950 • The Journal of Neuroscience, January 26, 2005 • 25(4):950 –961

Behavioral/Systems/Cognitive

Inositol 1,4,5-Trisphosphate-Dependent Ca2⫹ Threshold
Dynamics Detect Spike Timing in Cerebellar Purkinje Cells
Tomokazu Doi,1,2 Shinya Kuroda,3,4 Takayuki Michikawa,5 and Mitsuo Kawato1
ATR Computational Neuroscience Laboratories, Kansai Science City, Kyoto 619-0288, Japan, 2Graduate School of Information Science, Nara Institute of
Science and Technology, Ikoma-shi, Nara 630-0192, Japan, 3Undergraduate Program for Bioinformatics and Systems Biology, Graduate School ofInformation Science and Technology, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan, 4Precursory Research for Embryonic Science andTechnology, Japan Science and Technology, Bunkyo-ku, Tokyo 113-0033, Japan, and 5Division of Molecular Neurobiology, Department of Basic Medical
Science, Institute of Medical Science, The University of Tokyo, Minato-ku, Tokyo 108-8639, Japan
1

Large Ca 2⫹ signals essential for cerebellar long-term depression (LTD) at parallel fiber (PF)–Purkinje cell synapses are known to be
induced when PF activation precedes climbing fiber (CF) activation by 50 –200 ms, consistent with cerebellar learning theories. However,large Ca 2⫹ signals and/or LTD can also be induced by massive PF stimulation alone or by photolysis of caged Ca 2⫹ or inositol 1,4,5trisphosphate (IP3 ). To understand the spike-timing detection mechanisms in cerebellar LTD, we developed a kinetic model of Ca 2⫹
dynamics within a Purkinje dendritic spine. In our kinetic simulation, IP3 was first produced via the metabotropic pathway of PF inputs,
and the Ca 2⫹ influx in response to the CF input triggered regenerative Ca 2⫹-induced Ca 2⫹ release from the internal stores via the IP3
receptors activated by the increased IP3. The delay in IP3 increase caused by the PF metabotropic pathway generated the optimal PF–CF
interval. The Ca 2⫹ dynamics revealed a threshold for large Ca 2⫹ release that decreased as IP3 increased, and it coherently explained thedifferent forms of LTD. At 2.5 ␮M IP3 , CF activation after PF activation was essential to reach the threshold for the regenerative Ca 2⫹
release. At 10 ␮M IP3 , the same as achieved experimentally by strong IP3 photolysis, the threshold was lower, and thus large Ca 2⫹ release
was generated even without CF stimulation. In contrast, the basal 0.1 ␮M IP3 level resulted in an extremely high Ca 2⫹ threshold forregenerative Ca 2⫹ release. Thus, the results demonstrated that Ca 2⫹ dynamics can detect spike timing under physiological conditions,which supports cerebellar learning theories.
Key words: calcium; LTD; simulation; timing detection; inositol; feedback

Introduction
The Purkinje cells in the cerebellar cortex receive two major excitatory synaptic inputs, one of which is from a number of parallel
fibers (PFs) and the other of which is from a single climbing fiber(CF). Cerebellar long-term depression (LTD) is a persistent reduction in the efficacy of PF–Purkinje cell synapses that is induced by paired stimulation of PFs and CFs, and it is thought tobe one of the cellular bases of learning and memory (Marr, 1969;
Albus, 1971; Linden and Connor, 1995; Ito, 2001). Cerebellar
learning theories (Ito, 1970; Kawato, 1999; Yamamoto et al.,2002) state that PFs convey the information required for motor
command generation, whereas CFs carry error signals, which are
inevitably delayed compared with PF activation. Thus, the theories predict that PF activation preceding CF activation is a more
effective sequence to induce LTD than CF activation preceding
PF activation, and this has been supported by several experimental studies (Chen and Thompson, 1995; Schreurs et al., 1996;
Received July 8, 2004; revised Dec. 3, 2004; accepted Dec. 4, 2004.
This work was supported by the Human Frontier Science Program (M.K.) and the National Institute of Information
and Communications Technology.
Correspondence should be addressed to Tomokazu Doi, ATR Computational Neuroscience Laboratories, 2-2-2
Hikaridai, Kansai Science City, Kyoto 619-0288, Japan. E-mail: xtomdoi@atr.jp.
DOI:10.1523/JNEUROSCI.2727-04.2005
Copyright © 2005 Society for Neuroscience 0270-6474/05/250950-12$15.00/0

Wang et al., 2000). However, inconsistent PF–CF time windows
for LTD induction have frequently been reported, and it has beenfound possible to induce LTD in vitro by massive stimulation of a
PF bundle alone (Hartell, 1996), Ca 2⫹ uncaging (Miyata et al.,2000), or inositol 1,4,5-trisphosphate (IP3) uncaging (Finch andAugustine, 1998) instead of by conjunctive PF–CF stimulation,
thus raising doubts about LTD as one of the cellular bases oflearning (De Schutter, 1995; Llinas et al., 1997).
Among the many signaling molecules involved in LTD induction, Ca 2⫹ has been suggested as a candidate for the molecule that
detects the coincidence and the timing of PF- and CF-induced
spikes in Purkinje cell dendrites (Berridge, 1998; Wang et al.,2000). Multiphoton Ca 2⫹ imaging of dendritic spines in Purkinjecells has shown that PF stimulation followed by CF stimulation is
the most efficient sequence for inducing “supralinear” Ca 2⫹ signals (⬎10 ␮M) and subsequent cerebellar LTD (Wang et al.,2000). The term “supralinear” means that the Ca 2⫹ response to
conjunctive PF–CF inputs is greater than the sum of the responses to PF input alone and CF input alone. The IP3 receptor(IP3R) is a possible candidate for the coincidence detector forsupralinear Ca 2⫹ increases (Berridge, 1998). IP3Rs are intracellular Ca 2⫹ release channels that are coactivated by IP3 and Ca 2⫹,
and they are essential for the induction of LTD (Finch and Augustine, 1998; Inoue et al., 1998). Although PF inputs induce

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J. Neurosci., January 26, 2005 • 25(4):950 –961 • 951

production of IP3 via the metabotropic pathway (Finch and Augustine, 1998), CF input induces strong depolarization leading toa Ca 2⫹ influx through voltage-gated Ca 2⫹ channels (VGCCs) of
Purkinje cell dendrites at PF spines (Miyakawa et al., 1992). However, the idea that IP3Rs constitute the mechanism for coincidence detection needs to be further explored and tested to determine whether it is capable of explaining the temporal window
and the different forms of LTD.
To elucidate the dynamics of the cellular computations for
spike-timing detection, we developed a Ca 2⫹ dynamics model
based on known biochemical signal transduction pathways. We
first hypothesized that a delay in the increase in IP3 caused by the
PF metabotropic pathway generates the optimal PF–CF interval,that CF input after PF input triggers regenerative Ca 2⫹ releasefrom internal stores through IP3Rs, and that the Ca 2⫹ response
exhibited a threshold dynamics for large Ca 2⫹ release that decreases as IP3 increases. If this hypothesis is true, it would coherently explain the different forms of LTD.

Materials and Methods

Model structure and compartments of a Purkinje dendritic spine
We modeled a mature Purkinje dendritic spine and divided it into threecompartments [i.e., the cytosol, postsynaptic density (PSD), and endoplasmic reticulum (ER), with volumes of 0.1, 0.002, and 0.02 ␮m 3, respectively] based on serial electron microscopy observations (Harris andStevens, 1988), and the volume of the extracellular space was 10 ␮m 3.
Because of the small volume of the spine, we assumed that the molecules
it contains were well mixed and that their concentrations were uniform
within each compartment. All spatial diffusion of molecules between
compartments is mediated by channels and pumps, except for the diffusion of IP3 and Ca 2⫹ between the cytosol and PSD. The diffusion constants of the molecules are so high [the diffusion constants are 283 ␮m 2/s
for IP3 (Allbritton et al., 1992) and 220 ␮m 2/s for Ca 2⫹ (Klingauf and
Neher, 1997)] that diffusion between compartments in the small volumeof the spine should be completed within 1 ms. We therefore set that theconcentrations of IP3 and Ca 2⫹ would reach a state of dynamic equilibrium between the cytosol and PSD with a time constant of 1 ms, which ismuch faster than most chemical binding reactions in Ca 2⫹ signaling.
We did not model the Ca 2⫹ concentration in the dendritic shafts orsoma because dendritic spines are biochemically almost completely separated from their dendrites. Ca 2⫹ ions in the spines do not diffuse muchthrough the spine neck (Sabatini et al., 2002). Because cerebellar Purkinjecells contain higher concentrations of endogenous Ca 2⫹ buffers thanother neurons (Fierro and Llano, 1996; Maeda et al., 1999), for the sake ofsimplicity, all Ca 2⫹ entry from outside the spine can be assumed to be
attributable to Ca 2⫹ influx from VGCCs at the spine. Because action
potential activity in the soma does not propagate back to dendritic shafts
and spines in Purkinje cells (Stuart and Hausser, 1994; Vetter et al.,2001), Ca 2⫹ influx in the spine is mainly induced by PF and CF activation. We applied Ca 2⫹ pulse as Ca 2⫹ influx by PF and CF activation
instead of by modeling the kinetics of AMPARs and VGCCs. Althoughdense PF stimulation induces large Ca 2⫹ influxes (Wang et al., 2000), we
did not apply dense bundle PF stimulation. Ca 2⫹ influx induced byrepetitive PF activation may be strengthened or weakened because ofdepolarization and/or hyperpolarization as a result of the activity of othersynapses, but we ignored the effects of other synaptic activity as well as
short-term potentiation or depression in our model.

Block diagram of Ca 2⫹ signaling pathways

We constructed a block diagram of Ca 2⫹ signal transduction pathways ina Purkinje dendritic spine (Fig. 1), based on the literature (Fiala et al.,
1996; Kim and Thompson, 1997; Daniel et al., 1998; Rose and Konnerth,
2001; Ito, 2002). CF input generates a strong depolarization and opens
VGCCs, resulting in a Ca 2⫹ influx into the dendritic spine (Miyakawa etal., 1992). PF input raises the postsynaptic Ca 2⫹ concentration in the
Purkinje dendritic spine via two pathways, one mediated by AMPA re-

Figure 1. Block diagram of signaling pathways for Ca 2⫹ release at a dendritic spine in
Purkinje cells. Arrows and bars indicate stimulatory and inhibitory interactions. The dashedarrow indicates depolarization by CF stimulation. Molecular diffusion through the spine neck is
not considered.

ceptors (AMPARs) and the other mediated by metabotropic glutamate
receptors (mGluRs). In the AMPAR pathway, the activated AMPARs
induce depolarization of the membrane potential of the PF spine andopen VGCCs (Eilers et al., 1995), and extracellular Ca 2⫹ enters the cytosol through the VGCCs. In the mGluR pathway, glutamate released
from the PF terminal activates mGluRs, and the activated mGluRs in turninduce production of IP3 via G-proteins (Gq) and phospholipase C␤(PLC␤) at the PSD (Jiang et al., 1994; Sugiyama et al., 1999; Tanaka et al.,2000). IP3 then diffuses from the postsynaptic membrane into the cytosol. Finally, IP3Rs, which are IP3-gated Ca 2⫹ channels in the ER, releaseCa 2⫹ from the ER (Finch and Augustine, 1998).
Cytosolic IP3 is degraded by IP3 3-kinase and IP3 5-phosphatase withina few seconds (Dupont and Erneux, 1997; Irvine and Schell, 2001). Freecytosolic Ca 2⫹ is buffered by Ca 2⫹ endogenous buffers, and we included
parvalbumin, calbindin-D28k, and putative low-affinity buffers (Airaksinen et al., 1997; Maeda et al., 1999; Vecellio et al., 2000). We also included
an exogenous buffer, Magnesium Green 1 as a Ca 2⫹ indicator, to compare the results of simulation with the results of the Ca 2⫹ imaging experiments (Wang et al., 2000). Cytosolic Ca 2⫹ is pumped out by Na ⫹/Ca 2⫹ exchangers and Ca 2⫹-ATPase in the plasma membrane (PMCA)
and in the ER (SERCA) (Xu et al., 1997).

Representation of PF and CF inputs in the model
Instead of modeling the kinetics of AMPARs and VGCCs, a rectangularCa 2⫹ pulse was applied to the cytosol to achieve a Ca 2⫹ influx as a resultof VGCC opening. Instead of the CF input, an influx of 5000 Ca 2⫹ ions
was delivered over a period of 2 ms (in the form of a rectangular Ca 2⫹
entry waveform with 2500 ions/ms). PF input was represented by a burst
consisting of five pulses at 100 Hz. Instead of the AMPAR pathway, an
influx of 1500 Ca 2⫹ ions over a period of 1 ms per PF pulse was delivered.
This assumed degree of Ca 2⫹ influx is consistent with a previous estimatethat the Ca 2⫹ influx through VGCCs in the spines is ⬃1000 Ca 2⫹ ions
per depolarization (Sabatini and Svoboda, 2000). Glutamate was released
at the synaptic cleft during each PF pulse. Only 300 glutamate molecules
were assumed to be involved in the binding reaction of Glu and mGluRs,
because mGluRs localize at the edge of the PSD, where the neurotransmitter concentration is lower than at the center of the PSD (Mateos et al.,
2000; Franks et al., 2003). The glutamate was recycled by glutamate transporters with a decaying time constant of 5 ms.

Doi et al. • Spike-Timing Detection in Different Forms of LTD

952 • J. Neurosci., January 26, 2005 • 25(4):950 –961

Formulation of the reactions in the signaling pathways
All reactions in the signal transduction pathways were represented bybinding and enzymatic reactions. For example, a binding reaction inwhich A binds to B to form AB is expressed by the following equation:

kf
A⫹B|
0 AB,
kb

(1)

where kf and kb are the rate constants for the forward and backward
reactions. The rates kf and kb are determined by the dissociation constant
Kd and the time constant ␶. Kd is defined as Kd ⫽ kb/kf. ␶ indicates thereaction speed toward the equilibrium. Equation 1 is represented as adifferential equation with the following form:d 关AB兴/dt ⫽ kf 关A兴关B兴 ⫺ kb关AB兴.

(2)

Enzymatic reactions were modeled by the Michaelis–Menten
formulation:

k1
kcat
S⫹E|
-0 SE O¡ P ⫹ E,k⫺1

(3)where S, E, and P denote substrates, enzymes, and products, respectively.
The Michaelis constant Km is defined as Km ⫽ (k⫺1 ⫹ kcat)/k1. The
maximum enzyme velocity Vmax is defined as Vmax ⫽ kcat [E]total, where[E]total is the total concentration of the enzyme. Unless actual rate constants were provided in the literature, k⫺1 was assumed to be four times
larger than kcat. Additional simulations confirmed that the ratio of k⫺1 to
kcat had little effect on the Ca 2⫹ response. Graphic illustrations of all
reactions are provided in supplemental Figure 1 (available at www.
jneurosci.org as supplemental material). Details regarding parameter estimation, kinetic reactions, and the parameter table in the model are
given in supplemental Tables 1 and 2 (available at www.jneurosci.org as
supplemental material).

Schematic model of IP3R kinetics
We constructed a model of IP3R kinetics based on recent measurementsof Ca 2⫹ release (Marchant and Taylor, 1997) and a conceptual model(Adkins and Taylor, 1999). It was assumed that each IP3R possesses one
IP3 activation site, one Ca 2⫹ activation site, and four Ca 2⫹ inactivationsites. IP3 binding determines whether a Ca 2⫹ ion binds to the activationsite or to one of the four inactivation sites in the model (Fig. 2 A). In thebasal state, 90% of IP3Rs are not bound to any ligands (Fig. 2 A, R). IP3
binding to an IP3R exposes the Ca 2⫹ activation site. Thus, IP3R opening
requires sequential binding to one IP3 molecule and one Ca 2⫹ ion (Fig.
2 A, R3 RI3 RIC). On the other hand, if no IP3 binds to an IP3R, thefour Ca 2⫹ inactivation sites are exposed, and binding of one or moreCa 2⫹ ions to the inactivation sites of an IP3R prevents IP3 binding to theIP3 activation site (Fig. 2 A, B, R3 RC3 RC23 RC33 RC4) (Marchantand Taylor, 1997; Adkins and Taylor, 1999). The IP3R model reproduced
the equilibrium open probability of an IP3R that possesses a bell-shaped
Ca 2⫹ dependency with a peak free cytosolic Ca 2⫹ concentration([Ca 2⫹]i) of 0.2– 0.3 ␮M (Bezprozvanny et al., 1991; Fujiwara et al.,2001). The numbers of Ca 2⫹ activation and inactivation sites influence
the steepness of this open-probability function, and a curve fitting thebell-shaped open-probability function helped us to recognize that theexistence of one activation site and four inactivation sites in an IP3R
satisfied the experimental data (Fig. 2C). The number of inactivationsites is consistent with the fact that an IP3R consists of four identicalsubunits.

Estimation of kinetic parameters in the IP3R model
Biochemical reactions are generally characterized by two parameters, aforward rate constant (kf) and a backward rate constant (kb), and weestimated the dissociation constant Kd and time constant ␶, which
uniquely determine the value of kf and kb.
IP3-dependent activation. Distinct affinities of IP3R for IP3 have beenreported in in vitro and in vivo measurements. A purified IP3R in theconstituted lipid bilayer showed the affinity Kd for IP3 of ⬃1 ␮M (Bez-Figure 2. Model of IP3R kinetics used in the simulation. A, Most IP3Rs are not bound to any
ligands (R) in the steady state. Both IP3 and Ca 2⫹ are required to open IP3Rs. The Ca 2⫹ activation site does not emerge until IP3 binds to the IP3R. An IP3R opens when IP3 binds to the IP3R
and then Ca 2⫹ binds to the IP3R (R3 RI3 RIC). Ca 2⫹ also inactivates IP3Rs(R3 RC3 RC23 RC33 RC4). IP3 cannot bind to IP3Rs in any of these four inactivation states.
IP3Rs are activated and then inactivated by Ca 2⫹ elevation, because the inactivation of IP3Rs by
Ca 2⫹ is assumed to be much slower than the activation of IP3Rs by Ca 2⫹ for positive feedback
loop of Ca 2⫹-induced Ca 2⫹ release. B, Parameter reduction during the process of IP3R inactivation by Ca 2⫹ binding. White and black squares indicate naive and inactivated subunits,
respectively, of an IP3R. The eight parameters in the IP3R inactivation kinetics are reduced tothree parameters: kf , kb , and ⑀. C, Open probability of IP3Rs dependent on cytosolic Ca 2⫹
concentration (line) in the model at 10 ␮M IP3 and x in experiments with IP3Rs reconstituted
into planar lipid bilayers at 2 ␮M IP3 (Bezprozvanny et al., 1991). The difference in the IP3
concentrations is attributable to the lower IP3 affinity of IP3Rs in vivo than in the constructed
lipid bilayers (Khodakhah and Ogden, 1995; Fujiwara et al., 2001).

prozvanny et al., 1991), whereas Ca 2⫹ response to photolysis of caged IP3
in Purkinje cells suggested lower affinity of IP3R for IP3, because IP3induced Ca 2⫹ release required an IP3 concentration ⬎10 ␮M (Khodakhah and Ogden, 1995). From other in vivo measurements of IP3Rmediated Ca 2⫹ depletion of the Ca 2⫹ stores, the Kd value was estimatedto be 25.8 ␮M (Fujiwara et al., 2001), and we used this value. IP3 binding
must be slow to explain the delay of 50 – 400 ms in Ca 2⫹ release in
response to IP3 application, suggested in other in vivo measurements(Marchant and Taylor, 1997). In contrast, our model with rapid IP3
binding reproduced the delay in Ca 2⫹ release (see Fig. 8C, right, black),
and the delay resulted from a positive feedback of Ca 2⫹-induced Ca 2⫹
release. Because IP3R opening requires coactivation by Ca 2⫹ and IP3, IP3application alone does not instantaneously and fully activate Ca 2⫹ release through IP3Rs.
Ca2⫹-dependent activation. The dissociation constant Kd was estimated from the bell-shaped Ca 2⫹dependency measured by both theopen probability in lipid bilayers (Bezprozvanny et al., 1991) and theCa 2⫹ release in vivo (Fujiwara et al., 2001). We assumed that the activation is fast (␶ ⬍ 10 ms), because no studies have reported the speed of
Ca 2⫹ activation.
Ca2⫹-dependent inactivation. We assumed a multiple-step model for
Ca 2⫹-dependent inactivation, because multiple steps are required to reproduce the full inactivation of IP3Rs at high Ca 2⫹ concentrations (Bezprozvanny et al., 1991; Fujiwara et al., 2001). The four inactivation sites
are identical (i.e., only the number of inactivation sites occupied by Ca 2⫹
ions affects the sequential inactivation reactions, without regard to theactual positions occupied). The Ca 2⫹-binding reactions to the four inactivation sites are governed by a total of eight parameters [four forward
rate constants (kf) and four backward constants (kb), or four dissociation
constants (Kd) and four time constants (␶)] (see Fig. 2 B). For simplification, we assumed that binding of one additional Ca 2⫹ ion increases the kf
value ⑀-fold (⑀ is referred to as the “cooperative factor”). If ⑀ ⫽ 1, theinactivation reaction of each subunit is independent of the state of theother subunits. Note that this simplification is used just to reduce theparameter number and has not been experimentally confirmed.
Simplification reduced the eight parameters to three parameters (kf,kb, and ⑀). Curve fitting to the bell shape (Fig. 2C) yielded kb/kf ⫽ 2.25 ␮Mand ⑀ ⫽ 3. The time constant ␶ was estimated (kf ⫽ 2.22/s/␮M, kb ⫽ 5/s)

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J. Neurosci., January 26, 2005 • 25(4):950 –961 • 953

Table 1. Classification of parametersKnownorder of preference (Table 1) (for more detailed
information, see supplemental Table 3, available at www.jneurosci.org as supplemental maFrom
From
From reactions From otherterial). These above parameters, the values ofClasses of parameters
Purkinje cells other cells in test tubes
molecular subtypes Unknown Total
which were taken from the literature, are deTime constants (␶)
0
2
5
0
27
34
fined as known parameters. The remaining paDissociation constants (Kd)rameters are defined as unknown parameters.
These values were chosen for unknown paramand Michaelis
eters so that the simulation results did not conconstants (Km)
6
6
9
5
3
29
tradict the various known experimental findEnzyme turnovers (Vmax)
0
3
5
1
3
12
ings, as explained in detail below.
Initial concentrations ([])
7
7
3
1
3
21
Total
13
16
24
7
36
96
Time constants ␶, obtained from kf and kb,
are dominant in the unknown parameters. Slow
From Purkinje cells, Parameter values from measurements in cerebellar Purkinje cells; From other cells, parameter values from measurements in non-Purkinje
reactions in the signaling pathway often relay
cells, because data do not exist for Purkinje cells; From reaction in test tubes, parameter values from biochemical measurements of the reactions by purified
molecules, because no data have been measured in cells; From other molecular subtypes, parameter values from biochemical measurements of purified
the signaling stimulus with delay. We assumed
similar subtypes, because data do not exist for the subtype measured. We used the parameter values for PLC␤1 instead of those for PLC␤4; Unknown, theunreported ␶ values to be shorter than 10 ms toremaining parameters, the values of which are not found in the literature.
prevent them from significantly affecting thesimulation results without compelling reason.
2⫹
based on the inhibition of Ca release by previous incubation withThree values of the 29 dissociation constants Kd and Michaelis conCa 2⫹ (Adkins and Taylor, 1999).
stants Km were assumed (Kd of two putative low-affinity Ca 2⫹ buffers
We also assumed that the Ca 2⫹ inactivation reactions were slower
and Km of the binding of mGluR and Gq). The two buffers were assumed
than the Ca 2⫹ and IP3 activation reactions. Because of the difference into chelate free cytosolic Ca 2⫹ at ⬎10 ␮M [Ca 2⫹]i (Maeda et al., 1999).
2⫹
time constants for the Ca -dependent IP3R activation and inactivation,
The Kd of nonactivated mGluR for Gq was chosen so that half of theIP3Rs are transiently activated and then gradually inactivated by Ca 2⫹
mGluRs bound to Gq in the absence of Glu. A high Kd value tends toelevation.
allow mGluRs to bind Gq before glutamate release, whereas low Kd values
do not.
Initial state in the simulation
Three of the 12 parameter values of maximum enzyme velocities Vmax
We defined the baseline as the stable equilibrium point in the absence ofwere assumed (Gq activation by Glu-bound mGluRs, IP3 production by
any PF or CF inputs. All simulation started from the baseline. [Ca 2⫹]iCa 2⫹-bound PLC␤ without Gq, and IP3 degradation by IP3 3-kinase).
and [IP3] at baseline converged at 0.06 and 0.1 ␮M, respectively. The
The Vmax of Gq activation by Glu-bound mGluRs greatly influences thebaseline 0.1 ␮M [IP3] in our model is consistent with the 0.04 ␮M [IP3]peak amplitude of the IP3 response to the PF input in the positive direcmeasured in Xenopus oocytes (Luzzi et al., 1998).
tion. High IP3 peaks result in a large increase in Ca 2⫹ (⬎10 ␮M) to PF
input alone, whereas low IP3 peaks result in small increases in Ca 2⫹ (⬃2
Simulator and numerical integration
␮M), even when conjunctive PF and CF inputs are applied regardless of
We ran the model on a computer by using the GENESIS simulator withthe time difference between the PF input and the CF input. To reproduce
Kinetikit interface (Bhalla and Iyengar, 1999). The exponential Euler
the supralinearity of the Ca 2⫹ response to PF and CF inputs (Wang et al.,method (MacGregor, 1987) with a time step of 1 ␮s was used to solve the2000), the Vmax value was chosen. We also adjusted the Vmax value so thatdifferential equations.
the PF input would produce 3 ␮M IP3. The IP3 amplitude in our model isDefinition of time-window function
consistent with the amplitude reported in a study of Ca 2⫹ imaging and
The time window of the Ca 2⫹ response was defined as the average relaIP3 uncaging (Finch and Augustine, 1998). The investigators in thattive change in fluorescence (⌬F/F0) of Ca 2⫹ indicator as a function of thestudy compared Ca 2⫹ responses to photolysis of caged-IP3 and to PF
time difference between the PF input and the CF input. The indicator is
stimulation. They estimated that 16 PF stimuli at 60 Hz produce 10 –20
Magnesium Green 1 (250 ␮M; Kd ⫽ 19 ␮M; Fmax/Fmin ⫽ 2). The time
␮M IP3 in the presence of an AMPAR antagonist. In our model, mGluR
difference between the PF input and the CF input is identical to the time
activation with 16 PF stimuli at 60 Hz evokes 10.3 ␮M IP3. A balanceof the CF input because we applied the PF input at 0 ms. ⌬F/F0 was
between IP3 production and degradation that would maintain 0.1 ␮Maveraged from the first input (either PF or CF) to 500 ms after the final[IP3] at the baseline and reproduce the width of the time window (Wang
input of PF and CF. This time-window definition is as per the method
et al., 2000) was assumed.
used to analyze results of the Ca 2⫹ imaging experiment (Wang et al.,Of the 21 molecules in the model, concentrations of three have never2000) for direct comparisons. ⌬F/F0 in the simulation is calculated by the
been reported (SERCA, PMCA, and Na ⫹/Ca 2⫹). Their concentrations
following formula:were chosen to reproduce their relative contributions to intracellular
Ca 2⫹ clearance at different [Ca 2⫹]i (Fierro et al., 1998) and to maintain2⫹
关 MgGreen兴 ⫹ 关MgGreen–Ca 兴共Fmax/Fmin兲
thebaseline [Ca 2⫹]i at 0.06 ␮M.
⌬F/F 0 ⫽⫺ 1.
(4)
关MgGreen兴0 ⫹ 关MgGreen–Ca2⫹ 兴0 共Fmax/Fmin兲
The numerator and denominator are present fluorescence and basal fluorescence, respectively. MgGreen denotes free Magnesium Green 1, and
MgGreen–Ca 2⫹ is a binding form of MgGreen and Ca 2⫹. Fmax/Fmin is
the ratio of the maximum fluorescence to the minimum fluorescence.

Estimation of the values of unknown parameters
The schematic model as shown in Figure 1 illustrates the molecular interactions. The interactions were described by biochemical reactions
as graphically shown in supplemental Figure 1 (available at www.
jneurosci.org as supplemental material). Our model consists of 53 variables and 96 parameters (supplemental Tables 1 and 2, available at www.
jneurosci.org as supplemental material). To make the model as realistic
as possible, the values for the parameters were estimated from experimental data obtained in cerebellar Purkinje cells, if they were available.
When such data did not exist, experimental data from other cells, chemical reaction in test tubes, and other similar subtypes were used, in that

Results

Supralinear Ca 2ⴙ response induced by conjunctive PF–CF
input in the single-spine model
We confirmed that the supralinear Ca 2⫹ response to conjunctive
PF and CF stimulation was reproduced in the single-spine modelwith the parameter set shown in supplemental Tables 1 and 2(available at www.jneurosci.org as supplemental material). Figure 3A shows simulated Ca 2⫹ responses when PF and/or CF
inputs were applied at times 0 and 100 ms. PF input alone yielded
five Ca 2⫹ transients of ⬃1 ␮M (Fig. 3A, green, arrow 1) as a resultof Ca 2⫹ influx via the AMPAR pathway and a subsequent slow
Ca 2⫹ increase (arrow 2) because of IP3 production via themetabotropic pathway. CF input alone evoked a sharp Ca 2⫹transient with a peak of 2 ␮M (Fig. 3A, red). The Ca 2⫹ response to
conjunctive PF–CF inputs (Fig. 3A, blue) yielded a supralinear954 • J. Neurosci., January 26, 2005 • 25(4):950 –961

Doi et al. • Spike-Timing Detection in Different Forms of LTD

Figure 3. Time courses of simulated molecular concentrations in response to PF input(green), CF input (red), and conjunctive PF–CF inputs (blue). PF and CF inputs are applied at 0
and 100 ms, respectively. A, Free cytosolic Ca 2⫹ concentration ([Ca 2⫹]i ). B, IP3 concentration([IP3]). C, Percentage of open IP3Rs. D, Free Ca 2⫹ concentration in the ER ([Ca 2⫹]ER ). E, Percentage of inactivated IP3Rs.

signal (8.1 ␮M peak at 230 ms), and it returned to the basal levelwithin 1 s. The supralinearity of the simulated Ca 2⫹ responses issimilar to the supralinearity observed in the Ca 2⫹ imaging experiment (Wang et al., 2000), but not exactly the same. At ⬍1 ␮M[Ca 2⫹]i, the increase in fluorescence (⌬F/F0) is approximately
proportional to the increase in [Ca 2⫹]i, and the ratio is 0.2 ␮M⌬[Ca 2⫹]i per percentage of ⌬F/F0.
Cytosolic IP3 gradually accumulated after the PF inputs (3.1
␮M peak at 220 ms) (Fig. 3B, green), and the CF input had little
influence on the IP3 dynamics (Fig. 3B, compare blue, green).
The proportion of open IP3Rs in response to conjunctive PF–CF
inputs reached 30% (Fig. 3C, blue) as opposed to only 10% in
response to PF input alone (Fig. 3C, green) and 1.7% in response
to CF input alone (Fig. 3C, red). Therefore, CF-mediated Ca 2⫹transients triggered IP3R activation only during IP3 elevations.
The supralinear Ca 2⫹ response was induced by Ca 2⫹ releasefrom the ER Ca 2⫹ stores as a result of opening IP3Rs, as indicatedby the increase in open IP3Rs (Fig. 3C, blue) and the reduced
Ca 2⫹ concentration in the ER (Fig. 3D, blue). More than 90% of
the IP3Rs were in the inactivated state after the conjunctive
PF–CF input (Fig. 3E, blue), indicating that IP3R inactivation
terminated the supralinear Ca 2⫹ signal.
Ca 2ⴙ-dependent IP3R activation and inactivation are
essential for supralinear Ca 2ⴙ dynamics
Supralinear Ca 2⫹ dynamics may be mediated by a positive feedback loop(s) in the signaling pathways. We identified two candidate positive feedback loops for the supralinear Ca 2⫹ signal in theblock diagram (Fig. 1). One is the Ca 2⫹-dependent IP3R activation
loop. Ca 2⫹ elevation may open IP3Rs, resulting in additional Ca 2⫹elevation (Ca 2⫹3 IP3Rs3 Ca 2⫹). The other is the loop of Ca 2⫹dependent IP3 production by PLC␤. Ca 2⫹ elevation may enhance
the activity of PLC␤ for IP3 production, and IP3 elevation may open
IP3Rs and release Ca 2⫹ from the ER, resulting in additional Ca 2⫹elevation (Ca 2⫹3 PLC␤3 IP33 IP3Rs3 Ca 2⫹). The diagram also
contains two negative feedback loops for terminating Ca 2⫹ release.
The Ca 2⫹-dependent IP3R inactivation loop may inhibit IP3R
opening at high Ca 2⫹ concentration. Alternatively, Ca 2⫹ release
may continue until depletion of the Ca 2⫹ in the ER stores.
We deleted reactions to identify the dominant pathways
responsible for Ca 2⫹ dynamics (Fig. 4 A), by maintaining themolecular concentration on the upstream side of the reaction

Figure 4. Pathway deletion analysis. PF and CF inputs are given at 0 and 100 ms, respectively. Pathway deletion was started at the CF input. When a pathway in which molecule A(downstream) is dependent on molecule B (upstream) in Equation 1 was deleted, the concentration of B was fixed at the baseline in Equation 2, and changes in molecules A and AB were
calculated. If the upstream molecules were involved in multiple reactions (i.e., Ca 2⫹), we
maintained the molecular concentration constant only in the deleted reaction, and the molecular concentrations in the other reactions varied as usual. A, Illustration of the specific pathways
deleted in the diagrams below. B–F, The black and gray lines indicate time course of [Ca 2⫹]i inthe deleted pathway and under control conditions (identical to Fig. 3A, blue), respectively. B,
Deletion of the IP3-dependent IP3R activation pathway. C, Deletion of the [Ca 2⫹]i-dependent
IP3R activation pathway. D, Deletion of the [Ca 2⫹]i-dependent PLC␤ activation pathway. E,
Deletion of the [Ca 2⫹]i-dependent IP3R inactivation pathway. F, [Ca 2⫹]ER is held constant at
the baseline concentration.

constant at its basal concentration. For example, when the
IP3-dependent IP3R activation reaction is deleted (Fig. 4 B), theIP3Rs are exposed to IP3 at the basal concentration, whereas
the IP3-dependent reactions (degradation by IP3 3-kinase and5-phosphatase) react with variable IP3 concentration.
The first reaction we deleted was IP3 binding to IP3Rs. We
prevented IP3R activation from increasing IP3. There was no supralinear Ca 2⫹ response at all (Fig. 4 B), indicating that supralinear Ca 2⫹ signals require IP3 elevation. When IP3R activation wasdeleted by fixing the Ca 2⫹ concentration for the IP3R activation
alone, the supralinear Ca 2⫹ response was completely eliminated(Fig. 4C), whereas deletion of the pathway of PLC␤ activation by
Ca 2⫹ did not eliminate the supralinear Ca 2⫹ response (Fig. 4 D).
These results indicate that the supralinear Ca 2⫹ response to conjunctive PF-CF inputs is a regenerative process driven by the
positive feedback loop in which Ca 2⫹ induces Ca 2⫹ releasethrough the IP3Rs.
Deletion of the IP3R inactivation reaction dramatically prolonged and increased the [Ca 2⫹]i elevation (Fig. 4 E). Finally, weclamped [Ca 2⫹] in the ER ([Ca 2⫹]ER) at 150 ␮M, the basal con-

Doi et al. • Spike-Timing Detection in Different Forms of LTD

Figure 5. Regenerative Ca 2⫹ release dependence on the timing of PF–CF inputs. A, Ca 2⫹
responses to conjunctive PF–CF inputs with various intervals between the PF input and CF input.
The PF input is delivered at 0 ms, indicated by the five repetitive lines. CF input is delivered at⫺200 to ⫹ 400 ms, as indicated by the single line. B, Overplot of IP3 time courses in responseto the various PF–CF inputs.

centration. By maintaining the [Ca 2⫹]ER constant, we prevented
the gain and loss of [Ca 2⫹]ER attributable to Ca 2⫹ releasethrough the IP3Rs, Ca 2⫹ uptake by Ca 2⫹ pumps, and so on. The
constant [Ca 2⫹]ER did not significantly alter the time course ofchanges in [Ca 2⫹]i (Fig. 4 F). Thus, Ca 2⫹-dependent IP3R inactivation, not Ca 2⫹ depletion in the ER, is essential and solely
responsible for terminating the Ca 2⫹-induced Ca 2⫹ release.
PF–CF time window of the Ca 2ⴙ response is controlled by thechanges in IP3 concentration over time
To explore the molecular mechanisms responsible for the detection of spike timing, conjunctive PF–CF inputs were applied atvarious intervals between the PF input and the CF input, asshown in Figure 5A. The Ca 2⫹ response reached at least 5 ␮Mwhen the CF input followed PF input within intervals of 0 –300
ms (Fig. 5A). Application of PF–CF inputs in the reverse order
did not evoke regenerative Ca 2⫹ release. The time course of IP3
concentrations did not change much at different PF–CF intervals,
and it was always time-locked to the PF input (Fig. 5B).
To evaluate the timing-dependent Ca 2⫹ responses as a function of the PF–CF time interval quantitatively, the time-windowfunction of the Ca 2⫹ response was defined as the time-integrated⌬F/F0 from the first input to 500 ms after the last input (compare
Fig. 6 A) (see Materials and Methods), as per the method used toanalyze results of Ca 2⫹ imaging (Wang et al., 2000). The timewindow function is plotted as a solid curve in Figure 6 B.
The time-window function was large between 0 and 300 ms,
and Gaussian fitting to the time window yielded a peak time of
114 ms and half-width of 170 ms, consistent with the results of the

J. Neurosci., January 26, 2005 • 25(4):950 –961 • 955

Figure 6. Time window of the Ca 2⫹ response controlled by the IP3 time course. A, Examplesof the time-integral interval of the Ca 2⫹ response to PF–CF inputs. The PF–CF interval, ⌬t, isdefined as the time between the start of the PF input to the start of the CF input. Ca 2⫹ response
is defined as the average increase in fluorescence (⌬F/F0 ) from the first stimulus to 500 ms after
the final stimulus. The Ca 2⫹ indicator is Magnesium Green 1 (Kd ⫽ 19 ␮M; Fmax /Fmin ⫽ 2). B,
Time window of Ca 2⫹ response as a function of the PF–CF interval, controlled by the IP3 timecourse. Ca 2⫹ responses to the PF or CF input alone in the simulation and the Ca 2⫹ imaging
results are plotted on the right as closed and open circles, respectively. The simulated time
window (solid line) is compared with the Ca 2⫹ imaging data (open circles) (Wang et al., 2000).
The dotted and broken lines indicate time windows when IP3 peak time ( C) in response to the PF
input alone was shifted to earlier and to later times by modifying enzymatic parameters.

Ca 2⫹ imaging experiments (peak time, 92 ms; half-width, 212ms) (Wang et al., 2000). A very similar time window was obtained
when we used [Ca 2⫹]i instead of ⌬F/F0 as a measure of the Ca 2⫹
response (data not shown). Thus, it can be concluded that the
supralinear Ca 2⫹ response is not attributable to a saturating effect of the fluorescent indicator. Ca 2⫹ influx attributable to PF
and CF activation outside the above effective time window did
not trigger major Ca 2⫹ release from the ER.
Temporal similarity was found between the slow IP3 response
and the time-window function (Fig. 6 B, C, compare solid
curves), suggesting that the IP3 time course accounts for spiketiming detection. To systematically verify that the IP3 time course
actually does determine the time window of the Ca 2⫹ response,
we intentionally manipulated the IP3 time course by varying twoparameter values in the model as described below and then plotted the time windows. The GTPase activation protein (GAP) activity of PLC␤ promotes Gq inactivation, and the maximum
enzyme velocity of GAP was increased fivefold to shift the IP3
peak time to an earlier time. At the same time, the maximum

956 • J. Neurosci., January 26, 2005 • 25(4):950 –961

Doi et al. • Spike-Timing Detection in Different Forms of LTD

the inactivation of IP3R recovered at 500 ms. Elimination of parvalbumin, the other high-affinity buffer in the simulation, did nothave any significant effect on the Ca 2⫹ responses (Fig. 7B). We
therefore concluded that calbindin is essential for timing
detection.

Figure 7. Ca 2⫹ buffers contribute to shaping the time window of Ca 2⫹ response. A, Ca 2⫹
responses to conjunctive PF–CF inputs with various timings, identical to Figure 5A, except for in
the absence of the high-affinity Ca 2⫹ buffer calbindin. B, The time window of the Ca 2⫹ response without calbindin and/or paravalbumin.

velocity of enzyme catalysis of IP3 production by PLC␤ withCa 2⫹ and Gq was increased 3.5-fold to maintain the peak amplitude of the time window of Ca 2⫹ responses. As a result, the time
window narrowed and shifted to an earlier time (Fig. 6 B, C, dotted lines). To shift the IP3 peak to a later time, on the other hand,
the maximum enzyme velocities of GAP and IP3 production weredecreased 0.2- and 0.5-fold, respectively, and as a result, the time
window expanded and shifted to a later time (Fig. 6 B, C, brokenlines). Thus, the time-window function of the Ca 2⫹ response
depended on the IP3 time course.
Cerebellar Purkinje cells contain high concentrations of highaffinity buffers, such as calbindin and parvalbumin (Fierro andLlano, 1996; Maeda et al., 1999). These buffers hinder increases in[Ca 2⫹]i at ⬍1 ␮M [Ca 2⫹]i but saturate at ⬎1 ␮M [Ca 2⫹]i. We
explored the effect of eliminating the high-affinity buffers on thetime window of the Ca 2⫹ response. In the absence of calbindin,
PF or CF input produces a large Ca 2⫹ increase to ⬃5 ␮M (Fig.
7A). The time window of the Ca 2⫹ response had two peaks, atime difference between the PF and CF inputs of one ⬃0 ms andthe other ⬃500 ms (Fig. 7B). The first peak indicates that calbindin prevents only the Ca 2⫹ influx in response to conjunctive
PF–CF inputs from producing a regenerative increase in Ca 2⫹.
The second peak appeared because the IP3 elevation persisted andRegenerative Ca 2ⴙ release as a threshold phenomenon
regulated by [IP3]
Qualitatively different Ca 2⫹ dynamics at different IP3 levels weresystematically analyzed and visualized while simplifying the simulated PF input as a step-function elevation in IP3 (Fig. 8 A),
because the IP3 response to the PF input is much slower than the
Ca 2⫹ response (Fig. 5B). We analyzed Ca 2⫹ changes in responseto three different constant IP3 levels: 0.1 ␮M (actually no change),
2.5 ␮M, and 10 ␮M (Fig. 8 A). The first IP3 level, 0.1 ␮M, is identical to the baseline condition. The second IP3 level, 2.5 ␮M, corresponds to the IP3 concentration 100 ms after the PF input (Fig.
3B), and the third level, 10 ␮M, is excessively high and was obtained by IP3 uncaging (Khodakhah and Ogden, 1995; Finch andAugustine, 1998). From the baseline concentration of 0.1 ␮M,[IP3] was changed to one of the three levels in a step function andheld constant (Fig. 8 A). Synchronized with this IP3 step-up, an
influx of a different amount of Ca 2⫹ was delivered for 2 ms (Fig.
8 B), the same duration as the simulated CF input. The timecourses of [Ca 2⫹]i and IP3R inactivation are shown in Figure 8, C
and D, respectively, and they are also plotted as trajectories onphase planes in Figure 8 E. The different amounts of Ca 2⫹ influx
are color-coded in Figure 8 B, and their responses are correspondingly color-coded in Figure 8C–E.
At 0.1 ␮M IP3, [Ca 2⫹]i monotonically decreased after the
Ca 2⫹ influx (Fig. 8C,E, left), and larger Ca 2⫹ influxes produced
higher Ca 2⫹ peaks and stronger IP3R inactivation (Fig. 8C,D,
left). The trajectories were evenly distributed spatially in thephase plane (Fig. 8 E, left), indicating almost linear responses to
Ca 2⫹ influx.
At 2.5 ␮M IP3, [Ca 2⫹]i monotonically decreased after the influx when the Ca 2⫹ transients at the end of the Ca 2⫹ influx wereⱕ2 ␮M, whereas when they were larger than 3 ␮M, [Ca 2⫹]i
dropped very briefly and then increased back to ⬎10 ␮M (Fig. 8C,
middle inset). Thus, Ca 2⫹ dynamics can be classified into suprathreshold responses and subthreshold responses bounded bya threshold of 2–3 ␮M Ca 2⫹ transient. The trajectories aroundthis threshold were very sparsely spaced (Fig. 8 E, middle, compare the light and dark blue trajectories), which also confirms thenonlinear threshold phenomenon.
At 10 ␮M IP3, [Ca 2⫹]i increased to ⬎15 ␮M without any Ca 2⫹
influx (Fig. 8C, right, black line), indicating that the baseline[Ca 2⫹]i was already high enough to induce regenerative Ca 2⫹
release in the presence of high IP3. There was a latency of 50 ms in
the Ca 2⫹ response in the absence of a Ca 2⫹ influx (Fig. 8C, right,
black line), indicating that the positive feedback loop of regenerative Ca 2⫹ release requires a relatively long build-up time to be
fully activated at low [Ca 2⫹]i. A larger Ca 2⫹ influx elevated the[Ca 2⫹]i peak and reduced this latency (Fig. 8C, right, compare
the latency of different lines), which is consistent with the IP3induced Ca 2⫹ increase in experimental studies (Khodakhah andOgden, 1995; Finch and Augustine, 1998).
Figure 8 F is a color-coded contour map of the averaged Ca 2⫹
response over 500 ms as a function of the transient Ca 2⫹ peaks(see initial peaks in Fig. 8C, middle inset) along the ordinate andthe constant IP3 level along the abscissa. First, the averaged Ca 2⫹
response increased with the size of the transient Ca 2⫹ peak andthe concentration of IP3. However, the Ca 2⫹ dynamics wereDoi et al. • Spike-Timing Detection in Different Forms of LTD

J. Neurosci., January 26, 2005 • 25(4):950 –961 • 957

tour lines were densely packed in this region alone, indicating the threshold for alarge Ca 2⫹ release.
Robustness of the response of the spiketiming detection mechanism to values of
unknown parameters
The dependence of the supralinear Ca 2⫹
response on the timing of PF–CF inputs is
attributable to the delay between the PF
input and the IP3 peak (Fig. 6 B, C). We
explored the relationships between the robustness of the IP3 time course in responseto the PF input alone and against variations in each of the parameter values. The
value for each parameter was varied from1/100- to 100-fold the standard value withthe value for all other parameters constant.
The results of the sensitivity analysis of theIP3 time course in relation to each parameter are shown in supplemental Figure
2 A–E (available at www.jneurosci.org as
supplemental material). Although the IP3time course was robust to the variations in
the values of most of the parameters, theIP3 peak time was greatly affected by someparameters for maximum enzyme velocities (Vmax): Gq activation by Glu–mGluR(parameter ID in supplemental material:
a5; available at www.jneurosci.org), IP3
production by PLC␤ without Gq (b6) andwith Gq (b7), GAP activity of PLC␤ (b10,b11, b12), and IP3 degradation by IP3
3-kinase (c3). The values of only three ofthese critical parameters (a5, b6, and c3)
were unknown. Parameter a5 was chosen
so that the mGluR pathway produces 10
␮M IP3 at 16 PF pulses, based on Ca 2⫹
response to uncaging IP3 (Finch and Augustine, 1998). Parameters b6 and c3 were
Figure 8. Ca 2⫹ dynamics with threshold dependent on [IP3]. A, [IP3] was step-changed to 0.1, 2.5, or 10 ␮M from the steadychosen to maintain the basal [IP3] level at2⫹2⫹state, 0.1 ␮M, at time 0 ms. B, The 2 ms Ca influx started to be delivered at the same time, 0 ms. The amounts of Ca influx
0.1
␮M. Figure 9 shows time courses of IP3
2⫹
were adjusted so that the transient Ca peaks for 2 ms were 1 ␮M (violet), 2 ␮M (dark blue), 3 ␮M (light blue), 4 ␮M (dark green),
5 ␮M (light green), 6 ␮M (yellow), 7 ␮M (orange), 8 ␮M (red), 9 ␮M (magenta), and 10 ␮M (pink). The black lines indicate that no concentrations in relation to differences in
Ca 2⫹ influx was delivered. C, Response of [Ca 2⫹]i to the combination of step-changed [IP3] and Ca 2⫹ influx. Inset, Enlargement the values (1/10- to 10-fold) of the threeof the [Ca 2⫹]i response, the transient Ca 2⫹ peaks of which were 2 and 3 ␮M at 2.5 ␮M [IP3]. D, Response of IP3R inactivation rate unknown parameters. Gq activation by
to the combination of step-changed [IP3] and Ca 2⫹ influx. E, Trajectories in C (t-x plane) and D (t-y plane) were mapped on the Glu–mGluR (a5) had a major effect on thephase plane of [Ca 2⫹]i versus the IP3R inactivation rate (x-y plane). Arrows indicate the time directions in the trajectories. F, height of the peak of IP3 concentration but
Average Ca 2⫹ response over 500 ms after inputs is colored in the contour map as a function of step-changed [IP3] and the not on the time of the peak (Fig. 9A).
transient Ca 2⫹ peak. The bar at the left illustrates the color and contour scales of the averaged Ca 2⫹ response. The red area When we applied large V
max (10-fold theindicates suprathreshold responses, and the blue area indicates subthreshold responses. The three dotted lines approximately control) of IP production by PLC␤ in the32⫹indicate the maximum levels of the transient Ca peaks in response to CF and PF stimulation and the IP3 amplitude in response
absence of Gq (b6), we obtained a proto PF stimulation. See Discussion for the meaning of regions (i)–(vi).
longed IP3 response (Fig. 9B). IP3 degradation by IP3 3-kinase (c3) affected both thepeak amplitude and the time of the peakqualitatively different for small, medium, and large IP3, as already(Fig. 9C). A larger Vmax of IP3 degradation by IP3 3-kinase dedescribed in Figure 8 A–E. At 0.1 ␮M IP3, the contour lines werecreased the peak amplitude and shifted the peak to an earlier
time.
nearly horizontal, indicating that the average Ca 2⫹ response wasmainly determined by the amount of Ca 2⫹ influx. At 10 ␮M IP3,
Next, we examined the robustness of the time window of
the contour lines were nearly vertical, indicating that the average
PF–CF inputs and the time course of the concentration of the
Ca 2⫹ response was mainly determined by the IP3 level. The conchanges in IP3. Each parameter was varied from 1/100- to 100tour lines have negative diagonal slopes only in the 1– 4 ␮M IP3fold of its standard value with all other parameters constant, andregion, indicating that both the Ca 2⫹ influx and [IP3] signifithe behavior of the model was quantified in terms of the peakcantly influence the average Ca 2⫹ response. Moreover, the contime, half-width, and peak amplitude of the temporal window958 • J. Neurosci., January 26, 2005 • 25(4):950 –961

Doi et al. • Spike-Timing Detection in Different Forms of LTD

and make it possible to fully understand the mechanisms of theinformation processing performed by the molecular dynamic
networks. In this study, we modeled Ca 2⫹ dynamics within asingle spine to elucidate the actual mechanism of spike-timing
detection. Supralinear Ca 2⫹ responses were generated by Ca 2⫹dependent IP3R activation during IP3 elevation, and the PF–CF
time window was determined from the time course of changes in
IP3 concentrations. PF-mediated IP3 elevation lowered the Ca 2⫹
threshold, and CF-mediated Ca 2⫹ influx triggered Ca 2⫹ releasethrough the IP3Rs. Three Ca 2⫹ responses exhibit different dynamics at low, medium, and high IP3 concentrations.

Figure 9. Sensitivity of the IP3 time course to variations in parameter values. Three unknown
maximum enzyme velocities that affect the IP3 time course are shown as representative examples. The parameters were varied within the 1/10- to 10-fold range. A, Gq activation by Glu–mGluR complex. B, IP3 production by PLC␤ without Gq. C, IP3 degradation by IP3 3-kinase.

after Gaussian fitting (supplemental Fig. 2 F, G; available atwww.jneurosci.org as supplemental material). The results of thesensitivity analysis of the time window for each parameter are
shown in supplemental Figure 2 H–K (available at www.jneurosci.org as supplemental material). The parameters in the model
were classified into four categories: time constants, dissociationand Michaelis constants, maximum enzyme velocities, and initialconcentrations. Figure 10 shows graphs of the average changes
and SDs of the peak time (left column), half-width (middle column), and peak amplitude (right column) when the values werevaried as indicated along the abscissa of these four classes of parameters. Similar tendencies were shown between parameterswith values that are known (black) and unknown (gray). The
three characteristics of the temporal window changed little inresponse to small differences in the parameters (one-half or twotimes). Most importantly, even major changes in parameters (1/100 –100 times) had little effect on the peak time and half-width,
although peak amplitudes sometimes changed considerably. Because spike-timing detection is essentially characterized by theshape of the temporal window (i.e., the peak time and halfwidth), we concluded that the proposed model is very robust in
its most important characteristic (i.e., spike-timing detection fora large range of parameter variations).

Discussion
Because many signal transduction pathways interact with eachother and often exhibit nonlinear dynamics, it would be difficult
to understand complex cellular information processing on thebasis of experiments alone. Computational kinetic simulation
may make it possible to determine whether known signaling
pathways are sufficient to reproduce cellular events of interest

IP3 regulation of Ca 2ⴙ-induced Ca 2ⴙ release and the PF–CF
time window
In Figure 11 A, the IP3-dependent Ca 2⫹ threshold indicates the
Ca 2⫹ input required for IP3 concentration at the time, so that
averaged Ca 2⫹ response in the 500 ms after the CF input is ⬎3.16
␮M (Fig. 8 F, bold line between the blue area and red area), dependent on IP3 at this time (Fig. 5B). The Ca 2⫹ threshold represents the size of the Ca 2⫹ increase required for a large Ca 2⫹
response, if the CF input is applied at the time. The IP3dependent Ca 2⫹ threshold explains the optimal timing of the PF
input for the regenerative Ca 2⫹ response. CF input within 300 ms
after PF input induces a Ca 2⫹ transient and causes the Ca 2⫹
response to reach the threshold (Fig. 11 A, black line). In contrast,
when the CF input occurs before the PF input (Fig. 11 A, thickgray line) or occurs too late after the PF input (Fig. 11 A, dottedgray line), the CF-mediated Ca 2⫹ influx does not increase Ca 2⫹
concentrations enough to reach the threshold.
Regenerative Ca 2⫹-induced Ca 2⫹ release via IP3Rs accounts
for the supralinear Ca 2⫹ signals. First, Ca 2⫹-induced Ca 2⫹ release was initiated by CF-mediated Ca 2⫹ entry in the presence ofincreased IP3, then accelerated by the positive feedback loop of
Ca 2⫹-dependent IP3R activation, and finally terminated by thenegative feedback loop of Ca 2⫹-dependent IP3R inactivation(Fig. 11 B).
The results of the sensitivity analysis (Figs. 9, 10) (supplemental Fig. 2, available at www.jneurosci.org as supplemental material) reliably support the notion that the spike-timing detection
mechanism is not dependent on parameter values but on themodel structure, specifically on the difference in time scale between a rapid electrical reaction (Ca 2⫹ influx by depolarization)
and a slow biochemical reaction (IP3 production by the metabotropic pathway). However, it should be pointed out that although
the results of our simulation support our hypothesis, they do not
exclude possibilities of other functions of the pathways or additional pathways.
The temporal relationship between the coincidence of the PF
and CF inputs must be detected at some point of intersection
between the two signal transduction pathways downstream to the
PF and CF inputs, because we found only one possible intersection point in the MAP kinase cascade in addition to the IP3Rs
cascade in an LTD signaling model we previously proposed (Kuroda et al., 2001). CF input stimulates the MAP kinase cascade,whereas PF input activates the nitric oxide (NO)/cGMP pathway,
which plays a gating role in allowing activation of the MAP kinasecascade. However, because activation of the NO/cGMP pathway
is too slow (order of minutes) to detect subsecond timing (Hartell
et al., 2001), IP3Rs and the upstream pathways must detect thetiming of the PF and CF inputs. Wang et al. (2000) have proposedthat IP3Rs themselves may detect the temporal order of thechanges in IP3 and Ca 2⫹. IP3Rs may be inactivated by the CFmediated Ca 2⫹ influx before the PF-mediated IP3 elevation, be-Doi et al. • Spike-Timing Detection in Different Forms of LTD

J. Neurosci., January 26, 2005 • 25(4):950 –961 • 959

by measurement of the Ca 2⫹ response to
flash photolysis of caged IP3 and caged
Ca 2⫹. The caged compounds have different optimal wavelengths for uncaging thatenable the concentrations of IP3 and Ca 2⫹
to be controlled independently. We predict that IP3 uncaging will yield a delayed
Ca 2⫹ increase (Fig. 8C, right, black),
whereas Ca 2⫹ uncaging will yield an immediate and transient Ca 2⫹ increase (Fig.
8C, left). Conjunctive Ca 2⫹–IP3 uncaging
should induce a supralinear Ca 2⫹ response that is larger than the sum of the
response to uncaging of Ca 2⫹ alone anduncaging of IP3 alone. The largest Ca 2⫹
response would be expected in coincidental IP3–Ca 2⫹ conjunctive uncaging, unlike
conjunctive PF–CF stimulation, because,
unlike the metabotropic pathway, IP3 uncaging elevates [IP3] without any delay. If
delayed Ca 2⫹ photolysis after IP3 photolysis is found to be optimal, our model will
be refuted.
Various LTD forms unified within asingle model of Ca 2ⴙ dynamics
We previously modeled the signaling
pathways involved in LTD induction, including the NO/cGMP pathway, MAP kinase cascade, protein kinase C activation,
and AMPAR phosphorylation, which were
not incorporated in the current study, andfound that a supralinear Ca 2⫹ increase induced by conjunctive PF and CF inputs isrequired for induction of LTD (Kuroda etal., 2001). However, the Ca 2⫹ increase in
the previous model was based on experiFigure 10. Percentage of changes in the three characteristic quantities of the time window induced by changes in parameter
mental observations (Wang et al., 2000)
values. All parameters were classified as time constants (first row), dissociation and Michaelis constants (second row), maximum
enzyme velocities (third row), and initial concentrations (fourth row). Each parameter was varied within a range of 1/100- to and was not modeled in a kinetic simula100-fold with other parameters maintained constant. The time window was simulated for this set of parameters and was fitted by tion. Thus, the previous and current moda Gaussian function. The percentage of changes in the peak time (left), half-width (middle), and peak amplitude (right) of the els complement each other and could be
Gaussian function compared with those of the original time window in Figure 6 B were computed and then averaged over regarded as downstream and upstream
parameters within the four categories. These averaged changes in the characteristic quantities were plotted as a function of the models of the entire LTD signaling transmagnification (minification) factor of the changed parameter. The known and unknown parameters are marked in black and gray, duction. The previous downstream model
respectively. The bars represent ⫾SD. The asterisks in some figures mean that one or two of the computations could not be (Kuroda et al., 2001) computed AMPAR
completed because of numerical explosions.
phosphorylation dependent on Ca 2⫹ increase as one of the major inputs, whereas
cause IP3-evoked Ca 2⫹ release is inhibited by preincubation withthe current upstream model reproduced the Ca 2⫹ increase withCa 2⫹ (Khodakhah and Ogden, 1995; Adkins and Taylor, 1999)
PF and CF inputs. Therefore, although we only modeled theand thus can only be activated by the PF–CF order. However, theCa 2⫹ dynamics in this study, the current model is useful for
elucidating the critical role of Ca 2⫹ dynamics in different forms
Ca 2⫹ response to the CF input was much weaker in our simula2⫹
tion (Fig. 3A, red) than the strong Ca stimulation in theirof LTD as described below.
experiments (10 or 100 ␮M for ⬃1 s). Thus, this first possibility is
The various Ca 2⫹ dynamics dependent on IP3 are capable ofunlikely to account for the detection of the timing. Instead, thecoherently explaining various forms of LTD. We conceptuallykinetics of IP3 binding to IP3Rs may be slow and result in theillustrated six regions of different dynamics in Figure 8 F. First,
necessary time difference, and this seemed to be experimentally
region (i) at the baseline IP3 shows that CF input alone yields a
supported by a latency of ⬃50 ms in IP3-evoked Ca 2⫹ releasesmall Ca 2⫹ transient (⬃2 ␮M [Ca 2⫹]i) that does not enable the(Khodakhah and Ogden, 1995; Marchant and Taylor, 1997).
averaged Ca 2⫹ response to reach the Ca 2⫹ threshold. Region (i)also shows that LTD is eliminated when IP3 production via the
However, a similar latency was obtained in our simulation (Fig.
8C, right, black) with the fast IP3 binding in our IP3R kinetic
metabotropic pathway is inhibited. Mice lacking mGluR1 exhibit
model, and thus these experimental data cannot discriminate
impaired LTD induction and motor learning (Aiba et al., 1994;
between the two mechanisms of detecting the timing.
Conquet et al., 1994). Second, region (ii) illustrates that PF input
Clear discrimination between the two possibilities is possible
alone yields a Ca 2⫹ influx of ⬃1 ␮M and a subsequent IP3 in-Doi et al. • Spike-Timing Detection in Different Forms of LTD

960 • J. Neurosci., January 26, 2005 • 25(4):950 –961

Figure 11. Schematic model of PF–CF timing detection based on changes in the Ca 2⫹
threshold over time. A, Ca 2⫹ time courses with PF–CF input intervals of ⫺100 ms (thick grayline), ⫹200 ms (black line), and ⫹500 ms (dotted gray line). The threshold indicates theaveraged Ca 2⫹ response of 3.16 ␮M (Fig. 8 F) and was computed from the IP3 time course with
the PF input alone (Fig. 3B, green). If the transient Ca 2⫹ response to the CF input exceeds the
Ca 2⫹ threshold, regenerative Ca 2⫹ release is triggered, and a supralinear Ca 2⫹ response
appears. B, The ⌬ in the mGluR pathway indicates a time delay between PF input and IP3
accumulation. Ca 2⫹ release is both activated and inactivated by the Ca 2⫹ increase. In thepresence of sufficient IP3 , Ca 2⫹ influx triggers regenerative Ca 2⫹ release by the fast positive
feedback loop of Ca 2⫹-dependent IP3R activation. After the Ca 2⫹ increase, the slow negative
feedback loop terminates Ca 2⫹ release.

crease. The elongated region (ii) represents the transition of Ca 2⫹
dynamics after PF input, starting at the left side. As IP3 is increased by the PF input, the Ca 2⫹ dynamics move to the right
portion of region (ii). Ultimately, the Ca 2⫹ dynamics in region(ii) never reach the threshold. Third, the horizontal region (iii)
illustrates the dependency of Ca 2⫹ dynamics on the timing of
conjunctive PF–CF inputs. IP3 maximally increased to 3 ␮M andthen decreased to the baseline after PF input (Fig. 3B). Region(iii) represents these different IP3 at different times with respect
to the PF input, whereas the CF input induces a transient Ca 2⫹
peak at the same level (Fig. 5A). If the CF input is applied during
IP3 elevation [dynamics jump from the right side of region (ii) tothe red part of region (iii)], as when there is a delay of 0 –300 ms
in Figure 5A, the transient Ca 2⫹ peak crosses the threshold andinduces LTD. If the CF input and IP3 elevation do not coincide,the transient Ca 2⫹ peaks are insufficient to reach the threshold[dynamics jump from the left of region (ii) to the blue part ofregion (iii)]. Thus, conjunctive PF–CF inputs can induce LTD
only within the appropriate time window where PF-induced IP3
elevation provides a low Ca 2⫹ threshold. Fourth, region (iv)shows that LTD induction can be induced only by application ofa large Ca 2⫹ elevation in the spine, even without IP3 elevation.
This is consistent with the observation that repetitive Ca 2⫹ uncaging rescues the induction of LTD in Purkinje cells, the dendritic spines of which lack Ca 2⫹ stores (Miyata et al., 2000). Fifth,
region (v) may explain the finding that IP3 uncaging induces
Ca 2⫹ release and LTD without CF activation or depolarization(Khodakhah and Ogden, 1995; Finch and Augustine, 1998).
Sixth, and finally, region (vi) indicates that massive PF-bundle
stimulation is capable of providing large Ca 2⫹ transients thatinduce LTD without CF input (Hartell, 1996; Wang et al., 2000).
Although IP3 elevation contributes the Ca 2⫹ transients, the Ca 2⫹
influx is large enough to allow the Ca 2⫹ response to reach thethreshold.
The Ca 2ⴙ dynamics model and cerebellar learning theories
As stated in Introduction, according to cerebellar learning theories (Ito, 1970; Kawato, 1999; Yamamoto et al., 2002), LTD must
be induced when PF inputs precede CF inputs. Different groups
have used many different experimental paradigms and have re-ported a variety of time windows for LTD induction (Ekerot andKano, 1989; Karachot et al., 1994; Chen and Thompson, 1995;
Yamamoto et al., 2002). Furthermore, cerebellar motor learningtheories cannot easily explain the fact that LTD induction does
not necessarily require CF inputs. The IP3-dependent Ca 2⫹
threshold suggests that strong PF input alone triggers regenerative Ca 2⫹ release [Fig. 8 F, region (vi)]. LTD has been experimentally induced by strong PF activation without CF activation (Hartell, 1996), but this type of LTD apparently does not occur innatural response to stimuli or contribute to cerebellar learning.
However, De Schutter (1995) has proposed that LTD induced by
PF activation alone plays a role in normalizing the strength of the
PF-mediated AMPAR pathway. If PF activation alone is strong
enough to induce LTD, the AMPAR pathway is weakened until
the Ca 2⫹ response reaches the Ca 2⫹ threshold. This type of LTD
may function as a pretuning process for effective cerebellar
learning.
In conclusion, kinetic simulation has allowed us to understand IP3-dependent Ca 2⫹ dynamics, which are capable of coherently explaining various forms of LTD. Because the IP3dependent Ca 2⫹ threshold, the necessity of PF–CF conjunction,and the PF–CF time delay were reproduced only for physiologicalranges of synaptic inputs, the results of this study support conjunctive LTD as one of the cellular bases of cerebellar supervised
learning (Marr, 1969; Ito, 1970, 2001; Kawato, 1999).

References
Adkins CE, Taylor CW (1999) Lateral inhibition of inositol 1,4,5trisphosphate receptors by cytosolic Ca 2⫹. Curr Biol 9:1115–1118.
Aiba A, Kano M, Chen C, Stanton ME, Fox GD, Herrup K, Zwingman TA,
Tonegawa S (1994) Deficient cerebellar long-term depression and impaired motor learning in mGluR1 mutant mice. Cell 79:377–388.
Airaksinen MS, Eilers J, Garaschuk O, Thoenen H, Konnerth A, Meyer M(1997) Ataxia and altered dendritic calcium signaling in mice carrying atargeted null mutation of the calbindin D28k gene. Proc Natl Acad Sci
USA 94:1488 –1493.
Albus JS (1971) A theory of cerebellar function. Math Biosci 10:25– 61.
Allbritton NL, Meyer T, Stryer L (1992) Range of messenger action of calcium ion and inositol 1,4,5-trisphosphate. Science 258:1812–1815.
Berridge MJ (1998) Neuronal calcium signaling. Neuron 21:13–26.
Bezprozvanny I, Watras J, Ehrlich BE (1991) Bell-shaped calcium-response
curves of Ins(1,4,5)P3- and calcium-gated channels from endoplasmic
reticulum of cerebellum. Nature 351:751–754.
Bhalla US, Iyengar R (1999) Emergent properties of networks of biologicalsignaling pathways. Science 283:381–387.
Chen C, Thompson RF (1995) Temporal specificity of long-term depression in parallel fiber-Purkinje synapses in rat cerebellar slice. Learn Mem
2:185–198.
Conquet F, Bashir ZI, Davies CH, Daniel H, Ferraguti F, Bordi F, Franz-Bacon
K, Reggiani A, Matarese V, Conde F, Collingridge GL, Crepel F (1994)
Motor deficit and impairment of synaptic plasticity in mice lacking
mGluR1. Nature 372:237–243.
Daniel H, Levenes C, Crepel F (1998) Cellular mechanisms of cerebellar
LTD. Trends Neurosci 21:401– 407.
De Schutter E (1995) Cerebellar long-term depression might normalize excitation of Purkinje cells: a hypothesis. Trends Neurosci 18:291–295.
Dupont G, Erneux C (1997) Simulations of the effects of inositol 1,4,5trisphosphate 3-kinase and 5-phosphatase activities on Ca 2⫹ oscillations.
Cell Calcium 22:321–331.
Eilers J, Augustine GJ, Konnerth A (1995) Subthreshold synaptic Ca 2⫹ signalling in fine dendrites and spines of cerebellar Purkinje neurons. Nature
373:155–158.
Ekerot CF, Kano M (1989) Stimulation parameters influencing climbing
fibre induced long-term depression of parallel fibre synapses. Neurosci
Res 6:264 –268.
Fiala JC, Grossberg S, Bullock D (1996) Metabotropic glutamate receptor
activation in cerebellar Purkinje cells as substrate for adaptive timing of
the classically conditioned eye-blink response. J Neurosci 16:3760 –3774.

Doi et al. • Spike-Timing Detection in Different Forms of LTD
Fierro L, Llano I (1996) High endogenous calcium buffering in Purkinjecells from rat cerebellar slices. J Physiol (Lond) 496:617– 625.
Fierro L, DiPolo R, Llano I (1998) Intracellular calcium clearance in Purkinje cell somata from rat cerebellar slices. J Physiol (Lond) 510:499 –512.
Finch EA, Augustine GJ (1998) Local calcium signalling by inositol-1,4,5trisphosphate in Purkinje cell dendrites. Nature 396:753–756.
Franks KM, Stevens CF, Sejnowski TJ (2003) Independent sources of quantal variability at single glutamatergic synapses. J Neurosci 23:3186 –3195.
Fujiwara A, Hirose K, Yamazawa T, Iino M (2001) Reduced IP3 sensitivity ofIP3 receptor in Purkinje neurons. NeuroReport 12:2647–2651.
Harris KM, Stevens JK (1988) Dendritic spines of rat cerebellar Purkinjecells: serial electron microscopy with reference to their biophysical characteristics. J Neurosci 8:4455– 4469.
Hartell NA (1996) Strong activation of parallel fibers produces localized
calcium transients and a form of LTD that spreads to distant synapses.
Neuron 16:601– 610.
Hartell NA, Furuya S, Jacoby S, Okada D (2001) Intercellular action of nitricoxide increases cGMP in cerebellar Purkinje cells. NeuroReport 12:25–28.
Inoue T, Kato K, Kohda K, Mikoshiba K (1998) Type 1 inositol 1,4,5trisphosphate receptor is required for induction of long-term depressionin cerebellar Purkinje neurons. J Neurosci 18:5366 –5373.
Irvine RF, Schell MJ (2001) Back in the water: the return of the inositol
phosphates. Nat Rev Mol Cell Biol 2:327–338.
Ito M (1970) Neurophysiological aspects of the cerebellar motor control
system. Int J Neurol 7:162–176.
Ito M (2001) Cerebellar long-term depression: characterization, signaltransduction, and functional roles. Physiol Rev 81:1143–1195.
Ito M (2002) The molecular organization of cerebellar long-term depression. Nat Rev Neurosci 3:896 –902.
Jiang H, Wu D, Simon MI (1994) Activation of phospholipase C ␤4 byheterotrimeric GTP-binding proteins. J Biol Chem 269:7593–7596.
Karachot L, Kado RT, Ito M (1994) Stimulus parameters for induction of
long-term depression in in vitro rat Purkinje cells. Neurosci Res
21:161–168.
Kawato M (1999) Internal models for motor control and trajectory planning. Curr Opin Neurobiol 9:718 –727.
Khodakhah K, Ogden D (1995) Fast activation and inactivation of inositol
trisphosphate-evoked Ca 2⫹ release in rat cerebellar Purkinje neurones.
J Physiol (Lond) 487:343–358.
Kim JJ, Thompson RF (1997) Cerebellar circuits and synaptic mechanisms
involved in classical eyeblink conditioning. Trends Neurosci 20:177–181.
Klingauf J, Neher E (1997) Modeling buffered Ca 2⫹ diffusion near themembrane: implications for secretion in neuroendocrine cells. Biophys J
72:674 – 690.
Kuroda S, Schweighofer N, Kawato M (2001) Exploration of signal transduction pathways in cerebellar long-term depression by kinetic simulation. J Neurosci 21:5693–5702.
Linden DJ, Connor JA (1995) Long-term synaptic depression. Annu Rev
Neurosci 18:319 –357.
Llinas R, Lang EJ, Welsh JP (1997) The cerebellum, LTD, and memory:alternative views. Learn Mem 3:445– 455.
Luzzi V, Sims CE, Soughayer JS, Allbritton NL (1998) The physiologic concentration of inositol 1,4,5-trisphosphate in the oocytes of Xenopus laevis.
J Biol Chem 273:28657–28662.
MacGregor RJ (1987) Neural and brain modeling. San Diego: Academic.

J. Neurosci., January 26, 2005 • 25(4):950 –961 • 961
Maeda H, Ellis-Davies GC, Ito K, Miyashita Y, Kasai H (1999) Supralinear
Ca 2⫹ signaling by cooperative and mobile Ca 2⫹ buffering in Purkinje
neurons. Neuron 24:989 –1002.
Marchant JS, Taylor CW (1997) Cooperative activation of IP3 receptors by
sequential binding of IP3 and Ca 2⫹ safeguards against spontaneous activity. Curr Biol 7:510 –518.
Marr D (1969) A theory of cerebellar cortex. J Physiol (Lond) 202:437– 470.
Mateos JM, Benitez R, Elezgarai I, Azkue JJ, Lazaro E, Osorio A, Bilbao A,
Donate F, Sarria R, Conquet F, Ferraguti F, Kuhn R, Knopfel T, Grandes
P (2000) Immunolocalization of the mGluR1b splice variant of themetabotropic glutamate receptor 1 at parallel fiber-Purkinje cell synapses
in the rat cerebellar cortex. J Neurochem 74:1301–1309.
Miyakawa H, Lev-Ram V, Lasser-Ross N, Ross WN (1992) Calcium transients evoked by climbing fiber and parallel fiber synaptic inputs in guinea
pig cerebellar Purkinje neurons. J Neurophysiol 68:1178 –1189.
Miyata M, Finch EA, Khiroug L, Hashimoto K, Hayasaka S, Oda SI, Inouye M,
Takagishi Y, Augustine GJ, Kano M (2000) Local calcium release in dendritic spines required for long-term synaptic depression. Neuron
28:233–244.
Rose CR, Konnerth A (2001) Stores not just for storage. Intracellular calcium release and synaptic plasticity. Neuron 31:519 –522.
Sabatini BL, Svoboda K (2000) Analysis of calcium channels in single spines
using optical fluctuation analysis. Nature 408:589 –593.
Sabatini BL, Oertner TG, Svoboda K (2002) The life cycle of Ca 2⫹ ions in
dendritic spines. Neuron 33:439 – 452.
Schreurs BG, Oh MM, Alkon DL (1996) Pairing-specific long-term depression of Purkinje cell excitatory postsynaptic potentials results from a classical conditioning procedure in the rabbit cerebellar slice. J Neurophysiol
75:1051–1060.
Stuart G, Hausser M (1994) Initiation and spread of sodium action potentials in cerebellar Purkinje cells. Neuron 13:703–712.
Sugiyama T, Hirono M, Suzuki K, Nakamura Y, Aiba A, Nakamura K, Nakao
K, Katsuki M, Yoshioka T (1999) Localization of phospholipase C␤isozymes in the mouse cerebellum. Biochem Biophys Res Commun
265:473– 478.
Tanaka J, Nakagawa S, Kushiya E, Yamasaki M, Fukaya M, Iwanaga T, Simon
MI, Sakimura K, Kano M, Watanabe M (2000) Gq protein alpha subunits G␣q and G␣11 are localized at postsynaptic extra-junctional membrane of cerebellar Purkinje cells and hippocampal pyramidal cells. Eur
J Neurosci 12:781–792.
Vecellio M, Schwaller B, Meyer M, Hunziker W, Celio MR (2000) Alterations in Purkinje cell spines of calbindin D-28 k and parvalbumin knockout mice. Eur J Neurosci 12:945–954.
Vetter P, Roth A, Hausser M (2001) Propagation of action potentials in
dendrites depends on dendritic morphology. J Neurophysiol 85:926 –937.
Wang SS, Denk W, Hausser M (2000) Coincidence detection in single dendritic spines mediated by calcium release. Nat Neurosci 3:1266 –1273.
Xu T, Naraghi M, Kang H, Neher E (1997) Kinetic studies of Ca 2⫹ binding
and Ca 2⫹ clearance in the cytosol of adrenal chromaffin cells. Biophys J
73:532–545.
Yamamoto K, Kobayashi Y, Takemura A, Kawano K, Kawato M (2002)
Computational studies on acquisition and adaptation of ocular following responses based on cerebellar synaptic plasticity. J Neurophysiol 87:1554 –1571.