School of Computational Sciences and the Krasnow Institute for
Advanced Study, George Mason University, Fairfax, Virginia 22030
 |
INTRODUCTION |
Classical conditioning, a
form of associative learning, requires presentation of paired stimuli,
a conditioned stimulus (CS) and an unconditioned stimulus (US), within
a specific temporal interval. This implies that both the CS and the US
produce signals that interact to cause memory storage. Although the
requirement for paired stimuli has been known for decades, the identity
of the interacting signals is still unknown.
The sea slug, Hermissenda crassicornis, is an animal model
of classical conditioning and is an ideal model in which to evaluate the interaction of signals mediating classical conditioning.
Hermissenda learns to associate light (the CS) with
turbulence (the US). The memory of the association is stored in the
type B photoreceptors as an increase in input resistance and
excitability (Crow and Alkon 1980
; Farley
1987
; Farley and Alkon 1982
). Memory storage critically depends on an elevation of intracellular calcium
concentration (Matzel and Rogers 1993
;
Talk and Matzel 1996
), and activation of protein
kinase C (Alkon et al. 1988
; Farley and
Auerbach 1986
; McPhie et al. 1993
). Light leads
to production of diacylglycerol (DAG) (Talk et al.
1997
), an activator of protein kinase C (PKC), and inositol
triphosphate (IP3), which leads to release of
calcium from intracellular stores (Talk and Matzel
1996
).
The turbulence US causes hair cells to release
-amino butyric acid
(GABA) onto the terminal branches of the type B photoreceptor (Alkon et al. 1993
). The response to GABA stimulation
consists of an IPSP followed several seconds later by a small
depolarization lasting for several seconds (Matzel and Alkon
1991
). The hyperpolarization is caused by opening of
GABAA chloride channels and
GABAB potassium channels (Alkon et al.
1992
; Rogers et al. 1994
). The cause of the late
depolarization has not been determined, but observations are consistent
with a G-protein-dependent closure of potassium leak channels
(Matzel and Alkon 1991
; Rogers et al.
1994
).
Because light alone does not cause memory storage but does produce the
activators of PKC, it is imperative to identify which critical factors
are contributed by turbulence. Evidence suggests that calcium may be an
essential second-messenger contributed by turbulence to associative
memory storage. Support comes from the observation that dantrolene,
which prevents the propagation of calcium waves (Trafford et al.
1995
), prevents in vitro classical conditioning of
Hermissenda (Blackwell and Alkon 1999
).
Moreover, one experiment demonstrates that turbulence evokes a calcium
elevation that propagates from the terminal branches to the soma
(Ito et al. 1994
). However, another calcium-imaging
study has not observed calcium in the terminal branches (Muzzio
et al. 1998
).
Because the observation of a calcium wave has not been replicated, one
purpose of this modeling study was to determine if GABA stimulation can
contribute a calcium signal that propagates from the terminal branches
to the soma. The potassium channel underlying the late depolarization
has not been completely characterized, thus the second purpose of the
study was to determine whether the potassium leak channel is
responsible for the GABA-induced depolarization.
 |
METHODS |
The Hermissenda type B photoreceptor is modeled using
the GENESIS simulation software implemented on a UNIX workstation.
Chemesis, which consists of two additional libraries of GENESIS
objects, was used for simulating the biochemical reactions of the
GABAB synapse as well as calcium dynamics. One of
the Chemesis libraries has objects for simulating general biochemical
reactions, pools of molecules, and calcium release from intracellular
stores; the other Chemesis library has objects for simulating one- and
two-step ligand-gated channels and separable or nonseparable
formulations of calcium-dependent rate constants.
Morphology
The geometry of the type B photoreceptor model is an
approximation to the morphological features previously described
(Crow et al. 1979
; Eakin et al. 1967
;
Stensaas et al. 1969
) and is illustrated in Fig.
1B. The rhabdomere is a
cylinder 12 µm in diameter and 12 µm in length. The numerous
microvilli of the rhabdomere are taken into account by decreasing the
membrane resistance and increasing the capacitance proportional to the
surface area contributed by 5,000 microvilli of 0.16-µm diameter by
5-µm length. The diameter of the central core of the rhabdomere (the
part the microvilli are attached to) is 2 µm. The rhabdomere is
connected to the soma which is a cylinder 20 µm in diameter by 24 µm in length. The neurite, which functions as both an axon and
dendrite, is 100 µm in length; the elliptical cross section has a
long axis of 3 µm and a short axis of 1 µm. The neurite is
subdivided into four isopotential elliptic cylinders of 25 µm
(Fost and Clark 1996
). The neurite's terminal branches,
the site of all synaptic interactions, are modeled as two equivalent
cylinders, 15 µm in length. One cylinder represents the set of
nonsynaptic branches, and the other cylinder represents the set of
synaptic branches. The distal 10 µm compartment of the synaptic
branch contains the synaptic channels. Two variations on synaptic
connectivity are simulated by using two different radii of the terminal
branch cylinders. Under the assumption that 10% of the terminal
branches receive synaptic input (used for simulations unless otherwise indicated), the equivalent cylinder radius of the synaptic branch is
0.22 µm and the equivalent cylinder radius of the nonsynaptic branch
is 0.93 µm. Under the assumption that 50% of the terminal branches
receive synaptic input, the equivalent cylinder radius of both the
synaptic and nonsynaptic branches are 0.63 µm. The neurite and
terminal branch cylinders are subdivided into 1-µm-long compartments
for the purpose of modeling calcium concentration dynamics. Passive
membrane resistivity is 10 k
-cm2, membrane
capacitivity is 1 µF/cm2, axial resistivity is
100
-cm. A somatic shunt of 0.005 µS simulates the effect of a
sharp electrode. The resting potential of the cell is
57 mV; the
steady-state input resistance is 35 M
. These values are comparable
to the mean resting potential and input resistance
(RN) experimentally observed in
Hermissenda photoreceptors. The somatic shunt is required to
achieve the experimentally observed RN
using a physiologically realistic passive membrane resistivity (Rall and Agmon-Snir 1998
). The somatic shunt also has
the effect of increasing the resting potential by 5 mV.

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Fig. 1.
Model of type B photoreceptor. A: biochemical reactions,
ionic channels, and calcium regulatory mechanisms included in model.
Channels, receptors, and enzymes located within the dashed lines are
present only in the synaptic branch. The potassium leak channel, as
well as calcium pumps, calcium release channels, and buffers, are
located in all compartments of the model. B: morphology of
type B photoreceptor model. All compartments are modeled as equivalent
cylinders (the neurite is an elliptic cylinder). Membrane surface area
of the rhabdomeric microvilli is accounted for by a proportional
increase in the capacitance and decrease in the resistance.
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|
Channels
As illustrated in Fig. 1A, the model contains
GABAA synaptic channels,
GABAB synaptic channels, and a calcium-sensitive
potassium leak channel. Measurements of inhibitory synaptic input are
made in dark adapted photoreceptors at resting potential, thus it is not necessary to include channels and second-messenger pathways involved in phototransduction or the voltage-dependent channels that
are not active below
50 mV, i.e., the transient potassium channel
(Acosta-Urquidi and Crow 1995
), the calcium-dependent potassium channel (Farley 1988
; Sakakibara et al.
1993
), or the persistent calcium channel (Yamoah and
Crow 1994
).
Potassium leak channels are voltage-independent channels that are open
and conducting at rest. Neuromodulators coupled to phospholipase C
(PLC) cause the channels to close (Bayliss et al. 1994
;
Hsiao et al. 1997
; Jafri et al. 1997
;
Jones and Baughman 1992
; Lee and McCormick
1997
), and they are blocked by barium (Buckler
1999
). Two sets of experiments support the existence of a
potassium leak conductance in Hermissenda photoreceptors. First, the late depolarization following GABA stimulation is present at
potentials as low as
70 mV; and in 30 mM external
K+ artificial seawater (ASW), a late phase
outward current increases with more negative holding potential
(Rogers et al. 1994
). Second, light stimulation, which
causes an elevation in calcium (Muzzio et al. 1998
)
causes closure of potassium channels at potentials greater than
60 mV
(Alkon and Sakakibara 1985
; Blackwell
2000a
).
In the model, the potassium leak channels are distributed uniformly
throughout all compartments, with a maximal conductance of 300 µS/cm2, and are responsible for 75% of the
total leakage conductance. The reversal potential of this channel is
85 mV. It is assumed that one calcium ion binds to each of two
channel subunits to close the channel
|
(1)
|
where K*L is the open
state of a channel subunit, Ca-KL is
the closed state of a channel subunit,
= 0.45e-3
µM
1-ms
1, and
= 0.6e-3 ms
1. The
parameters are adjusted such that 92% of the channels are open at the
basal calcium concentration of 0.11 µM; less than 1% of the channels
are open at a 10 µM calcium concentration; and the time constant of
activation and decay is on the order of seconds, consistent with
voltage-clamp data of leak channels in carotid body cells
(Buckler 1999
) and corticocallosal neurons (Jones
and Baughman 1992
), and of the light-induced closure of potassium channels in Hermissenda photoreceptors
(Alkon and Sakakibara 1985
; Blackwell
2000a
). Qualitatively, the same results are produced if two
calcium ions bind to a single-channel subunit.
In Hermissenda, in response to mechanical stimulation, the
hair cells depolarize and generate action potentials that cause release
of GABA onto the type B photoreceptor terminal branches. In the model,
hair cell action potentials are modeled as Poisson distributed random
events with an initial rate of approximately 0.15 ms
1, and a rate that decreases exponentially
with a time constant of 1,000 ms (Alkon and Bak 1973
;
Schultz and Clark 1997
). In the model, for each action
potential produced by the hair cell, the GABA receptors are exposed to
a 1 mM concentration of GABA for a duration of 1 ms (Destexhe
and Sejnowski 1995
).
The GABAA channel is modeled as a ligand-gated
receptor channel with two bound states
|
(2)
|
(Bier et al. 1996
) where
RA is the unbound form,
GABA-RA is the bound and closed form,
and GABA-R*A is the open and
conducting form of the GABAA channel. The
GABAA channel transitions from the closed state
to the bound state with rate constants
k1 = 0.13e-3
µM
1-ms
1 and
k2 = 0.16 ms
1.
A second voltage-independent transition from the bound state to the
open state occurs with rate constants,
k3 = 0.019 ms
1
and k4 = 0.009 ms
1. The open state can return to the closed
state either through the bound state or directly with rate constants
k5 = 0.031e-3 µM
1-ms
1 and
k6 = 0.165 ms
1. These rate constants were obtained from
Destexhe et al. (1994)
and modified for the colder
Hermissenda temperature assuming a Q10 of 1.2 (ffrench-Mullen et al. 1988
). The current through the channel equals the fraction of channels in the open state
(GABA-R*A), times the maximal
conductance (95 nS), times the driving potential. The reversal
potential of chloride permeable GABAA channels in Hermissenda is
70 mV (Alkon et al. 1992
;
Rogers et al. 1994
). The maximal synaptic conductance of
95 nS is larger than that demonstrated by Rogers et al.
(1994)
for two reasons. One reason is the simplified morphology
of the synaptic branches and adjusting the parameters to match the
voltage response at the soma. Had a more realistic morphology been
implemented, the synaptic conductance could have been reduced to a more
realistic value with an equivalent voltage response at the soma;
however, this would not have changed the overall results. The second
reason for the large maximal conductance is that the kinetics of the
GABAA equations produce a small fraction of
channels in the open state. The observed
GABAA conductance is <14 nS, which is close to
the value observed by Rogers et al. (1994)
.
When the GABAB metabotropic receptor binds to
GABA, it catalyzes the activation of G protein
|
(3)
|
The rate constants describing GABA binding to the
GABAB receptor are
g1 = 0.06e-3
µM
1-ms
1 and
g2 = 0.5 ms
1.
The bound and active GABAB receptor binds to the
inactive G protein (composed of G
and
G
subunits) with rate constants
g3 = 2.0 µM
1-ms
1 and
g4 = 0.5 ms
1;
and catalyzes the exchange of GDP for GTP. These rate constants were
adjusted from those provided by Destexhe et al. (1994)
such that a single vesicle of neurotransmitter does not saturate the receptor in terms of G
-GTP produced or
GABAB postsynaptic current generated
(Tempia et al. 1998
). The active
G
subunit, G
-GTP, is
produced with rate constant g5 = 0.5 ms
1 (Mukhopadhyay and Ross
1999
); degradation of G
-GTP
(hydrolysis of the bound GTP) occurs with rate constant
g6 = 0.02 ms
1 (Biddlecome et al. 1996
) and
is the rate-limiting step for regeneration of inactive G protein,
G

. The total G protein concentration of
100 µM is conservatively estimated at 1/10th the concentration measured in photoreceptor membranes (Kahlert and Hofmann
1991
; Melia et al. 1997
; Nobes et al.
1992
).
The active G
subunit binds to the
GABAB potassium permeable channel
|
(4)
|
where KB is the unbound form,
G
-KB
is the bound and closed form, and
G
-K*B is the
open and conducting form of the GABAB channel.
The GABAB channel transitions from the closed
state to the bound state with rate constants
= 0.018 µM
1-ms
1 and
= 0.05 ms
1. A second voltage-independent
transition from the bound state to the open state occurs with rate
constants,
= 0.01 ms
1 and
= 0.002 ms
1. These parameter values were modified
from Destexhe et al. (1994)
using a Q10 of 2 (Otis et al. 1993
). The maximal conductance is 9.5 nS
and the reversal potential of potassium permeable
GABAB channels in Hermissenda is
85
mV (Alkon et al. 1992
; Rogers et al.
1994
).
Statocyst stimulation produces a train of action potentials
(Alkon and Bak 1973
; Schultz and Clark
1997
), resulting in a compound IPSP in the photoreceptor whose
size depends on temporal summation of GABAA and
GABAB responses. GABA receptors are located only in the terminal branches; none have been detected at the soma (Rogers and Matzel 1995
). Thus the somatic IPSP,
measured as the difference between resting potential and the maximal
hyperpolarization, is due to passive propagation of the compound IPSP
from the terminal branches along the neurite which functions as a
dendrite in this case. The maximal conductance of
GABAA and GABAB channels
has been adjusted to make the size of this compound IPSP equal to
4
mV, comparable to that observed in experiments (Blackwell and Alkon 1999
; Schultz and Clark 1997
).
Calcium release
In addition to its effect on the GABAB
channel, it is assumed that the active G
subunit binds to and activates PLC (Biddlecome et al.
1996
; Hahner et al. 1991
; Pfaff et al.
1999
; Suzuki et al. 1995
, 1999
)
|
(5)
|
where
= 0.1e-3
µM
1-ms
1, and
= 0.5 ms
1. The production of
IP3 from phosphatidyl inositol bisphosphate
(PIP2) by active PLC (G
-PLC*) is governed by Michaelis-Menten
dynamics
|
(6)
|
Vmax(PLC) for production of
IP3 ranges from 0.01 to 0.1 ms
1 (µmol IP3 · ms
1 · µmol
PLC
1) (Mitchell et al. 1995
;
Smrcka et al. 1991
; Suzuki et al. 1995
), and its value in the model is indicated in the figures. The affinity of
PIP2 for PLC (KM) is 10.0 µM (Rack et al. 1994
). The concentration of
PIP2 in the synaptic membrane is 100 µM
(Baba et al. 1986
; Szuts 1993
); the
concentration of PLC in the synaptic membrane is 10 µM.
IP3 produced by active PLC diffuses through the
cytosol (diffusion rate = 2.83e-9 cm2/ms)
and is degraded (DIP3) at a rate of
0.69e-3/ms (Allbritton et al. 1992
). The form of the
diffusion term is the same as given for calcium diffusion
(APPENDIX, Eq. A4).
Calcium release from the endoplasmic reticulum (ER) may occur via the
IP3 receptor channel
(IP3R), or the ryanodine receptor channel (RyR),
both of which are portrayed in Fig. 1A. The model of the
IP3R is from (De Young and Keizer
1992
; Li and Rinzel 1994
); the equations for
calcium flux through the IP3R, given in the APPENDIX, are the same as used in the type B photoreceptor
soma and rhabdomere model (Blackwell 2000b
). The release
of calcium from the ER via the IP3R increases
intracellular calcium concentration, which binds to the RyR and allows
release from the ER through that channel.
Release of calcium from the ER through the RyR was implemented using
the model of Tang and Othmer (1994)
. In this model, the RyR has two calcium binding sites: an excitatory site and an inhibitory site. The molecule may reside in one of four different states, depending on the occupancy of the calcium binding sites. The RyR molecule is in the open state (R*10)
when the excitatory site is bound. There are three closed states,
occurring when neither calcium binding site is occupied (R00), the inhibitory calcium binding
site is occupied (R01), or both
calcium binding sites are occupied
(R11):
|
(7)
|
where M1 = 0.015 µM
1-ms
1,
M2 = 0.8e-3
µM
1-ms
1,
L1 = 7.6e-3
ms
1, and L2 = 0.84e-3 ms
1.
R*10 is the open and conducting state.
The following equation describes calcium flux through the open
channels:
|
(8)
|
where the equilibrium value of [Ca
] is 20 µM, units of R*10 are fraction of RyR
in the open state, and the maximal rate of efflux,
Fmax(RyR), is 0.08 ms
1 unless otherwise specified.
Mechanisms serving to reduce or equilibrate calcium concentration
include diffusion (6e-9 cm2/ms), buffers, and
pumps. Equations and parameters for the calcium buffer are identical to
that described previously (Blackwell 2000b
), and are
included in the APPENDIX. Two different pumps, the smooth endoplasmic reticulum ATPase (SERCA) pump and the plasma membrane calcium ATPase (PMCA) pump (Morgans et al. 1998
), were
implemented in the present model. The equations used to describe
calcium flux due to the SERCA pump is:
|
(9)
|
where KD(SERCA) is 0.1 µM
(Li and Rinzel 1994
), and, unless otherwise indicated,
Vmax(SERCA) is 0.6 µM/ms. The square
power in this equation is the Hill coefficient. The second term on the left hand side is a compensatory leak;
JL-S has units of
ms
1, and its value is adjusted such that net
calcium flux from the ER is zero at the basal calcium value of 0.11 µM. The equation for calcium flux due to the PMCA pump is
|
(10)
|
where KD(PMCA) is 1.0 µM
(Enyedi et al. 1994
), area = surface area of the
cell membrane, vol = volume of the cytosolic compartment, and
JL-P is the compensatory leak adjusted
such that net flux across the plasma membrane is zero at basal calcium
concentration. The square power on this equation is the Hill
coefficient. For most simulations,
Vmax(PMCA) is 0 µMol/ms/cm2, but this equation is included
because one set of simulations evaluates the effect of a nonzero
Vmax(PMCA).
The equations for calcium flux due to diffusion, and the complete
equations for calcium concentration in the cytosol and the ER are given
in the APPENDIX.
 |
RESULTS |
Calcium waves
The first issues addressed by this study are whether a calcium
wave can propagate from the terminal branches to the soma and which
mechanisms are essential for wave propagation. The contributions of
IP3-induced calcium release (IICR) and
calcium-induced calcium release through the RyR (CICR) are evaluated by
simulations that vary the calcium flux due to each of these. The
mechanisms of calcium wave generation due to CICR are further explored
by inspecting the dynamics of RyRs during a calcium pulse. The role of
the PMCA and SERCA pumps is analyzed with additional simulations and by inspecting the calcium flux terms over time during the calcium wave.
In all simulations, a 3-s-duration mechanical stimulation of hair cells
is initiated 2 s after beginning the simulation. As illustrated in
Fig. 2A, the stimulus produces
an adapting train of action potentials between 2 and 5 s after the
simulation is initiated. Figure 2B shows the concentration
of G
-GTP produced by exposure of
GABAB receptors to a 1 mM concentration of GABA for a duration of 1 ms in response to each action potential. Due to the
dynamics of G-protein activation, the effects of individual "vesicles" of GABA are smoothed. The G
-GTP
binds to and activates PLC, whose concentration is portrayed in Fig.
2B. Both G
-GTP concentration and
active PLC concentration peaks at 0.54 s after the stimulus is
initiated. Active PLC catalyzes the production of
IP3 (illustrated in Fig.
3), which is required for IICR.

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Fig. 2.
Effect of hair cell stimulation on G -GTP and active
phospholipase C (PLC) concentration. A: adapting train
of action potentials simulating 3 s of hair cell stimulation
initiated 2 s after beginning the simulation. B:
concentration of G -GTP produced by GABAB
receptors bound to GABA and of active PLC produced by binding to
G -GTP. Concentration peaks at 0.54 s after the
stimulus is initiated.
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Fig. 3.
Role of inositol triphosphate (IP3)- and calcium-induced
calcium release (IICR and CICR) in generation of calcium wave. A
distance of 0 µm corresponds to the distal end of the neurite,
connected to the terminal branches; and a distance of 100 µm
corresponds to the proximal part of the neurite, connected to the soma.
Left: IP3 concentration.
Middle: calcium concentration for
Fmax(RyR) = 0.08 ms 1,
Vmax(SERCA) = 0.6 µM/ms.
Right: calcium concentration for
Fmax(RyR) = 0 ms 1,
Vmax(SERCA) = 0.3 µM/ms.
A: Vmax(PLC) = 0.1 ms 1. IP3 concentration reaches 0.2 µM as
far as 54 µm. With CICR, 2 waves propagate to the soma; with no CICR,
the wave propagates as far as 54 µm. B:
Vmax(PLC) = 0.03 ms 1.
C: Vmax(PLC) = 0.01 ms 1. IP3 concentration reaches 0.2 µM only
as far as 8 µm. With CICR, a calcium wave propagates to the soma;
with no CICR, calcium concentration is elevated 10 µm from the
terminal branches. Vmax(PMCA) = 0 for
all simulations.
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|
BOTH IP3R AND RYR ARE REQUIRED FOR THE
GENERATION OF CALCIUM WAVES.
Figure 3 shows that a calcium wave propagates from the terminal
branches to the soma and that both IICR and CICR are essential for wave
propagation. The requirement of CICR is seen by comparing Fig. 3,
middle and left columns, which illustrate calcium
concentration as a function of time and distance along the neurite for
Vmax(PLC) between 0.01 and 0.1 ms
1, values that encompass the range of
estimates of PI-specific PLC activity measured in photoreceptors
(Mitchell et al. 1995
; Rack et al. 1994
;
Smrcka et al. 1991
; Suzuki et al. 1995
).
A distance of 0 µm corresponds to the distal end of the neurite,
connected to the terminal branches; and a distance of 100 µm
corresponds to the proximal part of the neurite, connected to the soma.
If Fmax(RyR) = 0 (right), the calcium wave does not propagate all the way to
the soma; these plots show the distance of propagation due to IICR
alone, initiated by diffusion of IP3 toward the
soma. For Vmax(PLC) = 0.1 ms
1, IP3 concentration
reaches 0.2 µM (the threshold for IICR) as far as 54 µm from the
distal end of the neurite, whereas for
Vmax(PLC) = 0.01 ms
1, IP3 concentration
reaches 0.2 µM only as far as 8 µm. In all cases for
Fmax(RyR) = 0, the calcium wave
propagates to the distance at which IP3 reaches
0.2 µM. In contrast, for
Fmax(RyR) = 0.08, the calcium
wave propagates all the way to the soma; thus release through the
ryanodine receptor is responsible for calcium wave propagation the
remainder of the distance, which is substantial for
Vmax(PLC) = 0.01 ms
1.
The mechanism whereby CICR generates a calcium wave is further
illustrated in Fig. 4A, which
shows calcium concentration and the fraction of RyRs in each state. As
calcium concentration increases, the fraction of open channels (in the
R10 state) increases. This allows
calcium to flow from the ER to the cytosol and further increases the
calcium concentration. This positive feedback accelerates the rate at
which RyRs open and calcium concentration increases, analogous to the
activation of sodium channels by depolarization. Similar to sodium
channels, the RyRs inactivate when the calcium concentration is too
high. This is seen in Fig. 4A by the increase in the
fraction of inactive channels (channels in the
R11 and R01 states).

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Fig. 4.
Mechanisms of calcium wave generation. A: calcium
concentration and fraction of ryanodine receptor channels (RyRs) in
each state in compartment 22 µm from terminal branches vs. time for
Fmax(RyR) = 0.08 ms 1,
Vmax(SERCA) = 0.6 µM/ms,
Vmax(PLC) = 0.01 ms 1. As
calcium concentration increases, the fraction of open channels (in the
R10 state) increases, and the fraction of
unbound channels (in the R00 state)
decreases. Approximately 0.25 s after the initial increase in the
R10 state, while calcium concentration is
still increasing, calcium release peaks and the
R10 state begins to decrease. The fraction
of inactivated channels (R11 state and
R01 state) increases more slowly but remains
elevated for almost 2 s after calcium concentration returns to the
basal level. B: wave speed vs. distance as a function of
Vmax(PLC). The wave speed in the proximal
part of the neurite remains constant at 0.013 µm/ms, and as
Vmax(PLC) decreases to 0.01 ms 1, the region of the neurite in which the calcium wave
propagates with this speed increases from 20 to 80 µm.
C: peak calcium concentration versus distance as a
function of Vmax(PLC).
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A secondary calcium wave is seen for
Vmax(PLC) = 0.03 and 0.1 ms
1. The mechanism generating this secondary
wave is analogous to that generating multiple action potentials in
response to current injection. The RyR de-inactivates (referred to as
adaptation in the cardiac myocyte literature) (cf. Tang and
Othmer 1994
) once the calcium concentration returns to the
basal level. If the calcium influx remains elevated for 6-7 s (due to
IICR), the channel will activate again causing another calcium peak and
initiating another calcium wave. In the distal part of the neurite for
Vmax(PLC) = 0.1 ms
1, the calcium influx due to IICR is so high
that calcium concentration does not return to the basal level but
remains elevated at 1 µM. The RyR partially de-inactivates, and the
secondary wave is of lower amplitude in this part of the neurite.
In addition to CICR, IICR is essential for wave propagation, and
Vmax(PLC) has a dramatic effect on the
speed of wave propagation. First, IICR is the initiating stimulus for
the calcium wave. If Vmax(PLC) = 0.003, IP3 concentration is insufficient for IICR (<0.2 µM) and a calcium wave is never initiated (results not shown). Second, IICR is responsible for a portion of the calcium wave because
IP3 diffuses much farther than calcium
(Allbritton et al. 1992
). Wave propagation is faster in
regions where IP3 concentration exceeds 0.2 µM;
thus the calcium wave reaches the soma within 6.0 s after stimulus
onset for Vmax(PLC) = 0.1 but
requires 8.0 s for Vmax(PLC) = 0.01. The effect on wave speed is seen more clearly in Fig.
4B, which plots wave speed versus distance as a function of
Vmax(PLC). Wave speed is high in the
most distal part of the neurite and decreases to a plateau value
of 0.013 µm/ms in the proximal part of the neurite; as
Vmax(PLC) decreases to 0.01 ms
1, the region of the neurite in which the
calcium wave propagates at 0.013 µm/ms increases from 20 to 80 µm.
This suggests that the ryanodine receptor by itself supports calcium
wave propagation at a speed of 0.013 µm/ms. The distance at which
wave speed drops <0.02 µm/ms corresponds to the distance
IP3 exceeds 0.2 µM, the threshold for IICR
reported by Li and Rinzel (1994)
. A similar effect of
Vmax(PLC) is seen in the
plots of peak calcium concentration versus distance (Fig.
4C). In regions where IP3 exceeds 0.2 µM, IICR contributes to a peak calcium concentration >3 µM. The
correspondence between calcium concentration and wave speed suggests a
possible causal relationship, a concept that is explored further in the next sections.
ROLE OF DIFFUSION, PMCA, AND SERCA PUMPS.
In the absence of IICR (e.g., closer to the soma), the calcium wave due
to CICR depends on an initial increase in calcium concentration due to
diffusion from the adjacent compartment, although the wave itself is
not sustained by diffusion alone. The SERCA pump modulates the calcium
wave by directly changing both the calcium concentration increase due
to diffusion and the net flux of calcium out of the ER. Figure
5 demonstrates the interaction among
diffusion, release, and pump re-uptake by plotting various flux terms
versus time for several values of
Vmax(SERCA). All changes in
Vmax(SERCA) are accompanied by
compensatory changes in the JL-S to
maintain a constant basal calcium concentration. Figure 5A
shows that calcium flux due to diffusion increases first, at ~3.6 s,
in the compartment 22 µm from the terminal branches. The
concentration increase causes an increase in SERCA flux, which transfers calcium from the cytosol to the ER; thus the net flux out of
the ER becomes negative. The concentration increase also activates the
ryanodine receptor, and ~0.19 s after diffusion begins, the CICR flux
is large enough to change the net ER flux from negative to positive.
The SERCA pump affects this process by its control of the calcium flux.
A higher Vmax(SERCA) of 0.7 (Fig.
5B) reduces the net flux from the ER and opposes the
diffusive flux; calcium concentration is lower and the resulting
calcium flux due to diffusion is smaller. The calcium concentration
increases more slowly, thereby decreasing the rate of ryanodine
receptor activation and increasing the latency between the time of
diffusive increase and the time when the net ER flux changes sign.

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Fig. 5.
The regulation of wave speed by pumps, diffusion, and peak calcium
concentration. A-D: calcium flux vs. time for
Fmax(RyR) = 0.08 ms 1,
Vmax(PLC) = 0.01 ms 1
(same parameters as in Fig. 3C, middle).
A: Vmax(SERCA) = 0.5 µM/ms, compartment at 22 µm. Calcium flux due to diffusion
increases at 3.6 s. The resulting concentration increase causes an
increase in smooth endoplasmic reticulum calcium ATPase (SERCA) flux
(shown as a positive flux for transfer into the ER);
thus the net flux out of the ER (=RyR flux + leak flux SERCA
flux) is negative. The concentration increase also activates the
ryanodine receptor, and 0.19 s after diffusion begins, the RyR
flux is large enough to change the net ER flux from negative to
positive. B: Vmax(SERCA) = 0.7 µM/ms, compartment at 22 µm. The greater SERCA flux opposes
the diffusive flux and reduces the calcium concentration increase. The
rate of RyR activation is slower, which increases the latency between
the time of diffusive increase and the time when net ER flux changes
sign. C: Vmax(SERCA) = 0.8 µM/ms, compartment at 22 µm. Calcium concentration, RyR flux
and diffusive flux are slightly smaller than for
Vmax(SERCA) = 0.7 µM/ms, but the
calcium concentration increase is sufficient to produce fast release.
D: Vmax(SERCA) = 0.8 µM/ms, compartment at 38 µm. Calcium concentration has a peak of
0.44 µM, which produces a significantly slower and smaller calcium
release. E: peak calcium vs. distance along the neurite
as a function of Vmax(SERCA). As the wave
propagates to the soma, the peak calcium decreases to a plateau value
unless Vmax(SERCA) is too high.
F: wave speed vs. peak calcium concentration for all
parameters listed in Table 1.
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If Vmax(SERCA) is too high relative to
Fmax(RyR), the calcium wave does not
propagate to the soma. This phenomenon is explained in Fig. 5,
C and D, which illustrates calcium concentration
and flux terms for Vmax(SERCA) = 0.8 µM/ms at two locations along the neurite. In the compartment 22 µm from the terminal branches, the peak calcium concentration is 2.0 µM and the latency is 0.25 s. Although the rate of ryanodine
receptor activation is slower than for
Vmax(SERCA) = 0.7 µM/ms, flux
through the receptor reaches the same peak value of 10e-18 µMol/ms.
At 38 µm, the peak calcium concentration is only 0.44 µM, and the
latency is increased to 0.5 s. The calcium concentration in the
adjacent compartment is illustrated to show its relationship to
diffusive flux. Such a difference in peak calcium concentration between
two adjacent compartments is not seen for lower values of
Vmax(SERCA). The much lower diffusive
flux causes a slow activation of the RyR and allows the inactivation
process to proceed at a commensurate pace; thus peak flux through the
RyR is lower (7e-18 µMol/ms) than for other
Vmax(SERCA) values. All flux terms are
smaller and increase more slowly as compared with 22 µm.
Figure 5E summarizes the effect of the SERCA pump on peak
calcium concentration versus distance along the neurite. For
Vmax(SERCA) between 0.5 and 0.7 µM/ms, peak calcium concentration decreases to a plateau value.
For Vmax(SERCA) = 0.8 µM/ms, the calcium concentration does not reach a plateau value,
instead it decreases with distance, and the calcium wave dies out 40 µm from the terminal branches. For all values of
Fmax(RyR) tested, if
Vmax(SERCA) is too high, the peak
calcium concentration decrements with distance and does not reach a
plateau value.
The effect of the SERCA pump on calcium wave propagation is further
illustrated in Fig. 6A for
Fmax(RyR) = 0.16 ms
1 and PLC = 0.01 ms
1. For both values of
Vmax(SERCA), the wave from 0 to 20 µm propagates faster than the remainder of the wave due to IICR in
those compartments. The wave from 40 to 100 µm propagates at a
constant speed, with a higher speed and peak calcium concentration for
Vmax(SERCA) = 1.0 µM/ms as
compared with Vmax(SERCA) = 1.4 µM/ms.

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Fig. 6.
Calcium concentration as a function of time and distance along the
neurite. A: Fmax(RyR) = 0.16 ms 1, Vmax(PLC) = 0.01 ms 1, and Vmax(PMCA) = 0 µMol/ms-cm2. Vmax(SERCA)
is indicated in top left corner in µM/ms. The calcium
wave reaches soma at 8 s for
Vmax(SERCA) = 1.0 and at 12 s for
Vmax(SERCA) = 1.4. B:
Fmax(RyR) = 0.08 ms 1,
Vmax(PLC) = 0.03 ms 1, and
Vmax(SERCA) = 0.6 µM/ms.
Vmax(PMCA) is indicated in top left
corner in µmol/ms-cm2. The higher values prevent
or terminate secondary calcium waves.
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The PMCA pump has a lower affinity (KD = 1 µM) than the SERCA pump, yet it has an effect qualitatively
similar to that of the SERCA pump. Figure 6B illustrates the
effect of Vmax(PMCA) for
Vmax(PLC) = 0.03 ms
1, a value that produces secondary waves (see
Fig. 3). A moderate value of
Vmax(PMCA), equal to 4e-9
µMol/ms/cm2, slows the propagation of the
primary wave and stops the propagation of the secondary wave. A higher
value of Vmax(PMCA), equal to 8e-9
µMol/ms/cm2, prevents the secondary wave from appearing.
RELATIONSHIP BETWEEN CALCIUM CONCENTRATION AND WAVE SPEED.
Wave speed is inversely related to the latency between the influx of
calcium and the time when the net ER flux changes sign. Latency is
affected by the magnitude of diffusive flux (Tang and Othmer
1994
), which depends on calcium concentration in the adjacent compartment. Thus the SERCA pump affects latency by its control of
calcium concentration. Table 1, which
lists wave speed and peak calcium concentration in the distal half of
the neurite, shows that a higher
Vmax(SERCA) leads to faster wave
propagation.
Similarly, Fmax(RyR) modifies wave
speed through its effect on peak calcium concentration. If
Fmax(RyR) is doubled, the
Vmax(SERCA) also is increased to
maintain the next ER flux = 0 at the basal calcium concentration.
When both Fmax(RyR) and
Vmax(SERCA) are doubled, the net
calcium flux from the ER is doubled. Thus once the net ER flux becomes
positive, the calcium concentration increases faster and reaches a
greater peak value. For example, as shown in Table 1, doubling
Fmax(RyR) and
Vmax(SERCA) causes an increase in peak
calcium concentration comparable to reducing
Vmax(SERCA) from 0.6 to 0.5 µM/ms.
The change in peak calcium concentration due to a change in
Fmax(RyR) causes a change in latency
and wave speed.
Figure 5H shows that the relationship between peak calcium
concentration and wave speed is relatively independent of the various mechanisms that affect calcium concentration. The straight line is the
best linear fit to all the points, with an
R2 = 0.96. For no combination of
parameter values is a calcium wave sustained with a peak calcium
concentration <1.3 µM. Below this value, the diffusive flux is
insufficient to activate the ryanodine receptor channel to the extent
required for CICR-mediated amplification of the calcium signal
(Tang and Othmer 1994
).
Cause of late depolarization
The second issue addressed by this study is the origin of
the late depolarization and increase in
RN observed after GABA stimulation (Matzel and Alkon 1991
; Rogers et al.
1994
). A 200 ms,
0.5 nA current was injected every 800 ms,
before, during, and after simulated GABA stimulation to measure the
change in RN using the formula %
RN = 100 *
(
Vpost
Vpre)/
Vpre.
Figure 7A illustrates that both a late depolarization and an increase in
RN occur after GABA stimulation. The
late depolarization is observed between 4 and 10 s after the
beginning of the GABA stimulation and peaks at 6 s. The 1.3 mV
change in membrane potential is accompanied by a 4.5% increase in
RN; these changes are similar to the 2 mV depolarization and 3% input resistance increase observed by
Matzel and Alkon (1991)
. Figure 7B shows the
total conductance of the potassium leak channels in the synaptic
branch, nonsynaptic branch, and each of the four isopotential neurite
compartments. The Vmax(PLC) is 0.1 ms
1, thus two calcium waves are produced (see
Fig. 3) and cause two reductions in the potassium leak conductance in
the proximal neurite compartments. Comparison of 7B with
A, top reveals that the time course of the late
depolarization corresponds to the time course of potassium leak
conductance decrease. Similarly, the
RN increase during the late
depolarization is due to the closure of the leak potassium channels.
The onset of conductance decrease in the proximal neurite compartments
is delayed relative to that in the distal neurite compartments because
of the time it takes for the calcium wave to propagate to the proximal
compartments.

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Fig. 7.
Effect of GABA stimulation on membrane potential, leak conductance, and
synaptic conductance. A, top: membrane potential in soma
and terminal branches before, during, and after GABA stimulation.
Bottom: how hyperpolarizing current pulses are used to
measure RN. Both a late depolarization and
an increase in RN are apparent and last as
long as 10 s after the beginning of the GABA stimulation.
B: total conductance (channel density multiplied by
surface area) of the potassium leak channels in the synaptic branch,
nonsynaptic branch, and each of the 4 isopotential neurite
compartments. The smaller total conductance value in the terminal
branches is due to their smaller surface area, with the synaptic branch
being smaller than the nonsynaptic branch. C:
GABAA and GABAB conductance underlying the
hyperpolarization.
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The GABAA and GABAB
conductance underlying the hyperpolarization are illustrated in Fig.
7C. The GABAA conductance (offset by 7 nS in the figure) consists of multiple brief channel activations; in
contrast, the GABAB conductance shows the slow
and prolonged time course characteristic of G-protein-gated channels.
The large fluctuations in membrane potential caused by the fast
GABAA current (Fig. 7A) that appear in
the branch do not appear in the soma; they are averaged out by the
cable properties of the neurite. As previously observed (Matzel
and Alkon 1991
), RN decreases
by 23% (from 35 to 27 M
) during the hyperpolarization because of the increase in conductance of the GABA channels.
The size of both the late depolarization and the increase in input
resistance depend on parameters that affect calcium wave propagation.
Parameter values that cause a larger reduction in potassium leak
conductance (Fig. 8 left)
result in a larger late depolarization and increase in
RN (Fig. 8, right). This is
shown for Vmax(PMCA) (A),
Vmax(PLC) (B),
Vmax(SERCA) (C), for
Fmax(RyR) = 0.08 ms
1, and
Fmax(RyR) (D), for the
ratio of
Fmax(RyR)/Vmax(SERCA) = 0.133 µM
1. Despite the variation in the
magnitude of the effect, all parameters which support release of
calcium from intracellular stores also support an increase in membrane
potential and RN following GABA stimulation.

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Fig. 8.
The change in membrane potential and RN
(right) and potassium leak conductance
(left) as a function of
Vmax(PMCA) (A),
Vmax(PLC) (B),
Vmax(SERCA) (C) for
Fmax(RyR) = 0.08 ms 1, and
Fmax(RyR) (D) for ratio
Fmax(RyR)/Vmax(SERCA) = 0.133 µM 1. The potassium leak conductance is the sum
over all neurite and terminal branch compartments. Parameter values
that allow a greater elevation in calcium cause a larger reduction in
potassium leak conductance and result in a larger late depolarization
and increase in RN.
Fmax(RyR) = 0.08 ms 1 for
A-C, Vmax(SERCA) = 0.6 µM/ms for A and B,
Vmax(PMCA) = 0 µMol/ms-cm2 for
B-D, Vmax(PLC) = 0.03 ms 1 for A, C, and D.
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LATE DEPOLARIZATION IS NOT DUE TO VOLTAGE-DEPENDENT CURRENTS.
Hermissenda photoreceptors contain two other currents
that are partially active at rest. The transient calcium current,
ICaT, has a half activation of
40 mV
and half inactivation of
48 mV, and the hyperpolarization-activated
current, IH, is active at
60 mV. The
decrease in ICaT inactivation caused
by hyperpolarization may allow an increase in
ICaT following membrane
repolarization. This in turn may cause a small depolarization and
consequent inactivation of IH, causing
an increase in RN. This possibility
was investigated with one additional set of simulations using the
following modifications to the model: both
IH and
ICaT were implemented using activation and inactivation parameters presented in (Yamoah et al.
1998
). The maximal conductance was adjusted to produce a
current amplitude in voltage-clamp mode comparable to that recorded in
their experiments: maximal conductance of
IH = 833 nS/cm2
and maximal permeability of ICaT = 4e-7 cm/s. Activation of PLC by G
-GTP was
eliminated to determine if calcium influx through ICaT could activate CICR. The results
of these simulations did not reveal a delayed depolarization subsequent
to GABA stimulation. Examination of the currents showed that the
hyperpolarization due to GABA did not produce a significant rebound
activation of ICaT. Furthermore,
because IH has a reversal potential of
30 mV, any inactivation of IH
sufficient to cause an increase in RN
also would cause a hyperpolarization, which is not consistent with the
observations. No significant elevation in calcium, either synchronous
or as a wave, was observed; therefore no increase in
RN was observed due to potassium leak
channel reduction. Calcium influx through
ICaT channels was not sufficient to
activate CICR.
Size of synaptic branch affects IPSP but not calcium wave
All of the preceding simulations were performed with the size of
the synaptic branch much smaller than the size of the nonsynaptic branch (asymmetric). Simulations were repeated in which the synaptic branch and nonsynaptic branch sizes were equivalent (symmetric), implying that GABA synapses occur on 50% of the terminal branches. The
maximal conductance of the GABA channels was increased in the
asymmetric case to yield the same maximal conductance as in the
symmetric case. As illustrated in Fig.
9A, the IPSP measured in the
soma is
8 mV, larger than the
6 mV IPSP measured when the synaptic
branch is smaller than the nonsynaptic branch. The GABAB current in the symmetric case is very close
to that in the high-density asymmetric case (Fig. 9B,
top). However, the GABAA current in
the asymmetric case is significantly smaller than that in the symmetric
case because temporal summation is closer to linear in the symmetric
case. The total synaptic current hyperpolarizes the much smaller
asymmetric synaptic branch to a potential very close to the
GABAA reversal potential and reverses the driving potential for the GABAA current for several
hundred milliseconds (Fig. 9B, bottom).

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Fig. 9.
Simulations in which the synaptic branch and nonsynaptic branch are
equivalent (symmetric), implying that GABA synapses occur on 50% of
the terminal branches. Comparison with the asymmetric case in which
GABA conductance is equivalent to that of the symmetric case.
Fmax(RyR) = 0.08 ms 1,
Vmax(SERCA) = 0.6 µM/ms,
Vmax(PMCA) = 0, and
Vmax(PLC) = 0.01 ms 1.
A: membrane potential in the soma and terminal branches.
The inhibitory postsynaptic potential (IPSP) measured in the terminal
branches is greater in the asymmetric case as compared with the
symmetric case. In contrast, the IPSP in the soma is greater for the
symmetric case as compared with the asymmetric case. B:
the GABAB current in the symmetric case is very close to
that in the high-density asymmetric case. However, the
GABAA current in the asymmetric case is significantly
smaller than that in the symmetric case because membrane potential of
the asymmetric synaptic branch is close to the GABAA
reversal potential.
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