Ryanodine-receptor-driven intracellular calcium dynamics underlying spatial association of synaptic plasticity
Ryanodine-receptor-driven intracellular calcium dynamics underlying spatial association of synaptic plasticity
Daiki Futagi 0 1 2
Katsunori Kitano 0 1 2
Action Editor: Upinder Singh Bhalla 0 1 2
0 Department of Human and Computer Intelligence, Ritsumeikan University , 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577 , Japan
1 Graduate School of Information Science and Engineering, Ritsumeikan University , 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577 , Japan
2 Daiki Futagi
Synaptic modifications induced at one synapse are accompanied by hetero-synaptic changes at neighboring sites. In addition, it is suggested that the mechanism of spatial association of synaptic plasticity is based on intracellular calcium signaling that is mainly regulated by two types of receptors of endoplasmic reticulum calcium store: the ryanodine receptor (RyR) and the inositol triphosphate receptor (IP3R). However, it is not clear how these types of receptors regulate intracellular calcium flux and contribute to the outcome of calciumdependent synaptic change. To understand the relation between the synaptic association and store-regulated calcium dynamics, we focused on the function of RyR calcium regulation and simulated its behavior by using a computational neuron model. As a result, we observed that RyR-regulated calcium release depended on spike timings of pre- and postsynaptic neurons. From the induction site of calcium release, the chain activation of RyRs occurred, and spike-like calcium increase propagated along the dendrite. For calcium signaling, the propagated calcium increase did not tend to attenuate; these characteristics came from an all-or-none behavior of RyR-sensitive calcium store. Considering the role of calcium dependent synaptic plasticity, the results suggest that RyR-regulated calcium propagation induces a similar change at the synapses. However, according to the dependence of RyR calcium regulation on the model parameters, whether the chain activation of RyRs occurred, sensitively depended on spatial expression of RyR and nominal fluctuation of calcium flux. Therefore, calcium regulation of RyR helps initiate rather than relay calcium propagation.
Calcium signaling; Intracellular; Ryanodine receptor; Simulation
Synaptic modification is essential for developing and
maintaining functional neural circuits and depends on neuronal
activity (Bliss and Collingridge 1993; Katz and Shatz 1996;
Martin et al. 2000). Results of physiological studies focusing
on the dependence of synaptic plasticity on temporal activity
of neurons have demonstrated that synaptic efficacy can either
be long-term potentiated (LTP) or depressed (LTD) depending
on firing rate (Bliss and Lomo 1973; Sjöström et al. 2001) and
relative spike timing (STDP) (Markram et al. 1997; Bi and
Poo 1998) of pre- and postsynaptic neurons. In addition to the
temporal aspect of synaptic plasticity, some physiological
studies showed that synaptic modification induced at one
synapse is often accompanied by collective synaptic changes at
neighboring sites in a postsynaptic dendrite (Bi 2002). The
spatial synaptic plasticity association has also been observed
in a global area: for example, CA1 and CA3 of the
hippocampus (Lynch et al. 1977; Bradler and Barrionuevo 1989;
Nishiyama et al. 2000), cortex (Hirsch et al. 1992), and
amygdala (Royer and Paré 2003). This evidence implies that
individual synapses are not independent, and the outcome of
synaptic modification is partly determined by the association
between synapses. The spatial aspect of synaptic plasticity
provides the possibility to construct and stabilize selectivity of
synaptic input that functionally activates neuronal circuits.
According to a recent study, stimulus selectivity evolves by
clustering and segregating synapses on dendrites by
synchronized synaptic input (Takahashi et al. 2012). Therefore,
revealing a mechanism of synaptic plasticity association will
contribute to understanding the link between spatiotemporal input
patterns, synaptic modulation, and formation of a functional circuit.
Although exact spatial synaptic association has not been
clarified, it is assumed to be attributed to intracellular calcium
signaling. Within a cell, many enzymes and proteins are
affected by calcium ions, which are involved in multiple
molecular processes that are relevant to modulation of cell function
(Berridge et al. 2000). Over the years, molecular studies have
shown that increasing postsynaptic calcium is required for the
induction of bidirectional synaptic change. Increase in
postsynaptic calcium concentration generally triggers two types of
molecular signaling processes for the induction of synaptic
change: one process mediates LTP by the action of protein
kinases and the other process mediates LTD by the action of
phosphatases (Colbran and Brown 2004; Munton et al. 2004).
Based on experimental knowledge, it has been hypothesized
that the outcome of synaptic change is mainly determined by
the magnitude of postsynaptic calcium concentration
(Johnston et al. 2003; Graupner and Brunel 2012; Hulme et al.
2012). The pathways that increase postsynaptic calcium
concentration in cytoplasm are generally categorized into two
types: one is calcium influx through NMDA receptors
(NMDARs) and voltage-dependent calcium channels
(VDCCs) in the membrane and the other is calcium release
from intracellular calcium stores, such as the endoplasmic
reticulum (ER). Regarding the former pathway, it has been
suggested that calcium influx via NMDAR and VDCC is
necessary for the induction of synaptic plasticity (Nevian
and Sakmann 2006). Concerning the latter pathway, the
calcium release from the ER is mediated by calcium-dependent
receptors. Over the years, it has been shown that the calcium
release is relevant to the induction of bidirectional
homosynaptic change (Reyes and Stanton 1996; Raymond and
Redman 2002). In addition, recent physiological studies
implicate that calcium store regulates wave-like intercellular
calcium propagation (Watanabe et al. 2006) and the outcome of
collective synaptic changes at neighboring sites (Nishiyama
et al. 2000). Combined with the experimental suggestions that
the outcome of synaptic plasticity depends on postsynaptic
calcium concentration, it is hypothesized that calcium
propagation regulated by the ER plays an important role in the
mechanism of hetero-synaptic association.
The stored calcium ions in the ER are mainly released via
two types of receptors: ryanodine receptor (RyR) and inositol
1,4,5-trisphosphate receptor (IP3R). Binding with calcium
ions activates both types of receptors and induces calcium
release; increased cytoplasm calcium concentration induces
further calcium release from store. In addition, the magnitude
of calcium release via IP3R depends on cytoplasmic IP3
concentrations (Bezprozvanny et al. 1991). On the other hand,
RyR does not depend on IP3 and RyR-regulated calcium
release can be all or none (Usachev and Thayer 1997). The
combination of these receptors with their different
characteristics likely regulates intracellular calcium signaling (Rose and
Konnerth 2001). However, the role of these receptors in
calcium regulation and the link between ER-calcium regulation
and the outcome of collective synaptic change is not clear. In
particular, compared with IP3R, the role of RyR calcium
regulation is not well known. Some studies suggest that RyR is
not necessary for the triggering and calcium propagation along
the dendrite (Nakamura et al. 1999; Watanabe et al. 2006). In
contrast, the results of some physiological studies suggest that
RyR is expressed in the dendritic shaft (Seymour-Laurent and
Barish 1995; Khodakhah and Armstrong 1997; Hertle and
Yeckel 2007) and regulates calcium flux in postsynaptic
dendrites (Futatsugi et al. 1999; Fujii et al. 2000). Thus, the
specific role of RyR remains to be elucidated. Therefore, in this
study, we focused on RyR and assessed the behavior of
RyR-mediated spatiotemporal calcium flux. For the
observation, we simulated intracellular calcium dynamics at a
cellular level, using a computational model that integrates
the model of intracellular calcium dynamics (Keizer and
Levine 1996) and the multi-compartment neuron model
(Poirazi et al. 2003).
The calcium regulation underlying the association mechanism
depends on the electrical property and structure of soma and
dendrite that differ between neuron types. Regarding this
point, a hetero-synaptic association has often been observed
in CA1 pyramidal neurons so that we constructed a neuron
model based on the multi-compartment CA1 pyramidal
neuron model developed by Poirazi et al. (2003). In the following
sections, we describe the details of the mathematical neuron
model for our computational simulation. Our program for the
simulation was coded in C and the following differential
equations were solved by backward Euler method and exponential
2.1 Structure of computational neuron model
The neuron model consists of three cellular parts, which are
trunk and branch of apical dendrite shaft and soma as shown
in Fig. 1. Each cellular part is divided into cylindrical
compartments where the length L (μm) and the radius a (μm) are
different for the three cellular parts: for the proximal part, we
set Lsoma = 10 μm, Ltrunck = 1 μm, Lbranch = 1 μm, asoma = 5
μm, atrunk = 0.6 μm, and abranch = 0.4 μm. It should be noted
that spines are not explicitly included in the dendritic model.
Fig. 1 Structure of neuron
model. The length of soma,
dendritic trunk, and branch are
10 μm,150 μm, and 250 μm,
2.2 Electrical activity of computational neuron model In each compartment of cellular parts, the differential equation for the membrane voltage V is described as dV
Cm dt ¼ I Na þ I K þ I Ca þ I h þ I leak þ I coup þ I in;
I K ¼ I A þ I Kdr þ I AHP;
I Ca ¼ I CaL þ I CaT:
with the following steady-state equations and time constants
τX in ms for m, h, and s.
We set the membrane capacitance Cm to 1 μF/cm2. INa
denotes voltage-dependent sodium current. IK corresponds to
the sum of potassium A current IA, delayed rectifier potassium
current IKdr, and calcium-activated potassium
afterhyperpolarization current IAHP. ICa corresponds to the sum of
high-threshold calcium current ICaL, and low-threshold
calcium current ICaT. Ih denotes hyperpolarization-activated current.
Ileak denotes leak current. Icoup denotes the electrical coupling
between the compartments. Iin denotes current injections or
synaptic currents in mA/cm2. The forms of these currents
are mostly specified in Poirazi et al. (2003) and described in
the following sections.
2.2.1 Voltage-dependent sodium current INa
The model kinetics for the sodium currents INa is described as
I Na ¼ gNa m2 h s ðENa−V Þ;
where gNa is the maximal conductance of sodium currents INa
shown in Table 1. ENa is its reversal potential: ENa = 50 mV.
The parameters m and h are the activation and inactivation
gating variables for sodium currents, respectively. An
additional gating variable s is introduced to account for attenuation
of the sodium currents (Jung et al. 1997; Migliore et al. 1999).
The differential equation for the gating variables m, h, and s is
τ X dt ¼ X ∞−X ; X ¼ fm; h; sg:
0:00333expð0:0024ðV þ 60ÞQÞ :
1 þ expð0:0012ðV þ 60ÞQÞ
Naatt in Eq. (4) represents the degree of sodium current
attenuation and varies depending on the location
parameter dx (μm) as shown below, which denotes distance
Q in Eq. (5) is given by
where R is gas constant, F is Avogadro constant, T is the
absolute temperature and degC is the temperature in degrees
Celsius. In our simulation, degC was set to 34 °C.
2.2.2 Delayed rectifier potassium current IKdr
The model kinetics for the delayed rectifier potassium currents
IKdr is described as
Reversal potential of potassium currents EK is −80 mV. The
differential equation for the gating variables m is the same as
Eq. (3). The steady-state equation m∞ and time constant τm are
different in soma and dendrite:
where maximum conductance gA is shown in Table 1. The
differential equation for the gating variables m and h is the
same as Eq. (3). The steady-state equation and time constant
for m are different in proximal (soma or dx≤ 100) and distal
(dx> 100) compartments:
I A ¼ gA m h ðEK−V Þ;
2.2.3 A-type potassium current IA
The model kinetics for the A-type potassium currents IA is
¼ 1 þ expð0:001ζðV ÞðV −11ÞQÞ
expð0:00055ζðV ÞðV −11ÞQÞ
0:05qtð1 þ expð0:001ζðV ÞðV −11ÞQÞÞ
¼ 1 þ expð0:001ζðV ÞðV þ 1ÞQÞ
expð0:00039ζðV ÞðV þ 1ÞQÞ
0:1qtð1 þ expð0:001ζðV ÞðV þ 1ÞQÞÞ
where Q is specified in Eq. (7) and qt = 5(degC-24)/10. The
function ζ(V) in Eq. (11) and Eq. (12) differs depending
on the location of compartment as shown below.
On the other hand, the steady-state equation and time
constant for h are uniform:
h∞ ¼ 1 þ expð0:003ðV þ 56ÞQÞ
2.2.4 Calcium-dependent potassium current IAHP
The model kinetics for the calcium-dependent potassium
currents IAHP is described as
I AHP ¼ gAHP m ðEK−V Þ;
where maximum conductance gAHP is shown in Table 1. The
differential equation for the gating variables m is same as
Eq. (3). The steady-state equation m∞ and time constant τm
are given by
1 þ C0:a1c8yt expð−1:68V QÞ
where Cacyt (mM) is calcium concentration in cytoplasm and
Q is specified in Eq. (7).
0:001 þ Ccyt
GH K V ; Ccyt; Co ;
2.2.5 Voltage-dependent low-threshold calcium current ICaT
The model kinetics for the low-threshold calcium currents ICaT
is described as
where maximum conductance gCaL is shown in Table 1. The
differential equation for the gating variable m is same as
Eq. (3). The steady-state equation for m is same as Eq. (18).
α(V), β(V) and time constant for m are given by
where maximum conductance gCaT is shown in Table 1, z
denotes the valence of calcium ion and Co is the extracellular
calcium concentration. Co is set to 1 (mM). The differential
equation for the gating variables m and h are same as Eq. (3).
The steady-state equations and time constant for m and h are
βmðV Þ ¼ 0:046 exp − 22:73 ; βhðV Þ ¼
; αhðV Þ ¼ 0:00016 exp − V þ1957 ;
1 þ exp −
2.2.6 Voltage-dependent high-threshold calcium current ICaL
The model kinetics for the high-threshold calcium currents
ICaL is different in soma and dendrite. Kinetics equation for
somatic L-type calcium currents is given by
GH K V ; Ccyt; Co
I CaT ¼ gCaTm2h 0:001 þ Ccyt
GH K V ; Ccyt; Co ;
2.2.7 Hyperpolarization activated current Ih
The model kinetics for the hyperpolarization activated current
Ih is given by
where maximum conductance gh is shown in Table 1.
Reversal potential of currents Eh is −10 mV. The differential
equation for the gating variables m is same as Eq. (3). The
steadystate equation m∞ and time constant τ m are given by
1 þ exp −
exp − V 1þ7:1545
On the other hands, kinetics equation for dendritic L-type
calcium currents is given by
IdCeanLd ¼ gCaLm3hðECa−V Þ:
The differential equation for the gating variables m and h is
the same as Eq. (3). The time constants are τm = 3.6 ms
and τh=29 ms and steady-state equations are given by
2.2.8 Leak current Ileak
Stuart and Spruston suggested that membrane resistance Rm
follows a sigmoidally decreasing distribution from the soma to
the distal trunk of pyramidal neurons (Stuart and Spruston
1998). Therefore, membrane resistance is defined as follows
in accordance with Poirazi et al. (2003):
where Rsmoma ¼ 200 KΩcm2, Remnd ¼ 120 KΩcm2, dhalf=200
μm, steep=50 μm and dx (μm) is distance from soma. The
model kinetics for leak currents is described as
2.2.9 Electrical coupling between the compartments Icoup
The axial resistance Ra also decreases sigmoidally from the
soma to the apical trunk:
where Rsaoma ¼ 50 KΩcm, Reand ¼ 35 KΩcm, dhalf=210 μm,
steep=50 μm and dx (μm) is distance from soma. The axial
resistance in dendritic branch (dx>150) is set to the somatic
value, i.e., Ra = 50 KΩcm. The model kinetics for the coupling
currents of the i-th compartment is given by
I coup;i ¼ gi;iþ1ðV iþ1 − V iÞ þ gi;i−1ðV iþ1 − V iÞ;
gi;i 1 ¼ Li Li 1 Ra;i a1iaai2i2þ1 Li Ra;i ai2 1 ;
where ai and Li are radius and length of the i-th compartment,
respectively, and these structure parameters are set to the
values described in Section 2.1.
2.2.10 Synaptic current Iin
Iin is the current injection in the soma or NMDAR-mediated
synaptic currents from dendritic spine to the dendritic
compartment. For a diffused NMDA current from the spine, we
simply scaled the following double exponential model in
accordance with Zador et al. (1990):
I in;nmda ¼ αSsp gnmda 1 þ 0:00033 Mg2þ expð−0:06V Þ ðEnmda−V Þ;
τ 1 τ 2
where gnmda is maximum conductance of NMDA, ts,pre is
spike-time of postsynaptic neuron, τ1= 80 ms, τ1= 0.67 ms,
[Mg2+] =1.2 mM and Enmda=0 mV. Ssp and Ssh are surface
area of the spine head and dendritic compartment,
respectively: Ssp is set to 0.1 μm2 in accordance with Harris and Stevens
(1989). α is the current diffusion rate on the basis of the
physiological evidence that 1 % of calcium ions flowing into a
spine diffuse to the dendritic shaft (Sabatini et al. 2002): α is
set to 0.01. V is the actual membrane voltage of the spine, but
we consider V as the membrane voltage of the dendritic shaft
2.3 Intracellular calcium dynamics
The assumed intracellular calcium signaling is regulated by
the interaction between NMDAR, VDCC, and RyR. In order
to implement RyR-mediated calcium flux in the neuron
model, we modified the calcium dynamics of Poirazi et al. (2003)
with parameters proposed by Keizer and Levine (1996). In
each compartment, the behavior of intracellular calcium ions
is described as
¼ Jin− Jout þ Jdiff þ Jsoc;
¼ Jin− Jout þ Jdiff þ Jer;
Ctot − Ccyt ;
where Ccyt and Cer denote calcium concentration in the
cytoplasm and ER, respectively. Ctot denotes total free calcium
concentration in μM. r denotes effective volume ratio ER to
cytoplasm, and we set r = 0.02 in accordance with Keizer and
Levine (1996). J denotes calcium flux in μM/ms; these
calcium fluxes are defined in the following sections.
2.3.1 Calcium flux between intra- and extracellular space
In Eq. (33) and Eq. (34), Jin, Jout, and Jsoc denote calcium flux
between intra- and extracellular space. In particular, Jin
denotes the sum of calcium influxes through NMDAR and
VDCC, which is proportional to ion currents:
I Ca þ 0:1I in;nmda ;
where S and Vol denote surface area and volume of the
compartment, respectively. Jout denotes calcium efflux through
plasma membrane calcium ATPase- (PMCA-) type pump:
where vo= 0.024 μM/ms and Ko= 0.6 μM. Jsoc denotes
calcium influx through store-operated channels (SOCs) in the cell
membrane, which has the role of refilling the ER with calcium
ions and adjusting calcium concentration (Parekh and Putney
2005). The detail of calcium-regulation via SOC has not been
revealed, but it was observed that SOC-mediated calcium
refilling occurs when calcium stores in the ER become
depleted. Therefore, we define Jsoc as
2.3.2 Intracellular calcium diffusion
calcium flux in the i-th compartment due to diffusion is
J di f f;i ¼ πai2Li fδi;iþ1 Ccyt;iþ1 − Ccyt;i þ δi;i−1 Ccyt;i−1 − Ccyt;i g;
where Ccyt,i denotes the calcium concentration in the i-th
compartment. ai and Li are the radius and length of the i-th
compartment, respectively, and these structural parameters are set to the
values described in Section 2.1. D denotes the diffusion
coefficient of calcium ion and we initiated D = 13 μm2/s in accordance
with Allbritton et al. (1992). The value of the diffusion
coefficient implicitly takes the effect of calcium buffering into account.
2.3.3 Intracellular calcium flux between ER and cytoplasm
In Eq. (33) and Eq. (34), Jer denotes intracellular calcium flux
between the ER and cytoplasm, which is described as
Cer − Ccyt − vpump Cc2yt þ K2 :ð40Þ
On the right side of Eq. (40), the first term represents
calcium release through RyR and passive calcium leak from the
ER to cytoplasm. The second term represents calcium release
from the cytoplasm to ER through sarco/endoplasmic
reticulum calcium ATPase- (SERCA-) type pump. The behavior of
ER calcium regulation depends on parameter v; we initiated
vrel = 0.005ms−1, vleak = 0.00012 ms−1, and vpump= 0.12 μM/
ms. Popen denotes the open probability of RyR, which depends
on calcium concentration in cytoplasm. The temporal
development of Popen is described with the relaxation equation of w:
w 1 þ Cc3yt=Kb
Popen ¼ Ka=Cc4yt þ 1 þ Cc3yt=Kb
Ka=Cc4yt þ 1 þ Cc3yt=Kb
In Eq. (33) and Eq. (34), Jdiff denotes intracellular
calcium flux between the compartments. The intracellular
where the kinetics parameter K was specified in Keizer and
Levine (1996): Ka = 0.0192 μM4, Kb = 0.2573μM3, Kc=
0.0571 and Kd= 0.0001 ms−1.
Fig. 2 Model of CA1 pyramidal neuron and intracellular calcium
signaling. a Schematic figure for the protocol to induce calcium fluxes
by a presynaptic input and a postsynaptic action potential. b Schematic
figure for calcium signaling activated by spike-triggered calcium influx
via NMDAR and VDCC and by calcium release from the ER via RyR.
The spatial distribution of RyR in the dendrite shaft was assumed to be
uniform; the constant value of calcium releasing parameter in Eq. (40)
represented a uniform density of RyRs. The value of calcium pumping
and leaking parameters in Eq. (40) were similar to the releasing
parameter. The calcium signaling started at the dendritic compartment
of the target synapse (red circle in (a)), which was evoked by paired
firing of the pre- and postsynaptic neurons. When the postsynaptic
neuron fired, VDCC-mediated calcium influx (solid blue arrows in (b))
occurred within the range of the back-propagating action potential. On the
other hand, when the presynaptic neuron fired, NMDAR-mediated
calcium influx (dashed blue arrow in (b)) locally occurred at the target
spine, and a portion of the calcium ions diffused to the dendrite
compartment. It should be noted that spines were not explicitly
incorporated into our model, but the diffused calcium flux from a spine
to the dendritic shaft was modeled. The spike-evoked calcium influxes
via NMDAR and VDCC activated RyR and induced the first calcium
release from ER. Subsequently, the released calcium ions activated
neighboring RyRs and calcium ions propagated in the dendritic shaft
(purple arrow in (b))
A previous physiological study suggested that store-regulated
calcium signaling in dendrites underlies the mechanism of
hetero-synaptic association (Nishiyama et al. 2000). However,
it is not known how calcium store (ER) regulates calcium
signaling in the dendritic shaft. Figure 2 describes
hypothesized calcium signaling in the situation where hetero-synaptic
association occurs. Calcium signaling consists of two
significant processes that are mainly regulated by RyR on ER. The
first process is RyR-mediated calcium release from the ER
induced by spike-triggered calcium influx via NMDAR and
Fig. 3 Amplitude profiles of back-propagating action potential (b-AP) and calcium transients. a Change in the amplitude of b-AP along a dendrite. b
Peak of calcium transient induced by b-AP
VDCC. The second process is the propagation of RyR
activation; calcium ions released from the ER diffuse along
cytoplasm and then activate RyRs at neighboring sites. Therefore,
by using the described computational model, we simulated
spatiotemporal calcium dynamics along the following
protocols focusing on the two calcium-signaling processes and
investigated the potential role of RyR-calcium regulation.
3.1 The fundamental behavior of the computational model
Before performing the model simulation, we confirmed the
fundamental behavior of our model. We observed the change in
amplitude of dendritic membrane potential when b-AP was
generated all at once from the soma (Fig. 3(a)). In this test, b-AP is
generated by stimulating the soma with an external input current
Iin = 0.05 mA/cm2 for 5 ms. As shown in Fig. 3(a), the peak
amplitude of the dendritic b-AP decreases along the dendrite. The
simulation result reproduces experimental results measured by
patch clamp method in CA1 pyramidal neurons (Golding et al.
2001). Subsequent to b-AP, the membrane potential changed
toward depolarization, and calcium influx to cytoplasm through
VDCCs occurred in each dendritic compartment as shown in
Fig. 3(b). The compartment at a distance of 150 μm away from
soma is the dendritic junction of trunk and branch. Therefore,
150 μm away from soma, the level of calcium increase was
significantly attenuated. The simulation result qualitatively
matches the experimental result of Gasparini et al. (2007).
3.2 The spike-timing dependence of RyR-mediated
We confirmed the reaction of RyR to the temporal pattern of
the two calcium influxes evoked by a pair of pre- and
postsynaptic spikes. In the simulation, we observed the behavior
of RyR at a local site in a postsynaptic dendrite that was
200 μm away from the soma and received pre-synaptic input.
The spike timing of the presynaptic neuron was controlled by
tuning ts in Eq. (32). NMDAR-mediated calcium influx only
occurred at the induction site. The conductance of
NMDARmediated synaptic current gn in Eq. (32) was 0.118 mS/cm2.
The spike of postsynaptic neuron was generated and
propagated along the dendrite by giving an external input current
Iin = 0.05 mA/cm2 to soma; the b-AP induces
VDCCmediated calcium influx as shown in Fig. 3(b).
First, we controlled the order of pre- and postsynaptic
spikes. In the simulation, the spike-time of the presynaptic
neuron was fixed at 0 ms, and we adjusted the spike-time of
the postsynaptic neuron. Figure 4(a) shows the change of
cytoplasmic calcium concentration in the cases where the
postsynaptic spike occurred at −10, 0, 15, and 35 ms. It was
observed that RyR-regulated calcium release depends on the
order of the spikes. Supralinear calcium increase occurred when
the presynaptic spike-triggered calcium influx via NMDAR
preceded the postsynaptic spike-triggered calcium influx via
VDCC. In addition to the above result, we confirmed the
time-window in which calcium-release was induced by paired
calcium influxes. As shown in Fig. 4(b), the supralinear
calcium increase occurred when the presynaptic led the
postsynaptic spike by 11 to 27 ms. It showed that RyRs were activated
and released calcium from the ER under the specific condition
of synchronization of spike-triggered calcium influxes.
The above results presented in Fig. 4(a, b) were observed in
the case where the induction site of RyR-mediated calcium
release was fixed at 200 μm away from the soma. Subsequently,
we changed the location of the induction site and confirmed the
difference in the spike-timing dependence of calcium release.
For this simulation, we set the conductance gn in Eq. (32) to
the value that was necessary to induce calcium release at the
induction site. Consequently, NMDA-conductance per surface
area (red circles) was almost constant but slightly increased
along the dendrite (Fig. 4(c)). In contrast, the predicted
amplitude of synaptic current via NMDAR in the spine (Fig. 4(d)) was
constant at any location except around the dendritic junction.
Under this condition, the spike-time window became longer
and shifted slightly upwards along the dendrite (Fig. 4(e)).
3.3 The generation mechanism of supralinear calcium
Fig. 4 shows that the bifurcation occurring in calcium dynamics
depended on the temporal order and interval of calcium
influxes. In this section, we investigate the generation mechanism
of the bifurcation using phase plane analysis. The bifurcation
phenomenon was generated at a local site in the dendrite.
Therefore, we analyzed local calcium dynamics by
approximating and reducing the model equations described in Section 2.3.
We assumed that intracellular calcium flux between
neighboring compartments does not occur: Jdiff = 0 in Eq. (33) and
Eq. (34). Furthermore, the open probability of RyR (Popen)
relaxes into steady state instantly: w=w∞ in Eq. (41). Under
this condition, setting the right sides of Eq. (33) and Eq. (34)
equal to zero derives the following nullclines:
− 1 ;
Ctot ¼ ð1 þ rÞCcyt þ r
J out − J in þ vpump
vrel Po∞pen þ vleak
Figure 5 shows the phase plane that is the state space of Ctot
and Ccyt. In the figure, the nullclines of Eq. (44) and Eq. (45) are
described as green and red lines, respectively. By observing
these nullclines in the phase plane, we understand the
qualitative behavior of the model. For instance, when below the
Ctotnullcline (green line), the state of the model tends to shift
50 100 150
distance from soma (µm)
Fig. 4 Dependence of calcium release on timings of pre- and
postsynaptic spikes. a Calcium increase relies on the order of pre-and postsynaptic
spikes. Δtpre-post denotes spike-time interval between pre- and
postsynaptic neurons. The colored lines denote the change of cytoplasmic calcium
concentration at induction site 200 μm away from the soma. b Spike
timing dependent calcium increase. Δtpre-post denotes the pre-to-post
synaptic inter-spike intervals. The green and blue boxes are lower and upper
limit of the spike-time window. (c, d, e) Difference of spike-time window
for inducing calcium-release in each dendritic location. c Red circle
denotes the minimum NMDA-conductance in mS/cm2 to induce calcium
release from the ER at each site. d The estimated amplitude of
NMDAcurrent observed in a spine. The current is simulated under the condition
that membrane voltage is clamped to −80 mVand [Mg+] is set to 0 mM in
accordance with physiological experiments (Andrásfalvy and Magee
2001). e The green and blue circles denote lower and upper limit of
spike-time window in which calcium release could be induced
Fig. 5 Nullclines in phase plane consisting of Ctot and Ccyt. The green
and red lines denote the nullclines described in Eq. (44) and Eq. (45),
respectively. The green and red arrows indicate directions of changes of
Ccyt and Ctot in the areas divided by the nullclines, respectively
upward; this means that calcium concentration in the ER tends
to increase. In addition, when above the Ccyt-nullcline (red
line), the state of the model tends to shift to the right: this means
that calcium concentration in cytoplasm tends to increase. In
particular, since the Ccyt-nullcline indicates an increase or
decrease of cytoplasmic calcium concentration (Ccyt) it is
considered the boundary representing whether RyR-mediated
supralinear calcium increase occurs in cytoplasm.
Based on these observations, we confirmed the temporal
behavior of the model and the nullclines to paired calcium
influxes via NMDAR and VDCC. We then re-defined the
equation of calcium influx (Jin) with the simplification of the
calcium dynamics model, which is described in the following
tX − t
t = 260.0(ms)
t = 300.0(ms)
t = 650.0(ms)
t = 680.0(ms)
Fig. 6 Difference in model behaviors in the order of spike-evoked
calcium influxes. The temporal change of calcium concentration up to a
specific time shown in (a) and (c) are described as black lines in (b) and
(d) with nullclines, respectively. a Occurrence of calcium release. b
Trajectory of the state before the second spike (top), after the second
spike (middle), and in the steady state (bottom). c Failure of calcium
release. d Same as (b) but illustrating the trajectories in the case of
failure of calcium release shown in (c)
where Jin denotes the sum of the calcium influxes through
NMDAR and VDCC. The two calcium influxes are about the
same as described in Eq. (32) with different parameters: αnmda=
8.17, τ1 , nmda = 80 ms, τ2 , nmda = 0.67ms, αvdcc = 0.041, τ1 ,
vdcc=0.1 ms, and τ2,vdcc=0.05 ms. In Eq. (46), tX is the time
when NMDAR- or VDCC-mediated calcium influx occurs.
First of all, in Fig. 6(a) we show the time series of the model’s
state under the condition that an NMDAR-mediated calcium
influx precedes a VDCC-mediated influx: tnmda = 200 ms and
tvdcc = 250 ms. Figure 6(b) shows the temporal locus of the
model’s state and nullclines at 240, 260, and 650 ms in the
phase plane. As shown in Fig. 6(b) top, the locus of the model’s
state shifts upward in the phase plane with the occurrence of
NMDAR-mediated calcium influx; this means that fluxed
calcium ions are immediately pumped up by the SERCA-type
pump and stored in ER because NMDAR-mediated calcium
influx is slow. Subsequently, the model solution rises but does
not go over the Ccyt-nullcline, so that calcium release from ER
does not occur. However, the following VDCC-mediated
calcium influx causes the model solution to pass through the
Ccytnullcline (Fig. 6(b) center) and trigger the supralinear
cytoplasmic calcium increase mediated by RyR (Fig. 6(b) bottom). In
distinction to the above, Fig. 6(c) shows the results for the case
where a VDCC-mediated calcium influx precedes an
NMDAR-mediated influx: tnmda = 250 ms and tvdcc = 200 ms.
The preceding calcium influx through VDCC is fast and
increases cytoplasmic calcium concentration momentarily, so that
the model solution transferred rightward in the phase plane
(Fig. 6(d) top). However, this influx cannot induce calcium
release from the ER at that time because the ER does not
contain sufficient calcium ions. Subsequently, due to the following
NMDAR-mediated calcium influx, ER is filled with calcium
ions but the calcium release from the ER is not induced
(Fig. 6(d) center, bottom). This is because the following
cytoplasmic calcium increase is insufficient to activate RyR on ER
due to the SERCA-type calcium pump activity. According to
the phase plane analysis for the local calcium dynamics model,
we confirmed that RyR-mediated calcium release occurred
when the calcium concentration in cytoplasm sufficiently
increased under the condition of a high calcium concentration
in the ER. NMDAR-mediated calcium influx was slow and
persistent, consequently being responsible for maintaining a
high calcium concentration in the ER. In contrast, a
VDCCmediated calcium influx was rapid and transient, so that it
contributed to a rapid increase in calcium concentration in
cytoplasm. Taken together, it was found that supralinear calcium
increase tended to occur as shown in Fig. 4, when an
NMDAR-mediated calcium influx preceded a VDCC-mediated
influx, that is, a presynaptic spike preceded a postsynaptic spike.
3.4 Calcium propagation based on chain activation
After the paired spikes evoked a calcium increase at the local
dendritic site, neighboring RyRs were activated, and calcium
release from the ER was propagated in the dendrite.
Figure 7(a, b) shows the change of cytoplasmic calcium
concentration at each site of the dendrite, when supralinear
calcium increase occurred 200 μm away from the soma, as shown
in Fig. 4(a). The spike-like calcium-increase propagated
bidirectionally along the dendrite and its speed depended on its
direction. In the calcium propagation, the peak amount of
calcium increase was mostly constant and the propagation
suddenly stopped at a certain point (Fig. 7(c)); this is because
the degree of RyR-regulated calcium release was all-or-none.
In addition, the total amount of increased calcium
concentration was maximal at the induction site of the calcium
propagation. However, within the propagation range, except for the
induction site, there was no difference in the integrated
amount of increased calcium concentration (Fig. 7(d))
Although the propagation distance was about 30–50 μm
shown in Fig. 7(a, b), it differed depending on the location
where the first calcium release was inducted by the paired
calcium influxes through NMDAR and VDCC (Fig. 7(e)).
As the induction site of propagation moved closer to the soma,
the change of calcium concentration propagated more widely.
In the case where the induction site was closer to the dendritic
junction and at 150 μm away from soma, the propagation
range centering on the starting point was asymmetric. This is
because the spatial distribution of VDCC-expression was not
homogeneous along the dendrite, as shown in Fig. 3(b).
Fig. 7 Intracellular calcium propagation based on chain activation of
RyRs. (a, b) Propagation of calcium-increase in the dendrite, (a) and
(b) showing calcium propagation toward soma and toward end of the
dendrite, respectively. Top. Colored lines denote the change of calcium
concentration at different sites of the dendrite. Center. Calcium increases
at 5 (red) and 6 (green) μm away from the induction site toward soma (a)
and toward end of the dendrite (b). Bottom. Same as for center but for
distances of 25 and 26 μm away from induction site toward soma and
dendrite end. As the chain of calcium release propagates from the
induction site, the time interval of spike-like calcium increase becomes
longer and the shape of calcium increase tapers. (c) Peak amount of
calcium increase in the calcium propagation shown in (a) and (b). The
propagation suddenly stops at a certain site. (d) Level of calcium increase
at each site of the dendrite. The colored bars denote the integrated amount
of increased calcium concentration that is surplus above a certain
threshold. (e) The range of calcium propagation in the dendrite. The
circles represent an induction site of calcium propagation and the length
of the horizontal bars represents the sites where calcium release occurred.
Red represents calcium propagation induced by paired spikes of pre- and
postsynaptic neurons. Blue represents the case where the postsynaptic
spike does not occur, and calcium release is only induced by
Furthermore, in the case where the induction site was 180,
200, 220 μm away from the soma, the propagation toward
the soma stopped around the dendritic junction.
3.5 Effects of model parameters on calcium dynamics
The behavior of RyR calcium regulation depends on the three
model v parameters in Eq. (40). Therefore, we examined the
effect of these parameters on RyR-regulated calcium flux in
First, we observed the relation between model parameters
and peak amount of calcium increase at the induction site
200 μm away from the soma (Fig. 8(a, b)). In Fig. 8(a), as
the degree of calcium pumping is higher and calcium leak is
lower, the peak amount of calcium concentration increased
from 0.4 μM to 1.7 μM. This simulation result can be
accounted by the dependence of nullcline of the model on
the model parameters (Fig. 8(c)). For instance, as the degree
of calcium pumping (vpump) increases, the local maximum
point of Ccyt-nullcline shifts up in the phase plane. In this case,
the time locus of the equilibrium point cannot surpass the
nullcline and more calcium influx is required. In other words,
cytoplasmic calcium concentration is difficult to increase and
calcium release tends not to be triggered due to the high
degree of calcium pumping. Meanwhile, more calcium ions are
stored in the ER; therefore, the degree of calcium release is
increased as vpump increases.
Next, we examined the effect of the model parameters on
calcium propagation (Fig. 9(a)). In the color maps, spatial
change of brightness is discrete. It means that the parameter
set is mostly categorized into two regions: one represents
propagation that does not occur, and the other represents
propagation that does not stop. In addition, the range of calcium
propagation critically depends on the model parameters. For
1000 1200 1400 1600
1000 1200 1400 1600
instance, in the case where vleak and vpump are fixed to
0.00012 ms−1 and 0.12 μM/ms, respectively, and vrel is
changed from 0.004 to 0.006 ms−1, the range of calcium
propagation is highly dependent on vrel. Figure 9(b) shows that
although the peak amount of propagated calcium increase also
depends on the parameters, propagated calcium increase does
not tend to attenuate.
As a result of the simulation, we observed that paired spikes of
pre- and postsynaptic neurons induced RyR-regulated calcium
release at the induction site, and the released calcium ions
initiated intracellular calcium propagation based on the chain
activation of RyRs. In this section, based on the physiological
Fig. 8 Relation between model parameters of the ER and calcium dynamics
at the induction site. vrel and D are fixed to 0.005 ms−1 and 0.013 μm2/ms,
respectively. The location of the induction site is 200 μm away from the
soma. Paired spikes of pre- and postsynaptic neurons induce calcium
release; the spike interval is 15 ms. a Color brightness corresponds to the
peak amount of calcium increase at the induction site. b Time course of
cytoplasmic calcium concentration (left) and dependence of calcium
increase on the spike interval (right) using the parameter set surrounded by
yellow line in (a). c Relation between one model parameter and Ccyt-nullcline.
Ccyt-nullcline implies the threshold for calcium release: when time locus of
Ccyt and Ctot surpasses the nullcline, the model behavior is bifurcated and
and theoretical knowledge of calcium-dependent synaptic
plasticity, we discuss the role and effect of RyR-regulated
calcium dynamics on collective synaptic change.
4.1 Dependence of calcium release on spike timing
Because it has been suggested that synaptic plasticity underlies
formation and maintenance of the functional neural circuit, the
mechanism of activity-dependent synaptic plasticity has been
intensively studied. In particular, the experimental study by Bi
and Poo showed the temporal aspect of synaptic plasticity;
whether synaptic efficacy is potentiated or depressed depends
on spike timings of pre- and postsynaptic neurons (Bi and Poo
1998). In general, pre-to-post and post-to-pre spike timings
induce long-term potentiation and depression, respectively,
and the amount of change in synaptic efficacy depends on
the spike intervals. Furthermore, it has been observed that
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15
Fig. 9 Relation between model parameters of the ER and the range of
calcium propagation. a Color brightness corresponds to the total distance
of bidirectional calcium propagation from the induction site. Top, center,
and bottom figures show the difference in the case where vrel = 0.004,
spike-timing-dependent synaptic change at one synapse
(homo) induces synaptic change at other synapses (hetero),
and the changes in hetero-synapses also depend on the spike
timings at the homo-synapse (Nishiyama et al. 2000).
However, the detailed mechanism of such a spatial association of
synaptic plasticity is still unclear. The spatial association of
synaptic plasticity is presumably based on calcium propagation
from homo-synapse to hetero-synapses along dendrites. In
addition, it is suggested that calcium store (ER) mainly regulates
the dendritic calcium dynamics to control cytoplasmic calcium
concentration by storing and releasing calcium ions. Although
the calcium release might be regulated by the complex
interaction of IP3R and RyR on ER, we focused on the unidentified
function of RyR-calcium regulation in dendritic calcium
signaling, which underlies the hetero-synaptic association.
Assuming that the spatial association of synaptic plasticity
between synapses is mainly mediated by the dendritic calcium
dynamics, calcium store with RyR could play a role for linking
the temporal activity of pre- and postsynaptic neurons with
synaptic plasticity. Regarding this point, our simulation results
showed that the paired calcium influxes evoked by paired spikes
in pre-to-post order could induce calcium release via RyR from
0.005, 0.006 ms−1, respectively. b Peak amount of calcium increase
in each site of the dendrite using the parameter set surrounded by
red line in (a)
the ER (Fig. 4). The induction process of calcium release
consisted of the following two steps: the first step was filling
up the ER with calcium ions, and the second step was increasing
cytoplasmic calcium concentration sufficiently to cause
threshold-like behavior through the bifurcation as shown in
Fig. 6. NMDAR and VDCC contributed differently to these
steps; that is, a slow but persistent calcium influx via NMDAR
was responsible for the first step and contributed to bottom-up
cytoplasmic calcium concentration over the long term. However,
this mechanism was not responsible for the second step because
its low amplitude of calcium increase was insufficient to trigger
calcium release. On the other hand, transient but fast calcium
influx via VDCC caused a high calcium increase in the
cytoplasm as the second step but was not suitable for the first step
because its duration was too short for calcium ions to be pumped
into the ER. If the calcium pumping rates and RyR in Eq. (40)
are increased, VDCC could fill the ER with calcium ions and
NMDAR could trigger calcium release. However, in this case,
the segregation of the contributions is lost; either one of the two
could fill up the ER and trigger calcium release by itself. From
the processes described above, calcium release tended to occur
when a presynaptic spike preceded a postsynaptic one.
4.2 Dependence of the time-window on distance
from the induction site
Figure 4(c, d) suggests that the expression level of NMDAR
was almost constant but slightly increased along the dendrite
shaft except around the dendritic junction, and the predicted
peak amplitude of NMDAR-current was in accordance with a
physiological study focusing on the location dependence of
density of glutamate receptors (Pettit and Augustine 2000;
Andrásfalvy and Magee 2001). The almost uniform distribution
of NMDAR can be attributed to the fact that the peak amount of
calcium influx via VDCC attenuates along the dendrite
(Fig. 3(b)). At the distal site, where the amount of
VDCCmediated calcium influx is low, cytoplasmic calcium
concentration is required to be largely increased by NMDAR-mediated
calcium influx so as to induce calcium release from the ER, as
the phase plane analysis suggested (Fig. 6(b) and (d)). Even if
the conductance density of NMDAR-mediated current is
uniform, the increase in cytoplasmic calcium concentration brought
about by NMDAR-mediated current is larger at the thinner
dendrite because the constant of the conversion from current density
to concentration was a function of the inverse of the dendrite’s
diameter (Eq. (36)). Therefore, the almost uniform conductance
density of NMDAR-mediated current along the dendrite could
generate a sufficient increase in cytoplasmic calcium
concentration to induce calcium release even at a distal site. In general, as
the distance of a synapse from the soma increases, it takes more
time for a presynaptic input from the synapse to arrive at the
soma, and it also takes longer for the generated action potential
to propagate back to the synapse. Namely, the time interval
between a presynaptic input and the associated b-AP tends to
be longer for a distant synapse. If synaptic efficacy is modified
based on causal relations between presynaptic inputs and firings
of a postsynaptic neuron, the dependence of the time window on
location seems reasonable (Fig. 4(e)).
4.3 The behavior of calcium propagation based
on the chain activation of RyRs
We observed that the cytoplasmic calcium increase
propagated along the dendrite by the chain activation of RyRs (Fig. 7).
The speed of calcium propagation was several tens of
micrometers per second and the propagation range was several
hundred micrometers depending on the location of the induction
site. Although our model did not consider the behavior of
IP3R, these simulation results partly reproduce physiological
data (Nakamura et al. 1999; Watanabe et al. 2006).
Whether calcium release occurs at neighboring sites other than
the induction site or not depends on the timing of calcium influx
via VDCC and calcium flux propagating from the induction site.
Because propagation of a b-AP is much faster than diffusion of
calcium concentration, calcium influx via VDCC activated by a
b-AP occurred first and slightly increased calcium concentration
in the ER. However, the pumped calcium ions slowly leaked as
shown in Fig. 6(d). Thus, the calcium flux propagating from the
induction site arrived later at a neighboring site. Therefore, it took
more time to cause a calcium increase, in other words, it took
longer to surpass the threshold-like Ccyt-nullcline shown in
Fig. 6. Consequently, along with the calcium propagation, the
interval of the two fluxes was prolonged and temporal overlap
of calcium increases between adjacent sites was reduced
(Fig. 7(a, b)). As shown in the center and bottom part of
Fig. 7(a, b), calcium increase at one site (red line) is followed
by an increase at the adjacent site further away from the induction
site (green line). The falling phase after the transient calcium
increase (red-dashed line) was slowed down by calcium diffusion
from the adjacent site where the concentration was still relatively
high. As the overlap increased, the falling phase continued for a
longer time due to this effect. In contrast, if the overlap was small,
calcium concentration smoothly descended. Accordingly, closer
to the induction site, the calcium increase tends to last longer. The
comparison between Fig. 7(a) and 7(b) suggests that the
continuation of calcium increase depends on the direction of the
calcium propagation. This is because the calcium influx via VDCC
attenuates along the dendrite (Fig. 3(b)); the duration of calcium
increase becomes longer at a more distal site. This mechanism
underlies the result that the attenuation of the integrated amount
of calcium increase is relatively large at distal sites 200–235 μm
away from the soma, as illustrated in Fig. 7(d). In addition, we
observed that the calcium propagation from the dendritic branch
tended to stop at the joint of the dendritic trunk and branch. In
contrast, the propagation from trunk to branch was not prevented.
This phenomenon was observed in physiological experiments
(Nakamura et al. 2002). We also observed that the chain
activation of RyR is suddenly cut at a certain point in the dendrite
(Fig. 7, Fig. 9). These results suggest that the synaptic association
is compartmentalized at the branching point; although collective
synaptic change is localized within a compartment, it is able to
affect synapses at branched compartments. This structural
mechanism could contribute to the link between the local event of
synaptic change and the formation of selectivity to synaptic input.
4.4 Limitations of the modeling study
4.4.1 Exclusion of spine dynamics
It has been reported that RyRs are expressed not only in the
dendritic shaft but also in spines of CA1 pyramidal neurons
(Ellisman et al. 1990; Sharp et al. 1993). Furthermore, it has
been shown that calcium store with RyR in a spine contributes
to enhancement of calcium increase evoked by synaptic
activation (Emptage et al. 1999). In general, however, many
experimental studies have reported that RyRs have little influence on
the calcium increase in a spine (Kovalchuk et al. 2000; Nevian
and Sakmann 2006). The divergence of results likely derives
from differences in experimental and/or cellular conditions.
Therefore, we did not incorporate the calcium dynamics in a
spine and focused on revealing the spatiotemporal calcium
regulation at a cellular level by RyRs in the dendritic shaft in order
to exclude uncertainty of calcium signaling in a spine.
4.4.2 Dependence of outcomes on responsiveness of calcium
We observed that the dendritic calcium signaling consisted of
two steps. The first step was an initiation of calcium release in
the dendritic compartment of a homo-synapse, where the
occurrence of calcium release depended on spike timings of
preand postsynaptic neurons. However, the peak amount of
calcium increase was almost constant for any timing within the
spike-time window (Fig. 4(b)). The second step was the
propagation of spike-like calcium increase along the dendritic
shaft, at which there was no difference in the peak amount
of calcium increase within the propagation range (Fig. 7(c)).
How does the RyR-calcium regulation affect the outcome of
homo- and hetero-synaptic changes? The change in synaptic
efficacy actually results from complex interactions of calcium
signaling pathways (Berridge et al. 2000). Furthermore,
whether synaptic efficacy is potentiated or depressed depends on the
structure of a spine, subcellular distribution of endogenous
buffer, and other factors. However, if we assume that a sufficient
amount of released calcium ions diffuses from a dendritic shaft
to a spine, and the calcium increase in a dendritic shaft can be
regarded as the determinant of calcium-dependent synaptic
change, the amount of the synaptic change caused at a
homosynapse would be almost constant irrespective of the pre- and
postsynaptic spike timings. In addition, the calcium
propagation regulated by RyRs would induce similar changes among
hetero-synapses within the range of the calcium propagation.
These predictions are based on the result that the profile of the
calcium increase does not differ in intervals of pre- and
postsynaptic spikes and locations in a dendrite as mentioned above.
However, these predictions are partly inconsistent with the
experimental evidence on the spike-timing-dependent synaptic
plasticity at a homo-synapse (Bi and Poo 1998).
The characteristics mentioned above can be attributed to the
all-or-none response of the incorporated RyR-sensitive calcium
store model. In a CA1 pyramidal neuron, several different types
of receptors (subtypes of RyRs and IP3Rs) express on ER (Sharp
et al. 1993; Furuichi et al. 1994; Fitzpatrick et al. 2009). This
response property does not depend on the type but crucially
depends on many other factors, such as abundance and effect
of intracellular calcium buffers, activity of calcium-dependent
enzymes, and spatial formation of the receptor channels (Wehrens
et al. 2004; Cheng and Lederer 2008). Even if a single channel
exhibits an all-or-none response, the collective response of the
channels practically depends on such factors. For example,
because RyRs and IP3Rs form a cluster on ER, the spatial formation
of the channels is one of the important factors. Furthermore, the
actual activation of each channel is not deterministic but
stochastic. In the case that the stochastic nature is strong, the overall
response will be graded even if the response of each channel is
all-or-none. We assessed the case where the collective response
of RyRs could be well described by the deterministic model. In
the case where the responsiveness of calcium store is of the
graded type, the dependence of calcium release on spike intervals
and the attenuation of calcium propagation would be more
graded. Therefore, the distinct responsiveness would play different
roles for spatiotemporal calcium dynamics; the all-or-none type
would be responsible for detection of calcium influxes evoked by
paired spikes and initiation of calcium increase, whereas the other
type would contribute to adjusting the level of calcium
concentration and assisting calcium release at neighboring sites.
4.4.3 Parameter dependence
As described above, we quantitatively assessed characteristics
arising from the all-or-none response of the calcium store in the
spatial propagation of calcium release. The characteristics were
robust against changes in input conditions; the peak amount of
calcium release was almost constant independent of the spike
timing (Fig. 8(a, b), Fig. 9). In addition, at surrounding sites of
the induction site, the level of calcium increase barely changed
(Fig. 7(d)). However, RyR-regulated calcium propagation had a
sensitive aspect: its chain-like activation sensitively continued or
stopped depending on the properties of ER. Figure 9(a) implied a
relationship between RyR-mediated calcium propagation and the
spatial expression of RyR. Thus, assuming that the degrees of
leaking and pumping functions of ER did not vary across different
sites, and the rate of RyR (vrel in Eq. (40)) reflected the expression
level of RyR, the chain-like activation would transition to a failure
condition by decreasing the expression level of RyRs. In addition,
intracellular calcium flux dynamically fluctuates in vivo
depending on molecular processes, neural activities, and movement of
dendrite. If the spatial association of synaptic plasticity is brought
about by the chain-like activation of RyRs, it would be reasonable
to predict that the outcome is less robust against the fluctuations in
such related processes. Therefore, RyR calcium regulation likely
contributes to initiation rather than relay calcium propagation.
Acknowledgments This work was supported by JSPS KAKENHI
Grant Number 15H05877 (KK).
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of
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