Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, New Jersey 07102-1982
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ABSTRACT |
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Booth, Victoria and Amitabha Bose. Neural Mechanisms for Generating Rate and Temporal Codes in Model CA3 Pyramidal Cells. J. Neurophysiol. 85: 2432-2445, 2001. The effect of synaptic inhibition on burst firing of a two-compartment model of a CA3 pyramidal cell is considered. We show that, depending on its timing, a short dose of fast decaying synaptic inhibition can either delay or advance the timing of firing of subsequent bursts. Moreover, increasing the strength of the inhibitory input is shown to modulate the burst profile from a full complex burst, to a burst with multiple spikes, to single spikes. We additionally show how slowly decaying inhibitory input can be used to synchronize a network of pyramidal cells. Implications for the phase precession phenomenon of hippocampal place cells and for the generation of temporal and rate codes are discussed.
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INTRODUCTION |
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Place cells in region CA3 of rat
hippocampus have been observed to fire in a spatially specific and a
temporally specific manner. As the rat enters a place field, the
corresponding place cell, generally considered to be a pyramidal cell,
commences firing (O'Keefe and Dostrovsky 1971), and a
change in firing rate has been observed as the place field is crossed.
Two experimental studies observed an increase in the firing rate as the
center of the field was approached and then a decrease as the field was exited (O'Keefe and Recce 1993
; Skaggs et al.
1996
). This firing rate change has been modeled as a
two-dimensional Gaussian function of the animal's Cartesian location
in the environment (O'Keefe and Burgess 1996
). In a
study of CA1 pyramidal cells, a more monotonic increase in firing rate
was observed with lowest firing rate at the beginning of the place
field and highest rate at the end (Mehta et al. 2000
).
The preferential firing of place cells in place fields suggests a
firing rate code for location, and the observed changes in firing rate
as the field is crossed may code for location within the field. A
specific relationship between timing of place cell firing and the
hippocampal electroencephalogram (EEG), or theta rhythm, has also been
observed as a rat runs along a linear runway. Namely, the phase of the
theta rhythm at which a place cell fires systematically precesses as
the place field is crossed (O'Keefe and Recce 1993
;
Skaggs et al. 1996
). Each time the animal enters the
place field, firing begins at the same phase, and over the next 5 to 10 cycles of the theta rhythm it undergoes up to 360° of phase
precession. These findings suggest that there may also be a temporal
code for location in the phase of firing relative to the theta rhythm.
It has recently been shown that a rat's location can be more
accurately predicted when both rate and phase information are taken
into account (Jensen and Lisman 2000
).
Despite the fact that much of the hippocampal anatomy is known, the
neural mechanisms generating the changes in firing rate as the place
field is crossed and the phenomenon of phase precession have not been
completely determined. Place cells are known to receive excitatory
synaptic projections from dentate granule cells (Claiborne et
al. 1986) as well as cholinergic excitation from the medial
septum (Shute and Lewis 1963
). They also receive
synaptic inhibition from a variety of interneurons (Freund and
Buzsáki 1996
) that are influenced by GABAergic
projections from the medial septum (Freund and Buzsáki
1996
). In addition, place cells project to interneurons
(Csicsvari et al. 1998
) offering the possibility of
feedback inhibition after place cell firing (Karnup and Stelzer 1999
). The theta rhythm may further modulate place cell firing (Kamondi et al. 1998
).
Several models to account for place cell firing patterns have been
proposed (Bose et al. 2000; Jensen and Lisman
1996
; Kamondi et al. 1998
; Tsodyks et al.
1996
; Wallenstein and Hasselmo 1997
). In some of
these models, the phase of place cell firing within the place field is
essentially environment driven with precession occurring as a result of
recall of stored memories for neighboring locations (Jensen and
Lisman 1996
; Tsodyks et al. 1996
;
Wallenstein and Hasselmo 1997
). Alternatively, phase
precession has been achieved by varying the total amount of
depolarization to the cell (Kamondi et al. 1998
). The
models that address the firing rate changes within a place field rely
on increasing depolarization to the cell as the place field is crossed
to achieve the observed rate pattern (Kamondi et al.
1998
; Tsodyks et al. 1996
). We have previously proposed a minimal CA3 network model (Bose et al. 2000
)
that uses synaptic inhibition to control the timing of place cell
firing and generate the onset, occurrence, and end of phase precession. As only the burst envelope of place cell firing was modeled, mechanisms to account for the changes in firing rate as the place field is crossed
were not addressed.
It has been shown both in experiment and in models that inhibitory
input to pyramidal cells can alter their firing pattern. In CA3
pyramidal cells in vitro (Traub et al. 1994) and in
multicompartmental models (Kepecs and Wang 2000
;
Traub et al. 1994
), inhibition arriving at dendritic
locations could suppress the onset of bursting. It was additionally
observed in the model, that dendritic inhibition, when timed
appropriately, truncated the somatic burst envelope (Traub et
al. 1994
). In these pyramidal cells, complex burst firing depends on a dendritic Ca2+-based depolarization supporting
sodium action potentials initiated closer to the soma. Usually these
complex bursts are initiated by leading sodium spikes that
back-propagate to trigger the dendritic depolarization. Traub et
al. (1994)
surmised that the effects of inhibition on burst
waveform were a result of suppression of dendritic
Ca2+-based depolarization. Suppression of dendritic
Ca2+-based spikes by synaptic inhibition was observed in
dendritic recordings of hippocampal pyramidal cells in vitro
(Miles et al. 1996
; Tsubokawa and Ross
1996
) and in vivo (Buzsáki et al. 1996
). Furthermore, depending on the strength and timing of the inhibition, relative to leading sodium spikes, inhibition could delay the activation of the dendritic Ca2+-based spike, or it could
abort an already activated spike (Buzsáki et al.
1996
). While in some of these studies, suppression of the calcium spike resulted in no observed change in somatic firing (Tsubokawa and Ross 1996
), in cortical pyramidal cells,
when the dendritic calcium spike was inhibited, the associated action
potential burst was completely abolished (Larkum et al.
1999
). In a modeling study of a CA3 pyramidal neuron, dendritic
inhibition could modulate somatic firing in a more graded manner
(Kepecs and Wang 2000
).
In this paper, we investigate the effects of synaptic inhibition on
burst firing of a model CA3 pyramidal cell. We consider a
two-compartment model, developed by Pinsky and Rinzel
(1994, 1995
),
synaptically coupled to an excitable interneuron. We find that synaptic
inhibition can advance or delay the timing of burst firing with the
timing of inhibition determining the effect. As a result, periodically
timed inhibition can alter the frequency of burst firing, acting to
increase or decrease it in a range around the intrinsic burst
frequency. Furthermore, we find that, depending on its strength,
inhibition can modulate calcium influx into the dendrites, such that
the cell will fire full complex bursts, bursts with a small number of
spikes, or single spikes. This results in a mapping between synaptic
weight, firing frequency, and burst waveforms.
While the two-compartment model has sufficient detail to generate
complex bursts with compartmentally segregated mechanisms (Pinsky and Rinzel 1994) (and see Model), we
take advantage of its relative simplicity to analyze the effects of
inhibition on burst firing using phase plane methods. A result of this
analysis is a new insight into the mechanisms underlying complex burst generation. Briefly, in a repetitively bursting neuron, we find that
burst initiation does not strictly depend on dendritic mechanisms but
depends more on the interaction between somatic and dendritic voltages.
Dendritic inhibition has the effect of decoupling these two influences
to reveal the distinction. We also find that, when inhibition modulates
the dendritic active response, the interaction between somatic and
dendritic voltage, sometimes referred to as soma-dendritic
"ping-pong" (Wang 1999
), plays a crucial role in determining burst waveform.
Using the results of synaptic inhibition on a single pyramidal cell, we
additionally investigate the effects of inhibition on anatomically
isolated or weakly connected pyramidal cells. Previous modeling studies
have shown that a network of pyramidal cells with fast, recurrent
excitatory synapses display synchronous oscillations (Pinsky and
Rinzel 1994; Traub et al. 1993
). We show that
synaptic inhibition is an alternate mechanism for synchronizing a
network of pyramidal cells, as has been demonstrated in vitro in the
CA1 region by Cobb et al. (1995)
. Moreover, we show that bistability can be obtained between synchronous and out-of-phase network rhythms.
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MODEL AND METHODS |
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Model
The CA3 pyramidal cell model developed by Pinsky and
Rinzel (1994, 1995
) consists of a soma compartment electrotonically
coupled to a dendrite compartment. The soma compartment contains a
fast, inactivating sodium current and a potassium delayed-rectifier current and, when isolated from the dendrite compartment, repetitively fires action potentials at a range of frequencies (from very low up to
300 Hz) in response to maintained applied current
Is. The dendrite compartment contains a calcium
current and two potassium currents. A slowly activating
afterhyperpolarization (AHP) current has gating variable q
that depends only on Ca2+ concentration. The other
potassium current is fast activating with activation depending on
voltage, but the conductance also contains a saturating term that
depends on Ca2+ concentration. When isolated from the soma,
the dendrite compartment generates low-frequency Ca2+-based
spikes. When the compartments are coupled together electrotonically, the model displays a variety of firing patterns in response to somatic
applied current or dendritic synaptic input, including very
low-frequency bursting (less than 8 Hz), low-frequency bursting (8-20
Hz), and fast, periodic spiking (30 Hz). Bursting occurs in the model
due to interactions between the soma and dendrite compartments in
what has been dubbed soma-dendritic "ping-pong" (Wang
1999
). Specifically, a burst is initiated by a somatic
sodium spike that triggers a dendritic calcium spike. Successive
somatic spikes in the burst are caused by depolarization of the soma by the slower dendritic calcium spike. The burst ends when the dendritic calcium spike ends. The burst profile is not uniform but is
characterized by an interval of high-frequency, damped spiking in the
middle due to the large dendritic depolarization during the calcium
spike overdriving the somatic spike generator. We consider the model in
the very low-frequency bursting regime with somatic applied current
Is = 0.5 (µA/cm2). With this low
level of stimulation, the model displays periodic bursting at
approximately 1.5 Hz.
We synaptically couple this pyramidal cell to an excitable interneuron
(Fig. 1) that generates an action
potential in response to a brief applied current pulse or synaptic
excitation. The interneuron is modeled with the Morris-Lecar equations
(Morris and Lecar 1981). We consider two synaptic
architectures in this two-cell network. In the first network structure,
the only synaptic connection is between the interneuron and the
pyramidal cell with fast inhibitory synaptic current arriving to the
dendrite compartment. With this network, we consider the effect of
inhibition arriving during the interburst interval. In the second
network structure, in addition to the connection between interneuron
and dendrite compartment, the soma compartment of the pyramidal cell
makes a fast excitatory synaptic connection back onto the interneuron.
With this network structure, where the pyramidal cell receives feedback
inhibition from the interneuron, we consider the effects of inhibition
arriving during a burst. We do not include a synaptic delay in either
connection.
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The equations for this two cell network are
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The strength of synaptic current is governed by a maximal conductance
ginh for the synapse from interneuron to
pyramidal cell and gexc for the synapse from
pyramidal cell to interneuron. In the first synaptic architecture we
consider, gexc is set to zero, then it is made
nonzero for the second network structure (see figure captions for
values). The reversal potential in the synaptic current terms
determines whether the synapse is excitatory or inhibitory; we set
Vinh = 80 mV and
Vexc = 0 mV. The dynamics of the synaptic
currents are governed by equations of the form
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Methods
In RESULTS, we analyze the effect of inhibition on
burst firing using phase plane methods. Strictly speaking, phase plane methods are only appropriate for use on two-dimensional equation systems. We may apply these methods to the pyramidal cell model, however, by restricting the analysis to the silent phase of bursting. During the silent phase, several variables are approximately constant so that each compartment reduces to essentially a two-dimensional system. Specifically, in the soma compartment during the silent phase,
the sodium inactivation variable h is approximately 1 and does not change significantly until somatic voltage increases past 0 mV, near the peak of the leading sodium spike of the burst. Thus,
leading up to burst initiation, the soma dynamics are governed by the
Vs and n equations. Similarly, the
dendrite dynamics during the silent phase are essentially governed by
the Vd and q equations. The gating
variables s and c of the calcium current and the
voltage-activated potassium current, respectively, are approximately 0, as is Ca2+ concentration Ca. By restricting our
attention to the silent phase of bursting, leading up to burst
initiation, we may consider the soma trajectory in the
Vs n phase plane and the dendrite trajectory in the Vd
q phase plane.
In phase plane analysis of single compartment models, the trajectory is
governed by the position of the nullclines of each equation, and these
nullclines are stationary for fixed values of parameters. For our phase
plane analysis of this two-compartment model, we consider separate
phase planes for each compartment, and the trajectory for each
compartment is determined by the position of its respective nullclines.
But the compartments, and thus their phase planes, are linked through
the coupling current, Icoup = gc(Vd Vs)/p. As the voltages evolve, the
coupling current continuously changes. Hence the nullclines in each
phase plane are not stationary but are continuously moving. During the
silent phase, both voltages, and thus Icoup,
evolve slowly, and we can track trajectories in each phase plane
relative to slowly moving nullclines.
Another difference in the phase plane analysis of a two-compartment model compared to that of a single compartment model is the effect of brief synaptic current. In a single compartment model, inhibitory synaptic current, for example, shifts down the voltage equation nullcline, which generally has a cubic shape. When the synaptic current shuts off, the cubic nullcline returns to its original position. In the two-compartment model, inhibitory synaptic input to one of the compartments has a similar effect of shifting the cubic nullcline down. But when the synaptic current shuts off, the nullcline may not return to its original position because the voltages in each compartment, and thus Icoup, are not the same as before the synaptic input.
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RESULTS |
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Inhibition arriving before a burst delays burst firing
We consider the effect of a single dose of synaptic inhibition
arriving during the interburst interval on the timing of subsequent burst firing. We consider the pyramidal cell-interneuron network structure with one synaptic connection from the interneuron to the
dendrite compartment (ginh = 1 and
gexc = 0 mS/cm2). The
interneuron is made to fire by giving it a brief applied current pulse
of sufficient magnitude to generate a single action potential. The soma
voltage traces in Fig. 2,
A-C, show a delay in burst
firing caused by the inhibition. In Fig. 2A, no inhibition is given, and the pyramidal cell displays very low-frequency, periodic
bursting. In Fig. 2, B and C, the interneuron is
stimulated so that inhibition arrives before pyramidal cell firing. The
bottom traces show the inhibitory postsynaptic current
(IPSC) at the dendrite compartment. Since the synapse is fast and no
synaptic delay is modeled, there is only a 1- to 2-ms lag between the
applied current pulse to the interneuron and the IPSC in the dendrite compartment. Also note that the inhibition decays quickly. The synaptic
inhibition delays firing of the next burst with the amount of delay
depending on its time of arrival. Figure 2D shows a summary of this effect as a phase resetting curve where the phase that inhibition arrives is plotted on the x-axis, and the
resultant new phase after inhibition is plotted on the
y-axis. To interpret the diagram, we associate 360° in
phase to the intrinsic period of the pyramidal cell. We calculate the
time difference between when the inhibited pyramidal cell fires and
when it would have fired in the absence of inhibition. This time is
then converted to a phase
. The new phase is
. If
inhibition arrives immediately following a burst and up to
approximately 260° in the burst cycle, it has virtually no effect on
burst firing as seen by the new phase being approximately equal to the
old phase. When inhibition arrives closer to the time of burst firing,
it delays the next burst, thus shifting its phase back. The amount of
delay or phase shift increases as the timing of inhibition approaches
the time of burst firing with a maximal delay, with this set of model
parameters, of about 55° or 100 ms.
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A similar delay effect of inhibition is obtained when the intrinsic
frequency of the pyramidal cell is varied by changing the value of the
applied current to the soma compartment Is.
For Is values between approximately 0.2 and
1.3 µA/cm2, where the pyramidal cell displays very
low-frequency bursting in the range from 0.4 to 4 Hz, inhibition
arriving during the interburst interval delays the following burst, and
the amount of delay increases as the timing of inhibition approaches
the time of firing. The maximal delay obtained and the range of phases where the delay is observed depends on the intrinsic burst frequency. For example, for higher intrinsic frequencies obtained with larger values of Is, delays are first observed at
earlier phases, and the maximal phase delay is larger.
In this study, we consider the inhibition arriving to the dendrite compartment. The same delay effect is obtained if the inhibition arrives instead to the soma compartment. In fact, the phase where delays are first observed and the maximal phase delay are basically the same as those obtained above.
An interesting result of this effect is that if the inhibition is
periodic such that it always arrives at the same phase of the burst
cycle, pyramidal cell firing can be entrained to a lower frequency. For
example, if the interneuron is paced to fire with a slightly longer
period than the intrinsic pyramidal cell burst period and the
interneuron fires at a phase between 260° up to 360° of the burst
cycle such that the phase delay is equivalent to the difference in
periods, then the pyramidal cell will always be inhibited at the same
phase, and firing will be entrained to the lower interneuron frequency.
We have previously proved that fast synaptic inhibition can entrain a
simple neuron model to a lower frequency than its intrinsic firing
frequency and that the lower frequency firing is a stable periodic
orbit (Bose et al. 2000). A similar proof may be applied
to the present network model to show that the lower frequency firing is
a stable state. With these values of model parameters, periodic
synaptic inhibition arriving once during the interburst interval can
entrain the pyramidal cell to fire at frequencies in the range from
almost 1.3 to 1.5 Hz, where the intrinsic burst frequency is
approximately 1.5 Hz (simulations not shown). We note that if the
inhibition arrives at a sufficiently high frequency, the pyramidal cell
frequency can be made arbitrarily small or even completely suppressed.
In the following paragraphs, we analyze how synaptic inhibition causes
the delay in firing in the two-compartment pyramidal cell model with
phase plane methods, starting with an analysis of burst initiation. An
interesting insight revealed by this analysis is a subtle difference in
the mechanism of burst initiation than is described in the original
model paper (Pinsky and Rinzel 1994). In the very
low-frequency bursting regime, Pinsky and Rinzel (1994)
describe that the duration of the silent phase is determined by the
potassium AHP current in the dendrite compartment. In particular, they
propose that when the slow gating variable, q, for this
current passes below a threshold value, a somatic sodium spike is
triggered, thus initiating a burst. Our analysis shows that while the
decay of the gating variable q governs the duration of the
silent phase, it is actually the resulting slow rise in
Vd that leads to burst initiation. So, instead
of there being a threshold value for q, there is a
Vd threshold,
V*d, that must be crossed to
trigger the leading sodium spike of the burst. In a normal burst cycle,
the slow rise in Vd is determined primarily by
the decrease in q, but inhibition arriving before a burst
decouples Vd from q, thus revealing
this distinction.
For our phase plane analysis, we refer to the soma compartment
Vs n and dendrite compartment
Vd
q phase planes shown in Fig.
3. In both phase planes, the burst
trajectory of Fig. 2C is shown by the heavy curve (arrows
indicate flow direction). At the beginning of the silent phase
(indicated by "sp" in Fig. 3, A and B),
Vs and Vd are
hyperpolarized resulting in a hyperpolarizing coupling current in each
compartment that pushes the Vs and
Vd cubic nullclines down to the positions at the
lower boundaries of the shaded regions. In the soma compartment
Vs
n phase plane, the local minima
or left knee of the Vs cubic is below the
n-nullcline (thin dashed curve). This results in a fixed
point on the lefthand branch of the Vs cubic at
the intersection of the two nullclines, which prohibits the soma from
spiking. Because of the fast dynamics in the soma compartment, the soma
trajectory moves to within a small neighborhood of this fixed point at
the beginning of the silent phase.
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During the silent phase, the electrotonic coupling between compartments
causes a slow evolution of the cubic nullclines in each phase plane. As
Vs and Vd depolarize
during the silent phase (solid portion of trajectory curves), the
coupling current increases and the cubic nullclines slowly move up
through the shaded regions (direction indicated by long arrows). The
dendrite trajectory moves down the lefthand branch of the slowly rising
Vd cubic and the soma trajectory tracks the
fixed point at the intersection of the slowly rising
Vs cubic and the n-nullcline. Note
that the trajectory in the Vd q
plane remains away from the knees of the Vd
cubic nullcline (Fig. 3C) as is expected since firing is initiated in the soma compartment.
Let Icoup = gc(Vd Vs)/p denote the coupling current in
the soma compartment. For the uninhibited trajectory,
Icoup reaches a critical value
I*coup when the left knee of the
Vs cubic becomes tangent to the n-nullcline. The position of the cubic nullclines at this
point is indicated by the upper boundaries of the shaded regions in Fig. 3. Let V*s be the value
of Vs at the point of tangency of the Vs and n nullclines, also called a
saddle-node point. When Icoup increases through
I*coup, the fixed point on the
Vs nullcline disappears and, provided that
Vs is near
V*s, the soma can trigger the
leading spike of the burst. In the uninhibited trajectory, since the
soma voltage is at V*s when
Icoup = I*coup, this defines a threshold value
for dendrite voltage at burst initiation,
V*d.
We can summarize by stating two conditions for burst initiation: 1) Vs must be sufficiently close to V*s and 2) Icoup must be greater than I*coup. Since condition 1 is satisfied in the silent phase, condition 2 reduces to Vd > V*d. Pinsky and Rinzel's description of burst initiation caused by q decreasing below a threshold value is consistent with these two conditions. In the intrinsic burst trajectory, since decreasing q governs the rise in Vd, an equivalent q threshold can be defined as the q value on the upper boundary of the shaded region in Fig. 3B at Vd = V*d. However, for the case of synaptic inhibition given before the burst, we will show that a true threshold in q does not exist.
The cause of delay in burst firing when inhibition is given during the
silent phase before a burst can now be described (Fig. 3B,
dotted curve shows inhibited trajectory). In response to inhibition, the Vd cubic nullcline is quickly shifted down.
Depending on the strength of inhibition, the cubic may be shifted to a
position below or within the shaded region (not explicitly shown in the figure). The Vd trajectory approximately follows
the shifted nullcline (note the sharp Vd
decrease in Vd q trajectory in
Fig. 3B). This decrease in Vd, and
hence Icoup, shifts the
Vs cubic down to a position below or within the
shaded region (not explicitly shown). The soma trajectory continues to
track the fixed point as it is moved to lower Vs
(since the trajectory moves back along the same path that it had been
traveling, this shift in Vs is not apparent in
the figure). The slow variable q is not dependent on
Vd and thus continues to decrease. When the
inhibition shuts off, the Vd nullcline will
quickly rise. It does not return to its position before the synaptic
event, however, because now Vs, Vd, and hence the coupling current are at
different values than before the inhibition. The overall effect of the
inhibition is to shift both the Vd and
Vs cubic nullclines down, forcing the trajectories to evolve along the cubic nullclines as they slowly move
up through the shaded region. Since the soma trajectory remains close
to the fixed point on the Vs cubic,
condition 1 for burst firing is still satisfied after the
inhibition. The delay in firing is caused by the additional time needed
for condition 2 to be satisfied, namely for
Vd to increase past
V*d. During this time,
q continues to decrease, passing below the nominal threshold
value determined by the intrinsic trajectory. Thus it is clear that it
is more appropriate to think about a threshold value for
Vd for burst initiation rather than a threshold
value for q.
We note that due to this delay effect of inhibition, this model cell is not able to fire via postinhibitory rebound in the strict sense. Namely, the cell does not immediately fire when it is released from inhibition, regardless of when the inhibition is given.
Inhibition arriving during burst causes advance in burst firing and modulates burst waveform
We now consider the effect of inhibition arriving during a burst, specifically arriving just following the leading spike of the burst. We achieve this timing for the inhibition with the network structure in which there is an excitatory synaptic connection from the soma compartment of the pyramidal cell to the interneuron (gexc nonzero) and an inhibitory synaptic connection from the interneuron back to the dendrite compartment (ginh nonzero). In this network, pyramidal cell firing causes the interneuron to spike, thus sending inhibitory synaptic current back to the dendrite compartment of the pyramidal cell (Fig. 1). Since there is no synaptic delay modeled, the inhibition arrives just after the leading spike of the burst.
As shown in Fig. 4, A and B, such feedback inhibition during a burst causes an advance in the firing of the subsequent burst. When the pyramidal cell is repetitively bursting, as it is in Fig. 4 (uninhibited firing shown in A), the feedback inhibition advances each burst and higher frequency repetitive bursting can be obtained (Fig. 4B, top trace). The brief IPSC in the dendrite compartment as a result of interneuron firing is shown in the bottom trace of Fig. 4B. The amount of phase advance depends on the strength of inhibition that is controlled by ginh. As ginh increases from zero to 0.45 mS/cm2, the phase advance increases to, in this case, approximately 120°, equivalent to decreasing the period by 215 ms. This phase advance by inhibition is summarized in the phase response plot shown in Fig. 5A. In this figure, the new phase resulting from the inhibition is plotted versus strength of inhibition, ginh. The data points marked with "C" (for complex burst) show the increase in phase advance of bursting as ginh is increased from 0 to 0.45 mS/cm2. Another way to view these results is in the plot of steady-state burst frequency shown in Fig. 5B. In this plot, the "C" data points show the increase in burst frequency from approximately 1.5 to 2.3 Hz as ginh is increased from 0 to 0.45 mS/cm2.
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When ginh is increased past 0.45 mS/cm2, the phase advance continues to increase, and,
furthermore, the burst waveform is modified. We find a smooth
transition from a full complex burst (0 ginh
0.45, Fig. 4, A and
B) to bursts consisting of 4 spikes
(ginh = 0.5, Fig. 4C) and 3 spikes (ginh = 0.51, D), to
bursts with 2 spikes or spike doublets
(ginh = 0.53, E), to single
spikes (ginh
0.57, F).
Occurring with these changes in burst profile is an increase in phase
advance of the subsequent burst, from approximately 120° for the
complex burst when ginh = 0.45 to 310° for the single spike when ginh = 1 mS/cm2. When the pyramidal cell is repetitively bursting,
as it is here, these phase advances correspond to an increase in burst
frequency from 2.3 Hz when ginh = 0.45 to
over 11 Hz when ginh = 1 mS/cm2. We again refer to Fig. 5, A and
B, to summarize the increases in phase advance and
frequency, respectively, with increasing ginh.
In the figures, the number at each data point indicates the number of
spikes per burst. Combining these results with those from the previous
section, inhibition can entrain the pyramidal cell to fire in a range
from arbitrarily low frequencies up to 11 Hz.
We obtain similar phase advances and modulation of waveform for different intrinsic burst frequencies of the pyramidal cell when the applied current to the soma Is is changed. But the values of ginh where the effects occur are different. For example, when the intrinsic burst frequency of the pyramidal cell is higher (Is = 0.75 µA/cm2), the transitions in waveform occur at lower values of ginh, and the phase advances occurring with the waveform changes are larger. The transitions in waveform, however, are not smooth. As ginh is increased, complex bursting gives way to irregular firing of 3 spike bursts and doublets. Periodic firing of bursts with 4 or 3 spikes are not observed in this case. But as ginh is increased further, periodic firing of spike doublets and single spikes is obtained. In general, for all values of Is, regular spike doublet and single spike firing is obtained for high values of ginh.
We also obtain similar phase advances and similar modulation of waveform as inhibition is strengthened when the inhibition arrives at the soma compartment instead of at the dendrite. Again, the values of ginh where similar results are observed are different, but phase advances and regular firing of the same types of burst waveforms are obtained. For example, in the case shown here, if inhibition arrives to the soma compartment, the transition from complex bursting to regular firing of bursts with 4 spikes occurs at a higher ginh value, and the range of ginh values over which 4-spike bursts are obtained is larger. As ginh is increased further, regular firing of 3-spike bursts and spike doublets are obtained over larger ranges of ginh values.
To understand the phase advance caused by inhibition, we recall that in
this model a burst is generated by dendritic, calcium-based depolarization supporting somatic, sodium spiking. The general effect
of inhibition arriving just after burst initiation is to lessen
dendritic depolarization. This attenuation of peak
Vd when ginh = 0.4 mS/cm2 compared to the uninhibited case can be seen when
both trajectories are plotted in the dendrite compartment
Vd q phase plane (Fig. 6A) and in a time plot (Fig.
6G, bursts are offset for comparison purposes). We note that
strict phase plane analysis of solutions is only appropriate during the
silent phase of bursting, but we find that plotting the trajectories in
their phase planes is helpful in understanding effects of inhibition
during the burst. As a result of the inhibition, Ca2+
influx is suppressed during the burst (Fig. 6B), which
causes less activation of the slow potassium AHP current with gating variable q (Fig. 6C). The inhibition acts during
the active phase of the burst (Fig. 6G, heavy bars under
bursts indicate duration of IPSC) and has decayed away by the time the
cell returns to the silent phase. Thus, in the silent phase (Fig.
6A, beginning indicated by "sp"), the cell will track
the nullclines corresponding to the intrinsic burst case in the
Vd
q phase plane (heavy portion of
dotted curve). As discussed in the previous section, when no inhibition
is given, the control of the rise of Vd by
q during the silent phase determines the duration of the
interburst interval. The inhibition induced attenuation of the
Ca and q peaks shortens the subsequent silent
phase of the burst cycle since, as can be seen in the
Vd
q phase plane, the inhibited
trajectory begins the silent phase (heavy portion of solid curve) at a
lower value of q and higher value of
Vd. Thus it takes less time for
Vd to cross
V*d and trigger a somatic
spike.
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The change in burst profile with increasing ginh
occurs similarly by an attenuation of dendritic depolarization. When
ginh is small (ginh = 0.4 mS/cm2, for example), the inhibition is not strong
enough to prevent a Vd Ca2+-based
spike. So while peak Vd is attenuated compared
to the uninhibited case, it is still large enough to overdrive the soma
spike generator and create a complex burst (Fig. 6G, 1st 2 bursts). If, however, ginh is large
(ginh 0.57, for example), the inhibition
abolishes the Vd Ca2+-based spike,
and only the leading sodium spike of the burst is realized. As can be
seen in the Vd
q phase plane (Fig.
6D, solid curve) and in a time plot (Fig. 6G,
last burst), there is a Vd spike due to
backpropagation of the leading sodium spike, but the dendritic response
has been inhibited. The leading spike allows a minimal Ca
increase (Fig. 6E, solid curve) and thus a small increase in
q (Fig. 6F, solid curve).
For intermediate values of ginh (0.5 ginh
0.56 mS/cm2), the
dynamics are more interesting. In these cases, the inhibition is strong
enough to prevent the initiation of a full Vd
Ca2+-based spike, but is not strong enough to completely
hyperpolarize the dendrite. The partial depolarization of the dendrite
allows soma-dendritic ping-pong interactions to support a burst. For example, when ginh = 0.51 mS/cm2, the leading Vd spike, shown
in the Vd
q phase plane (Fig. 6D, dashed curve) and in a time plot (Fig. 6G,
4th burst), is the same as for larger ginh, but
the weaker inhibition allows the dendrite to remain sufficiently
depolarized to support another sodium spike. The backpropagation of
this second sodium spike again sufficiently depolarizes the dendrite
providing for a third sodium spike. This ping-pong effect does not
continue indefinitely since dendritic depolarization and calcium influx
(Fig. 6E, dashed curve) trigger the
IK
C current, which ultimately ends
the burst. Similar dynamics account for the 4-spike bursts (Fig.
6G, 3rd burst) and spike doublets that are observed for
ginh values in this intermediate range.
Pinsky and Rinzel (1994) showed that by increasing
applied current to the soma Is, to mimic
N-methyl-D-aspartate (NMDA) excitation, the
pyramidal cell changes from a bursting mode to a single spiking mode
with frequencies between 20 and 30 Hz. The changes effected by
increasing Is resulted from an essential
shutdown of the dendritic calcium-based mechanisms. At high values of
Is, the dendrite and soma spike at the same
frequency, causing low-amplitude, fast oscillations in calcium
concentration. These changes in Ca are sufficiently fast
such that the slow gating variable q of the AHP current
remains at a constant level. Even though the constant q
level is relatively high, near peak values in the bursting mode, the
AHP current does not participate in the afterhyperpolarization of
spikes or affect the interspike interval. Also, any hyperpolarizing effect on the soma compartment by the AHP current is counteracted by
the high somatic applied current. To summarize their results, tonic
somatic depolarization weakens the effect of the dendritic compartment.
In our model, fast synaptic inhibition to the dendrite provides a
similar regulation of dendritic calcium mechanisms. However, the
frequency of single spikes that we obtain is significantly lower than
what Pinsky and Rinzel (1994)
obtain with large
Is. The reason is that even though calcium
concentration remains at low levels, the AHP gating variable
q is activated with every spike and participates in spike
afterhyperpolarization and determination of the interspike interval
since somatic applied current levels are low.
Synchronous and out-of-phase oscillations in networks of pyramidal cells
Pinsky and Rinzel (1994) show that a network
consisting of a number of their two-compartment pyramidal cells can
exhibit synchronous or near-synchronous oscillations if the pyramidal
cells have recurrent AMPA-mediated excitatory connections. This
mechanism for synchrony was also previously observed by Traub et
al. (1991)
in their large-scale computational model. By
blocking AMPA in those studies, the cells desynchronize. These studies
did not include the effect of inhibition in promoting relevant rhythmic
patterns. We show here that there is an alternate mechanism that may
promote synchrony among the pyramidal cells; namely slowly decaying
inhibitory input can be used to synchronize pyramidal cells that are
weakly connected or even unconnected.
We consider two pyramidal cells that each have reciprocal synapses onto the same interneuron (Fig. 7A). The interneuron makes an inhibitory synaptic connection to the dendrite compartment of each pyramidal cell and an excitatory synaptic current is sent from the soma compartments of each pyramidal cell back to the interneuron. The pyramidal cells initially are assumed to have no excitatory connections between themselves. We focus on two specific rhythm patterns; synchronous and out-of-phase oscillations.
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Figure 7B shows the voltage traces of the two pyramidal
cells P1 and P2. The
simulation shows that the cells start out-of-phase with one another,
but that their burst envelopes quickly synchronize after the first few
cycles. Both cells receive a common slowly decaying inhibition from the
interneuron I (decay rate of synaptic gating decreased to
= 0.1 mS
1). Slowly decaying inhibition is known
to synchronize cells when both cells receive the inhibition from a
common inhibitory cell, but in situations where neither cell synapses
back to the inhibitory cell (Terman et al. 1996
). If the
cells do synapse on the interneuron, for one-compartment models, a
delay to the onset of inhibition is necessary for synchrony
(Rubin and Terman 2000
; Terman et al. 1998
; van Vreeswijk et al. 1994
). In our model,
we do not require an explicit delay, because the separation of the soma
and dendrite provides an "effective delay" to the onset of
inhibition. In particular, the inhibition acts to hyperpolarize the
dendrites, which then hyperpolarizes the soma via the coupling current.
The speed with which the inhibition ultimately affects the soma is
dependent on the intrinsic properties of the dendrite (the rate at
which it hyperpolarizes) and the strength of the electrical coupling. If either of these quantities is small, then there effectively will be
a small window of time from when the inhibitory cell fired to the time
the soma receives the full impact of this input. During this time, the
soma will be able to fire a sodium spike, thus initiating the burst.
This delay plays the same role that explicit synaptic delays play in
Terman et al. (1998)
and Rubin and Terman (2000)
. The slowly decaying inhibition has the effect of
keeping the dendritic voltages of each cell close to one another. As we showed earlier, the dendrite voltage must cross
V*d in order for the soma to
fire. The slowly decaying inhibition allows the dendritic voltages of
each cell to compress and cross this threshold within a small time
window of one another. By varying the maximal conductance of the
inhibitory synapse, we also obtain synchrony between the cells during
complex burst mode, single spike mode, and multiple spike mode
(simulations not shown). We note that to achieve synchrony in the burst
mode, a much slower decay of inhibition is necessary relative to the decay rate for synchrony of single spikes. In the burst mode, the
dendrite voltage trajectories return to the silent phase with low
Vd and high q values. If the
inhibition decays quickly, the dendrite trajectories will quickly
follow that of an uninhibited dendrite, as depicted in Fig.
2B. In this case, there will be no compression between
Vd trajectories. Alternatively, this type of
inhibition can synchronize single spiking cells since they return to
the silent state with higher Vd and lower
q values. Thus the quickly decaying inhibition will now act
to delay the next spikes, as in Fig. 2B, which allows
compression of the Vd trajectories to occur.
Figure 7C shows the two cells oscillating in anti-phase with one another. The parameter values are the same as those for the synchronous oscillations. Thus the network exhibits bistability of periodic solutions. The bistability is produced much in the same way as for one-compartment relaxation oscillator models. The synchronous solution arises as discussed above if the dendritic voltages start out sufficiently close together. If these voltages are not close, then an anti-phase, and, more generally, an out-of-phase solution arises. In this case, the firing of P1, for example, causes P2 to receive an inhibitory input before it were to fire. This, as depicted in Fig. 2B, may cause a delay of the P2 burst. Thus during the time P1 is active, P2 is moved away from its firing threshold. Similarly when P2 fires, it delays the onset of a P1 burst. Moreover, depending on the strength of the inhibition, the firing pattern of each cell within its burst is also modulated as described previously.
The addition of weak excitatory coupling between the pyramidal cells does not change the qualitative behavior described above. However, it can change the firing pattern within a burst of each cell. Weak excitation lessens the overall level of hyperpolarization due to the interneuron, if both excitation and inhibition wear off on similar time scales. As seen above, the firing pattern of the cells is sensitive to small changes in the maximal conductance of the inhibitory synapse. As a result, weakly connected pyramidal cells that may have fired spike doublets while unconnected, may now fire bursts with more spikes due to the changed balance between excitation and inhibition (simulations not shown).
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DISCUSSION |
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We have shown that, depending on its timing, synaptic inhibition
may delay or advance burst firing in a model pyramidal cell. As a
result, periodically timed inhibition can either increase or decrease
the firing frequency of a repetitively bursting neuron within a range
around the intrinsic bursting frequency. Increasing burst frequency
requires inhibition to arrive at the beginning of a burst, following
the initial sodium spike. In this case, we generate periodic inhibition
as a result of the reciprocal synaptic connections between the
pyramidal cell and the interneuron. Thus the source of periodic
inhibition is the repetitive firing of the pyramidal cell itself.
Decreasing burst frequency, on the other hand, requires that inhibition
arrive before burst firing, at a constant phase of the burst cycle.
This can be achieved if the interneuron fires periodically at a lower
frequency than the intrinsic frequency of the pyramidal cell. A
possible source for periodic inhibition in this case may be the theta
rhythm. It is known that cells in the medial septum make GABAergic
connections to interneurons in CA3 and act as a pacemaker drive for the
theta rhythm (Green and Arduini 1954). If the intrinsic
frequency of the pyramidal cell is greater than theta frequency,
interneurons driven to fire at theta frequencies could provide the
appropriate periodic inhibition to decrease pyramidal cell frequency.
Our results show that slowly decaying inhibition can be used to
synchronize the activity of pairs of pyramidal cells. The results
suggest an alternative to the explanation of Pinsky and Rinzel
(1994) and Traub et al. (1991)
that fast
excitatory AMPA-mediated synapses between these cells are responsible
for synchrony. The synchronization of actual CA3 cells could be
reflective of a two-step process; the first is recruitment of co-active
place cells due to excitatory synapses; the second is maintenance of
the synchrony due to slowly decaying inhibition. Synchrony ranges from
complex bursts to single spikes, thus encompassing the full range of
firing behaviors detailed in Fig. 4. Moreover, there is no difficulty in generalizing the synchrony result to larger networks of pyramidal cells.
An experiment that would investigate whether inhibitory interneurons
participate in synchronizing pyramidal cells in CA3 would be to
stimulate a single interneuron and record from two or more of its
target pyramidal cells. It has been shown in CA1, that a single
interneuron can entrain the firing of two target pyramidal cells in
vitro (Cobb et al. 1995). Similarly, in CA3, we may
expect that if AMPA and NMDA receptors are blocked, we would observe synchronous inhibitory postsynaptic potentials (IPSPs) in the target
pyramidal cells entraining their firing. Additional experiments where
certain synaptic receptors are blocked and then washed out could
further indicate the contributions of both inhibitory and excitatory
inputs to network synchrony.
The delay/advance of bursts due to inhibitory input can also be
achieved in simpler mathematical models, such as relaxation oscillators. With these models, inhibition applied during a burst tends
to shorten the burst length. However, in the present model, the length
of the burst changes in less intuitive ways. For example, in Fig. 4, we
note that the length of the burst increases as inhibition is
increased from ginh = 0.4-0.5
mS/cm2, and then decreases in response to further increase
in ginh. The initial burst lengthening results
from a modulation of the ping-pong interaction between dendrite and
soma by inhibition. If inhibition is weak, dendritic voltage is high
during the burst leading to fast activation of
IKC, which ends the burst. If
inhibition is strong, the dendrite is completely suppressed and does
not support a burst of multiple spikes. For intermediate values, the
ping-pong effect allows the IK
C current to build up slowly, thereby elongating the burst.
Relation to experimental observations
When synaptic inhibition arrives during a burst, our modeling
results suggest that it can modulate the dendritic calcium spike, and
thus the somatic firing pattern, in a graded manner. Several experimental studies show complete abolition of dendritic calcium spikes (Buzsáki et al. 1996; Miles et al.
1996
; Traub et al. 1994
; Tsubokawa and
Ross 1996
), although an already activated calcium spike could
be aborted by inhibition resulting in a shorter calcium spike
(Buzsáki et al. 1996
). A graded response, however, is suggested by the dendritic recordings of Kamondi et al.
(1998)
when current pulses of different amplitudes are injected
into the dendrite. In their Fig. 7B, a strong current pulse
evokes a large-amplitude calcium spike. In response to a weaker current pulse, the dendritic voltage displays a group of fast spikes of decrementing amplitude. We compare these results with our model results
when no inhibition arrives during the burst and when the dendrite
receives a moderate level of inhibition during a burst (Fig.
6G). When no inhibition is given, Vd
displays a full Ca2+-based spike (1st burst in figure).
With moderate inhibition (ginh = 0.5 mS/cm2, 3rd burst in figure), Vd
displays a group of back-propagated spikes similar to the initial group
of spikes in the Kamondi et al. (1998)
dendritic voltage
trace. In the Kamondi et al. (1998)
trace, the higher
frequency of the initial group of spikes, compared to the low-amplitude
spikes seen later on during the applied current pulse, seems to suggest
a partial depolarization due to dendritic mechanisms that may support
somatic spikes in a ping-pong fashion.
In the model, as a result of inhibition arriving during a burst and attenuating the dendritic calcium spike, the following burst was advanced due to less activation of the potassium AHP current. We further found that complex bursts brought on by a distinct dendritic calcium spike occurred at lower frequencies than bursts with fewer number of spikes and that single spikes fired at the highest frequency. A prediction of our modeling results is that, in pyramidal cell firing, bursts consisting of a few spikes will be followed by shorter interburst intervals than bursts consisting of a larger number of spikes. This firing pattern would be a result of less activation of the potassium AHP current with a shorter burst and thus less hyperpolarization of dendritic voltage, leading to crossing of the Vd threshold more quickly. Similarly, the model results predict that single spikes should be followed by the shortest interval to next firing.
In this paper, we have modeled the interneuron as a single cell, and
the strength of the synaptic input to the pyramidal cell is determined
by the maximal conductance of the synaptic current. As the dendrite
compartment represents the lumped distal dendrites of the pyramidal
cell, similarly the interneuron could be thought of as representing a
pool of interneurons impinging on the distal dendrites, and the maximal
conductance could be thought of as a measure of the net inhibitory
input. In this way, the model results are not inconsistent with the
experimental results that found negligible effect of unitary IPSPs on
somatic firing patterns (Karnup and Stelzer 1999;
Tsubokawa and Ross 1996
). If the single IPSP was
received in a background of excitatory input, its effect would not
influence the net dendritic depolarization significantly. Our model
results suggest that an attenuation of net dendritic depolarization
during burst generation may result in modulation of burst waveform.
In this model, the firing frequency and burst waveform observed in the
soma compartment were modulated by net dendritic depolarization, which
to some extent depended on dendritic calcium concentration. For
example, the occurrence of a full dendritic calcium-based spike and the
accompanying large increase in intracellular Ca2+ resulted
in a complex burst in the soma followed by a long interburst interval.
But when the dendritic calcium spike was attenuated by inhibition and
calcium influx was suppressed, shorter bursts or even single spikes
were obtained in the soma and were followed by shorter intervals until
the next firing event. These results suggest a dependence of firing
pattern and frequency on dendritic calcium concentration, similar to
the encoding of firing frequency of cortical layer V pyramidal neurons
by dendritic intracellular calcium suggested experimentally
(Helmchen et al. 1996) and in models (Wang
1998
).
Firing rate changes and phase precession of place cells
Our motivation for studying the effects of synaptic inhibition on
pyramidal cell firing is to understand the neural mechanisms responsible for the firing patterns of place cells in hippocampal region CA3. The results presented here suggest that synaptic inhibition may be able to modulate firing rate in a way that is consistent with
the experimental observations. Also, the results suggest that firing of
the bursting cell can be modulated to produce the phase precession
phenomena as modeled in our previous work (Bose et al.
2000). There we proposed that a minimal model consisting of one place cell P, one interneuron I, and one
theta pacemaker T could accurately describe the phase
precession phenomena. Briefly, we argued that when the animal is
outside the place field of the pyramidal cell, the pacemaker drives the
interneuron, which in turn entrains the pyramidal cell at the
theta rhythm. Within the place field, the pyramidal cell instead drives
the interneuron, and they both phase precess relative to the theta
pacemaker. The change in control of the interneuron from the pacemaker
to the place cell is initiated by the dentate gyrus, which sends an
excitatory input to the place cell at the beginning of the place field.
The phase precession of P and I ends after 360°
of precession when I returns to a phase at which
T can recapture control of it. Two of the major predictions
of that model are that some interneurons phase precess, and that the
minimal network could determine the end of the place field with no
additional external input. There are two drawbacks of that work,
however. One is that within the place field, we only modeled phase and
not firing rate. Second, the model predicts an out-of-place field
firing rate that is too high. The model presented in this work
eliminates both of these concerns. As demonstrated, the interneuron can
entrain the pyramidal cell at very low frequencies when the inhibition
arrives prior to the burst. This behavior is similar to out-of-place
field activity. Additionally, when the pyramidal cell drives the
interneuron, the burst frequency is dramatically faster and can exhibit
precession. Moreover, changes in firing rate due to changes in the
overall level of inhibition versus excitation can be achieved as the
animal passes through the place field. For example, a monotonic
decrease in the net inhibitory input to a given pyramidal cell would
have the effect of increasing its firing rate as seen by Mehta
et al. (2000)
. This decrease could result from added excitation
obtained due to recruitment of co-active place cells. The mechanisms
for change of control of the interneuron from T to
P and back to T remain as before. We shall
demonstrate the viability of these ideas in future work.
Our work also can be generalized to show how the same cells can participate in multiple, disjointed place fields. For example, in our two pyramidal cell, one interneuron network, bistability between synchronous and anti-phase or out-of-phase solutions exists. The synchronous solution represents firing of place cells that share the same place field. However, these cells might also be part of other cell assemblies that code for other spatial locations. That the cells can also oscillate out-of-phase, for the same parameter values and synaptic connections, implies that the cells have the potential to participate in multiple cell assemblies. What keeps the cells of different cell assemblies from synchronizing is the level of inhibition versus excitation that any particular cell receives. The analysis of this paper provides a framework to understand the balance between these two effects. We propose that depending on the timing of the inputs, excitation and inhibition can compete or cooperate to produce multiple types of behaviors.
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ACKNOWLEDGMENTS |
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We thank M. Recce and G. Buzsaki for many helpful discussions.
This research was supported by National Science Foundation Grants IBN-9722946 (to V. Booth) and DMS-9973230 (to V. Booth and A. Bose) and New Jersey Institute of Technology Grants 421590 (to V. Booth) and 421540 (to A. Bose).
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FOOTNOTES |
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Address for reprint requests: V. Booth (E-mail: vbooth{at}m.njit.edu).
Received 18 September 2000; accepted in final form 14 February 2001.
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REFERENCES |
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