A Ca2+-Based Computational Model for NMDA Receptor-Dependent Synaptic Plasticity at Individual Post-Synaptic Spines in the Hippocampus

Abstract

Associative synaptic plasticity is synapse specific and requires coincident activity in pre-synaptic and post-synaptic neurons to activate NMDA receptors (NMDARs). The resultant Ca(2+) influx is the critical trigger for the induction of synaptic plasticity. Given its centrality for the induction of synaptic plasticity, a model for NMDAR activation incorporating the timing of pre-synaptic glutamate release and post-synaptic depolarization by back-propagating action potentials could potentially predict the pre- and post-synaptic spike patterns required to induce synaptic plasticity. We have developed such a model by incorporating currently available data on the timecourse and amplitude of the post-synaptic membrane potential within individual spines. We couple this with data on the kinetics of synaptic NMDARs and then use the model to predict the continuous spine [Ca(2+)] in response to regular or irregular pre- and post-synaptic spike patterns. We then incorporate experimental data from synaptic plasticity induction protocols by regular activity patterns to couple the predicted local peak [Ca(2+)] to changes in synaptic strength. We find that our model accurately describes [Ca(2+)] in dendritic spines resulting from NMDAR activation during pre-synaptic and post-synaptic activity when compared to previous experimental observations. The model also replicates the experimentally determined plasticity outcome of regular and irregular spike patterns when applied to a single synapse. This model could therefore be used to predict the induction of synaptic plasticity under a variety of experimental conditions and spike patterns.

Full text

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 1
SYNAPTIC NEUROSCIENC
E
Original research article
published: 21 July 2010
doi: 10.3389/fnsyn.2010.00031
to vary Ca2+ influx through NMDARs by fixing the post-synaptic
membrane potential during low frequency synaptic stimulation
(Isaac et al., 1995; Daw et al., 2000).
Spike timing-dependent plasticity (STDP) is a form of Hebbian
synaptic plasticity that incorporates a temporal specificity to coin-
cident pre- and post-synaptic activity. In the hippocampus, STDP
was originally thought to be induced by single pairs of pre- and
post-synaptic action potentials such that if the pre-synaptic action
potential occurs before the post-synaptic action potential LTP is
induced whereas if the order of action potentials is reversed then
LTD is induced (Bi and Poo, 1998; Debanne et al., 1998; Nishiyama
et al., 2000; Campanac and Debanne, 2008; Kwag and Paulsen,
2009). Other data have proposed this model should include a
requirement for bursts of post-synaptic action potentials for the
induction of LTP although LTD may be induced by single pairs
(Pike et al., 1999; Wittenberg and Wang, 2006; Buchanan and
Mellor, 2007) reviewed in (Buchanan and Mellor, 2010). This is
a divergence from the situation at cortical synapses where single
pairs of action potentials can induce both LTP and LTD (Sjostrom
et al., 2001; Sjostrom and Nelson, 2002; Froemke et al., 2006, but
see Nevian and Sakmann, 2006).
IntroductIon
Hebbian synaptic plasticity is the cellular and molecular correlate
of associative learning in the brain. During presentation of infor-
mation that needs to be retained for future use, specific synapses
are subjected to activity patterns that induce a long-term change
in synaptic strength. For Hebbian synaptic plasticity at Schaffer
collateral synapses in the hippocampus, these patterns require
coincident activity in pre- and post-synaptic neurons to activate
NMDA receptors (NMDARs) present on the membrane of the
post-synaptic dendritic spine. The resulting Ca2+ influx through
NMDARs is the critical trigger for induction of synapse specific
plasticity (Lisman, 1989).
Classically, high frequency synaptic stimulation induces long-
term potentiation (LTP) whereas low frequency stimulation induces
long-term depression (LTD) suggesting that brief high concentra-
tions of Ca2+ in the post-synaptic spine induce LTP whereas pro-
longed lower concentrations of Ca2+ induce LTD (Bear et al., 1987;
Hansel et al., 1996). This hypothesis is supported by measurements
of Ca2+ concentration during plasticity induction (Hansel et al.,
1997; Cho et al., 2001; Cormier et al., 2001; Ismailov et al., 2004;
Gall et al., 2005) and by plasticity induction protocols designed
A Ca2+-based computational model for NMDA receptor-
dependent synaptic plasticity at individual post-synaptic
spines in the hippocampus
Owen J. L. Rackham1, Krasimira Tsaneva-Atanasova 2, Ayalvadi Ganesh3 and Jack R. Mellor4*
1 Department of Engineering Mathematics, Bristol Centre for Complexity Sciences, University of Bristol, University Walk, Bristol, UK
2 Department of Engineering Mathematics, Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, University Walk, Bristol, UK
3 Department of Mathematics, Bristol Centre for Complexity Sciences, University of Bristol, University Walk, Bristol, UK
4 Department of Anatomy, Medical Research Council Centre for Synaptic Plasticity, University of Bristol, University Walk, Bristol, UK
Associative synaptic plasticity is synapse specific and requires coincident activity in pre-synaptic
and post-synaptic neurons to activate NMDA receptors (NMDARs). The resultant Ca2+ influx is
the critical trigger for the induction of synaptic plasticity. Given its centrality for the induction
of synaptic plasticity, a model for NMDAR activation incorporating the timing of pre-synaptic
glutamate release and post-synaptic depolarization by back-propagating action potentials could
potentially predict the pre- and post-synaptic spike patterns required to induce synaptic plasticity.
We have developed such a model by incorporating currently available data on the timecourse
and amplitude of the post-synaptic membrane potential within individual spines. We couple this
with data on the kinetics of synaptic NMDARs and then use the model to predict the continuous
spine [Ca2+] in response to regular or irregular pre- and post-synaptic spike patterns. We then
incorporate experimental data from synaptic plasticity induction protocols by regular activity
patterns to couple the predicted local peak [Ca2+] to changes in synaptic strength. We find that
our model accurately describes [Ca2+] in dendritic spines resulting from NMDAR activation during
pre-synaptic and post-synaptic activity when compared to previous experimental observations.
The model also replicates the experimentally determined plasticity outcome of regular and
irregular spike patterns when applied to a single synapse. This model could therefore be used
to predict the induction of synaptic plasticity under a variety of experimental conditions and
spike patterns.
Keywords: synaptic plasticity, hippocampus, dendritic spines, NMDA receptor, spike timing-dependent plasticity
Edited by:
Per Jesper Sjöström, University
College London, UK
Reviewed by:
Harel Z. Shouval, University of Texas
Medical School at Houston, USA
Thomas G. Oertner, Friedrich Miescher
Institute for Biomedical Research,
Switzerland
*Correspondence:
Jack R. Mellor, Department of
Anatomy, Medical Research Council
Centre for Synaptic Plasticity,
University of Bristol, University Walk,
Bristol, UK. e-mail: jack.mellor@bristol.
ac.uk

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 2
Rackham et al. Model for hippocampal synaptic plasticity
Since Ca2+ influx through NMDARs is pivotal for LTP and LTD,
this suggests the induction of synaptic plasticity can be predicted
by NMDAR opening kinetics in response to pre-synaptic glutamate
release and post-synaptic depolarization. This approach has been
adopted for the modeling of post-synaptic calcium dynamics in
response to synaptic stimulation or back-propagating action poten-
tials (Franks et al., 2002; Grunditz et al., 2008; Keller et al., 2008)
and to STDP induction protocols (Shouval et al., 2002; Rubin et al.,
2005; Graupner and Brunel, 2007; Helias et al., 2008; Urakubo et al.,
2008; Castellani et al., 2009). However, these STDP models are limited
by the experimental data used to determine their parameters and,
in addition, ought to accurately predict the plasticity outcomes of
a variety of induction protocols. Recent advances in dendritic spine
imaging provide data on spine depolarization and Ca2+ concentrations
in response to pre- and post-synaptic action potentials (Sabatini et al.,
2002; Nevian and Sakmann, 2006; Bloodgood and Sabatini, 2007;
Canepari et al., 2007; Palmer and Stuart, 2009) that potentially greatly
increase the accuracy of such models of plasticity induction.
We have developed a computational model of synaptic plasticity
induction based on one originally described by Shouval et al. Our
model incorporates the latest experimental data on dendritic spine
depolarization and Ca2+ dynamics. We also test the predictive power
of our model on many plasticity induction protocols by calculating
continuous Ca2+ concentrations during long induction periods. We
find that our model accurately predicts the experimental data tested
and we hypothesize that it can thus be used to search for instances of
synaptic plasticity induction during continuous activity at Hebbian
synapses in the hippocampus.
MaterIals and Methods
We use a physiologically plausible model based on intracellular Ca2+
dynamics caused by NMDAR activation during the induction of syn-
aptic plasticity to predict the plasticity outcome of any set of pre- and
post-synaptic activity patterns that occur at the Schaffer collateral
synapse in the hippocampus. Since we are interested in studying
experimental spike trains we modify a model originally proposed by
Shouval et al. (2002) to allow us to carry out such analysis. We make
a number of critical modifications to analyze the Ca2+ dynamics in
individual dendritic spines during long periods of irregular spiking
activity. This is illustrated using short epochs of overlapping hip-
pocampal place cell activity (Figure 1) (Isaac et al., 2009).
Essential components of the experimental spike trains for the
activation of NMDARs and therefore the induction of synaptic
plasticity are (i) the pre-synaptic release of glutamate that dic-
tates the binding of glutamate to the NMDARs and (ii) the post-
synaptic membrane potential that determines the relative blockade
of NMDARs by Mg2+. For the purposes of this model, we have
assumed the two events that determine the post-synaptic mem-
brane potential within an individual dendritic spine are excita-
tory post-synaptic potentials (EPSPs) and back-propagating action
potentials (BPAPs). We start by modeling the BPAPs as follows:
BPAP ,
max
bs
f
bs i
post
f
bs
i
post
() exp()
tVItt
I
tt
=−−








+
<
∑τ
ss
bs i
post
s
bs
exp()−−





tt
τ
(1)
FIGURE 1 | Calculating [Ca2+] in dendritic spines during continuous pre-
and post-synaptic activity. The model initially calculates the membrane
potential during continuous activity by summating the membrane potential
changes due to EPSPs and BPAPs from a resting membrane potential of
−65 mV. The Ca2+ current passing through synaptic NMDARs is then calculated
from the membrane potential and glutamate binding kinetics. Finally spine
[Ca2+] is calculated from Ca2+ buffering and diffusion kinetics. Left hand panels
show the post-synaptic responses to an epoch of place cell activity spanning
3500 ms. Right hand panels show a 200 ms excerpt from this epoch.
where
Vmax
bs
is the maximum depolarization due to the BPAP,
If
bs
, and
Is
bs
are the relative magnitudes of the fast and slow components of
the BPAP, respectively, that sum to one, and the integration time step
δ is 0.1 ms. Due to the slower (and much smaller) after- depolarizing

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 3
Rackham et al. Model for hippocampal synaptic plasticity
EPSPNMDA f
f
ep s
s
ep
m
() expexp (
tI tItVV
=−




+−









−
ττ
rr1
rest
NMDA
f
f
ep
f
ep s
s
ep
EPSP
)
() expex
V
dt
dt ItI
⇒
=− −




−
11
τττpp ()
max( ). (max
−









−
=⇒
=⇒
tVV
V
t
τs
ep
mr1
rest
EPSP
0
00924 (()).
..
t
Nn
=⇒
=
00812
5
00812
If
and
Is
are the relative magnitudes of the fast and slow component
of the NMDAR current as a result of glutamate binding, respec-
tively, that sum to one, and Θ is the Heaviside (unit) step function.
The voltage dependence of the current that takes into account Mg2+
block of the receptor (Jahr and Stevens, 1990) is represented by
the term B(Vm)(Vm−Vr1)/Vrest, where (Vm−Vr1) is the driving force
determined by the reversal potential, Vr1 (0 mV), and
BV KV
() exp( )([ ]/ .)
.
m
Mm Mg
=+−
1
1357
We then calculate the spine membrane potential as the summa-
tion of BPAP and EPSPAMPA and EPSPNMDA:
Vt Vt tt
mrestAMPANMDA
BPAP EPSP EPSP() () () (),=+ ++
(4)
where Vrest is set at −65 mV unless otherwise stated. An example
of the predicted spine voltage can be seen for a sample epoch of
overlapping place cell activity in Figure 1.
Since NMDARs provide the major source of Ca2+ influx into
post-synaptic dendritic spines (Bloodgood and Sabatini, 2007), we
incorporate in our model the Ca2+ current through NMDAR that
takes the following form (Shouval et al., 2002):
It PG It tt
I
i
tt
NMDA NMDA f
i
f
s
i
pre
() ()exp()
(
=−−








+
<
∑0Θ
Θ
τ
tt tt BV VV
i)exp ()
()(),
−−






−
i
s
mm r2
τ
(5)
This is similar to Eq. 3 except for the terms P0 and GNMDA that
represent the open channel probability and NMDAR Ca2+ con-
ductance respectively and Vr2 is the reversal potential for calcium
(130 mV).
Next, the rate of change of the [Ca2+] inside the post-synaptic
spine is governed by:
d
dt I
[] []
,
Ca Ca
NMDA
Ca
22++
=−ατ
(6)
where α is a factor that converts current to flux and τCa is the calcium
passive decay time constant. An example of the Ca2+ current flow
through NMDARs and the resulting predicted [Ca2+] in the spine
can be seen in Figure 1.
Finally, we assume that spine [Ca2+] is the trigger for synaptic
strength change. For the purposes of our study the continuous
model for synaptic strength used in Shouval et al. is modified to act
as a Ca2+-gated function based on local peaks in [Ca2+] as follows:
potential, if two spikes happen near enough to each other that the
first spike is still decaying, the effect of the BPAPs is additive. Since
we are modeling the BPAP at the spine
Vmax
bs
is set at 67 mV in line
with experimental data measuring membrane potential in spines
with voltage-sensitive dyes (Canepari et al., 2007; Palmer and Stuart,
2009). This is smaller than the maximum BPAP amplitude found at
the soma used by Shouval et al. An example of the modeled BPAP
during place cell activity can be seen in Figure 1.
The equation that governs the behavior of AMPAR-mediated
EPSPs in the model is similar to (1) having a slow and a fast expo-
nential component:
EPSPAMPA
a
i
pre
s
ep
i
pre
f
ep
()
exp()
exp()
t
Ntt tt
=−−




+−−



ττ







−
<
∑
tt
VV
V
i
pre
mr1
rest
()
,
(2)
where the parameter Na reflects the maximum effect that a sin-
gle AMPAR-mediated EPSP can have. The value of Na can vary
depending on the number of synapses activated. Activation of a
single synapse results in a membrane depolarization in the spine of
approximately 10 mV (Palmer and Stuart, 2009). Again this deviates
from the value of 1 mV recorded at the soma and used by Shouval
et al. Assuming that the maximum depolarization that a single EPSP
can generate is 10 mV we define Na in the following way:
EPSPAMPA
f
ep
s
ep
mr1
() expexp ()
tttVV
V
=−




+−









−
ττ
rrest
AMPA
f
ep
f
ep
s
ep
s
ep
EPSP
dt
⇒
=− −




−−

dt tt() expexp
11
ττττ








−
=⇒
=− +
()
(log log)
VV
V
t
mr1
rest
f
ep
s
ep
s
ep
f
ep
f
0
ττ ττ
τeep
s
ep
EPSP
+




⇒
=⇒ =⇒
=
τ
max( ). (max()).
.
tt
Na
00128 06968
10
069668.
EPSPAMPA also depends on the membrane potential, Vm. This
dependence is represented by the term (Vm−Vr1)/Vrest where Vr1 is
the reversal potential for AMPARs (0 mV) and Vrest is the resting
membrane potential (−65 mV).
The equation that governs the behavior of NMDA-mediated
EPSPs in the model has the following form:
EPSPNMDA n
f
i
f
s
i
pre
() ()exp()
(
tN It tt
It
tt
i
i
=−−








+
<
∑Θ
Θ
τ
))exp ()
()
()
,
−−






−tt BV VV
V
i
s
m
mr1
rest
τ
(3)
where the parameter Nn reflects the maximum effect of the
NMDAR-mediated component of the EPSP. This is calculated
in a similar fashion to Na for EPSPAMPA using a value of 5 mV
for the NMDAR-mediated EPSP at −65 mV in the absence of
Mg2+ measured by dendritic recordings (Fernandez de Sevilla
et al., 2007).

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 4
Rackham et al. Model for hippocampal synaptic plasticity
Table 1 | Parameter values of the synaptic model.
Parameter Value Parameter Value
If
bs
0.75 α1 0.3
τf
bs
3 ms α2 0.45
τs
bs
25 ms β1 80
τf
ep
5 ms β2 80
τs
ep
50 ms P1 100 ms
If 0.5 P2 0.02 ms
τf 50 ms P3 4
τs 200 ms P4 1000 ms
τCa 50 ms P0 0.5
Vmax
bs
67 mV GNMDA 0.002 μM/ms mV
Vrest −65 mV KM 0.092 mV−1
Vr1 0 mV Na 14.35 mV
Vr2 130 mV Nn 61.58 mV
Having validated our model for the observed Ca2+ influx at
dendritic spines we next asked the question if the model could
replicate experimental data for the induction of synaptic plastic-
ity using a variety of protocols. We have restricted our model to
comparisons with experimental data from the Schaffer collateral
synapse of the hippocampus and not considered other synapses
in other brain regions.
spIke tIMIng-dependent plastIcIty wIth paIrs of pre- and
post-synaptIc spIkes
To model STDP with pairs of pre- and post-synaptic spikes we
initially assumed single synaptic activation and varied ∆t between
−20 and + 100 ms at intervals of 0.1 ms measuring the peak [Ca2+]
at each value of ∆t (Figure 3A). [Ca2+] rose from its baseline of
72 nM (the peak [Ca2+] attained for a single EPSP in isolation) to
a peak of 230 nM at ∆t ≈10 ms (Figure 3B). Experiments such as
these have been shown to generate no significant synaptic plasticity
(Buchanan and Mellor, 2007) whereas those using larger amplitude
EPSPs have been shown to generate LTD (Wittenberg and Wang,
2006). We estimated the activation of multiple synapses at the same
time would increase the depolarization within a single dendritic
spine during an EPSP from 10 to 20 mV based on experimental
predictions (Palmer and Stuart, 2009). Thus we have estimated that
the activation of other spines will contribute an additional 10 mV
of depolarization within an activated spine above and beyond the
experimentally determined 10 mV for activation of a single synapse.
This doubling of the EPSP amplitude resulted in an increase in
peak [Ca2+] at all values of ∆t with a peak of 279 nM occurring at
∆t ≈10 ms (Figure 3C).
It has also been shown that the frequency of spike pairing is
important for the induction of plasticity such that at higher fre-
quencies (>5–10 Hz) LTP can be induced (Wittenberg and Wang,
2006; Buchanan and Mellor, 2007). We varied the frequency of
spike pairings in our model for 10 mV EPSPs over a range of
frequencies from 1 to 100 Hz (Figure 3D). Summation of Ca2+
transients was found to occur at frequencies greater than ∼5 Hz
indicating that increasing the frequency will shift the STDP pro-
tocol towards larger [Ca2+] and therefore LTP in line with the
experimental data.
Ca Ca :
Ca Ca
∀′=∧ ″<
=
+
()()
++
+
++
[] []
[][],
22
1
22
00
1
W
WW
j
j
j
jj
ηΩ Ca 0
Ca Ca Ca
Ω
ΩΩ
[]
[][],[
j
jjjj
W
2
22 2
1
+
++ +
()
>
+
()()



η]]
()
≤






 0
(7)
where
Ω[]
exp( ())
exp( ()).exp( (
Ca Ca
Ca
C
j
j
j
22
2
2
2
2
2
1
1025
+
+
+
=−
+−
−
βα
βα
βaa
Ca
Ca Ca C
j
j
j
j
ff
2
1
1
2
1
2
2
1
1
+
+
+
+
−
+−
=
α
βα
η
))
exp( ()),
[]
()
,(
aa Ca
j
j
P
P
PP
21
2
24
3
+
+
=++)() .
The critical target for Ca2+ influx through NMDARs is the
enzyme CAMKII. Due to its ability to autophosphorylate, the acti-
vation of this molecule can be long lived and the level is determined
by local peak [Ca2+]. Thus, synaptic weight change is determined at
local peak [Ca2+] (Miller et al., 2005; Graupner and Brunel, 2007;
Helias et al., 2008; Urakubo et al., 2008; Castellani et al., 2009).
Since there is no noise associated with our model, these peaks are
measured instantaneously without smoothing. Experimentally,
increases in synaptic weight tend towards saturation as synaptic
weight increases. In addition, decreases require synaptic weight to
always be >0. These constraints explain the form of Eq. 7.
Numerical integration was performed using forward Euler
method implemented in MATLAB.
The parameter values used in the simulations are given for com-
pleteness in Table 1.
results
Our starting point for developing a model for the induction of
synaptic plasticity was to incorporate the most recent and accurate
measurements of voltage changes within dendritic spines using data
from measurements of voltage-dependent dyes (Canepari et al.,
2007; Palmer and Stuart, 2009). We model the membrane poten-
tial at the spine rather than the soma because this is the site of the
NMDARs critical for the induction of synaptic plasticity. This shifts
the determination of membrane depolarization away from BPAPs
and towards EPSPs since the former attenuate as they pass along the
dendrite and the latter are now measured at their site of origin. This
is a departure from previous models that used values for BPAPs and
EPSPs recorded at the soma (Shouval et al., 2002). With this change,
our model predicts that an EPSP resulting from the activation of a
single synapse is sufficient to cause a significant Ca2+ influx through
NMDARs (Figure 2A) in line with experimentally observed data
(Bloodgood and Sabatini, 2007; Canepari et al., 2007; Sobczyk and
Svoboda, 2007). The pairing of a BPAP with a single EPSP with a
time delay of 10 ms produces 3–4 times the Ca2+ influx (Figure 2A)
that again agrees qualitatively with experimentally observed data
(Bloodgood and Sabatini, 2007). For comparison we changed the
maximal EPSP and BPAP amplitudes to those known to occur at
the soma (∼1 and ∼100 mV respectively). With these parameters, a
single EPSP produces limited Ca2+ influx whereas pairing an EPSP
with a BPAP produces a large Ca2+ influx (Figure 2B).

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 5
Rackham et al. Model for hippocampal synaptic plasticity
plasticity is induced (Buchanan and Mellor, 2007). When we used
theta burst stimulation with five stimuli to only the pre-synaptic
input, the model predicted peak [Ca2+] within the spine to be
325 nM (Figure 5) and with four stimuli 250 nM. The value for
four stimuli is more physiologically relevant since the probability
of neurotransmitter release at any one Schaffer collateral synapse
is considerably less than 1. Therefore it is highly unlikely that
an experimental theta burst will ever generate five EPSPs at an
individual synapse.
paIrIng post-synaptIc depolarIzatIon wIth
pre-synaptIc stIMulatIon
Other common synaptic plasticity induction protocols have dis-
pensed with the need for post-synaptic spikes altogether and use
voltage clamp to depolarize the post-synaptic membrane and allow
NMDAR activation. This technique neatly demonstrates the bidi-
rectional nature of NMDAR-dependent plasticity since depolariza-
tion to moderate levels (−40 mV) produces LTD whereas higher
depolarization (0 mV) produces LTP (Isaac et al., 1995; Daw et al.,
2000). We tested this with our model by clamping the membrane
potential (Vm) at either −40 or 0 mV (Figure 6). Peak [Ca2+] in
response to EPSPs were 336 nM and 2.43 μM respectively, which
when compared to peak [Ca2+] produced by other protocols would
be expected to induce LTD and LTP respectively in agreement with
experimental data.
the ca2+ hypothesIs can explaIn prevIous experIMental data
The Ca2+ hypothesis states that brief high concentrations of Ca2+
in the post-synaptic spine induce LTP whereas prolonged lower
concentrations of Ca2+ induce LTD (Bear et al., 1987; Hansel et al.,
1996). This is expressed graphically in Figure 7. Points are indi-
cated representing the predicted [Ca2+] from our model for specific
plasticity inducing protocols. STDP with single pairs of BPAPs and
small EPSPs do not induce plasticity (Buchanan and Mellor, 2007)
but when large EPSPs are used LTD is induced (Wittenberg and
Wang, 2006) and STDP with triplets of single EPSPs and bursts of
BPAPs produces LTP (Pike et al., 1999; Wittenberg and Wang, 2006;
Buchanan and Mellor, 2007). When the post-synaptic membrane
potential is set at −40 mV during pre-synaptic stimulation LTD is
induced (Daw et al., 2000) whereas at 0 mV LTP is induced (Isaac
et al., 1996). Theta burst pairing also induces LTP (Frick et al.,
spIke tIMIng-dependent plastIcIty wIth trIplets of spIkes
Post-synaptic burst firing has been shown to be important for the
induction of LTP at Schaffer collateral synapses in the hippocampus
where burst firing in this instance refers to any number of spikes
greater than one (Pike et al., 1999; Wittenberg and Wang, 2006;
Buchanan and Mellor, 2007). We tested this on our model using
triplets of spikes composed of one pre-synaptic spike and two post-
synaptic spikes where ∆t is the time between the pre-synaptic spike
and the first post-synaptic spike and ∆s is the delay between the two
post-synaptic spikes (Figure 4A). We first used 10 mV EPSPs with a
constant ∆s of 10 ms and varied ∆t between −20 and +100 ms. This
produced a peak [Ca2+] of 420 nM at ∆t = 4 ms which increased to
a peak [Ca2+] of 475 nM when 20 mV EPSPs were used (Figure 4B)
confirming that spike triplets produce higher peak [Ca2+] than spike
pairs and therefore are more likely to induce LTP.
We next varied ∆s whilst maintaining ∆t constant at 10 ms for
both 10 and 20 mV EPSPs revealing a decrease in peak [Ca2+] as ∆s
increases (Figure 4C). Finally, we varied the frequency of triplets
for 10 mV EPSPs over a range of frequencies from 1 to 100 Hz
whilst keeping ∆t and ∆s constant at 10 ms each (Figure 4D).
Summation of Ca2+ transients was found to occur at frequencies
greater than ∼4 Hz.
theta burst plastIcIty
We now moved away from STDP to look at other common syn-
aptic plasticity induction protocols. The theta burst protocol was
developed to mimic the activity patterns believed to occur at hip-
pocampal synapses during learning and consists of bursts of four
or five spikes at 100 Hz with an interburst interval of 200 ms.
These can either be applied to the pre- and post-synaptic neuron
coincidentally (Frick et al., 2004) or to just the pre-synaptic neuron
(Larson et al., 1986). The latter then leads to post-synaptic spikes
through EPSP summation if the initial EPSP amplitude is suf-
ficiently large (Buchanan and Mellor, 2007). We used our model
to mimic coincident theta burst activity in pre- and post-synaptic
neurons using 10 mV EPSPs and found that this type of synap-
tic stimulation produces very large peak [Ca2+] within dendritic
spines (Figure 5) indicating that this protocol is very efficient at
producing LTP in agreement with experimental data. Experimental
data also shows when theta burst stimulation is given to only the
pre-synaptic neuron without initiating action potentials then no
FIGURE 2 | Comparison of predicted [Ca2+] dynamics in dendritic spines and in the soma. [Ca2+] profiles in response to a 10 mV EPSP at the spine (A) or a 1 mV
EPSP at the soma (B) on their own (gray) or in combination with a BPAP (black) of amplitude 60 mV at the spine (A) or 100 mV at the soma (B). The delay between
EPSP and BPAP initiation is 10 ms.

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Rackham et al. Model for hippocampal synaptic plasticity
InductIon of synaptIc plastIcIty by hIppocaMpal place cell
fIrIng patterns
One of the main purposes for developing a model that is capable
of continuously monitoring [Ca2+] in spines and therefore predicts
changes in synaptic strength is to scan long periods of neuronal activ-
ity for epochs that would be expected to induce plasticity without
having to directly measure synaptic strength. To test if the model could
perform this task we used data from experiments where long sections
of hippocampal place cell activity were replayed into single hippoc-
ampal synapses to test the plasticity outcome (Isaac et al., 2009).
2004; Buchanan and Mellor, 2007) whereas theta burst to only
pre-synaptic inputs does not (Buchanan and Mellor, 2007). In this
instance the absolute [Ca2+] values predicted by the model are not
as important as the relative magnitudes between plasticity induc-
tion protocols. However, it is interesting to note that the absolute
[Ca2+] values predicted by the model broadly agree with those
measured experimentally for the induction of synaptic plasticity at
Schaffer collateral synapses on CA1 pyramidal neurons (Cormier
et al., 2001). Thus the predictions from our model support the
Ca2+ hypothesis for synaptic plasticity induction.
FIGURE 3 | [Ca2+] dynamics in response to paired pre- and post-synaptic
spikes. (A) The model calculates [Ca2+] within a spine from the membrane
potential resulting from a pair of pre- and post-synaptic spikes. Gray line shows
EPSP in the absence of BPAP. Varying ∆t shows that [Ca2+]max is greatest when
0 ≤ ∆t ≤ 30 ms for 10 mV (B) or 20 mV (C) EPSPs. (D) The frequency of spike
pairings given at ∆t = 10 ms determines [Ca2+]max.

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 7
Rackham et al. Model for hippocampal synaptic plasticity
robust LTP induction in agreement with the experimental data
(Figures 9A–D). We also tested two further pairs of place cells with
non-overlapping or adjacent place fields (1A and 1C, 2E and 2D) and
found the model predicted only a small LTD for the non- overlapping
pair and a small LTP for the adjacent pair (Figures 9E–F).
Our original experimental data also tested a pair of place cells
that had an asymmetric cross-correlation such that cell 1A prefer-
entially fired just before cell 1B. Because classical STDP rules state
that the temporal order of pre- and post-synaptic spikes controls
the direction of synaptic plasticity (Bi and Poo, 1998; Song et al.,
2000), the existence of this asymmetry suggested that when cell
1A was pre-synaptic and cell 1B post-synaptic then LTP would be
induced but if the cells were reversed then LTD would be induced.
We first took an ∼16-min period of activity from a pair of place
cells (1A and 1B) that had overlapping place fields and therefore
would be expected to fire at approximately the same time (Isaac
et al., 2009). During the ∼16-min period short coincident bursts of
activity could be seen in the two place cells that the model predicted
would produce large [Ca2+] sufficient to induce LTP (Figures 1
and 8). This LTP was initiated in the first few minutes of activity
and eventually reached a plateau.
We tested a set of four further pairs of place cells (2A and 2B, 2C
and 2D, 3A and 3B, 4A and 4B) with overlapping place fields but with
strikingly different spiking characteristics [for a full description of
the place cell spike pattern characteristics and plasticity outcomes
see Isaac et al. (2009)] and found in each case the model predicted
FIGURE 4 | [Ca2+] dynamics in response to triplets of one pre- and two
post-synaptic spikes. (A) The model calculates [Ca2+] within a spine from
the membrane potential resulting from a triplet of pre- and post-synaptic
spikes. Gray line shows EPSP in the absence of BPAP. (B). Varying ∆t shows
that [Ca2+]max is greatest when 0 ≤ ∆t ≤ 30 ms for 10 or 20 mV EPSPs for ∆s = 10
ms. (C) Varying ∆s shows that [Ca2+]max decreases as ∆s increases for 10 or
20 mV EPSPs and ∆t = 10 ms. (D) The frequency of spike pairings given at
∆t = 10 ms and ∆s = 10 ms determines [Ca2+]max.

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 8
Rackham et al. Model for hippocampal synaptic plasticity
post-synaptic then LTD would be induced but the model predicted
only marginal LTD (Figure 10B) in line with the experimental data
(Isaac et al., 2009).
Finally we have compared the experimentally determined plas-
ticity outcome from nine pairs of place cells with the outcome
predicted by our model. We find that the correlation between the
predicted and observed values is significant (Figure 10C, r2 = 0.58,
P < 0.05 by linear regression) and therefore conclude that the model
successfully predicts the induction of synaptic plasticity by irregular
activity patterns.
dIscussIon
The model described in this study incorporates two important
components of Ca2+ dynamics in dendritic spines that are neces-
sary for the induction of synaptic plasticity. Firstly, our model is
capable of analyzing Ca2+ influx and concentration continuously
and therefore it can determine the plasticity outcome of multiple
synaptic events that occur in vivo in an irregular pattern. Secondly,
[Ca2+] is modeled at the synapse in dendritic spines rather than at
the soma. This is important since the critical Ca2+ signal for the
induction of synaptic plasticity occurs at the spine. It also changes
the relative importance of EPSP vs BPAP depolarization which
has major implications for the predicted induction of STDP. This
approach is validated by comparison of the predicted vs observed
Ca2+ transients in response to either a single EPSP or coupled with
a BPAP (Figure 2) (Bloodgood and Sabatini, 2007).
The absolute values for [Ca2+] within the dendrite required for
the induction of synaptic plasticity have been estimated as 150–
500 nM for LTD and >500 nM for LTP (Cormier et al., 2001).
However, other researchers have estimated [Ca2+] within a spine
FIGURE 5 | Theta burst pairing produces large spine [Ca2+]. The model
calculates [Ca2+] within a spine from the membrane potential resulting from
coincident theta burst stimulation of pre- and post-synaptic neurons (black) or
only pre-synaptic neuron (gray).
FIGURE 6 | Post-synaptic voltage clamp paired with pre-synaptic
stimulation determines spine [Ca2+]. The model predicts that voltage clamp
of the post-synaptic membrane potential at −40 mV produces a much smaller
spine [Ca2+] than 0 mV when paired with a single pre-synaptic stimulation.
However, when we reversed the place cell firing patterns such that
cell 1B was pre-synaptic and cell 1A post-synaptic the model pre-
dicted LTP (Figure 10A) that corroborates the experimental results
and closely reproduces the experimentally determined timecourse
of LTP development (Isaac et al., 2009). We also manipulated the
spike patterns in cell 1B to remove all spikes that occurred less than
100 ms after a spike in cell 1A leaving only spikes that occurred
before any spike in cell 1A. Classical STDP rules would again pre-
dict that if cell 1A was pre-synaptic and the modified cell 1B was

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 9
Rackham et al. Model for hippocampal synaptic plasticity
FIGURE 7 | Spine [Ca2+] determines the direction and magnitude of synaptic
weight change. (A). The Ω−function describes the relationship between peak spine
[Ca2+] and synaptic weight change. Symbols represent the peak [Ca2+] produced by a
single application of the plasticity induction protocols shown in Figures 3–6 and
indicate the resulting predicted synaptic weight change. (B) The η-function describes
the learning rate for synaptic weight change as a function of peak spine [Ca2+].
FIGURE 8 | Example of predicted synaptic weight change during overlapping place cell activity. The model calculates spine [Ca2+] during a ∼16-min period of
activity from two place cells (1A and 1B) with overlapping place fields. The synaptic weight change is then calculated from the peak spine [Ca2+] and shows a robust,
rapidly developing potentiation.

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 10
Rackham et al. Model for hippocampal synaptic plasticity
in response to a single EPSP at 700 nM and a much higher 12 μM
during pairing of post-synaptic depolarization with synaptic stimu-
lation (Sabatini et al., 2002). This discrepancy could be explained
in a number of ways. The [Ca2+] in a dendritic spine in response
to synaptic stimulation could be considerably higher than in the
dendritic shaft because of the diffusion barrier created by the spine
neck. In addition, accurate absolute values for [Ca2+] measured by
fluorescent Ca2+ indicators are difficult to achieve and therefore
most studies are restricted to ratiometric measurements of tran-
sient [Ca2+] increases. For the purposes of synaptic plasticity this
is sufficient since the increase in [Ca2+] triggers induction. Here,
we have calculated the [Ca2+] based on a number of assumptions
for channel conductance and Ca2+ diffusion. More importantly,
we have modeled the relative [Ca2+] increases caused by various
induction protocols and used these to define the graph in Figure 7
that predicts the plasticity outcome.
Inhibitory synaptic transmission has a major role regulating the
induction of synaptic plasticity in the hippocampus. The transient
depression of inhibition induced by activation of pre-synaptic
cannabinoid or GABAB receptors facilitates the induction of LTP
(Davies et al., 1991; Chevaleyre and Castillo, 2004). This modulation
of synaptic plasticity is not included in our current model but incor-
poration of the hyperpolarizing effects of GABAergic transmission
would be an important future improvement and might, for example,
contribute to the frequency dependence of STDP induction.
NMDARs are not the only sources of Ca2+ within dendritic spines
but are certainly the most important for the induction of synaptic
plasticity. A role has also been demonstrated for Ca2+ stores present
in the endoplasmic reticulum in dendrites and spines (linked to Ca2+
influx through NMDARs or mGluRs) and also voltage-dependent
FIGURE 9 | Predicted synaptic weight changes for overlapping and non-overlapping place cell activity. Calculated synaptic weight changes for four pairs of
overlapping place cells 2A, 2B (A), 2C, 2D (B), 3A, 3B (C), and 4A, 4B (D) as well as one pair of non-overlapping place cells 1A, 1C (E) and one pair of adjacent place cells
2E, 2D (F).
FIGURE 10 | Predicted synaptic weight changes for place cell activity with
specific spike patterns. Calculated synaptic weight changes for a pair of
overlapping place cells with an asymmetric cross-correlation 1B, 1A (A) and a pair
of place cells where all spike intervals with positive ∆t less than 100 ms have
been removed 1A, 1B (B). (C) A comparison between the induced plasticity
predicted by the model and the observed plasticity from experimental data
(Isaac et al., 2009).

Frontiers in Synaptic Neuroscience www.frontiersin.org July 2010 | Volume 2 | Article 31 | 11
Rackham et al. Model for hippocampal synaptic plasticity
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Conflict of Interest Statement: The
authors declare that the research was
conducted in the absence of any com-
mercial or financial relationships that
could be construed as a potential conflict
of interest.
Received: 21 January 2010; paper pend-
ing published: 22 February 2010; accepted:
27 June 2010; published online: 21 July
2010.
Citation: Rackham OJL, Tsaneva-
Atanasova K, Ganesh A and Mellor JR
(2010) A Ca2+-based computational model
for NMDA receptor-dependent synaptic
plasticity at individual post-synaptic spines
in the hippocampus. Front. Syn. Neurosci.
2:31. doi: 10.3389/fnsyn.2010.00031
Copyright © 2010 Rackham, Tsaneva-
Atanasova, Ganesh and Mellor. This is an
open-access article subject to an exclusive
license agreement between the authors
and the Frontiers Research Foundation,
which permits unrestricted use, distribu-
tion, and reproduction in any medium,
provided the original authors and source
are credited.