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The Journal of Neuroscience, April 1, 2002, 22(7):2963-2976
Activity Patterns in a Model for the Subthalamopallidal Network
of the Basal Ganglia
D.
Terman1,
J. E.
Rubin2,
A. C.
Yew1, and
C. J.
Wilson3
1 Department of Mathematics, The Ohio State University,
Columbus, Ohio 43210, 2 Department of Mathematics, The
University of Pittsburgh, Pittsburgh, Pennsylvania 15260, and
3 Division of Life Sciences, University of Texas at San
Antonio, San Antonio, Texas 78249
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ABSTRACT |
Based on recent experimental data, we have developed a
conductance-based computational network model of the subthalamic
nucleus and the external segment of the globus pallidus in the indirect pathway of the basal ganglia. Computer simulations and analysis of this
model illuminate the roles of the coupling architecture of the
network, and associated synaptic conductances, in modulating the
activity patterns displayed by this network. Depending on the
relationships of these coupling parameters, the network can support
three general classes of sustained firing patterns: clustering, propagating waves, and repetitive spiking that may show little regularity or correlation. Each activity pattern can occur continuously or in discrete episodes. We characterize the mechanisms underlying these rhythms, as well as the influence of parameters on details such
as spiking frequency and wave speed. These results suggest that the
subthalamopallidal circuit is capable both of correlated rhythmic
activity and of irregular autonomous patterns of activity that block
rhythmicity. Increased striatal input to, and weakened intrapallidal
inhibition within, the indirect pathway can switch the behavior of the
circuit from irregular to rhythmic. This may be sufficient to explain
the emergence of correlated oscillatory activity in the
subthalamopallidal circuit after destruction of dopaminergic neurons in
Parkinson's disease and in animal models of parkinsonism.
Key words:
basal ganglia; subthalamic nucleus; globus pallidus; computational models; oscillations; synchrony; Parkinson's disease
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INTRODUCTION |
Most current models of the basal
ganglia are static models, in that they represent the inputs and
outputs of the component nuclei as firing rates. For example, the
Albin et al. (1989) model, commonly used to explain the
symptoms of Parkinsonism, views the interactions of the direct and
indirect pathway as constant in time and explains the symptoms of
Parkinson's disease in terms of changes in mean rate of the basal
ganglia output (Wichmann and DeLong, 1996). In contrast,
recent experimental studies have not strongly confirmed the predicted
changes in mean rate in these structures under dopamine depletion, but
have instead revealed prominent low-frequency periodicity (4-30 Hz) of
firing and dramatically increased correlations among neurons in the
external segment of the globus pallidus (GPe) and the subthalamic
nucleus (STN) (Bergman et al., 1994; Nini et al.,
1995; Magnin et al., 2000; Raz et al., 2000; Brown et al., 2001). It is remarkable that
the changes in firing pattern seen in those structures do not appear to
be attributable to comparable changes in the firing patterns of
striatal output cells, although cholinergic striatal interneurons show
changes comparable with those seen in the globus pallidus (Raz
et al., 1996). The authors of those studies have proposed that
a rate model of the basal ganglia is inadequate to capture the dynamic interaction of the STN and GPe that may generate these pathological changes.
In particular, such dynamic interactions may lead to oscillatory
activity patterns. Indeed, the connections between GPe and STN neurons
show a number of features common to central pattern generators, which
suggests that these nuclei might together be capable of self-sustained
oscillatory activity. The GPe cells project to the subthalamus and to
other basal ganglia structures and inhibit the cells in those regions
via GABAA-receptor mediated inhibition. In addition, the
GPe cells inhibit each other via recurrent axon collaterals
(Stanford and Cooper, 1999; Ogura and Kita,
2000). Subthalamic cells excite neurons in several basal ganglia structures, including the GPe (Kitai and Kita,
1987). Furthermore, these cells show powerful and long-lasting
rebound excitation after episodes of hyperpolarization generated by
synaptic inhibition from the GPe (Bevan et al., 2000).
These properties suggest the possibility for oscillatory rhythms in
which GPe neurons are excited by activity in the STN neurons, in turn
inhibit the STN cells, and are again excited by the resulting rebound
excitation of STN cells that follows the inhibition. A mechanism of
this kind has been proposed by Plenz and Kitai (1999) on
the basis of their studies of basal ganglia organotypic tissue cultures.
To explore such dynamic interactions, we have performed computer
simulations of conductance-based models of the subthalamopallidal circuit, within the limits of our knowledge of the topography of the
synaptic connections and the cellular properties involved. These
simulations reveal a wide variety of oscillatory patterns, depending on
the arrangements and strengths of synaptic connections within and
between the cellular populations. We classify these patterns as
clustering, propagating waves, and repetitive spiking (which may show
little regularity or correlation); each activity pattern can occur
continuously or in discrete episodes. The mechanisms underlying these
patterns can be understood in terms of the intrinsic properties and
synaptic interactions of the cells in our simulations. The network can
be switched from irregular uncorrelated spiking to correlated rhythmic
patterns through an increase of incoming striatal input together with a
weakening of intrapallidal inhibition, suggesting that these changes
may underlie the emergence of correlated rhythmic activity in the
subthalamopallidal circuit in pathological states.
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MATERIALS AND METHODS |
Single-compartment conductance-based biophysical models of cells
from the STN and the GPe were developed, based on recent voltage-clamp
and current-clamp data (Figs. 1, 2). Simulations of these models and of
model networks composed of synaptically coupled STN and GPe cells were
performed using XPPAUT, developed by G. B. Ermentrout and available at
ftp://ftp.math.pitt.edu/pub/bardware. A copy of the XPPAUT file
containing the models constructed is available from D.T. on request.
STN cells. The STN model includes a set of currents and
corresponding kinetics that are based on recent experimental data (Bevan and Wilson, 1999; Bevan et al.,
2000). In particular, the model features spike-producing
currents (IK and INa), a
low-threshold T-type Ca2+ current
(IT) and a high-threshold
Ca2+ current (ICa)
(Song et al., 2000), a
Ca2+-activated, voltage-independent
"afterhyperpolarization" K+ current
(IAHP), and a leak current
(IL). The membrane potential of each STN neuron
obeys the current balance equation:
The leak current is given by IL = gL(v  vL), and the
other voltage-dependent currents are described by the Hodgkin-Huxley formalism as follows: IK = gKn4(v vK), INa = gNam (v)h(v vNa), IT = gTa(v)b (r)(v vCa), and ICa = gCas(v)(v vCa). The slowly operating gating variables n,
h, and r are treated as functions of both time and
voltage and have first-order kinetics governed by differential
equations of the form dX/dt =  X
[(X (v) X)/ X(v)] (where X can be
n, h, or r), with X(v) =  +  /[1 + exp[(v   )/ ]]. Using this
formulation, activation (and inactivation) time constants have a
sigmoidal dependence on voltage, 0/ is the minimum,
and (0 + 1)/ is the maximum time
constant. The voltage at which the time constant is midway between its
maximum and minimum values is  , and
is the slope factor for the voltage dependence of
the time constant. Activation gating for the rapidly activating
channels (m, a, and s) was treated as
instantaneous. For all gating variables X = n, m, h, a,
r, or s, the steady-state voltage dependence was
determined using X(v) = 1/[1 + exp[(v X)/X]], where X is
the half activation (or inactivation) voltage for gating variable X, and X is the slope factor for that
variable. For the T current inactivation variable
b, we used b(r) = 1/[1 + exp[(r b)/b]] 1/[1 + exp[b/b]].
The unusual way of modeling T current inactivation here
[with b(r) instead of simply r]
was designed to combine the effects of a hyperpolarization-activated inward (sag, or H) current with those of a T current, making
the rebound bursts of an STN cell more prominent. These rebound bursts (see Fig. 1e,f) agree with experimental data (Bevan
and Wilson, 1999; Bevan et al., 2001).
As the final intrinsic current, we take
IAHP = gAHP(v vK)([Ca]/ ([Ca] + k1)) where [Ca], the intracellular
concentration of Ca2+ ions, is governed by [Ca]' =  (ICa IT kCa[Ca]). The constant combines the
effects of buffers, cell volume, the molar charge of calcium, and is in
units of (mole-sec)/(coulombs-liter). The constant
k1 is the dissociation constant of the
calcium-dependent AHP current. The constant kCa
is the calcium pump rate constant and is in units of
(coulombs-liter)/(moles-sec). The current IG S that represents synaptic input from the GPe to STN is modeled as
IGS = gGS(v vGS) sj. The summation is
taken over the presynaptic GPe neurons, and each synaptic variable
sj solves a first-order differential equation
s'j =  H(vgj g)(1 sj)  sj. Here vgj is the
membrane potential of the GPe neuron j, and
H(v) = 1/(1 + exp[(v  )/]).
Parameter values for STN cells used in the simulations are given in
Table 1. The capacitance
Cm was normalized to 1 pF/µm2. Also note that we scaled our model such
that the currents have units of picoamperes per square
micrometer, to be consistent with the data of Bevan and
Wilson (1999) and Bevan et al. (2000).
GPe cells. Our model for single GPe cells is similar in form
to that of the STN neurons. The equation for the GPe neurons is:
where Iapp represents a constant external
applied current. IL, IK
(Baranauskas et al., 1999; Hernandez-Pineda et
al., 1999), INa,
ICa (Surmeier et al., 1994;
Stefani et al., 1998), and IAHP are modeled with the same formulas and equations given above for the
STN cells, whereas the low-threshold calcium current takes a simpler
form: IT = gTa(v)r(v vCa), where r satisfies a first-order
differential equation. Here we assume r(v)
 r, a constant. This has the effect of reducing the size of the posthyperpolarization rebound in GPe cells
compared with STN cells.
The experimental literature suggests that the GPe neurons have similar
ionic currents to STN cells, but in different proportions. Parameter
values for GPe cells used in the simulations are given in Table
2. In particular, parameters associated
with the GPe potassium channels allow for fast spiking of GPe cells
(Hernandez-Pineda et al., 1999); we have not
included a second slowly deactivating component of the potassium
current (Baranauskas et al., 1999). These parameters
were selected to match the data for spontaneous firing patterns,
rebound response to offset of hyperpolarization, spike frequency
adaptation, and frequency-intensity curves obtained from the study of
GPe cells in slices (Kita and Kitai, 1991; Nambu and Llinãs, 1994; Cooper and Stanford,
2000).
Two different synaptic currents are included in the GPe model.
ISG represents excitatory input from the STN,
and IGG represents the inhibitory influence
coming from other GPe cells. These are modeled by the same type of
expression as used for IGS, with
appropriately renamed synaptic parameters.
The input to the GPe cells from the striatum is represented by a
constant hyperpolarizing current Iapp common to
all the GPe cells. This current is not specified in Table 2 because in
our network simulations it will be one of the main parameters that we
vary. The use of this unpatterned inhibitory influence as a parameter
was not intended as a realistic representation of the pattern of
activity in the striatopallidal pathway, but rather as an approximation
to the overall effect of greater or lesser striatal inhibition to the GPe.
Synaptic connectivity. Currently, the details of connections
between STN and GPe cells are poorly understood. It is known that STN
neurons provide one of the largest sources of excitatory input to the
globus pallidus and that the GPe is a major source of inhibitory
afferents to the STN (Kitai and Kita, 1987). However, the spatial distribution of axons in each pathway, as well as the
number of cells innervated by single neurons in each direction, are not
known to the precision required for a computer model. Some early
reports suggested that the STN cells project to the GPe in a diffuse
manner (Hazrati and Parent, 1992), whereas more recent
studies indicate that the two are more tightly interconnected, with a
connectivity that may be precisely topographical (Shink et al.,
1996) or more heterogeneous (Sato et al., 2000).
It is certain from the existing studies that the connection is sparse; that is, each GPe neuron makes synaptic contacts with few STN neurons
and vice versa. But the number of synapses made by single neurons or
the proportion of neurons contacted by an axon within the arborization
cannot currently be estimated. Therefore, we consider multiple
architectures based on low numbers of contacts from STN to GPe cells,
from GPe to STN cells, and within the GPe. In the model networks, each
GPe neuron sends inhibition to the entire GPe population or some part
of it, as well as to one or more STN neurons. Each STN neuron sends
excitation to one or more GPe neurons. The intrinsic parameters
associated with the STN and GPe cells are the same for each simulation;
these are the parameter values of the individual model STN and GPe
neurons given in Tables 1 and 2. We used networks of between 8 and 20 cells of each type for the simulations described in this paper.
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RESULTS |
Firing properties of model STN and GPe neurons
The model STN neurons were adjusted to exhibit properties that are
characteristic of the firing of STN neurons in experimental studies on
slices (Bevan and Wilson, 1999; Bevan et al.,
2000; Beurrier et al., 2000). The model neurons
are spontaneously active with a firing rate of ~3 Hz. When current is
injected, the cells achieve firing rates on the order of 200 Hz with
little change in the voltage range of the membrane potential traversed
by the action potentials. Figure
1a presents currents generated
by voltage-clamp simulations. Figure 1b displays the
membrane potential of a model STN cell as a function of constant
current injection, after setting gNa = 0. The frequency-current (f-I) relation for a model STN cell
is shown in Figure 1c. Figure 1d shows the
duration of the afterhyperpolarization after a 500 msec period of
high-frequency spikes evoked by different levels of current pulses.
Note that the afterhyperpolarization increases smoothly over a wide
range of firing rates during the pulses. Figure 1, e and
f, demonstrates that STN neurons generate rebound bursts
after the removal of negative current. The duration of the rebound
depends on the degree of hyperpolarization. The rebound bursts
typically last up to 200 msec and are then followed by spontaneous
firing.
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Figure 1.
Properties of STN model neuron. a,
Current as a function of voltage. For fixed voltages, steady-state
currents were computed with slow gating variables set to their limiting
values [X  X(v); see Materials and
Methods]. In this and all subsequent figures, omitted units are as in
Tables 1 and 2. b, Membrane potential of a model STN cell
under various current injections. The parameter
gNa has been set to 0 to mimic the behavior of
an STN cell in the presence of sufficient concentration of TTX to block
spiking. c, Spike frequency as a function of injected
current (solid line, full model; dotted line,
gAHP = 0; dashed line,
gCa = gT = 0).
d, Duration of afterhyperpolarization after high-frequency
spiking. A constant current pulse was applied to a model STN cell for
500 msec. After this, a prolonged afterhyperpolarization occurred
before the cell returned to regular spiking. Its duration is plotted
against the strength of applied current. e, f, STN rebound
bursts after hyperpolarizing injections. e, Model responses
of STN cell to currents of varying duration: 25 pA/µm2 of current applied for 300 (top), 450 (middle), and 600 (bottom)
msec. Longer current application augments deinactivation of
IT, enhancing rebound. f, Responses
to currents of varying magnitude: 20 (top), 30 (middle), and 40 (bottom)
pA/µm2 of current applied for 300 msec. Stronger
current application augments deinactivation of
IT, enhancing rebound.
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Sample responses of model GPe neurons to current injections are shown
in Figure 2. High-frequency repetitive
firing does not show strong accommodation and is followed by a
pronounced hyperpolarization (Fig. 2). After a hyperpolarizing current
pulse, the neuron can exhibit rebound firing. We note, however, that
the ability of GPe neurons to rebound is not essential for the
generation of rhythmic population activity. For weak hyperpolarizing
applied currents, the neuron exhibits oscillations in which an active spiking phase alternates with a silent phase of near resting behavior. This activity pattern, which is common to fast-spiking neurons in the
GPe (Cooper and Stanford, 2000) and elsewhere, has
important implications for the model. In our implementation, it arises
because of the slow time course of calcium buildup and decay in the
neurons. At higher levels of current, the model cells show continuous
firing in response to current pulses. Similar dynamics can arise from other mechanisms (Rush and Rinzel, 1995), and the
mechanism of this kind of firing in GPe neurons is not known with
certainty. The network dynamics that arise from this fire-and-pause
mode of activity in our model do not depend critically on the cellular mechanism. We also note that several papers have reported various types
of neurons within the GPe. Our model GPe neurons, as shown in Figure 2,
were adjusted to display properties similar to the type 2 neurons
described by Nambu and Llinãs (1994), the
type A neuron described by Cooper and Stanford (2000),
and the repetitive firing neurons described by Kita and Kitai
(1991). Recent experiments have suggested that this is the most
dominant type of neuron within GPe. The episodic firing behavior seen
at weakly negative applied currents resembles the high-frequency
discharge with pause described by DeLong (1971). In our
simulations of the STN-GPe network, Iapp is
taken to be a small hyperpolarizing (negative) current and, as
described in Materials and Methods, will approximate input from the
striatum.
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Figure 2.
Properties of GPe model neuron. a, Top
three time traces show firing profiles of a model GPe cell under
depolarizing, zero, and small hyperpolarizing input currents
(Iapp in pA/µm2),
respectively. Bottom trace shows afterhyperpolarization of a
model GPe cell after injection of a depolarizing current pulse.
b, Membrane potential of a model GPe cell as a function of
injected current, with gNa = 0. c, Frequency of GPe spiking as a function of injected
current.
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The prototype networks
In the following sections we shall describe the network activity
generated by three prototype network architectures, as well as how this
activity depends on the synaptic conductances
gSG, gGG, and the
applied current Iapp to GPe. Each of the three prototype architectures includes a distinct type of connectivity between STN and GPe cells. In all of these, each cell only contacts a
small number of other cells. However, within the simulated networks, the density of connections relative to the size of the network varies
across architectures, and they are named accordingly: in sparsely
connected architectures, each cell sends out connections to a small
subset of the STN and GPe populations; in tightly connected architectures, the connectivity patterns yield localized circuits of
reciprocally connected GPe and STN cells, in which each cell sends
inputs to, and receives inputs from, a relatively large subset of the
other cells in the circuit.
Random, sparsely connected architecture
The simplest class of networks consistent with low connectivity is
one in which each GPe cell sends inhibitory input to a small proportion
of the STN neurons selected randomly, and the STN cells also make
sparse and random connections. The chances of reciprocal connections
between any STN-GPe cell pair is low. This class of networks was
represented using the network shown in Figure
3a. The subthalamic neurons
could fire spontaneously by their usual pacemaker mechanism
(Bevan and Wilson, 1999; Beurrier et al.,
2000) but the GPe neurons were inhibited by the application of
Iapp (representing striatal inhibition) to a
level just adequate to prevent their spontaneous firing.
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Figure 3.
Activity patterns in a random, sparsely connected
architecture. a, Arrangement of the model network. Each STN
neuron excites a single GPe neuron selected at random, and each GPe
neuron inhibits three randomly chosen STN cells. GPe cells also inhibit
each other through all-to-all connections. b, Dependence of
activity patterns on coupling strengths gGG
and gSG. Weak STNGPe excitation or strong
GPeGPe inhibition leads to sparse irregular firing patterns.
Intermediate values yield episodic patterns, whereas high levels of
excitation and low levels of GPe mutual inhibition give rise to
continuous uncorrelated activity. c, Membrane potential (in
millivolts) as a function of time (milliseconds) for individual cells
in each of the activity patterns: sparse activity
(gGG = 0.06 nS/µm2;
gSG = 0.03 nS/µm2;
gGS = 2.5 nS/µm2;
Iapp = 1.2 pA/µm2),
episodic, almost-synchronized spiking
(gGG = 0 nS/µm2;
gSG = 0.016 nS/µm2;
gGS = 2.5 nS/µm2;
Iapp = 1.2 pA/µm2),
and continuous, irregular spiking (gGG = 0.02 nS/µm2; gSG = 0.1 nS/µm2; gGS = 2.5 nS/µm2; Iapp = 1.2 pA/µm2). d, Network activity in
various patterns. In each plot, 10 rows show the voltage traces of 10 cells, with time evolving over 2000 msec to the right along
each row. Voltage is coded in grayscale as shown. Because
they are so brief, individual action potentials (dark gray line
segments) are not prominent, but are more clearly indicated by
their afterhyperpolarization (white bars).
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The activity patterns that this network displays are summarized in
Figure 3b-d. In these patterns, the firing of each cell is
typically irregular and weakly correlated with the activity of other
cells. Dependence of activity on gSG and
gGG is illustrated in Figure 3b.
As with all the architectures, if gSG is too
small, then the GPe cells cannot respond to STN firing, and there is no
network activity, only the pacemaking of the STN cells. For larger
values of gSG, the occasional firing of GPe
cells produces an irregular background of inhibition that deregularizes
the STN neurons, resulting in sparse irregular firing in both
structures (Fig. 3c, first panel). GPe neurons fire in a correlated (but not perfectly one-to-one) way with the single
STN neuron to which each is connected, but correlations are weak among
randomly selected cell pairs. The correlations that do occur have a
very narrow time scale, with connected GPe and STN cells firing almost
simultaneously as the GPe neuron fires on the EPSP generated by the STN
neuron or not at all. With still larger values of
gSG, the network displays episodic activity, as shown in Figure 3, c and d. During each
episode, the cells fire irregularly, and there is an increased
synchronization both within and among nuclei. The episodes of firing
involve all neurons in the network. The episodes last for ~300 msec,
and their durations increase with increasing
gSG; the silent periods between the episodes
remain approximately constant at 500 msec. When
gSG becomes sufficiently large, the network
activity switches to continuous, irregular spiking as depicted in
Figure 3, c and d. There are weak or no
within-nuclei correlations of firing in this continuous regime, and
across-nuclei correlations are rare because they are restricted to
cells receiving direct interconnections (which are sparse). This is
shown in Figure 3d, but was also tested by calculating the
cross-correlations among and between the two groups of cells.
Episodic rhythms
To understand the neuronal mechanism underlying episodic rhythms,
suppose that one or more STN cells fire action potentials. The
resulting excitation may induce the corresponding GPe cells to respond
with their own spikes. This leads to hyperpolarization of STN cells,
which resets the pacemakers of these. This is the same mechanism
illustrated experimentally by Bevan et al. (2001) after
stimulation of inhibitory inputs to the STN. The postinhibitory firing
of the STN neurons induces another round of firing in the GPe neurons.
So long as there is some divergence in one or both directions (as in
the GPeSTN connection of Fig. 3a), activity in some
neurons will tend to recruit others, and the entire network will
ultimately be recruited. This causes T currents within the STN cells to deinactivate so that activity during the episode can
be sustained through postinhibitory rebound. The termination of an
episode of activity is caused by activation of the outward K+ current, IAHP, in the GPe
cells as calcium builds up with each additional spike. This is
illustrated in Figure 4. Once
IAHP is sufficiently activated, a GPe neuron is
no longer able to respond to excitation from STN. When GPe activity
terminates, STN cells may fire one last burst of spikes caused by
postinhibitory rebound. Activity then recurs when STN cells recover
enough to fire again, by which time the GPe cells can again respond to
the STN.
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Figure 4.
Mechanisms underlying episodic activity patterns.
The gray trace in the top box shows the evolution
of voltage over time for a single GPe cell in an episodic pattern,
whereas the black trace shows the voltage for a single STN
cell. The boxes below show the intracellular calcium
concentration of each cell as a function of time. Initially, GPe spikes
closely follow STN spikes. Here Iapp is
sufficiently strong such that the build-up of calcium terminates the
GPe activity of the cell eventually, after which the STN cell fires one
last volley of rebound spikes until about 2600 msec. Subsequent decay
of calcium allows STN activity to resume after time 3200 msec; this
recruits the GPe cell again.
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As described earlier, if the overall current input to a GPe cell is
weakly inhibitory, then the calcium of the cell can reach a level that
inhibits firing (Fig. 2a); thus, the neuron fires in the
episodic mode. In our simulations, we assume that the hyperpolarizing applied current from striatum to GPe (Iapp) is
strong enough such that GPe cells are not spontaneously active. When
the excitatory input from STN to GPe is relatively weak, the total
input to GPe is appropriate to trigger calcium-regulated episodic
firing. On the other hand, if the excitation from STN to GPe is
sufficiently strong, the calcium-dependent AHP current in GPe cells
will achieve an equilibrium level that will slow but no longer
terminate GPe firing. GPe cells will always be able to respond to the
firing of STN cells in this case, and a continuous rhythm results.
Increasing gSG transforms an episodic rhythm
into a continuous one; such a transition may also be achieved by
decreasing the magnitude of Iapp from a large
hyperpolarizing level.
The duration between active episodes is determined primarily by the
IAHP current in STN neurons. During an episode,
STN neurons fire rapid action potentials resulting in an increase of
their intracellular calcium. Once the event terminates, there is a
prolonged afterhyperpolarization before the cells can resume their
pacemaking spiking activity (Bevan and Wilson, 1999)
(Fig. 1d) and initiate a new episode. The time between
episodes does not depend significantly on Iapp
or the synaptic parameters gSG,
gGG.
The duration of each episode is determined by two factors. The first is
how long it takes for calcium to build up and hyperpolarize GPe neurons
so that they no longer respond to excitation from STN. This depends on
the strength of STNGPe synaptic excitation, as well as on
Iapp. The second factor is the duration of any
rebound bursting generated by STN neurons at the end of the episode.
Both of these depend on the rate of firing within the episode.
Firing rate within the episode is determined by the strength of
synaptic connections and the time between onset of GPe inhibition in an STN neuron and the postinhibitory response.
All of the above description holds regardless of the action of the
intra-GPe inhibition. Because it is all-to all, the intra-GPe inhibition in this network acts in proportion to the total activity in
the GPe, and so opposes the STNGPe excitation. For larger values of
gGG, the network requires increased STN
activity to support organized firing. The value of
gSG at which the activity switches from
episodic to continuous firing increases, in an approximately linear
manner with gGG. When
gGG becomes too large, the network exhibits
asynchronous, irregular activity, as occurs when
gSG is small.
Structured, sparsely connected architecture
We next consider the off-center architecture depicted in Figure
5a. Although the GPeSTN
connection is more structured in this model network, it is so in a way
that avoids direct reciprocal connections between GPe and STN. The
nearest-neighbor intra-GPe inhibition is structured to create a lateral
inhibition among GPe neurons that have overlapping projections in the
STN.
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Figure 5.
Activity patterns in a structured, sparsely
connected architecture. For all simulations in this Figure,
vGG = 85 mV and = 0.04 msec 1 for the GPe cells. a, Arrangement
of the model network. Each GPe neuron inhibits its two immediate GPe
neighbors; it also inhibits two STN neurons, skipping the three located
nearest to it. Each STN cell sends excitation only to the nearest, in
register GPe cell. Spatially periodic boundary conditions were imposed.
b, Dependence of activity patterns on coupling strengths
gGG and gSG when
gGS = 4.5 nS/µm2
and Iapp = 1.0
pA/µm2. The parameter regime and initial
conditions used favor formation of clusters rather than waves.
Increases in gSG lead to continuous activity;
increases in gGG weaken activity.
c, Voltage (in millivolts) as a function of time (in
milliseconds) for individual cells in the three clustered activity
patterns: weak and irregular clustered activity
(gGG = 0.06 nS/µm2;
gSG = 0.2 nS/µm2;
gGS = 4.5 nS/µm2;
Iapp = 1.0 pA/µm2),
episodic clustered oscillations (gGG = 0.06 nS/µm2; gSG = 0.56 nS/µm2; gGS = 4.5 nS/µm2; Iapp = 1.0 pA/µm2), and continuous clustering
(gGG = 0.06 nS/µm2;
gSG = 0.72 nS/µm2;
gGS = 4.5 nS/µm2;
Iapp = 1.0 pA/µm2).
d, Network activity in various patterns, as in Figure
3d but with eight rows shown.
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This network can give rise to more varied network dynamics than the
unstructured network described above. Most patterns feature clustering,
in which each structure is divided into subsets of neurons that become
highly correlated with each other (Fig. 5b-d). The most
commonly observed clustered pattern consists of two clusters, with
alternating pairs of cells belonging to opposite clusters (Fig.
5c,d). Different clusters alternate firing, and in this pattern, cluster membership is persistent over time. The switch of
activity from one cluster to another can involve either an overlap of
firing or a brief interval of network quiescence, depending on
parameters. There is a tight synchrony of firing among cells of the
same cluster.
Clustered rhythms
To understand the neuronal mechanism underlying clustered rhythms,
consider an example of an activity pattern consisting of two clusters.
Suppose that one subpopulation of STN neurons, which we denote as
S1, excites its corresponding
subpopulation of GPe neurons, call them
G1, to initiate an episode of firing. If
the inhibition from G1 is powerful enough, then
it prevents the remaining STN neurons, in a different subpopulation
S2, from firing. Eventually, cells in
S2 escape from their suppressed state and fire.
This induces the remaining GPe cells, group
G2, to fire, and the resulting inhibition
terminates the activity of S1. The roles of the
S1/G1 and
S2/G2 clusters are now reversed.
The primary reason why cells in S2 are able to
escape is that although S2 cells are
hyperpolarized, their inward IT currents deinactivate. A second reason is that there is some slight adaptation of firing rates of cells in S1 as their
IT currents inactivate and their AHP currents
accumulate. Unlike the episodic firing described earlier, cluster
alternation is not primarily driven by accumulation of AHP currents,
but by persistent inhibition and the resulting removal of inactivation
of rebound currents. These factors appear in Figure
6. As long as S1
cells maintain a high firing frequency, then so will cells in
G1. This results in a tonic level of inhibition
to cells in S2, preventing them from
firing. A decrease in the firing rate of S1
caused by adaptation helps allow S2 to become
active, and its excitability is enhanced by rebound currents
accumulated during the persistent inhibition. The length of time for
which one cluster can fire before another one takes over, and hence the
population bursting frequency, is mostly set by the rate of
deinactivation of IT in inactive STN cells
relative to the level of inhibition they receive. For the parameters
used here (based on studies from brain slices) the cluster alternation
rate ranges from 4 to 6 Hz.
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Figure 6.
Mechanisms underlying clustered activity patterns.
The top box shows the superimposed voltage time courses for
an STN cell (dotted trace) and a GPe cell (solid
trace) from a single, tightly synchronized pair belonging to the
same cluster in a clustered rhythm; the middle box shows the
same for a single pair from a different cluster. The bottom
box shows the availability level of the IT
currents for the STN cells in the two different pairs (solid
curve corresponds to middle box; dashed
curve to top box). When availability of
IT becomes sufficiently large, the suppressed
cluster is able to escape and fire; this then suppresses the previously
active cluster.
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We note that other mechanisms besides deinactivation of the
IT current may also promote escape, thus
contributing to the generation of clustered rhythms. For instance, it
is possible that short-term synaptic plasticity plays a role in escape
mechanisms; in fact, Hanson and Jaeger (2002) have
recently demonstrated that the STN to GPe pathway exhibits short-term depression.
A key point is that network activity segregates cells in such a way
that at any given time, neurons within silent STN clusters are
receiving more inhibition from active GPe cells than are the active STN
neurons. Hence, inhibition may play the dual role of maintaining the
active STN bursts (by deinactivating IT), while suppressing the silent STN neurons.
Additional clustered patterns also arise from this architecture, as do
propagating waves. Many of these patterns are dynamic, with cluster
membership changing over time. Examples of alternative cluster patterns
include a two-cluster pattern in which every cell is in a different
cluster from its two immediate neighbors and a four-cluster pattern in
which cells in the same cluster are separated by three other cells.
Clustered activity leads to discrete phase differences among neurons.
The range and distribution of those differences is determined by the
number of clusters, with the simplest outcome, consisting of two
phases, described here. In a larger network of this kind, it is
feasible that a large number of clusters could arise, giving a wide
range of phase relationships. A continuous distribution of phases among
STN and GPe neurons, however, would require traveling waves of activity
within the STN and GPe.
As in the less structured architectures, intermediate values of
gSG give rise to episodic activity in the
network as a whole (Fig. 5c,d). This occurs for the same
reason (accumulation of Ca-dependent K current), and because this is
slower than the process responsible for cluster alternation, it affects
all clusters equally. At higher values of excitation and/or lower
levels of intra-GPe inhibition, clustered activity becomes continuous,
because the effect of the AHP current is overcome (Fig.
5b-d). Episodic clustering is less robust than continuous
clustering, because in a sparsely connected network, it is difficult
for calcium to build up sufficiently throughout the GPe population to
turn off activity across a large segment of the network. In fact, for
sparsely connected networks in general, episodic rhythms are more
robust when the architecture is random as opposed to structured (random connections allow activity to spread across the network more quickly).
Structured, tightly connected architecture
The third network to be considered is the structured, tightly
connected architecture represented by the model system in Figure 7a. Figure 7b
illustrates regions in the (gSG,
gGG) parameter plane that support each
activity pattern, whereas Figure 7, c and d,
shows how cellular activity varies over time in a variety of these
patterns. Again, Iapp is set at a level so that isolated GPe neurons are silent.
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Figure 7.
Activity patterns in the structured, tightly
connected architecture. a, Network used for these
simulations. Each GPe neuron contacts the five closest STN neurons, as
well as all of the GPe cells. Each STN cell sends excitation to the
three closest GPe cells. Spatially periodic boundary conditions were
imposed. b, Dependence of activity patterns on coupling
strengths gGG and
gSG when gGS = 1.0 nS/µm2 and Iapp = 1.2 pA/µm2. Increases in
gSG lead to continuous activity; increases in
gGG yield a transition to waves followed by
sparse, irregular firing. The value of gGG at
which each of these transitions occurs rises with
gSG. c, Voltage (in millivolts) as
a function of time (in milliseconds) for individual cells in various
activity patterns: episodic, almost-synchronized spiking
(gGG = 0.0 nS/µm2;
gSG = 0.013 nS/µm2;
gGS = 1.0 nS/µm2;
Iapp = 1.2 pA/µm2
for GPe), episodic wave (gGG = 0.02 nS/µm2; gSG = 0.013 nS/µm2; gGS = 1.0 nS/µm2; Iapp = 1.2
pA/µm2), and continuous wave
(gGG = 0.1 nS/µm2;
gSG = 0.03 nS/µm2;
gGS = 1.0 nS/µm2;
Iapp = 1.2 pA/µm2).
d, Network activity in various patterns, featuring STN cells
during episodic spiking, episodic wave, continuous wave, and sparse
irregular spiking (gGG = 0.23 nS/µm2; gSG = 0.03 nS/µm2; gGS = 1.0 nS/µm2; Iapp = 1.2
pA/µm2). The GPe cells exhibit voltage patterns
very similar to the STN cells.
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Let us consider the network behavior at various different levels of
intra-GPe inhibition. First suppose that there is virtually no
intra-GPe inhibition (gGG  0). In
this case, if gSG is very small, the STN
neurons spike slowly via their pacemaker mechanism, the GPe neurons are
mostly silent, and interaction between the GPe and STN populations is
too weak to generate any discernible network rhythm. Increasing
gSG gives rise to an episodic pattern, with
every cell participating in events of repetitive spiking activity,
separated by periods of quiescence, that repeat periodically at 1-2
Hz. When gSG gets above a certain level, the
population activity switches to a continuous mode; all the STN and GPe
cells exhibit tonic spiking, with the spiking frequency growing to
~25 Hz as gSG is increased. The
transformation from episodic to continuous activity achieved by
increasing gSG can also be observed after
decreasing the magnitude of the hyperpolarizing
Iapp. Recall that weakening Iapp (making it less negative) transforms a GPe
neuron from firing in the episodic mode to spiking continuously. The
mechanism underlying the transformation from episodic to continuous
activity in the network is similar to that in an isolated GPe neuron,
and in both of the architectures described earlier.
Higher levels of the intra-GPe inhibition have the effect of producing
traveling waves, within both the episodic and continuous firing modes
(Fig. 7c,d). These waves correspond to solitary pulses with
no activity in the wake of the wave (note that the appearance of
multiple "bands" of activity in Figure 7d is caused by
the periodic boundary conditions imposed; at any one time, only one localized group of cells is active). The propagation structure of a
wave was clearly discernible in cross-correllograms calculated from the
simulations (data not shown).
If gGG is increased to even higher values,
the waves lose their shape and firing activity eventually becomes
sparse, irregular, and uncorrelated. For higher levels of excitatory
drive from STN to GPe, the intra-GPe synaptic conductance
gGG needs to be stronger before waves are
able to form, but these waves exist for a larger interval of
gGG values (Fig. 7b).
This network does not exhibit clustered patterns. Moreover, the
existence of waves here depends on a rather large GPeGPe synaptic
footprint. If each GPe neuron sends inhibition only to its immediate
neighbors, then activity becomes distributed throughout the populations
rather than localized in the form of waves.
Propagating waves
To understand the neuronal mechanism underlying propagating wave
activity, we first consider a simple network with each neuronal population represented by a one-dimensional array, indexed from left to
right by i = ... , 2, 1, 0, 1, 2, 3, ... Each GPe cell Gi sends inhibition to one STN
cell Si, as well as every GPe cell, and each STN
cell Si sends off-centered excitation to the two GPe cells Gi1 and Gi+1.
We assume that a wave has been generated and is propagating to the
right, and we suppose that at some time, say t = 0, the
GPe cell G0 starts to burst. We shall describe
how the activity continues to propagate to the right in a lurching manner.
As G0 fires, it inhibits the STN cell
S0 and deinactivates the T current in
S0. When S0 escapes or
becomes released from inhibition, it fires a volley of spikes, in turn
exciting the GPe cells G1 and
G1. At this point G1 is
more excitable than G1, because G1 had recently fired and is still in a
partially refractory state. Hence, G1 will
tend to fire in response to S0 before
G1 does. Once G1 fires,
it sends inhibition to all the other GPe cells. In particular, this
inhibition will prevent G1 from firing.
Furthermore, when G1 fires, it inhibits
the STN cell S1; thus the entire process
repeats, except it is now shifted over by one STN and one GPe cell.
We have explored numerous related architectures and have found that the
existence of propagating waves is a very robust activity pattern. Waves
can exist if the STNGPe connectivity footprint is broader than the
GPeSTN footprint, and also in the reverse situation; in addition,
both on-centered and off-centered footprints can support propagating
waves. The mechanism underlying wave activity in various different
network architectures is essentially the same as described above.
Typically, a group of STN cells fires synchronously because of
postinhibitory rebound, and this causes excitation to be sent to a
corresponding group of GPe cells. The GPe cells just ahead of the
leading edge of the wave will be the first cells to fire in response,
because other nearby GPe cells have just fired and hence are in a
refractory mode. This firing will inhibit the other GPe cells that had
received excitatory input, suppressing their activity; at the same
time, it will inhibit a group of STN cells. This group of STN neurons
will eventually fire when they are able to escape, or become released
from, the inhibitory influence from GPe. The process then continues as before.
We note that to generate a wave, it is important that the GPe cells at
the leading edge of the wave are able to inhibit those GPe cells behind
them. It is therefore necessary to have gGG sufficiently large. Decreasing gGG often
transforms wave activity to clustering or rapid spiking (either
episodic or continuous). For solitary waves to arise, the GPeGPe
footprint should also be larger than the STNGPe footprint;
otherwise, activity spreads quite rapidly throughout the network and
full-population spiking behavior or spatially periodic waves (data not
shown) typically result.
The speed with which the waves propagate is dependent on the size of
the STNGPe and GPeSTN connectivity footprints. Other synaptic
parameters, such as gSG, also play a role in
determining wave speed. Moreover, the average active (and silent) phase
duration of STN neurons is directly correlated with the length of time it takes the wave to travel from one end of the array to the other, and
hence inversely correlated with the speed. We observed that at fixed
levels of gSG, stronger intra-GPe inhibition
produces slower waves; whereas at fixed levels of
gGG, varying gSG has two competing influences on wave speed, caused by different mechanisms coming into play for the release or escape from inhibition of inactive STN cells.
Irregular and uncorrelated activity
These computational studies suggest that there are several sources
leading to irregular and uncorrelated activity patterns in the STN-GPe
network. An unstructured and sparsely connected topography is one
possible source; however, certain choices of the synaptic parameters
and applied currents also promote irregular behavior regardless of
network architecture. For example, we observed that if
gGG is sufficiently large and
Iapp is above (i.e., less negative than) some
fixed hyperpolarizing level, then the network activity is irregular.
The implications will be discussed in the next section. Here we
describe both the topographic and the input-related neuronal mechanisms
underlying irregular uncorrelated behavior.
We saw earlier (Figs. 5, 7) that a structured architecture is required
for the generation of both clustered and propagating wave solutions. In
a clustered solution, the network organizes itself so that each STN
cell in an inactive cluster receives approximately the same degree of
inhibitory input. For a wave to be propagated, the active STN and GPe
cells must have structured footprints to spread activity to cells ahead
of the leading edge. These patterns are not possible in a randomly
connected network. In such a topography, the firing of a small number
of cells tends to spread activity (which may be either episodic or
continuous) efficiently throughout the network. Once one STN cell
fires, it spreads inhibition to other STN cells via connections with
the GPe cells, thus delaying or possibly preventing activity of other
STN cells. Hence, one expects at most a loose synchronization between
the firing of STN cells.
We next discuss why weakening striatal inhibition of the GPe
(represented here as the magnitude of Iapp)
promotes irregular behavior, especially if
gGG is sufficiently large. Recall that making
Iapp less negative allows the GPe cells to fire
tonically (Fig. 2a). This leads to tonic inhibitory input to
the STN. If this input is sufficiently strong, then it will completely
suppress STN activity. For moderate levels of tonic inhibition, the STN cells may still be able to fire action potentials caused by
deinactivation of IT; however, in the tonic
firing regime, the GPe neurons are less sensitive to excitation from
STN. This means that, for both moderate and strong levels of striatal
inhibition, excitation from the STN is too weak to organize the GPe
population into distinct clusters or propagating waves. In this case,
strong intra-GPe inhibition may further desynchronize oscillations by
delaying, weakening, or suppressing the firing of some GPe cells
immediately after other GPe cells fire. This type of network behavior
is consistent with that reported in related excitatory-inhibitory
networks (Hansel and Mato, 2001).
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DISCUSSION |
The subthalamic nucleus and the globus pallidus are heavily
interconnected and should have a strong tendency to entrain each other.
Given that neurons in both areas have membrane properties that
predispose them to rhythmic firing, it is somewhat surprising that they
are not engaged in rhythmic firing all the time. All available
evidence, however, indicates that these structures do not generate
correlated spontaneous rhythmic activity (or even show internally
correlated firing) under physiological conditions encountered in
extracellular recording experiments (Nini et al., 1995;
Magill et al., 2000; Raz et al., 2000;
Urbain et al., 2000).
After experimental dopamine depletion, and in patients with
Parkinson's disease, strongly correlated rhythmic activity can be
observed in both the subthalamic nucleus and in the globus pallidus
external segment (Bergman et al., 1994; Nini et
al., 1995; Magnin et al., 2000; Raz et
al., 2000; Brown et al., 2001). The subthalamic
nucleus is required for the oscillations in the globus pallidus, as STN
lesions abolish pallidal burst firing (Ni et al., 2000);
similarly, compromise of GPe interferes with parkinsonian activity of
STN (Chesselet and Delfs, 1996; Hassani et al.,
1996). Plenz and Kitai (1999) have shown that in
organotypic cultures, correlated activity can arise in both structures
and is caused by the interaction between the STN and GPe rather than being driven by an external source. In that reduced preparation, powerful excitation from the rebound burst firing of STN neurons produced increased activity in the GPe population, which then inhibited
the subthalamic cells and set them up for another rebound.
We have shown in a biophysical, conductance-based model that the
cellular properties of STN and GPe cells can give rise to a variety of
rhythmic or irregular self-sustained firing patterns, depending on both
the arrangement of connections among and within the nuclei and the
effective strengths of the connections. The model was based on a
simplified representation of the properties of STN and GPe neurons in
slices and does not include all the influences that may act in
vivo. The dependence on network architecture points out the
importance of certain missing pieces of anatomical information. It is
critical to know the spatial extent of the recurrent collateral
connections among GPe neurons and whether they are spatially organized
or diffuse. Likewise, it is important to determine the precision of the
spatial organization of GPeSTN and STNGPe projections and whether
the two nuclei project on each other in a reciprocal or out of register
manner. In one anatomical study, dense, precisely reciprocal
connections between the subthalamic nucleus and the globus pallidus was
reported (Shink et al., 1996), but when individual axons
are stained, the connectivity pattern has appeared much more diffuse
(Sato et al., 2000). In the absence of definitive
anatomical information, our model can be of some benefit by comparing
the expected activity patterns emerging from various likely
connectivity architectures and by characterizing conditions under which
the network may exhibit either synchronous or asynchronous oscillations.
An emerging body of evidence suggests that even in the pathological
states caused by dopamine denervation, global synchronous oscillatory
activity is not common. A key finding of the studies of synchronous
oscillations in 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP)-treated monkeys was the wide range of phase relationships among neurons in the globus pallidus, indicating that not all the cells
oscillate synchronously (Raz et al., 2000). This is in
contrast with the synchronous episodes seen in our random, sparsely
connected network. We note that this architecture lacks the structure
required to support spatially organized activity, which could produce
stable phase shifts among neurons. The rest tremor in Parkinson's
disease also shows a variety of phases in different parts of the body,
suggesting that it is not driven by a single oscillator (Hurtado
et al., 2000; Ben-Paz et al., 2001). In the
reduced preparation described by Plenz and Kitai (1999),
some STN-GPe pairs of cells showed in-phase slow oscillations, whereas
others bursted out of phase. These results suggest that different parts
of the STN-GPe system may oscillate separately from others. In this
case, one possibility is that the rhythm-generating mechanisms that we
have elucidated in our model network could be played out among multiple
oscillatory subnetworks. In the model network, the structured
architectures could produce spatial clustering and waves, which are
consistent with the range of phase relations of cellular activity seen
in experimental dopamine depletion studies. The network also produced
the slow oscillations observed by Plenz and Kitai (1999)
in culture. Our results suggest that the absence of rhythmic activity
at tremor frequency in their preparation may occur because the STN-GPe
network in their experiment lacks the structured architecture (Fig.
5a) needed for the generation of a clustered rhythm. The
dynamic clustering rhythms seen in our network are also reminiscent of
behavior reported in several experimental studies. Hurtado et
al. (1999) recorded neuronal activity from awake Parkinson's
disease patients undergoing stereotaxic pallidotomy. They found that
some paired recording sites within GPi showed periods of transient
synchronization. Dynamic synchronization of pallidal activity in
MPTP-treated monkeys was also reported by Bergman et al.
(1998).
Roles of inhibition and the function of the indirect pathway
According to recent studies, correlated oscillatory activity in
the GPe and STN neurons is closely related to the generation of the
symptoms of Parkinsonism. The origin of the oscillatory activity or the
correlations among the neurons is not obvious, because the same neurons
do not show strong correlations in untreated animals (Raz et
al., 2000). The firing rate model holds that during Parkinsonian states, an increased level of inhibition from the striatum
to GPe causes a decrease in the activity of GPe. This in turn would
send less inhibition to STN, thus increasing STN activity and
ultimately leading to increased inhibitory output from the basal
ganglia to the thalamus (DeLong, 1990; Wichmann and DeLong, 1996; Obeso et al., 1997). In our
model network, a more complex picture emerges, in which the STN and GPe
are spontaneously oscillatory and synchronous, whereas intra-GPe
inhibition and an appropriate level of input from the striatum can act
to suppress rhythmic behavior. Note that inhibition plays multiple
roles in the generation of each of the activity patterns we observed.
In the clustered rhythm, for example, active STN neurons need moderate levels of feedback inhibition from the GPe to synchronize among themselves. Silent STN neurons, on the other hand, are prevented from
firing because they receive more powerful tonic inhibition. For the
generation of propagating waves, intra-GPe inhibition is needed to
prevent activity from persisting in the wake of the wave. Hence, this
inhibition helps to organize the network into a structured activity
pattern. If one increases the intra-GPe inhibition, this can
desynchronize the GPe oscillations, and irregular firing may result.
The role of striatal inhibition is the most pivotal in generating or
suppressing the organized oscillatory activity, because it
simultaneously controls the inhibitory feedback to the STN through the
GPe cells and influences the intra-GPe inhibition, which controls
spatial patterning of activity in the network.
Some papers have questioned the role of the so-called indirect pathway
(Parent and Hazrati, 1995; Levy et al.,
1997; Parent and Cicchetti, 1998). These
arguments point to experiments demonstrating that GPe activity does not
decrease substantially in a Parkinsonian state, as well as results
indicating that the overall level of GABAA received by GPe
cells (from both striatal and intrinsic sources) may hold steady or
even decrease. The analysis and simulations given in this paper suggest
that to account for the new experiments, one does not need to diminish
the role of the indirect pathway. Instead, our analysis
demonstrates that the diverse contributions of inhibition to indirect
pathway firing patterns can shift the network between rhythmic and
irregular modes of firing. In Figure 8,
we suggest an alternative interpretation of the roles of inhibition and
excitation in allowing the indirect pathway to generate tremor-like activity. In the normal state, there is strong GABAA
synaptic inhibition among GPe neurons, making their output to the STN
asynchronous, and effectively weakening the synaptic interactions
between GPe and STN. After dopaminergic denervation, an increased level
of inhibition from the striatum to GPe is combined with the release of enkephalin and dynorphin, which acts presynaptically to weaken the collateral connections among GPe cells (Stanford and Cooper, 1999; Ogura and Kita, 2000). From our results,
this could strengthen and synchronize the interactions between STN and
GPe and shift the network into an oscillatory mode.
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Figure 8.
Schematic diagram of the indirect pathway
connections in the basal ganglia in normal (left column) and
parkinsonian (right column) states. Minus symbols
denote inhibitory connections; plus symbols denote
excitatory ones. In the parkinsonian regime, the combination of
weakened intra-GPe connections and strengthened striatal input set the
stage for synchronous GPe-STN oscillations and correlated rhythmic STN
output.
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FOOTNOTES |
Received Aug. 21, 2001; revised Jan. 15, 2002; accepted Jan. 23, 2002.
This work was supported by the National Science Foundation Grants
DMS-0103822 (D.T., A.C.Y.) and DMS-9804447 (J.E.R.) and National
Institutes of Health, National Institute of Neurological Disorders and
Stroke Grant NS26473 (C.J.W.). We thank Mark Bevan for providing direct
access to his data on STN and GP cell activity.
Correspondence should be addressed to David Terman, Department of
Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus,
OH 43210. E-mail: terman{at}math.ohio-state.edu.
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ABSTRACT
INTRODUCTION
