of the steepness of the IKS activation function
(
KS), its threshold (
KS),
or the general excitatory synaptic efficiency
(
exc), although its magnitude depends on these
parameters. We also verified that the stabilizing effect of
IKS,max reduction still holds when
IKS is inserted into the distal dendritic
compartment in addition to its proximal placement.

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Figure 4.
Effects of DA-induced variations in
IKS,max on the stability of TPs (as measured by
Iaff,crit). Dashed vertical lines
indicate the parameter range within which
IKS,max during full network simulations varied.
All other network parameters had the baseline values given in Table 1.
OVL, Overlap; CON, connectivity. A,
Reduction of IKS,max leads to higher stability
for different overlaps, more pronounced at low overlaps
(CON = 25%). B, Reduction of
IKS,max leads to higher stability for different
connectivities, and this trend is more pronounced at low connectivities
(OVL = 15%).
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Contribution of the general excitatory synaptic efficiency to
stabilization of delay activity
The effect of a reduction of the general excitatory synaptic
efficiency (
exc), which mimics the EPSP reduction
observed after DA application in vitro (Pralong and Jones,
1993
; Law-Tho et al., 1994
), depends on the settings of the other
network parameters and on the connectivity between the TP and the
intervening stimulus. In particular, with the network model being in a
low DA configuration (i.e., with all parameters assuming baseline
values as given in Table 1), a reduction of
exc only has
a slightly stabilizing effect at different levels of overlap between
the target and intervening pattern (Fig.
5A), and only at degrees (up
to 50%) of connectivity that are not too high (data not shown).
However, with all other parameters having values of a high DA
configuration (i.e., assuming maximal DA unit output,
fDA = 1.0 in Equation 6), reduction of
exc has a strongly stabilizing effect at different
degrees of overlap (Fig. 5B). This effect is much more
pronounced for low degrees of connectivity, and nearly absent at very
high connectivities (Fig. 5C). Thus, reduction of the
general excitatory synaptic efficiency has a consistently stabilizing
effect only in concert with other DA-induced parameter changes. This is
shown in more detail in Figure 5D,E, which demonstrates for
a moderate degree of overlap (15%) and connectivity (25%) that the
stabilizing effect of
exc reduction increases with
decreasing values of the INaP activation
threshold (
NaP) and decreasing values of the
maximum IKS amplitude
(IKS,max). Only for very low values of
NaP does the effect reverse for the same reason
mentioned in the section on INaP. At some point,
decreasing
NaP results in a decline of the difference in
INaP activation between the TP and the non-TP units. The interaction of the DA-induced changes in
exc
with changes of INaP or
IKS is related to the fact that
INaP amplifies and IKS
diminishes excitatory synaptic input. Mathematically, this relationship
can be understood by explicating the stable frequency equation of the
delay activity (see Appendix, Eq. 11), where
exc enters
in a divisive fashion into the VNaP and
VKS terms.

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Figure 5.
Effects of DA-induced variations in the general
excitatory synaptic efficiency ( exc) on the
stability of TPs (as measured by Iaff,crit).
Dashed vertical lines indicate the parameter range within
which exc during full network simulations varied.
OVL, Overlap; CON, connectivity. A,
With all other network parameters being in a low (baseline) DA
configuration, reduction of exc has only a slightly
stabilizing effect at different overlaps (CON = 25%).
B, With all other network parameters in a high DA
configuration, reduction of exc has a >10-fold greater
stabilizing effect (CON = 25%). C, In a
high DA configuration, reduction of exc has a
stabilizing effect at all except very high connectivities
(OVL = 15%). D, The effect of a variation
in exc is more pronounced at low than at high, except
very low INaP activation thresholds
( NaP) (OVL = 15%;
CON = 25%). E, The effect of a variation in
exc is more pronounced at low values of
IKS,max (OVL = 15%;
CON = 25%). F, In the high DA
configuration, a reduction of exc only for PFC
internal excitatory synapses but not for afferent synapses
has a slightly stabilizing effect for high but not for low degrees of
overlap (CON = 25%).
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Figure 5F demonstrates that in the high DA configuration the
reduction of
exc exerts its effect mainly but not
exclusively by diminishing the impact of intervening stimuli on current
network activity. Reducing the excitatory synaptic efficiency for the internal connections only while leaving it constant for the
afferent input still leads to a small stabilizing effect for higher
degrees of overlap, whereas the effect reverses at low overlaps. Thus, the change in the internal network dynamics produced by
exc reduction itself does not have a consistent effect
on stability (although it would if INaP,max
would be set very high).
Contribution of the coupling strength to stabilization of
delay activity
The decrease of the coupling strength between the distal and
proximal model compartment, which was meant to represent the dopaminergic modulation of a dendritic HVA Ca2+
current (see Materials and Methods), not surprisingly strongly diminishes the impact that intervening stimuli have on current network
activity. More interestingly, this effect is much more pronounced for
short-lasting distal afferent inputs than for longer-lasting inputs
(Fig. 6A), at various
degrees of overlap and connectivity (data not shown). Hence, decreasing
