JOIIKNAI OF NF.IIK(
Vd. 63. No. 5. Ma)
Insights Into Associative Long-Term Potentiation From
Computational Models of NMDA Receptor-Mediated Calcium
Influx and Intracellular Calcium Concentration Changes
WILLIAM
R. HOLMES
AND
WILLIAM
B. LEVY
Mathematical
Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases,
National Institutes ofHealth, Bethesda, Maryland 20892; and Department of Neurosurgery,
Universit .17of. Virginia School ofkfedicine, Charlottesville, Virginia 22908
*
SUMMARY
AND
CONCLUSIONS
I. Because induction
of associative long-term potentiation
(LTP) in the dentate gyrus is thought to depend on Ca2+ influx
through channels controlled by N-methyl-D-aspartate
(NMDA)
receptors, quantitative modeling was performed of synaptically
mediated Ca” influx as a function of synaptic coactivation. The
goal was to determine whether Ca”+ influx through NMDA-receptor channels was, by itself, sufficient to explain associative
LTP, including control experiments and the temporal requirements of LTP.
2. Ca” influx through NMDA-receptor
channels was modeled
at a synapse on a dendritic spine of a reconstructed hippocampal
dentate granule cell when I- 1 15 synapses on spines at different
dendritic locations were activated eight times at frequencies of
lo-800 Hz. The resulting change in [Ca”] in the spine head was
estimated from the Ca” influx with the use of a model of a
dendritic spine that included Ca2’ buffers, pumps, and diffusion.
3. To use a compelling model of synaptic activation, we developed quantitative descriptions of the NMDA and non-NMDA
receptor-mediated
conductances consistent with available experimental data. The experimental
data reported for NMDA and
non-NMDA
receptor-channel
properties and data from other
non-LTP
experiments
that separated the NMDA
and nonNM DA receptor-mediated
components
of synaptic events
proved to be limiting for particular synaptic variables. Relative to
the non-NMDA
glutamate-type
receptors, I) the unbinding
of
transmitter from NMDA receptors had to be slow, 2) the transition from the bound NMDA receptor-transmitter
complex to the
open channel state had to be even slower, and 3) the average
number of NMDA-receptor
channels at a single activated synapse
on a single spine head that were open and conducting at a given
moment in time had to be very small (usually < 1).
4. With the use of these quantitative synaptic conductance descriptions, Ca’ ’ influx through NMDA-receptor
channels at a
synapse was computed for a variety of conditions. For a constant
number of pulses, Ca” influx was calculated as a function of
input frequency and the number of coactivated synapses. When
few svnapses were coactivated, Ca” influx was small, even for
high-frequencv
activation.
However, with larger numbers of
coactivated synapses, there was a steep increase in Ca2’ influx
with increasing input frequency because of the voltage-dependent
nature of the NM DA receptor-mediated
conductance. Nevertheless, total Ca” influx was never increased more than fourfold by
increasing input frequency or the number of coactivated synapses. Such increases are too small to explain the selective induction of associative LTP in light of the sine qua non control experiment.
5. The three- to fourfold increases in Ca” influx obtained by
increasing input frequency were amplified into 20- to 30-fold
increases in free Ca2+ concentration
in the spine head. This amplification of small changes in influx to large changes in concentration occurred because of transient saturation of the fast Ca2+
buffering systems. Varying the concentration and location of Ca2+
binding sites in the spine head provided insight into the range of
Ca2+ binding site concentration
values for which this amplification occurs.
6. The experimentally
described temporal requirements
of
dentate gyrus associative potentiation
studied with two stimulating electrodes, one activating a weakly excitatory input and the
other activating a strongly excitatory input, were examined in the
context of the present model. The temporal input patterns that
produced the best potentiation
of the weak input response in
previous experiments were the same ones that produced the largest Ca’+ influx and the largest peak spine head values of [Ca2’] in
the model. Parameter values estimated independently
in the development of the quantitative synaptic conductance descriptions
were critically responsible for this correspondence;
the timing results in the model depended on two slow rate constants in the
activation of the NMDA receptor-mediated
conductance.
7. In conclusion,
the nonlinearity
in Ca” influx through
NMDA-receptor
channels because of increases in input frequency
and intensity, although very important, was not sufficient to explain associative LTP. However, the transient saturation of the
fast Ca’+ buffering systems significantly amplified the nonlinearity in Ca’+ influx to provide a large, steep nonlinearity
in the
alteration of spine head Ca” concentration
as a function of convergent coactivity. Successive calcium-triggered
reactions could
further amplify this nonlinearity.
1NTRODUCTION
Long-term
potentiation
(LTP) is a modification of synaptic strength studied at excitatory synapsesof the hippo-
campus and its dentate gyrus (Bliss and Gardner-Medwin
1973; Bliss and Lomo 1973; Douglas and Goddard 1975;
Lomo 1966). As a synaptic phenomenon, LTP is the leading candidate for the subcellular basis of associative memory because of its longevity and because of the associative
coactivity requirement for its induction (Barrioneuvo and
Brown 1983; Lee 1983; Levy and Steward 1979;
McNaughton et al. 1978). There is much evidence suggesting that in the dentate gyrus and CA1 region of hippocampus, Ca” influx through N-methyl-o-aspartate
(NMDA)-receptor channels is the trigger event for the induction of LTP (Collingridge et al. 1983; Harris et al. 1984;
Lynch et al. 1983; MacDermott et al. 1986; Turner et al.
INSIGHTS
INTO
LTP
FROM
1982; Wigstrom
and Gustafsson
1984). Moreover,
this
conductance
offered by NMDA-receptor
channels is
blocked in a voltage-dependent
manner by Mg2+ (Mayer et
al. 1984; Nowak et al. 1984) in such a way that this voltage
dependence could explain the associative nature of LTP
(Herron et al. 1986; Wigstrom and Gustafsson
1985; see
also almost all of the articles in Landfield and Deadwyler
1988, each of which discusses the appeal of this hypothesis).
Although this voltage-dependent
conductance leads to
an associative property at synapses with NMDA receptors,
it is not at all clear that the characteristics
of this voltage
dependence are sufficient to explain the associative nature
of synaptic potentiation found in the hippocampus.
To
appreciate the problem, we should realize that the demonstration of associative LTP required a pair of experiments:
the demonstration
of LTP itself and a control experiment
in which activation of a weakly excitatory input failed to
show any potentiation
(Levy
and Steward
1979;
McNaughton
et al. 1978). In fact, when using a very weak
test stimulus, afferent activation at moderate-to-low
frequencies can occur as often as desired without producing
potentiation
(Levy and Steward, unpublished
observations). This means associative LTP depends on a high degree of nonlinearity. It can even be argued that this nonlinearity is, in a qualitative sense, as steep as possible. That is,
if repeated activation of a single synapse leads to truly zero
potentiation while coactivation of some larger number of
synapses leads to potentiation, then the nonlinearity is, at
least in part, a jump or step function of the number of
active afferents.
The search for such a nonlinearity is, in part, the goal of
the biochemical
model studied by Gamble and Koch
(1987); it is important to the biochemical theory of Lisman
(1985), and it also is implicit in the ideas of Miller and
Kennedy (1985, 1986) concerning the role of autophosphorylations. On the other hand, Herron et al. (1986) and
Wigstrom and Gustafsson (1985) are, at least implicitly,
proposing that the NMDA receptor can provide this nonlinearity.
To decide if the voltage dependence
of the
NMDA receptor-mediated
conductance is sufficient to explain synaptic associativity,
including the sine qua non
control experiment, we turned to quantitative modeling.
There were three stages to the modeling in this research.
In the first stage, we developed quantitative descriptions of
the NMDA and non-NMDA
receptor-mediated
synaptic
conductances
that reproduced experimentally
observed
NMDA and non-NMDA
receptor-mediated
components
of the somatic excitatory postsynaptic potentials (EPSPs).
Important considerations
were EPSP amplitude and time
course. In the second stage, these quantified synaptic conductances were used to simulate synaptic activation and
Ca’+ influx at dendritic spines of a hippocampal dentate
granule cell. We varied input frequency and the number of
coactivated synapses with the hope of finding a nonlinear
relationship between these variables and Ca2+ influx. A
nonlinear relationship occurred, but it was not sufficient to
explain both associative LTP and the requisite control experiment, the control being that repetitive activation of a
single synapse does not produce as large an increase in Ca2+
influx as in the associative activation case. Because of this
COMPUTATIONAL
MODELS
1149
result, we began the third stage of the modeling. Taking a
suggestion from the model of Gamble and Koch ( 1987), we
investigated a possible role for intracellular Ca2+ reactions
as an amplifier of the small nonlinearity provided by the
voltage dependence of the NMDA receptor-mediated
conductance. It was at this stage that a large amplitude nonlinearity was found. In fact, something as simple as buffer
saturation kinetics was capable of providing a significant
amplification of the nonlinearity arising from properties of
the NMDA-receptor
channel.
METHODS
Cell used in the simulations
The cell used in the simulationswasthe rat ventral leaf hippocampal dentate granule cell pictured in Fig. 1. The dendritic
lengths and diameters were quantified by Dr. N. L. Desmond and
were judged to be representative
(Desmond and Levy 1982,
1984). Because of variations in diameter, each interbranch segment was divided into subsegments in which the diameter was
approximately
constant. As a consequence, no dendritic subsegment used in the model was longer than 40 pm in length.
Dentate granule cells are covered with dendritic spines. Spine
counts, dimensions, densities, and types of spines for proximal,
mid-dendritic,
and distal locations on ventral leaf hippocampal
dentate granule cells were taken from previous studies (Desmond
and Levy 1985). This information
was used to include 4,304
dendritic spines in the model of this cell. These spines provided
37% of the total membrane area of the cell. Spines were modeled
explicitly at over 100 different locations on the dendrite indicated
by an arrow in Fig. 1. The model allowed many spines to be
activated simultaneously
at each location. To include the thousands of spines not explicitly
included in the model, specific
membrane resistance (R,) was reduced and specific membrane
capacitance (Cm) was increased on each dendritic segment in direct proportion to the increase in total membrane area provided
by spines on that particular dendritic segment. (For example, if a
dendritic segment excluding spines had a membrane area of 1,000
pm2, and if the spines on the dendritic segment also had a membrane area of 1,000 pm2, then this segment would be modeled
with R, halved and C, doubled from the assumed base-line
values given below.) EPSPs computed using this means of implicitly including spines were found to be virtually identical to EPSPs
computed
when all spines were explicitly
modeled (Holmes
1986a). Furthermore,
we found that the first several hundred
equalizing time constants of the cell (Rall 1969) computed with
the use of a compartmental
model (Perkel et al. 198 1) were identical to two to three digits whether one implicitly
or explicitly
included the dendritic spines. (The constant multipliers of the
exponential terms also were virtually identical except when they
were very small, i.e., ~0. 1% of the signal.)
In the simulations, R,, Cm, and the intracellular resistivity (Ri),
were assumed to be 12,000 Q-cm’, 1.O pF/cm’, and 70 Q-cm,
respectively. Input resistance was 72.4 MQ. The resting potential
of the cell was assumed to be -70 mV. These values are consistent
with those estimated in other studies for these cells (Brown et al.
198 1; Durand et al. 1983; Fricke and Prince 1984). The cell was
modeled without an axon, and no attempt was made to include
the voltage-dependent
conductances usually associated with action-potential
generation
and propagation.
Action potentials
would not propagate far enough up the dendrites to affect the
region of interest.
Synapses and synaptic conductances
Dentate granule cells receive over 95% of their input from layer
II stellate cells in entorhinal cortex (Steward and Vinsant 1983).
W. R. HOLMES
1150
AND
W. B. LEVY
Y
\
FIG. 1. Ventral
leaf hippocampal
dentate granule cell used in the simulations
(courtesy
of Dr. N. L. Desmond).
Dendritic spines were explicitly
modeled on
the dendrite indicated
by arrow. Synapses
activated
in the simulations
were all located within the boxed region.
\\
50 pm
Intracellular
and extracellular
studies demonstrate
that entorhinal cortex-dentate
gyrus (EC-DG) synapses are of the classic
excitatory type (Lomo 197 l), and a variety of results are consistent with glutamate or a glutamate-like
substance as being the
neurotransmitter
(Crunelli et al. 1982; Nadler et al. 1977; White
et al. 1977). NMDA receptors in particular have been found on
these cells (Monaghan
and Cotman 1985). In the model it was
assumed that glutamate is the neurotransmitter
at EC-DG synapses and that it interacts with NMDA receptors and non-NMDA
receptors to produce two different kinds of ionic conductance
changes (Wigstrom and Gustafsson 1987). The conductance mediated by non-NMDA
receptors was assumed to be a mixed
Na+/K+ conductance having a fast time course and a reversal
potential of 0 mV (Ascher and Nowak 1988a: Crunelli et al. 1984;
Mayer et al. 1984; Nowak et al. 1984). The channels opened by
interaction of glutamate at NMDA receptors were assumed to be
permeable to Ca2+, Na+, and KS with the relative permeability,
Pc,:PNa:PK, being 10.6: 1.O: 1.O (Mayer and Westbrook
1987).
This conductance had a reversal potential of 0 mV (Cull-Candy
and Ogden 1985; Mayer and Westbrook 1987; Nowak et al. 1984)
and had a fast rise time (but much slower than the non-NMDA
receptor-mediated
conductance) and slow decay (see below). Unlike the non-NMDA
receptor-mediated
conductance, this conductance was subject to voltage-dependent
Mg2+ block.
No voltage-dependent
Ca2 + channels were included in the
model. There were several reasons for this. First, LTP in the
dentate is induced by activation of NMDA-receptor
channels and
not by activation of voltage-dependent
Ca2+ channels. Depolarization alone, which would selectively activate the voltage-dependent Ca2’ channels, does not induce LTP. Second, if significant
numbers of voltage-dependent
Ca2’ channels exist on dentate
granule cells, then their role in neuronal function is much less
evident, at present. than, for example, in CA1 pyramidal cells.
The slow depolarizations
and burst discharges generated by voltpyramidal
age-dependent
Ca2’ channels in CA 1 hippocampal
cells are not seen in dentate granule cells (Fricke and Prince
1984). Third, there is very little data describing voltage-dependent
Ca2+ channels on dentate granule cells. We did not believe we
could justify assumptions about possible types (i.e., N, L, or T as
categorized by Nowycky et al. 1985), locations, or activation and
inactivation kinetics of such channels at this time.
Mathematical
procedure
The mathematical procedure used to model the voltage changes
at different locations within the cell is one described previously
(Holmes 1986b). Recently it was shown that the algorithm used in
this procedure compares favorably with other methods for solving
stiff systems of differential equations (Luxon 1987). The major
restriction of this procedure as applied to the present study was
that continuously
varying conductances had to be modeled by
piecewise constant functions. However, this was overcome by
taking small time steps when approximating
the smooth conductance changes mediated by the NMDA and non-NMDA
receptors.
Synaptic inputs and input ./iequency
The cell received synaptic inputs at anywhere from 1 to 115
different synapses on spines all located within the boxed region of
Fig. 1. Each synapse was activated a total of 8 times at frequencies
ranging from 10 to 800 Hz. Activation at 800 Hz is nonphysiological, but results obtained at this frequency were included for
comparison. Calcium influx at one synapse in the midst of those
activated was determined as a function of the number of coactive
inputs and the activation frequency of these coactive inputs.
Modeling
the synaptic conductances
It was very important that the NMDA and non-NMDA
receptor-mediated
synaptic conductances were modeled in an appropriate manner. Conductance descriptions were chosen to be consistent with published experimental
data, while at the same time
being simple enough to be used in a model as detailed as the one
INSIGHTS
INTO
LTP
FROM
used here. The a-function (Jack and Redman 197 1; Rall 1967) is
often used as a simple, biologically reasonable approximation
of
the synaptic conductance. However, the a-function is not appropriate for modeling the NMDA receptor-mediated
conductance
because it does not give the correct shape for the NMDA receptor-mediated
component of the EPSP. Consequently, to describe
the NMDA and non-NMDA
receptor-mediated
synaptic conductances, we used equations derived from a simple kinetic
model.
NON-NMDA
TANCE.
RECEPTOR-MEDIATED
SYNAPTIC
CONDUC-
The kinetic model used for the non-NMDA
receptormediated conductance is one that has been studied quantitatively
previously (Magleby and Stevens 1972; Perkel et al. 198 1)
AfRk?ARcAR*
I
1
B
(2)
a
The differential equations for AR and AR* obtained
kinetic model can be solved to give
AR(t)
= [ l/(r,
- r2)] { (AR(O)[r,
+ ~1 + aAR*(O))
-(AR(O)[r*
AR*(t)
= [ l/(vl
- r2)]
{(-AR*(O)[r,
MODELS
1151
duration in normal Mg2+ is less than the mean channel open time
in Mg2+-free medium) and because a bimodal distribution
of
channel open times has been obtained in some preparations (Jahr
and Stevens 1987). This representation
of the second path away
from AR* is a simplification
of an unknown pathway; the actual
mechanism is not important for the model. Second, NMDA-receptor channels are blocked by Mg2+ in a voltage-dependent
manner.
To account for these differences, the kinetic model for the
NMDA receptor-mediated
conductance was modified from the
non-NMDA
receptor-mediated
conductance
model (Eq. 2)
giving
P
where A, R, AR, and AR* are, respectively, the number of neurotransmitter molecules, receptor channels in the closed state, receptor-transmitter
complexes in the closed channel state, and receptor-transmitter
complexes in the open channel state. The rate
constants kl and kvl describe the binding and unbinding of neurotransmitter to the receptor, and p and cy describe the conformational transition between the closed and open states of the channel. Following Perkel et al. (198 l), we assume that neurotransmitter I) arrives at the postsynaptic
membrane
in a wave or a
&function (thus removing the reaction A + R + AR after t = 0),
2) some of the transmitter immediately binds a certain number of
receptors to form the complex AR, and 3) any remaining unbound transmitter or transmitter that becomes unbound at a later
time is immediately removed from the synaptic cleft via uptake
processes, inactivation, or diffusion. As a result, Eq. I reduces to
A+RzAR=AR*
COMPUTATIONAL
exp(W)
+ CY] +
@AR*(O)) exp(r2t>}
+ CU]+ DAR(O)) exp(rd)
- (-AR*(O)[r,
from this
+ a] +
and
AR* + Mg*+
where AR(O) and AR*(O) are the initial conditions for AR and
AR* (for the first input AR(O) = AR0 and AR*(O) = 0), r1 = -u +
(a2 - b)‘/‘, y2 = -u - (a2 - @l/2, a = (cu + p + k- ,)/2, and b =
cuk- 1. The non-NMDA
receptor-mediated
synaptic conductance
was then computed by multiplying
AR*(t) by 5 pS, the value
assumed for the single channel conductance for the non-NMDAreceptor channel. Values reported for this single channel conductance vary from 140 fS to 18-35 pS (Ascher and Nowak 1988a;
Cull-Candy and Ogden 1985; Cull-Candy et al. 1988; Nowak et
al. 1984; Tang et al. 1989; Trussell and Fischbach 1989); 5 pS was
used as an average value. It was not important to have a precise
value for the single channel conductance.
A different choice
would require a compensating
change in the value chosen for
ARo, for example, to match the experimentally observed EPSP (as
described below). The results would be almost identical.
NMDA
RECEPTOR-MEDIATED
CONDUCTANCE.
The kinetic
model used to determine the NMDA receptor-mediated
synaptic
conductance differed in two ways from the model used for the
conductance mediated by non-NMDA
receptors. First, a second
path leading away from AR* was included. This was done because
the channel burst duration
data (Ascher and Nowak 1988b;
Nowak et al. 1984) suggested that another path should exist (burst
s R*Mg*+
W)
where again A, R, AR, and AR* are, respectively, the number of
agonist or neurotransmitter
molecules, receptor channels in the
closed state, receptor-transmitter
complexes in the closed channel
state, and receptor-transmitter
complexes in the open channel
state. R* is the receptor channel in the open state, and R*Mg2+
and AR*Mg2+ represent open but magnesium-blocked
states. The
equations for magnesium block are based on the ion channel
block model used by others (Neher and Steinbach, 1978; Woodhull 1973). The scheme given by Eqs. 4a and 4b may be an
oversimplification
of the real situation, because there is evidence
suggesting that glycine and at least two glutamate molecules are
needed to reach the open state (Howe et al. 1988; Johnson and
Ascher 1987), but the true kinetic scheme remains to be determined. Nevertheless, use of this model gave results consistent
with a large body of experimental
observations
as discussed
below.
If it is assumed that the transition R* + R is extremely fast and
that the transitions R + R* and A + R* + AR* rarely if ever
occur, then the kinetic model reduces to
(3)
PAR(O)) exp(r&}
= AR*Mg*+
R* + Mg*+
A+Rp4R
B
1
=AR*zA+R
al
and
AR*
+ Mg’+
k+
F AR*Mg*
+
We assume that the blocking and unblocking of the channel by
Mg2+ occurs much faster than the transitions AR* --) A + R or
AR* + AR; thus Eqs. 5u and Sb can be considered separately.
Doing this, we find that the solutions of the differential equations
for AR and AR* based on the kinetic model of Eq. 5~ reduce to
those in Eq. 3 where cx = cyI + cy2as long as ,0 + k-i or cy. Because
we were forced to choose such a p as described below, Eq. 3 was
used to determine AR(t) and AR*(r) for the NMDA receptor-mediated conductance with CY= cyI + cy2.
To solve the differential equation corresponding
to Eq. 5b, we
note that the blocking and unblocking reactions are fast on the
time scale given by cyand ,0. As a result, this reaction is essentially
at equilibrium,
and the proportion
of open, unblocked channels
can be obtained from
mklblocked/A KLl = (k/k
+ )/((k/k
+ ) + [Mg*‘])
(6)
An equation expressing k-/k+ as a function of voltage is given by
Ascher and Nowak (1988b).
With the use of Eq. 6 for the percentage of unblocked channels,
we computed the chord conductance and current-voltage
(I-V)
1152
W. R. HOLMES
B
A
Voltage
-80
-60
-40
-20
0
Current
(mV)
I/
J.
+20
Voltage (mV)
FIG. 2.
Chord conductance
(block function)
and I-V plot. A: block
function computed
from Eq. 6 as a function of voltage. Ordinate
gives the
fraction of channels that are not blocked. One-half
of the open channels
are blocked by Mg2+ at -25 mV. [Mg2’] was 1.2 mM. Chord conductance
is obtained if the ordinate is multiplied
by a constant (given by the number
of open channels multiplied
by the single channel conductance).
B: I-V
plot corresponding
to the chord conductance
function
given in A. Ordinate scale is arbitrary
because it depends on the constant used to compute
the chord conductance.
plots shown in Fig. 2. These plots, based on the Ascher and
Nowak expression for k-/k+, have the same general shape as
those obtained from experiments
with cultured spinal cord
neurons (Mayer and Westbrook
1984, 1985), although differences, possibly resulting from the high concentration
of extracellular calcium used in those studies, do exist at more hyperpolarized potentials. The NMDA receptor-mediated
synaptic conductance was computed
from the number of open, unblocked
channels, AR&rocked (obtained from Eqs. 3 and 6) multiplied by
50 pS, the single channel conductance for the NMDA-receptor
channel (Ascher et al. 1988; Cull-Candy et al. 1985; Cull-Candy
and Usowicz 1987; Nowak et al. 1984).
Choosing synaptic conductance parameter
AND
W. B. LEVY
1984) and in the hippocampus.
(Monaghan
and Cotman 1985;
Monaghan et al. 1985; Rainbow et al. 1984).
5) The affinity of glutamate for NMDA receptors is two orders
of magnitude stronger than that for non-NMDA
receptors, i.e.,
IQ are on the order of 3 and 300 PM, respectively (Mayer, personal communication).
6) The time course of the non-NMDA
receptor-mediated
component of the EPSP has a fast rise and a fast decay, whereas
the time course of the NMDA-receptor
component (as revealed in
Mg*+-free medium) has a slower rise and can take over 100 ms to
decay (Collingridge
et al. 1988a,b; Dale and Roberts 1985; Forsythe and Westbrook 1988; Wigstrom et al. 1985).
NON-NMDA
ETERS.
CONDUCTANCE
PARAM-
1.
Non-NMDA and NA4DA receptor-mediated
conductanceparameters
TABLE
Parameter
Description
A. Non-NMDA-receptor
Value
reaction
P
A+RzAR=AR* I2
I
values
To use Eq. 3, values for A&, cy,kAl, and ,0 had to be chosen for
both conductances. The parameter values used in the simulations
are given in Table 1. We chose values that would I) give nonNMDA and NMDA receptor-mediated
components of the somatic EPSP with a time course and amplitude similar to that
found experimentally,
and 2) satisfy a number of other experimental observations. Our choices were either fixed or severely
constrained by the following experimental observations:
I) The mean channel open time for the non-NMDA
receptormediated conductance is < 1 ms (Jahr and Stevens 1987) whereas
the mean channel open time for the NMDA receptor-mediated
conductance is on the order of 5-7 ms (Ascher et al. 1988; CullCandy and Usowicz 1987; Nowak et al. 1984). Jahr and Stevens
(1987) report a bimodal
distribution
of open times for the
NMDA-receptor
channel with peaks at l-3 and lo- 15 ms.
2) The amplitude at the soma of an EPSP generated at a single
synapse has been estimated to be on the order of 100-500 PV
depending
on distance from the soma (Jack et al. 197 1;
McNaughton
et al. 198 1).
3) Although the relative amplitudes of the non-NMDA
and
NMDA receptor-mediated
components of the EPSP have been
found to vary widely, the amplitude of the NMDA receptor-mediated component is usually smaller than that of the non-NMDA
receptor-mediated
component (Collingridge
et al. 1988a,b; Dale
and Roberts 1985; Forsythe and Westbrook 1988; Lambert and
Jones 1989; Wigstrom et al. 1985).
4) The number of NMDA
receptors has been found to be
approximately
the same as or greater than the number of nonNMDA receptors in brain postsynaptic densities (Fagg and Matus
RECEPTOR-MEDIATED
By assuming a value of 0.5 ms for the mean channel
open time, we chose CY,which is the inverse of the mean channel
open time, to be 2.0 ms-‘. The dissociation constant, k-, , was also
chosen to be 2.0 ms-I; that is, 1OO-fold larger than the km1of the
NMDA-receptor
reaction (which was constrained by the falling
phase of the EPSP as described below) in accordance with observation 5). The other two parameters, /3 and ARO, were chosen to
make the amplitude and shape of the non-NMDA
receptor-mediated component
of the EPSP approximately
correct. Unique
;
k-1
AR0
Inverse of the mean channel open time
Transition
to the open channel state
constant
Receptor-transmitter
dissociation
constant
Receptor-transmitter
complexes
after a
pulse of transmitter
Single channel conductance
B. NMDA-receptor
2.0 ms-’
2.0 ms-’
2.0 ms-’
200
5 PS
react ion
B
A+RrAR+AR*zA-kR
’ I
aI
ARzlblocked
/A Rhll = (k- lk t MU- lk +) + wk2’-11
Ql
a2
P
k-1
AR0
k- lk+
Inverse of the mean channel open time
Inverse of the mean channel open time
Transition
to the open channel state
constant
Receptor-transmitter
dissociation
constant
Receptor-transmitter
complexes
after a
pulse of transmitter
Single channel conductance
KMS for Mg2+ block reaction
Extracellular
See text for references
Mg2-‘- concentration
for some parameter
values.
0.17 ms-’
0.17 ms-’
0.0006 ms-’
0.02 ms- ’
200
50 ps
8.8 x 1o-3
exp(V/ 12.5)
1.2 mM
10.6: 1.O: 1 .O
INSIGHTS
INTO LTP FROM COMPUTATIONAL
valuescould not be chosenfor thesetwo parameters.We chose
values of 2 ms-’ and 200 for p and A&, respectively,although
doubling AR0 and halving p (or vice versa)gave a similar EPSP.
The exact numbersfor p and A& are not important so long as
their product is maintained.Thesevaluesarecollectedin Table 1.
The effect of theseparameterchoiceswasa synapticconductance
that had a peak of 0.28 nS with a time to peak of 0.42 ms and a
half-width of 1.47ms.
NMDA
TERS.
RECEPTOR-MEDIATED
CONDUCTANCE
PARAME-
From observation I, above, there may be one or two
meanchannelopen times for NMDA-receptor channels.If there
is only onemeanchannelopentime of 5.9 ms,then cyland cy2are
0.17 ms-‘, and their sum, cy,is 0.34 ms-‘. If there are two mean
channel open times of 3.6 and 15 ms, then cyis still 0.34 ms?
Becauseit is the sum cyl-t-cy2that matters,many paired valuesof
cyland cy2are mathematicallyequivalent.
The constant, kml, had its primary effect on the decay of the
EPSP.Becausethe decay of the EPSPhasbeen observedto last
100msor more, we chosek_l to be 0.02 ms-I.
A& was chosento be 200, the sameas for the non-NMDA
receptor-mediatedconductance,basedlargely on observation4).
One could argue that the number should be either larger or
smaller.Observation 5 arguesthat the number should be much
larger (however, the forward binding constants, if available,
would be more informative than Kd values).On the other hand,
both observation3 and the suggestionthat two or more glutamate moleculesmay bind to an NMDA receptor arguethat the
number shouldbe smaller.
The main effectsof p and A& were on the amplitude of the
NMDA receptor-mediatedcomponent of the EPSP.TO havethe
correct amplitude, p waschosento be 0.0006 ms? As with the
non-NMDA receptor-mediated conductance, the particular
valueschosenfor p and AR0 were not important aslong astheir
product was maintained. The constraint applied here was that
AR0 shouldbe the samefor both reactions.
These parameterchoicesare also collected in Table 1. Peak
conductance,time to peak, and half-width for this synaptic conductancewere not computedbecauseof the dependenceof these
parameterson local depolarization asdiscussed
below.
Modeling
MODELS
1153
puted wasnot a pure calcium current. The relative permeability
of Ca2+,Na+, and IV at thesechannels,Pc,:PNa:PK,has been
estimatedto be 10.6:1.O:1.O(Mayer and Westbrook 1987).Using
thesevaluesand standardvaluesfor intracellular and extracellular
ion concentrationsalong with activities (as in Mayer and Westbrook 1987),we computedthe Ca2+,Na+, and K+ componentsof
the synaptic current through NMDA-receptor channels.Total
Ca2+influx wasthen determinedby integrating the calcium current over time.
Determining
[Ca2+] in the spine head
A compartmentalmodel of Ca2+flux in a dendritic spinethat
includedconsiderationof Ca2+binding reactionswasusedto calculatethe Ca2+concentration in the spineheadover time (Fig. 3).
Th e spine head and neck and the dendrite were representedas
cylinders. There were 11 compartmentsin this model: 4 for the
spine head, 3 for the spine neck, and 4 for the dendrite. Each
compartment was a short cylinder or disc. A model with eight
additional dendritic compartmentswasusedto extend the length
of the dendritic segmentunder considerationwhen [Ca2+]in the
dendrite wasthe variable of interest. Theseadditional dendritic
compartmentsdid not affect the computedvalue for [Ca2+]in the
spinehead. Compartments 1 and 2 were both 50 nm thick and
representedthe immediatesubsynapticvolume. The sizesof the
other compartmentsare drawn to scalein Fig. 3.
The calcium current at one synapse, computed from the
NMDA receptor-mediatedsynaptic current as describedabove,
producedan influx of Ca2+through the outer surfaceof the end
spineheadcompartment (compartment 1 of Fig. 3). We did not
include voltage-dependentca2+ channelsin the spinebecause,as
repetitive synaptic activation
Each synaptic activation was assumedto add AR0 receptortransmitter complexesto the total numberof receptor-transmitter
complexes,AR(t). This is equivalentto sayingthat the number of
receptorsis much largerthan the number of receptorsbound to
transmitter (at leastduring the 8 synaptic activations modeled
here).This assumptionwill not affect the non-NMDA receptormediatedconductancebecausethis conductancehassucha fast
time course.On the other hand, the NMDA receptor-mediated
conductancehasa slowtime course,and if the numberof NMDA
receptorsweresmall,then nearly all would be bound to transmitter with the first or secondafferent activation. However, such
receptor saturation was not found in twin pulse experiments
(Muller et al. 1988;Muller and Lynch 1988, 1989), suggesting
that the number of NMDA receptorsis much larger than A&.
IO
8
Determining CaZt influx from the NMDA
receptor-mediated current
FIG. 3. Dendritic spine model. A dendritic spine was modeled as a
series of compartments. Calcium entered through the tip of compartment
apsewas obtained from ZNMDA= g( V - VreV)where g was the 1. Once in the spine, Ca2+ could diffuse into neighboring compartments,
NMDA receptor-mediatedconductanceand I& wasthe reversal become buffered, or be pumped out of the cell. Spine dimensions were
potential. However, becausethe NMDA-receptor channelsare 0.55 X 0.55 pm for the spine head and 0.73 X 0.1 pm for the spine neck.
permeableto Na+ and K+ as well as to Ca2+,the current com- Section of dendrite was 2.0 X 1.O pm. Spine neck resistance was 70 MQ.
Current,
&WID~ 9 through NMDA-receptor channelsat a syn-
1154
W. R. HOLMES
AND
mentioned earlier, it is believed that LTP in the dentate is induced
by Ca2+ influx through NMDA-receptor
channels, not Ca2+ influx
through voltage-dependent
Ca2+ channels.
Once in the spine head, Ca2+ could diffuse into a neighboring
compartment,
become bound to buffer, or be removed from the
cell by a Ca2+ pump. These processes were modeled by the use of
the compartmental
approximation
to the one-dimensional
diffusion equation that resulted in the following equation for compartment i
d[Cali/dl= -ul c:((A/6h,i-*(LCali
- [W-J + (Al~)i,i+,([Cali
- [Wi+h}
+ influx
(for compartment
1 only)
- kpi( [Cal; - [Cal,)
- kbfLCaliLBli
+ libb(LBtli- IBli)
d[B]i/dr
= -khf[Ca]i[B]i
+ I\-&([Bt]i
(0
- [B];)
where [Cali, [B]i, and [Bt]i are the concentrations
of Ca2+, free
buffer, and total buffer in the ith compartment,
[Cal, is the resting
Ca” concentration
(assumed to be 20 nM), D is the diffusion
constant for Ca’+ (0.6 pm2/ms), Vi is the volume of the ith compartment, and (nl6)i.j is the coupling coefficient between compartments
i and j. This coupling coefficient is computed
as
2(Ai li + /Ij /j)/( /i + I’j)” where Ai and Aj are the cross-sectional areas
and li and lj are the lengths (or thicknesses) of compartments i and
j. The constants kpi, khf, and khb are the rate constants for the
Ca2+ pump and for the association and dissociation of Ca2+ and
buffer, respectively. Values used for the various parameters in the
spine model are described below and are summarized in Table 2.
The constant liyi was computed from the velocity of the ATPdependent Ca” pump ( 1.4 X 10m4 cm/s) (Blaustein 1976) multiplied by Ai/ I/i where Ai is the membrane area of compartment
i.
This pump has low capacity (200 fmol/cm2-s)
(DiPolo and
Beauge 1983). At the [Ca2’] where it reaches maximum capacity,
the low-affinity, high-capacity Na+-Ca2+ exchange pump begins
to operate. This exchange has a similar velocity over the range of
intracellular [Ca2’] studied here (see Fig. 2 of DiPolo and Beauge
1983). Thus both transport mechanisms were lumped into a single term in Ey. 7.
The forward and backward rate constants, kbf and kbb, for the
buffer are unknown.
Based on a Kd of 1 PM for calmodulin
(Connor and Nikolakopoulou
1982; Klee et al. 1980; Olwin and
Storm, 1985) we chose khb to be 0.5 ms-’ and kbf to be 0.5
PM-‘mss,
values similar to those used by others (Connor and
Nikolakopoulou
1982; Gamble and Koch 1987; Simon and
Llinas 1985). The rate constants were varied an order of magnitude with and without changing the Kd value to test the robustness
of the results to such variations.
Parameter
D
4)
khh
A-i+-
WI
[Cal,
Description
Value
Ca’ ’ diffusion
coefficient
Pump rate constant
Backward
buf%er rate constant range of
values tested
Forward
buft‘er rate constant range of
values tested
Total buff‘cr concentration
(near PSD)
range of values tested
(Compartments
3-l 1, away from PSD)
range of values tested
Resting Ca’+ concentration
Spine head dimensions
(1 X d)
Spine neck dimensions
(1 X d)
See text for references
tic density.
for particular
parameter
values.
0.6 pm2/ms
1.4 X 10e4 cm/s
0.5 ms-’
0.054 .o
0.5 PM-’ ms-’
0.05-5.0
200 PM
100-400
100 PM
50-200
20 nM
0.55 X 0.55 pm
0.73 X 0.10 pm
PSD, postsynap-
W. B. LEVY
The Ca2+ buffer concentration
in each compartment
is also
unknown. We initially chose 200 PM in compartments
1 and 2
(the first 100 nm from the tip of the spine head) and 100 PM
elsewhere, because calmodulin
is part of the postsynaptic density
(Grab et al. 1979, 1980; Wood et al. 1979). The 200 PM estimate
for Ca2+ binding sites was based on an assumed calmodulin concentration of 20-40 PM (Carafoli 1987) with four independent
sites per molecule, and the presence in smaller unknown concentrations of calbindin, calcineurin,
other Ca2+-binding
proteins
and phospholipids.
The effects of Ca2+ sequestering organelles
and other possible mechanisms for long-term
buffering were
omitted because of the short time period of the simulations (< 1 s;
usually ~200 ms). Because actual buffer concentration
in dendritic spines is unknown,
we modeled several different buffer
concentrations.
RESULTS
This section is divided into four parts. In the first part,
quantitative descriptions of the NMDA and non-NMDA
receptor-mediated
synaptic conductances are used to produce NMDA and non-NMDA
receptor-mediated
components of a somatic EPSP similar to those observed experimentally. To do this, while at the same time satisfying the
constraints imposed by other reported experimental observations, some of the rate constants given in the synaptic
conductance equations had to assume values within narrow ranges. In the second part, these synaptic conductance
descriptions are used in a model of a dentate granule cell to
compute Ca2+ influx into a spine head as a function of
input frequency and the number of coactivated synapses.
Because the results given by this model fail to fit a critical,
experimentally observed characteristic of associative LTP,
the analysis is extended, in the third part, to allow calculation of Ca2+ concentration in the spine head as a function
of repeated synaptic activation. This is done with the use of
the Ca2+ influx calculated in the second part along with the
additional considerations of Ca2+ diffusion, pumps, and
binding in the spine head. This more comprehensive
model succeeds where the previous model failed. In the
fourth part, it is shown that the model successfully predicts
and explains the effects of asynchronous synaptic activation of weak and strong afferent pathways on the induction
of LTP as observed experimentally.
Fitting the components ofthe somatic EPSP
The first step in modeling Ca2+ influx through NMDAreceptor channels at a synapse was to obtain descriptions of
the NMDA and non-NMDA
receptor-mediated
synaptic
conductances that would produce NMDA
and nonNMDA receptor-mediated
components of the somatic
EPSP similar to those found experimentally. To do this we
used Eqs. 2 and 3 to describe the non-NMDA
receptormediated conductance and Eqs. 5, 6, and 3 to describe the
NMDA receptor-mediated
conductance. Initial values for
the unknown conductance parameters were chosen based
on the experimental
observations given in METHODS.
These initial values were varied, within the constraints imposed by the experimental observations, until the separate
NMDA and non-NMDA
receptor-mediated
components
of the somatic EPSP appeared to have the same time course
as those observed experimentally
(under conditions in
INSIGHTS
INTO
LTP
FROM
which inhibition
was blocked). The final values for the
conductance parameters were those specified earlier in
Table 1.
With the use of these conductance descriptions and conductance parameter values, five synapses on dendritic
spines located within the boxed region of Fig. 1 were coactivated to produce a composite EPSP. We activated five
synapses instead of one to facilitate the comparison with
reported experimental
data, because the experimental
EPSPs used for comparison were usually composite EPSPs.
[The amplitude of an EPSP evoked by activation of a single
synapse is quite small (100-500 pV)] (Jack et al. 197 1;
McNaughton
et al. 198 1). Pharmacologic manipulations
can be used to separate the NMDA and non-NMDA
receptor-mediated
components of the somatic EPSP. The
use of such manipulations
was simulated in the model by
activating the two synaptic conductances first separately, at
the five synapses, and then together, assuming 0 and 1.2
mM extracellular [ Mg*+].
The somatic EPSP produced by the non-NMDA
receptor-mediated conductance rose and fell quickly as shown
in Fig. 4A. Figure 4B illustrates the somatic EPSP produced
by the NMDA receptor-mediated
conductance in 0 extracellular [Mg*+]. This EPSP rose and fell slowly; the decay
back to rest took over 100 ms, which is much longer than
expected based on the membrane time constant. Coactivation of the two conductances in 0 extracellular [Mg*+] gave
the EPSP in Fig. 4C. These results were very similar to
those obtained experimentally in a number of preparations
in which inhibition and one component of the EPSP was
blocked pharmacologically
(Collingridge et al. 1988a; Dale
and Roberts 1985; Forsythe and Westbrook 1988; WigStrom et al. 1985).
When both conductances were activated in 1.2 mM extracellular [Mg*+], the resulting somatic EPSP was nearly
identical to that given in Fig. 4A for activation of the nonNMDA receptor-mediated-conductance
alone. The depolarization resulting from the activation of both conductances at five synapses was too small (3-4 mV in the spine
head) to relieve the voltage-dependent magnesium block of
the NMDA-receptor
channels. Thus, with this amount of
activation, the NMDA receptor-mediated
conductance
contributed only negligibly to the EPSP.
SENSITIVITY
TER CHOICES.
OF
THE
MODEL
TO CONDUCTANCE
PARAME-
Because A& and the rate constants in the
conductance equations have not been measured precisely
in experiments, we examined the effects of variations in the
chosen parameters on the shape and amplitude of the somatic EPSP components. TO do SO,we doubled and halved
the value of each parameter. We found that small variations in the individual values of cy,& k-, , and AR0 used for
each conductance were not very important, and that, at
least for the non-NMDA
receptor-mediated
conductance,
certain large variations could be made provided that constraints, as given in METHODS,
were satisfied.
For the non-NMDA
receptor-mediated
conductance,
the major change seen with doubling or halving a parameter was a change of scale. Halving cy, the inverse of the
mean channel open time, or k-, , the receptor-transmitter
COMPUTATIONAL
MODELS
1155
A
non-NMDA
NMDA
20
40
60
80
100
120
r
I.U -
non-NMDA
0
20
40
60
Time
+ NMDA
80
100
120
(ms)
NMDA
and non-NMDA
receptor-mediated
components
of
FIG. 4.
the somatic EPSP. A: non-NMDA
component.
Superimposed
on this
curve is the sum of the NMDA
and non-NMDA
components
of the
somatic EPSP in 1.2 mM [Mg2’].
Difference
was too small to be detected
in this figure. B: NMDA
component
in 0 [Mg’+].
C’: sum of the NMDA
and non-NMDA
components
given in A and B.
dissociation constant, or doubling 0, the constant reflecting
the allosteric transition to the open channel state, shifted
the peak of the EPSP (as given in Fig. 4A) slightly to the
right, but otherwise these changes merely increased the
amplitude of the EPSP less than twofold. (Doubling CYor
k-, , or halving ,6, produced complementary
changes.) As
might be expected, doubling or halving A&, the number of
receptor-transmitter
complexes formed after a pulse of
transmitter, nearly doubled or halved the peak amplitude
of the EPSP without changing the shape. The fact that
twofold changes in the parameter values produced less than
twofold changes in EPSP amplitude
and only minor
changes in EPSP shape indicates that small variations in
parameter values are not important. As for large variations
in parameter values, one could double ,6 and halve k-, , for
example, and still have a non-NMDA
receptor-mediated
EPSP component with a shape and amplitude in accordance with the experimental observations. One can compensate for a large change in one parameter with a large
complementary
change in another as long as constraints
imposed by other experimental observations are satisfied.
For the NMDA
receptor-mediated
conductance,
changes in p or A& scaled the amplitude of the EPSP,
whereas changes in -CYand k__, primarily
affected EPSP
shape. Doubling or halving ,6 or A& almost exactly doubled or halved the NMDA receptor-mediated
component
of the EPSP (as given in Fig. -4B). The small size of ,6
prevented the doubling or halving of this parameter from
affecting the shape of-the EPSP iwith p small, its major
1156
W. R. HOLMES
AND
contribution comes in the ,&AR(O) term in Eq. 31. Doubling
or halving CYfor this conductance produced changes in the
EPSP analogous to those found for doubling or halving CY
on the non-NMDA
receptor-mediated
EPSP. The peak of
the EPSP was shifted slightly to the right when CYwas
halved and to the left when CYwas doubled, with the peak
amplitude changing less than twofold. Because of the small
size of k- 1, halving or doubling this parameter had comparatively small effects on the peak amplitude of the EPSP,
but it did have a major effect on the falling phase of the
EPSP. Doubling k-, made the EPSP fall much faster,
whereas halving k-, produced an EPSP that decayed very
slowly. These results indicate that, as found above for the
non-NMDA
receptor-mediated
conductance, small variations in the conductance parameters will not severely distort the shape and amplitude of the EPSP.
However, there was less flexibility for large parameter
changes for this conductance than there was for the nonNMDA receptor-mediated
conductance. Here, one could
double 0 and halve A& and still have the same EPSP; but
even if ,0 were doubled several times, its value would still be
small, and one would have the same EPSP with a much
smaller A&. As for the other parameters, the experimentally reported mean channel open time fixed CY,and the
slow falling phase of the EPSP forced kel, the receptortransmitter dissociation constant, to assume a small value
in a certain range.
To illustrate the effects of the NMDA receptor-mediated
conductance parameters k-, and ,@(or equivalently, A&)
on the somatic EPSP, we doubled ,0 in one simulation and
halved k-, and reduced p by - 16% (to 0.0005 ms-‘) in
another (p was reduced to keep the peak amplitude at a
level similar to that seen in Fig. 4B). The results are given
in Fig. 5. The effect of doubling /3 was to double the NMDA
receptor-mediated component of the EPSP. This produced
a somatic EPSP with a very slow initial rate of decay (compare Fig. 54 with Fig. 4C) not unlike some EPSPs observed
by others experimentally (Dale and Roberts 1985). When
k- 1 was halved to 0.0 1 ms-’ and 6 was reduced to 0.0005
ms I, the falling phase of the EPSP was much prolonged
(Fig. 5B). In fact, such very slowly decaying EPSPs have
been observed in some preparations (i.e., Forsythe and
Westbrook, 1988).
To summarize, the purpose of this section was to produce a relevant biophysical model of NMDA and nonNMDA receptor-mediated
synaptic currents. To do this,
we quantified the various synaptic conductance parameters
in the multistate receptor-transmitter
reactions given in
METHODS
in such a way that somatic EPSPs computed
with the model were similar to those observed experimentally. For some parameters, large changes away from the
values given in Table 1 ruined the fit of amplitude and time
course between theoretical and experimentally
observed
synaptic events. For other parameters, the synaptic event
was unaffected by a wide range of changes in one parameter provided there was a compensating change in a different
parameter; for these parameters, the actual values were
irrelevant to the modeling that followed because only the
fitting of the synaptic event, and not the individual parameters, was critical.
W. B. LEVY
A
1.0
0.8
0.6
0.4
sE
w
0.2
0.0
Q) B
z
%
>
1.0
0.8
non-NMDA
+ NMDA
0.6
Time (ms)
FIG. 5. Somatic
EPSPs with different
,8 and k-r values. EPSP and the
NMDA
receptor-mediated
component
of the EPSP in 0 extracellular
[Mg2’] are given (A) when ,8 was doubled to 0.00 12 and (B) when k-, was
reduced to 0.0 1 and ,8 was reduced to 0.0005. Values of the other parameters were those given in Table 1. Compare
these EPSPs with those in Fig. 4,
B and C. Non-NMDA
receptor-mediated
EPSP component
was the same
as given in Fig. 4A.
Ca2’ influx as a .function oj’input frequency and the
number ofcoactivated inputs
In the second stage of the simulations, the conductance
descriptions given above were used in a model to compute
Ca2+ influx through NMDA-receptor
channels as a function of input frequency and the number of coactivated
inputs. NMDA and non-NMDA
receptor-mediated
synaptic conductances on different numbers ( I- 115) of dendritic spines, located on the dentate granule cell as indicated in Fig. 1, were activated eight times at frequencies of
lo-800 Hz. Ca2+ influx was computed at one synapse on a
dendritic spine located in the midst of all of the coactivated
synapses. This Ca2+ influx was plotted in Fig. 6A as a function of frequency for different numbers of coactivated synapses.
When the number of coactivated synapses was large,
there was a nonlinear, sigmoid increase in the level of Ca2+
influx with increasing input frequency (up to 200 Hz) that
was not observed with lower levels of activation (Fig. 6A).
For one or five coactivated synapses, increases in input
frequency had little effect on Ca2’ influx. For 10 or 20
coactivated synapses, there were small increases in Ca2’
influx with increasing frequency. However, for 58 or more
coactivated synapses, there was a nonlinear, sigmoid increase in Ca2+ influx with increasing frequency. The steepness of this increase grew larger with larger numbers of
coactivated synapses. Above 200 Hz, Ca2+ influx reached a
plateau and then decreased slightly with further increases
in frequency. Despite the steep rise in Ca2+ influx observed
with increasing input frequency when the number of coactivated synapses was large, the difference between Ca’+ influx at 10 Hz and 200 Hz was never more than fourfold.
The graph in Fig. 6A also shows that, for Ca2+ influx to be
INSIGHTS
6.0
INTO
LTP
FROM
Number
of
co-activated
synapses
Y
115
COMPUTATIONAL
B
96
g
*
-5
77
;
=I
=c
.6
s
39
20
IO
-1
6.0
Frequency
-
4.0
*
*0
3.0
2.0
0
Frequency
(Hz)
1157
5.0
I
58
MODELS
Number
20
40
60 80
of co-activated
100
(Hz)
800
400
200
100
50
25
10
120
synapses
FIG. 6. Ca” influx as a function
of input frequency
and the number of coactivated
synapses. Ca2’ influx (in femto-coulombs) at a single synapse is plotted (A) as a function
of input frequency
for different
numbers of coactivated
synapses and
(B) as a function
of the number
of coactivated
synapses for different
frequencies.
Bottom CUYW in B is a reference curve
corresponding
to the influx resulting
from a single activation
of l-l 15 synapses multiplied
by 8 (the total number
of
activations
in the stimulus train). Data is the same for both plots except for the bottom czuvc in B.
doubled by increasing input frequency, there must be a
sufficiently large number of coactivated synapses. When
there were 20 or fewer coactivated synapses, Ca*+ influx
could not be doubled regardless of input frequency. With
58 coactivated synapses, Ca*+ influx doubled when input
frequency was increased from 10 to 200 Hz.
The data in Fig. 6A were replotted in Fig. 6B to show
Ca*+ influx as a function of the number of coactive inputs
for different frequencies. Here we found that Ca*+ influx
increased up to fourfold as the number of coactivated synapses increased from 1 to 115. At 10 Hz, there were only
small increases in Ca*+ influx with increasing numbers of
coactivated synapses. At 25 Hz, Ca*+ influx almost doubled as the number of coactivated synapses increased from
1 to 115. The steepest increases in Ca*+ influx were seen at
frequencies of 100 and 200 Hz, where Ca*+ influx rose
threefold or more as the number of coactivated synapses
increased. At frequencies of 400 Hz or more, the increases
in Ca’+ influx were still steep with increasing numbers of
coactivated synapses, but only as long as the number of
coactivated synapses was small. Once the number of coactivated synapses exceeded 58, the rate of increase in Ca*+
influx declined to that seen at 25 Hz.
To illustrate further the separate contributions of input
frequency and number of coactivated synapses to the nonlinearity observed in Ca*’ influx, an additional reference
curve was added at the hottom of Fig. 6B. Ca*+ influx
resulting from a single coactivation of 1- 115 synapses was
multiplied by 8 and plotted in this curve (8, because the
other curves represented the coactivation of I- 115 synapses a total of 8 times at the different frequencies). There
was no effect of input frequency on this additional curve,
and Ca’+ influx increased -40% as the number of coactivated synapses increased from 1 to 115. The Ca*+ influx
given by this curve was only slightly less than that observed
with increasing numbers of synapses coactivated eight
times at 10 Hz. Thus an input frequency of 10 Hz did not
significantly augment Ca*+ influx per synaptic activation.
A small increase in Ca*+ influx because of frequency was
seen at 25 Hz. However, only when large numbers of synapses were coactivated eight times at 50 Hz, or above, did
Ca*+ influx approach being double that observed with a
single coactivation multiplied by 8.
An understanding of the sigmoid relationship, illustrated
in Fig. 6, between Ca*+ influx and input frequency when
large numbers of synapses were coactivated came from
studying the changes in spine head potential and NMDA
receptor-mediated
conductance over time. These changes
are illustrated in Fig. 7 for 96 synapses coactivated at 50
and 200 Hz. In Fig. 7A, the potential in one spine head is
plotted over time. At 50 Hz, each of the eight synaptic
activations produced almost identical changes in the spine
head potential. The potential changes were too widely separated in time to sum to a voltage effective for relieving
Mg*’ block of the NMDA-receptor
channels. As a result,
conductance through NMDA receptor-channels was low
(Fig. 7B), and Ca2+ influx was small. In contrast, at 200 Hz,
activation was fast enough to produce temporal summation of the spine head potential. This increased level of
depolarization,
compared with that produced at 50 Hz,
relieved the voltage-dependent Mg*+ block of the NMDAreceptor channels. Consequently, there was a large increase
in the NMDA receptor-mediated
conductance (Fig. 7B)
and a much larger Ca*+ influx than at 50 Hz.
Thus we have a quantitative assessment of the voltagedependent feedback process as often hypothesized as the
explanation for associativity. With high-frequency, highintensity afferent activation, spatial-temporal
summation
of synaptic activation was translated via dendritic depolarization into activation of the NMDA receptor-mediated
conductance.
Ca2’ concentration as a function of input frequency and
the number of coactivated inputs
Although Ca*+ influx through NMDA-receptor
channels
increased in a nonlinear, sigmoid manner with increasing
1158
W. R. HOLMES
A
-10
T
-70
f
0
Hz
-
:
20
40
60
80
Time
3
10
E
2
0
W. B. LEVY
of a dendritic spine. These processes were 1) diffusion, 2)
Ca2+ binding, and 3) pumped Ca2+ removal. On a time
scale of hundreds of milliseconds, high affinity binding was
the most important of the three.
The dendritic spine model had 11 compartments as pictured in Fig. 3. The time course of Ca2+ influx into the
outermost compartment,
for different input frequencies
and numbers of coactivated synapses, was obtained from
the simulations described in the previous section. Once
Ca2+ entered the cell, it could diffuse into neighboring
compartments, be pumped out of the spine, or be bound.
Unless noted otherwise, the concentration of Ca2+ binding
sites was 200 PM within 0.1 pm of the surface of the spine
head at the synapse (in the area of the postsynaptic density)
and 100 PM elsewhere.
When l-l 15 synapses were coactivated once, the concentration of Ca2+ in one spine head rose from the 20 nM
resting level to a transient peak of 89- 118 nM. When these
same synapses were coactivated 8 times, much larger increases in Ca2+ concentration occurred, and the magnitude
of these increases grew steeply with increasing input frequency and increasing numbers of coactivated synapses.
The Ca’+ concentration at the tip of the spine head (compartment 1 in Fig. 3) is plotted in Fig. 8 for 1,20,58, and 96
synapses coactivated at 50 and 200 Hz. When a single
synapse was activated eight times at 50 Hz, peak [Ca2’] in
the spine head was 0.92 PM. When this one synapse was
coactivated with 95 other synapses eight times at 50 Hz, the
peak [Ca”] in the spine head was 7.1 PM. A twofold difference in Ca2+ influx between these two cases was amplified into over a sevenfold difference in peak Ca2+ concentration. This amplification
occurred because the buffer in
the spine head was transiently saturated. An even greater
difference was seen when input frequency was 200 Hz. At
this frequency, peak [Ca2’] near the tip of a spine head was
0.97 PM for activation of a single synapse but was 28.4 PM
when 96 synapses were coactivated. Even though there was
only a 3-fold difference in Ca2+ influx between these two
cases, this difference was amplified into nearly a 30-fold
difference in peak Ca2+ concentration near the postsynaptic density because of transient saturation of the Ca2+
buffer.
Large differences in peak [Ca’+] were also seen when
frequency was increased for a given number of coactivated
synapses. Figure 8 shows a fourfold increase in peak [Ca2’]
when input frequency was raised from 50 to 200 Hz. This
contrasts with a less than twofold increase in Ca2+ influx.
Not shown in Fig. 8 is [Ca2’] when 96 synapses were coactivated at 10 Hz; in this case, peak [Ca2’] was only 1.1 PM.
Thus, when 96 synapses were coactivated, there was more
than a 25-fold increase in peak [Ca”] in the spine head
when input frequency was raised from 10 to 200 Hz.
In these simulations there was a slight gradient of [Ca”‘]
through the rest of the spine head and a steep gradient
through the spine neck. In all cases there were comparatively small changes in [Ca2’] in the parent dendrite over
the time observed; for example, the concentration in the
dendrite (compartment 8 in Fig. 3) rose from 20 nM to 122
nM by 1,000 ms when 96 synapses were coactivated eight
times at 200 Hz. However, [Ca”‘] in the dendrite (but not
z:::;:;:;:200
50 Hz
-60
AND
Time
100
120
140
160
(ms)
(ms)
FIG. 7. Spine head potential
and NMDA
receptor-mediated
synaptic
conductance:
96 spine synapses were coactivated
at 50 and 200 Hz. A:
potential at 1 spine head is given as a function
of time. t3: NMDA
receptor-mediated
synaptic conductance
at the synapse on the same spine head
is plotted over time.
input frequency for large numbers of coactivated synapses,
the size of this increase was never more than fourfold. A
fourfold increase would not seem to provide the amplitude
(or “safety factor”) or the specificity for a process thought
to trigger the induction of associative LTP. If the number
of impulses delivered to a single synapse at low frequency
were increased fourfold, from 8 to 32, the total Ca2+ influx
would be the same as that produced with eight pulse associative coactivity. However, such extended solitary activation does not produce LTP (Harris et al. 1978). This motivated the third stage of the simulations.
Because the philosophy of this research was to keep the
model as simple as possible, we hypothesized that just the
next reaction(s) after Ca2’ influx would provide an additional nonlinearity sufficient to explain associative LTP.
The next reaction is Ca2+ binding to calmodulin and molecules having similar affinity for Ca2+. These reactions
would precede the Ca2+ -dependent enzymatic activations
that might actually lead to LTP. Thus the modeling shifts
from studying just Ca2+ influx to studying Ca2+ concentration. Specifically, we modeled changes in the concentration
of Ca’+ over time in different parts of a dendritic spine as a
function of input frequency and the number of coactivated
synapses.
To predict free Ca”+ concentration required us to consider three processes in addition to Ca2+ influx in a model
INSIGHTS
INTO
LTP
FROM
COMPUTATIONAL
MODELS
1159
thus we examined the steepness and amplitude of the nonlinearity as a function of the values of the binding reaction
50 Hz
96
rate constants and the location and concentration of Ca2+
T
binding sites in the spine head.
When the ratio of the buffer reaction rate constants kb,,/
kbf (or &) was kept constant, large changes in kbb and kbf
produced only small changes in the time course and peak
of the Ca2+ concentration. For example, for 96 synapses
coactivated at 200 Hz, doubling or halving both kbb and kbf
changed the peak [Ca”] value by < 1%. Furthermore, the
new time courses were nearly indistinguishable
from that
given in Fig. 8. With a IO-fold increase in both rate constants, the increase in the peak [Ca”] value was still < l%,
0
50
100
150
200
250
300
and again, the time course was virtually unchanged. When
Time (ms)
the rate constants were both reduced an order of magniB
tude from 0.5 to 0.05, the peak Ca2+ concentration fell by
96
5-6%, and one could see small differences in the time
200 Hz
course of the Ca2’ concentration. The reason that these
large changes in the Ca2+ -binding reaction rate constants
n
produced such small changes in the Ca2+ concentration
was that the Ca2+-binding reaction occurred on a much
faster time scale than Ca2+ influx, diffusion, or removal.
58
Thus, as long as kbb and kbf were large enough, it was not
5 10
important
what particular values were used in the model
1
for
a
given
&.
\
20
When the buffer reaction rate constants, kbb and kbf,
1
I
.
I
.
were changed in a way that their ratio, Kd, also changed,
0
1'1'1'
II
'.
1I
'.
0
20
40
60
80
100
120
140
the results remained qualitatively the same, but there were
Time (ms)
quantitative differences, especially when Kd became large.
With the use of the same example of 96 synapses coactiFIG. 8. Time course of Ca*+ concentration
as a function
of input frevated
at 200 Hz as was used above, the effects of fivefold
quency and the number
of coactivated
synapses. [Ca*‘] in 1 spine head
variations
in Kd were examined by changing either kbf or
(compartment
1, Fig. 3) is given for 1, 20, 58, and 96 synapses coactivated
kbb by a factor of 5 (see Eq. 7).
at (A) 50 Hz and (B) 200 Hz. Buffer concentration
was 200 PM near the
postsynaptic
density (compartments
1 and 2, Fig. 3) and 100 PM elseWhen Kd was increased from 1 to 5 PM, peak Ca2+ conwhere.
centration increased 12- 13% regardless of whether kbb was
reduced or kbf was increased. Larger differences might be
possible for different numbers of coactivated synapses, but
in the spine head) depended on the length of the dendrite
qualitatively the results would be unchanged. Increasing
used in the model and on the number of spines with coac- the unbinding rate constant or reducing the binding rate
tivated synapses sharing this length of dendrite. The den- constant meant that more free ionized calcium ions were
drite was modeled with four compartments and was 2 pm present, and consequently the concentration was increased.
long with closed ends in all of the simulations whose results With the larger Kd, the [Ca*‘] began rising a little earlier
are illustrated. In one simulation we lengthened the den- and decayed more slowly than shown in Fig. 8, but otherdrite to 5 pm by increasing the number of dendritic com- wise the time courses were similar. Quantitatively, a larger
partments. In this case, [Ca2’] in the dendrite at the base of Kd would mean that slightly fewer synapses would need
the spine peaked at 89 nM at nearly 120 ms when 96 to be coactivated at a given frequency to get a given
synapses were coactivated eight times at 200 Hz. The peak [ Ca2+].
longer length of the modeled dendrite allowed Ca2+ to difReducing Kd had a smaller quantitative effect. When Kd
fuse into other dendritic compartments. However, when 5 was 0.2 instead of 1.O, because of reducing kbb or raising kbf
of the 96 coactivated synapses were on spines located on by a factor of 5, peak [Ca”.] was reduced by 2%. The [Ca”]
this 5-pm-long dendrite, peak dendritic [Ca2’] was ~400
began rising slightly later and fell a little more quickly than
nM. Nevertheless, even with this high density of coacti- in Fig. 8, making the change in [Ca2’] a little sharper; but
vated synapses, peak dendritic [Ca2’] was <2% of the peak again, all of the qualitative features seen in Fig. 8 were
spine head [Ca”‘].
preserved.
Changes in the location and concentration of Ca2’-bindDEPENDENCE
OF RESULTS
ON CA2+-BINDING
PARAMETER
VALUES.
Unlike the nonlinearity in Ca2+ influx, the non- ing sites in the spine head, besides changing [Ca2’], did
have an effect on the steepness and amplitude of the nonlinearity observed in peak Ca2+ concentration approached
both the steepness and the amplitude necessary to trigger a linearity in peak Ca2+ concentration. However, the nonlinselective potentiation process. However, some of the Ca2+- earity remained steep and the amplitude remained large as
long as the concentration of Ca2+-binding sites was not too
binding parameter values are only known approximately;
A
8
1160
W. R. HOLMES
large. This is shown in Table 3. In this table the amplification in [Ca”‘] that occurred when the number of synapses
coactivated at 200 Hz was increased from 1 to 96 is compared for five different buffer distributions.
The middle
distribution gives the standard buffer concentration values
used in the model.
In the two cases when buffer concentration was lower
than the standard model values, the buffer in the spine
head saturated rapidly when large numbers of synapses
were coactivated. Peak [Ca”] was 1% and 29,fold larger
with 96 synapses coactivated at 200 Hz than with 1 synapse
activated at 200 Hz. The l&fold difference was smaller
than that found with the buffer concentrations used in the
simulations because of the large value for peak [Ca2’] when
one synapse was activated. The 29-fold difference was the
same as that seen with the standard buffer concentration
values used in the model.
In the two cases when buffer concentration was higher
than the standard model values, Ca2+ concentration
peaked at a much lower level because more calcium could
be buffered. Peak [Ca”‘] was 9- and 13-fold larger with 96
synapses coactivated at 200 Hz than with 1 synapse activated at 200 Hz. Although these increases were still large,
they were less than one-half those seen with lower buffer
concentrations. As the number of coactivated synapses was
increased above 96 (i.e., to 1,008), Ca2+ influx approached
an asymptotic
value. Thus Ca2’ concentration
could
always be kept at a low level with a sufficiently large concentration of binding sites (although this sufficiently large
value might not have any physiological reality).
In summary, the qualitative results given by the present
simulations were not affected by large changes in Ca2+binding parameter values. Changes in the rate constants
associated with Ca2’ binding produced no qualitative
changes in the results, and in many cases even the quantitative changes were very small. The results were more sensitive to changes in Ca2+ -binding site concentration. When
buffer concentration was low compared with hypothesized
values, large changes in [Ca”] were not selectively dependent on coactivity; there were large changes in [Ca”‘] with
one or just a few coactivated synapses. At very high buffer
concentrations, changes in [Ca”‘] were small. However,
there was a wide intermediate range of buffer concentration values that produced the desired steep increases in
[Ca2’]; that is, [Ca2+] increases were large, and they were
large only when large numbers of synapses were coactivated. Thus, although buffer concentration was an important variable for producing the sought after selectivity,
TABLE 3.
AND W. B. LEVY
these simulations showed that having precise buffer concentration values was not particularly critical in a qualitative sense.
DEPENDENCE
PARAMETER
OF RESULTS ON SYNAPTIC CONDUCTANCE
VALUES.
To determine if the large, steep
nonlinearity
in peak [Ca”] with increasing numbers of
coactivated synapses was dependent on the conductance
parameter values used in the conductance equations, we
used the parameter values that generated the different
EPSPs illustrated in Fig. 5 in one set of simulations. When
,B was doubled (as was done to get Fig. 54, Ca2+ influx
increased four- to fivefold as input frequency and the number of coactivated synapses were increased. These four- to
fivefold increases in Ca2+ influx were amplified into 20- to
30-fold increases in peak spine head [Ca2’]. This amplification factor was similar to that illustrated in Fig. 8, even
though there were quantitative differences in Ca2+ influx
and [Ca2’]. When k+ was reduced (as in Fig. 5B), the
increases in Ca2+ influx were less than threefold. These
increases in influx were amplified into lo- to 15.fold increases in peak [Ca”‘] in the spine head. Thus, although
there were quantitative changes in Ca2+ influx and peak
spine head [Ca2’], the qualitative role of high affinity Ca2+
buffering systems as an amplifier of the nonlinearity
in
Ca2+ influx was robust with respect to these variations in 0
and k-, .
Asynchronous synaptic activation: LTP timing studies
Up to this point all of the synapses activated in the simulations were activated simultaneously. These simulations
were direct analogues of experiments that examined associativity by varying the intensity of stimulation from a single electrode during the conditioning
periods (i.e.,
McNaughton et al. 1978; Wilson 198 1). However, in other
experimental studies, two stimulating electrodes are used
to allow experimentally
controlled, asynchronous activation of a pair of inputs, one weakly excitatory and the other
strongly excitatory (i.e., Gustafsson and Wigstrom 1986;
Levy and Steward 1983). In these studies, potentiation of
the weakly excitatory input response occurs only when the
weak and strong inputs are activated together in certain
temporal sequences.
For example, consider results obtained in the dentate.
Dentate granule cells receive a strongly excitatory ipsilatera1 projection from entorhinal cortex that converges with
a weakly excitatory contralateral projection. Stimulation of
Amplificution of[Ca*‘] in compartmentI
Total [Buffer],
PSD, PM
Total [Buffer],
Elsewhere, PM
Peak [Ca’],
1 Activated Synapse,
PM
Peak [Ca’+], 96
Coactivated Synapses,
PM
100
100
200*
200
400
50
100
100*
200
200
2.81
1.24
0.97
0.48
0.40
50.5
35.8
28.4
6.2
3.6
PSD refers to compartments 1 and 2; elsewhere to compartments 3- 11. *Standard values used in the model.
Amplification
Ratio
18
29
29
13
9
INSIGHTS
INTO LTP FROM COMPUTATIONAL
MODELS
1161
synapse activated eight times. Stimulation of the small conthe weak contralateral projection alone (8 pulses delivered
at 400 Hz) does not potentiate the contralateral response. tralateral projection was modeled as the coactivation of
Activation of the strong ipsilateral projection produces a five synapses in the same manner. The simulations used
powerful postsynaptic event. When both projections are the same temporal sequences as in the experimental study.
stimulated (8 pulses at 400 Hz), potentiation of the weak Levy and Steward found that the weak contralateral projeccontralateral input response can occur depending on the tion had to be stimulated before-or simultaneously with
temporal relationship between the two inputs during con- -the strong ipsilateral projection to potentiate the contraditioning stimulation (Levy and Steward 1983).
lateral response. The model gave results in qualitative
agreement with the Levy and Steward study as is shown in
This study was repeated in a set of simulations. Stimulation of the large ipsilateral projection was modeled as Fig. 9. Figure 9A gives the timing sequences and compares
coactivation of 50 different synapses at 400 Hz with each the modeled Ca*+ influx and peak spine head [Ca”] with
Experimental
Result
Modeled
Ca influx (fC)
Depression
2.33
1.7
Depression
2.52
2.1
Depression
2.78
2.8
Potentiation
4.14
9.6
w+s
1 ms
Potentiation
5.18
28.1
w-s
Potentiation
4.80
23.1
Potentiation
4.18
14.8
t-
A
10ms
Weak (5)
Strong (50)
s-w
Modeled
Peak [Cal (PM)
ZOms
+I ! ! ! + ! !
s-w
W
S
8ms
s-w
1 ms
s+
w
8ms
w-s
20ms
C
W 8msS
-20 T
S 8ms W
W 8ms S
-60
a
0
IO
20
30
40
50
Time (ms)
60
70
80
0
lo
20
30
Time
40
50
(ms)
FIG. 9. Modeling asynchronous synaptic activation experiments from the dentate gyrus. Weak (contralateral) inputs
were paired with strong (ipsilateral) inputs in the indicated timing sequences. Weak and strong inputs were represented in
the model by coactivation of 5 and 50 synapses,respectively. A: comparison of modeled results with experimental finding of
potentiation or depression. B: NMDA receptor-mediated conductance at a weak input synapse is plotted when the strong
input train preceded the weak input train by 8 ms and vice versa (2nd and 6th timing sequences in A). C: spine head
potential over time is plotted for the synapses whose conductances were plotted in B. Note that with S 8 ms before W, the
large depolarization had substantially decayed before weak input synapses were glutamate-activated and, hence, before
NMDA-receptor channels at these synapses could open; however, with W 8 ms before S, the large depolarization came after
the weak input synapses were glutamate-activated and at a time when the NMDA-receptor channels at these synapsescould
open.
60
70
80
1162
W. R. HOLMES
AND W. B. LEVY
the experimental observation of potentiation
or depres- happened to Ca2+ after it entered the spine head. We found
sion. The largest Ca2+ influx and largest peak spine head that transient saturation of fast Ca2+ buffering systems am[Ca”] occurred when the weak contralateral input pre- plified the associatively induced 3- to 4-fold increases in
ceded the strong ipsilateral input by 1 ms. The smallest
Ca2+ influx into 20- to 30-fold increases in spine head
Ca”+ influx and smallest peak spine head [Ca”‘] occurred
[Ca”]. Such large increases via buffer saturation kinetics
when the strong ipsilateral input preceded the weak input
easily could be amplified further by nonlinearities in subby 20 ms. Thus the variation in [Ca”‘] was qualitatively
sequent Ca2+ -dependent enzyme reactions. To examine
the fuller relevance of the model to associative LTP, we
consistent with the experimental results.
The reasons for the different Ca2+ influx with different
also simulated the temporal characteristics of associative
timing patterns can be understood by studying Fig. 9B. The LTP and found that the parameter values inferred during
top czlrve in this figure gives the NMDA receptor-mediated
development
of the synaptic conductance descriptions
conductance at a synapse activated by the weak contralatwere suitable ones for simulating the temporal requireera1 input when the weak input train preceded the strong ments of associative LTP.
by 8 ms. The conductance was small during the first 25 ms
when only the weak input had been activated. Once the Conductance description and parameters
strong input was activated, the conductance increased draThe equations describing the NMDA and non-NMDA
matically. During the time that the weak input was acticonductances were based on a comvated, transmitter became bound to the NMDA receptors receptor-mediated
(A + R + AR, see Eq. 5). However, because of the slow monly used kinetic scheme (Magleby and Stevens 1972).
kinetics of the transition to the open state (AR --) AR*), few Although this scheme may accurately describe the nonreaction, it is obviously a
channels opened, and the ones that did were subject to NMDA receptor-transmitter
of the NMDA receptor-transmitter
reacvoltage-dependent Mg’+ block (AR* --) AR*Mg2+). At the simplification
time that the strong input was activated, there were still a tion. Our equation did not include the modulatory effects
large number of receptor channels in the AR state because of glycine (Johnson and Ascher 1987) or the three open
of the extremely slow back transition A + R + AR. These and three closed channel states proposed by others (Howe
receptor channels could move to the AR* state, but now the et al. 1988). Nevertheless, this simple kinetic equation provided a conductance representation that reproduced the
depolarization
brought about by the strong, ipsilateral
major characteristics of the NMDA receptor-mediated
input (Fig. 9C) relieved much of the voltage-dependent
Mg’+ block of these channels. The result was that, even component of the EPSP revealed experimentally in Mg2+though there was no new, additional transmitter arriving at free medium.
Whatever the true kinetic scheme may be, the parameter
the synapses of the weak input afferents, the NMDA receptor-mediated conductance increased dramatically. The values chosen in the simple kinetic equation, as conbottom c’urvc in Fig. 9B shows the conductance when the strained by the experimental data, imply that there must be
strong input preceded the weak by 8 ms. In this case, when a reaction in which the unbinding of glutamate is slow, and
the weak input was activated, the depolarization caused by there must be a reaction that makes the transition to the
the strong input had decayed significantly (Fig. 9C) before open channel state very slow. These slow transitions are
the weak input channels reached the open state. Conse- represented by k-, and ,@in our equation. The fact that
quently, once these channels moved to the AR* state, there fitting the synaptic conductances required a small dissociation constant, k-, , for the NMDA receptor-transmitter
rewas not enough depolarization to relieve the Mg2+ block
action, is consistent with the fact that the affinity of glutaand allow a large Ca”+ influx.
mate for NMDA receptors is very high. The very small ,0
we
were forced to choose reflects the fact that when I)
DISCUSSION
receptors are plentiful in number, 2) receptor affinity for
To model Ca’+ influx through NMDA receptors, it was transmitter is very high, and 3) the single channel conducfirst necessary to develop quantitative descriptions of the tance of the open, unblocked channel is large, a tremenNMDA and non-NMDA
receptor-mediated
conducdously large activation will occur unless something keeps
tances. Although these descriptions did not include every the channels from reaching the open state. Channel block
by magnesium ions is one means to limit the NMDA rehypothesized open and closed channel state, they produced
EPSPs comparable with those found experimentally while ceptor-mediated
conductance, but experiments done in
being consistent with numerous other pieces of experimenMg2+-free medium indicate that this, by itself, is not suffital data (observations 1-6, METHODS). Given these de- cient. There must be a step in the reaction scheme that
scriptions of the synaptic conductances, it was then possi- proceeds very slowly and thus prevents many channels
ble to model Ca”’ influx at a spine head on a dentate
from becoming open. The role of this slow transition is
granule cell as a function of the number of coactivated
played by the very small ,B in our kinetic scheme.
synapses and input frequency. We found that the voltageA number of factors could influence the values of the
dependent properties of the NMDA-receptor
channel al- parameters that describe the conductances. First, the equalowed a steep, nonlinear increase in Ca2+ influx to occur tions for Mg2+ block were based on expressions derived
with increasing input frequency when the number of coac- from data in which extracellular [Mg2+] was much lower
tivated synapses was large. However, the size of this in- than that thought to exist in vivo (maximum of 150 PM vs.
crease was too small if associativity is to be dependent
1.2 mM in vivo). Although extrapolating these results to
solely on this Ca2+ influx. Consequently, we looked at what physiological [Mg2+] seems reasonable, the actual effect on
INSIGHTS
INTO
LTP
FROM
the block function remains unknown. Second, and perhaps
more importantly,
our estimates of the conductance parameter values were based largely on data from experiments performed at room temperature. One would expect
faster kinetics and, consequently, a faster NMDA receptor-mediated EPSP at physiological temperatures. In fact,
shorter NMDA-receptor
channel mean open times have
been obtained from data taken at higher temperatures
(Nowak et al. 1984). Ascher et al. (1988) have estimated
values of the temperature coefficient, QIO, to be 1.6 for the
NMDA channel conductance and 2.0 for the channel
mean open time for an increase in temperature between 14
and 24OC. Based on some of the results presented here and
results of a preliminary model (Holmes and Levy 1988),
the major effect of faster kinetics on the modeled results
would be a small increase in the steepness of the nonlinearity in Ca” influx through NMDA-receptor
channels.
However, none of the qualitative conclusions presented
here would be changed by faster kinetics. Quantitatively,
the fourfold difference between lowest and highest Ca2+
influx shown in Fig. 6A would increase to about fivefold;
the slope of the transition would be steeper, and peak Ca2+
influx would occur at a higher frequency. These differences
in Ca2+ influx would translate into a steeper nonlinearity in
spine head Ca2+ concentration as a function of frequency,
with the largest change occurring at a higher frequency
than that found with the kinetic parameters used here.
Once the effect of temperature on both the kinetic parameters and the components of the EPSP is better described
quantitatively, the model can be modified accordingly.
COMPUTATIONAL
MODELS
1163
conductance at a middendritic
synapse, lasting for 20 ms,
can produce a similar amplitude of depolarization at the
soma as a 5-nS synaptic conductance lasting 0.2 ms.
[Quantitatively,
many factors influence this relationship
(Holmes 1989)]. However, NMDA-receptor
channels have
both a large single channel conductance and a long mean
open time. By comparison, non-NMDA-receptor
channels
have a small single channel conductance and a short mean
open time-average
values of both parameters being about
one order of magnitude smaller than for NMDA-receptor
channels. Furthermore, NMDA receptors are plentiful on
dentate granule cells, and the affinity of glutamate is much
higher for NMDA receptors than for non-NMDA
receptors. Thus, with equal numbers of NMDA and nonNMDA-receptor
channels reaching the open, conducting
state, one would expect the non-NMDA
receptor-mediated component of the EPSP to be totally overshadowed
by the NMDA receptor-mediated
component in Mg2+-free
medium. Because this does not happen, few NMDA-receptor channels can be open and conducting simultaneously at a synapse.
Second, large changes in [Ca”.] in the spine head would
occur with activation of a single synapse if many NMDAreceptor channels were conducting simultaneously
at a
synapse. It has been estimated that an influx of only 50 free
calcium ions would be required to increase the calcium ion
concentration in a Purkinje cell dendritic spine to 5 PM if
there were no buffering of this influx (Andrews et al. 1988).
The spine head in Fig. 3 has a volume of 1.31 X lo-l6 1,
which is larger than that in the Purkinje cell spines studied
by Andrews et al., but the situation is similar. Because of
buffering mechanisms, the resting concentration of free
Very few NMDA-receptor channels are simultaneously
ionized calcium is quite low; a [Ca”] of 20 nM would
open-and conducting at a synapse
imply that there are only one or two free calcium ions in
channels are open
An interesting consequence that followed from the the spine head. If two NMDA-receptor
quantitative description of the NMDA receptor-mediated
for 6 ms, the driving force is 70 mV, and the Ca2+ component of the current (all but 5% of which is blocked by Mg2+)
conductance was that only small numbers of NMDA-receptor channels were ever simultaneously open and con- is IO%, then there would be a Ca2’ influx of 0.21 fC
ducting at a single synapse. This is in agreement with the (50 pS X 70 mV X 10% X 5% X 6 ms X 2) or 600-700 Ca2+
suggestion of Huettner and Bean (1988) that the probabilions. Unbuffered, this influx would raise [Ca”] to 8.3 PM
ity of NMDA-receptor
channel opening is very small. In in the spine head given in Fig. 3. Now, if instead, 20
channels were opened, and these were
retrospect, one can understand why this has to be true by NMDA-receptor
considering the converse. If, at one synapse, many chan- opened 10 times a second ( 100-Hz activation), it is easy to
nels became open simultaneously,
then 1) the resulting
see that Ca2+ influx could become tremendously large. This
influx could overwhelm the capacity of fast buffering
conductance change would produce an EPSP in Mg2+-free
medium that would be much larger and much different in mechanisms in the cell and cause [Ca”] to rise steeply and
shape than has been observed, and 2) there would be no to a high level with only one activated synapse. The capacity of the fast Ca2’ buffering mechanisms in the spine head
associativity because large changes in spine head [Ca”]
could occur with activation of a single synapse. These two is unknown, but it would have to be extremely large to
avoid saturation in such a circumstance. There would be
points will be discussed in turn.
no associative aspect to LTP if such large Ca’+ concentraFirst, if many NMDA-receptor
channels were conducting simultaneously, the NMDA receptor-mediated
com- tions in the spine head could be produced by activation of a
ponent of the somatic EPSP would totally dominate the single synapse. For associativity to exist as a Ca2+-depenchannels can be
non-NMDA
component in Mg2+-free medium. Two im- dent phenomenon, few NMDA-receptor
portant properties of a synaptically induced ionic conducopen and conducting simultaneously at a single synapse.
tance are its strength, as measured by the peak conducOther models of spine head [Ca”] also argue against
tance, and its duration. To produce an EPSP with a given there being more than a couple of NMDA-receptor
chanamplitude, a synapse can have a conductance with a large nels open simultaneously at a given synapse. Gamble and
amplitude and short duration or a conductance with a Koch (1987) modeled changes in spine head [Ca2’] resultsmall amplitude and long duration. For example, in un- ing from activation of voltage-dependent Ca2’ channels
published simulations we have found that a 50-pS synaptic
and obtained significant changes in [Ca”] with a conduc-
W. R. HOLMES
tance change equivalent to a single 0.5-pS channel being
open for 2 ms (see their Fig. 2). Although the duration of
the NMDA receptor-mediated
conductance is much
longer than 2 ms, their study and ours are in agreement in
showing that very few open channels are needed to get
significant changes in spine head [Ca”‘].
Should the NMDA receptor-mediated
synaptic
conductance be modeled stochastically?
In Fig. 7B, the NMDA receptor-mediated
conductance
at one synapse was given for two simulated conditions.
Although the single channel conductance for an NMDAreceptor channel is 50 pS, the NMDA receptor-mediated
conductances given in Fig. 7B were always ~60 pS. It
might seem that channel opening should be modeled stochastically instead of deterministically,
as was done here. If
channel opening were modeled stochastically, the NMDA
receptor-mediated
conductance would always be a multiple of 50 pS. However, at physiological concentrations of
[Me’+], there is little reason to consider going to stochastic
models because of the fast flicker between the open and
Mg’+-blocked states of the NMDA-receptor
channel (i.e.,
see Fig. 1 of Ascher and Nowak 1988b). The fast flicker
between the open and open-blocked states of the channel
creates a conductance that, if averaged over very short time
intervals (tenths of milliseconds), would closely resemble
that given by the deterministic model used here. With this
fast flicker between the open and closed states, it is easy to
understand how the computed average conductance could
be less than that for a single open channel.
In Mg’+-free medium, however, a stochastic description
of the NMDA receptor-mediated
conductance should be
used to model Ca2+ influx. With 0 extracellular Mg2+, the
time course of Ca2+ influx would be different if channel
opening were modeled stochastically rather than deterministically, although one might not expect to see much of a
difference in total Ca2+ influx. Because such differences
might exist, we did not report values for Ca2+ influx or peak
[Ca”] computed with our deterministic model when there
was 0 extracellular [Mg2’]. Modeling channel opening stochastically instead of deterministically
would not affect the
somatic composite EPSPs computed for Mg2+-free medium illustrated in Figs. 4 and 5, however, because the
activated synapses were far from the soma (electrotonic
distances ranging from 0.3 to 0.6). The effects of stochastitally modeled variations in the conductance time course
on the EPSP would be smoothed by the time the EPSP
reached the soma.
Ca” influx through NMDA-receptor channels is not
su$kiekt
to explain associative LTP
.
Over a certain voltage range (Fig. 2), the voltage dependence of Mg” block results in positive feedback of the
NMDA receptor-mediated
conductance upon itself. That
is, each increment of depolarization increases the conductance that leads to further depolarization, and so on. As
expected in the simplest theory of NMDA receptor-mediated associativity, the simulations showed a steep, nonlinear increase in conductance and, therefore, in Ca2+ influx, as input frequency was increased-provided
that a
AND
W. B. LEVY
sufficient number of synapses were activated simultaneously.
However, Ca2+ influx at a synapse was only fourfold
larger when 115 synapses were coactivated eight times at
200 Hz than when one synapse was activated eight times at
10 Hz. This is certainly a significant amount, but it is not
enough of a difference to explain the experimentally
observed intensity dependence of LTP. In particular, weak
low-frequency activation can be extended not just fourfold,
but to 40-fold more pulses without inducing LTP in the
experimental situation (Harris et al. 1978).
There is one additional point concerning Fig. 6 that deserves comment. When the number of coactivated synapses was large, Ca2+ influx declined at very high frequencies. There are two reasons for this decline. First, when
large numbers of synapses were coactivated at high frequencies, spine head depolarization approached the synaptic reversal potential. Although this depolarization
increased the NMDA-receptor
channel conductance, the
driving force for synaptic current was much reduced. The
increase in conductance was overwhelmed by the reduced
driving force, and thus Ca2+ influx was smaller. Second, the
mean channel open time for NMDA-receptor
channels in
the model, 5.9 ms, was longer than the interval between
successive synaptic activations at 400 Hz. With large numbers of coactivated inputs, the additional depolarization
provided by the high-frequency input was less important
than the interval between synaptic activations in the eightpulse stimulus train. A shorter time period between synaptic activations prevented Ca2+ influx from taking full advantage of the 5.9-ms mean channel open time. Taken
together, these factors caused Ca2+ influx to decline at very
high input frequencies.
Intracellular Ca 2+ reactions ampl(jj the nonlinearity
provided by NMDA-receptor channels
Because the nonlinear increase of Ca”+ influx as a function of increasing numbers of coactivated synapses and
input frequency was not large enough to explain the induction of associative LTP and its control, we investigated
Ca2+ concentration changes in the spine head. There are a
number of Ca2’-dependent reactions that may occur in the
spine head, and each of these may or may not contribute to
the nonlinearity required for the induction of LTP. However, instead of looking at these reactions, we looked at
reactions that both control spine head [Ca”] and precede
many, if not all, of these Ca2+-dependent reactions as a
possible source of nonlinearity.
Control of [Ca”] is very delicate in a dendritic spine
because of its small volume. When a single synapse was
activated once, Ca2’ influx was 0.20 fC. When a single
synapse was activated eight times, Ca2+ influx was
1.67- 1.68 fC, depending on input frequency ( lo-800 Hz).
If all of this Ca2+ remained unbound, the concentration of
Ca2+ in the spine head would rise to 7.8 PM with the single
activation and to 65 PM with eight activations (regardless
of frequency). Such large changes in [Ca”] would be likely
to activate every Ca2+-dependent mechanism in the spine.
This should not occur every time a single synapse is activated, because then there would be no associativity for the
INSIGHTS
INTO
LTP
FROM
induction of LTP. Thus, to keep [Ca”] from rising too fast
or too high, there must be at least a fast, brief duration
mechanism that prevents significant [Ca’+] increases after
nonassociative activation of a single synapse.
After introducing
CaZi buffers, pumps, and diffusion
into the dendritic spine model, we found that the concentration of Ca’+ near the postsynaptic density was kept low
after single, isolated synaptic activations. However, large,
steep, and transient increases in [Ca”] occurred after highfrequency activation of large numbers of synapses because
the fast Ca’+ buffer became saturated. Although transient,
this saturation of the fast Ca” buffer in the spine head was
able to amplify the small nonlinearity
of Ca” influx
through NMDA-receptor
channels into a large nonlinearity in Ca’+ concentration.
If the short-term buffer capacity in the dendritic spine is
primarily due to Ca’+ binding proteins such as calmodulin,
calbindin, or calcineurin, then the same substances that
trigger many Ca” -dependent reactions help produce a
steep nonlinearity just by buffering calcium. Any further
nonlinearity inherent in subsequent calcium-dependent
reactions (e.g., cooperative enzymatic reactions, autophosphorylating properties of CAM-kinase
II, or even the differential time course of CAM-kinase
II functionally countered by a calmodulin-activated
phosphodiesterase)
would
only amplify the sensitivity and the safety factor of the
processes involved in the induction of associative LTP.
Perhaps several amplification steps are required to induce
associative LTP. If our assumed values for Ca”+ binding
site concentration are close to correct, then transient saturation of the fast Ca’+ buffer is the first, and possibly the
most important, of such amplification steps.
The
AFFECTING
THE QUAN’I‘lTATIVEl
RESULTS.
quantitative results depended on parameters used to describe the kinetics of Ca” buf5ering reactions, pumps, and
diffusion in the spine head. Because values for these parameters are unknown
for spine heads, we had to use
values estimated from the squid and other preparations. Of
these parameters, the rate constants associated with highaffinity Ca’j binding are most important because of the
fast time scale of these binding reactions. Fortunately, the
qualitative and quantitative features of our results are robust to variations in these rate constants and their ratio, or
&. Similarly, short-term buffer capacity and distribution is
not known, and thus several direrent values were used in
the simulations. Not surprisingly,
the buffer was most effective in limiting spine head [Ca’-‘] when the highest concentration was located within 100 nm of the location of
calcium entry or within the area of the electron-dense
postsynaptic
density. As for the concentration
values
themselves, a buffer concentration
of lo-20 PM would
become saturated with every input: [Ca”] would always
grow large. A l-2 mM buffer concentration
would never
become saturated; [Ca”]
would change little [thus explaining why intracellular
injections of ethylene glycolbis(P-aminoethylether)-N,N,
N’, N’-tetraacetic
acid (EGTA)
and 1,2-bis( c+aminophenoxy)ethane-N,N,N’,N’-tetraacetic
acid (BAPTA) block LTP]. However, as shown in Table 3,
there is a sensitive quantitative dependency of [Ca”] on
buffer concentrations
within the range of 50-200 PM. Al-
FACTORS
COMPUTATIONAL
MODELS
though free Ca”+ concentration was found to be sensitive to
the concentration
and localization of calcium buffers, the
qualitative features of the results presented here will remain valid unless the true values for buffer concentration
are much different from those considered here.
DENDRITIC
SPINES.
The suggestion has been made that
thin spine necks might isolate metabolic events within the
head of a dendritic spine (Brown et al. 1988; Coss and
Perkel 1985; Harris and Stevens 1988; Shepherd 1979;
Wickens
1988). Our simulations
suggest that a primary
function of spines in dentate granule cells may be to provide a locus for physiologically
important,
transient increases in calcium concentration.
Because of the small volume of the spine head and the constricting spine neck diameter, much larger calcium concentrations
can build up
near the postsynaptic density of a synapse on a spine than
near the postsynaptic density of an identical synapse on the
dendritic shaft. Results of preliminary simulations support
this idea. These observations imply a simple explanation
for the function of spines: by virtue of hindering diffusion,
spines enable the local concentration
of Ca” to become
larger than would otherwise be possible.
Other mdw7ism.s
.t;,r produc~ing the nonlincarit .v
There are other possible mechanisms that might produce
the desired nonlinearity.
One could hypothesize a dendritically localized, voltage-triggered conductance that further depolarizes a dendrite after some threshold level of
synaptically driven depolarization.
However, there is no
evidence for such a process in the dentate gyrus in tissue
slice experiments
that use physiological
bathing media.
The slow depolarizations and burst discharges generated in
hippocampal pyramidal cells resulting in large part from
inward Ca3+ currents are not elicited in dentate granule
cells in standard Ringer solution. The experimental evidence suggests that these large inward Ca” currents are not
readily generated in dentate granule cells (Fricke and
Prince 1984).
As.wdw0n02r.s s wzuptic uct ivut ion
The experimentally
determined temporal window of associativity, in which potentiation of a weak input response
is produced when a powerfully depolarizing event follows,
but not when it precedes, weak input synaptic activation,
was reproduced with the computer model. This temporal
requirement may initially seem counterintuitive
until one
considers what does and does not occur at the receptor
level.
For many, the intuitive timing rule is the reverse of the
actual rule. If the strong input comes before the weak
input, one might imagine that the strong input would provide a high level of depolarization that would last until the
weak input arrives. This depolarization would relieve Mg”
block at the weak input synapses; Ca” influx would be
large, and the weak input response would be potentiated.
Conversely, if the weak input comes first, one might imagine that any open channels at weak input synapses would
be blocked by Mg’+ because of insufficient depolarization.
However, this is not what happens.
The- kev
--I factors
----a- explaining ” the
- - actual
--- - - --~ timing
-~
” rule are- the
1166
W. R. HOLMES
AND
W. B. LEVY
reactions that are
two slow rate constants, 0 and k- 1, in the description of the the complex of calmodulin-dependent
NMDA receptor-mediated
conductance. With small & the localized to the postsynaptic density.
transition to the open channel state, AR + AR*, procedes
slowly; thus, even when the weak input was activated as
We thank N. Desmond
for providing
the morphological
data for the cell
soon as 1 ms after the end of the strong input train, most of used in this study. We acknowledge helpful discussions with M. Mayer and
the depolarization resulting from activation of the strong I. Forsythe, and we thank M. Mayer, I. Segev, W. Rall, N. Desmond, C.
Colbert, and M. Bear for comments
and suggestions on preliminary
verinput had decayed before the NMDA-receptor
channels at sions
of this manuscript.
We thank the Frederick
Cancer Research Facility
weak input synapses reached the open state. Consequently,
for the use of the CRAY X-MP/24
and VAX 8600 computers
to do many
during most of the time when the channels at weak input
of the simulations.
This research was supported,
in part, by National Institute of Neurologisynapses were open, they were blocked by Mg2+, and Ca2+
and Communicative
Disorders
and Stroke Grants NS-07845
(to W. R.
influx was small. With small k-, , dissociation of the recep- cal
Holmes) and NS- 15488 and NIMH
RSDA K02-MH00622
and the Unitor-transmitter
complex, AR + A + R, procedes slowly; versity of Virginia, Dept. of Neurosurgery
(to W. B. Levy).
thus, even when the strong input was activated 20 ms after
Address for reprint
requests: W. R. Holmes,
Mathematical
Research
the weak input, many NMDA-receptor
channels at weak Branch, Bldg. 3 1 Rm. 4B-54, National Institute of Diabetes and Digestive
input synapses were still glutamate-activated
and capable and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892.
of moving to the open state, AR + AR*. The large depolarReceived
12 October
1989; accepted in final form 5 January
1990.
ization provided by the strong input relieved the Mg2+
block; Ca” influx was large, and the experimental result
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