INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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 GABA_A chloride channels and GABA_B 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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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 GABA_B 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 (R_N) experimentally observed in Hermissenda photoreceptors. The somatic shunt is required to achieve the experimentally observed R_N 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.

View larger version (61K): [in this window] [in a new window]   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.

Channels

As illustrated in Fig. 1A, the model contains GABA_A synaptic channels, GABA_B 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-K_L is the closed state of a channel subunit,  = 0.45e-3 µM¹-ms¹, and  = 0.6e-3 ms¹. 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 ms¹, 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 GABA_A channel is modeled as a ligand-gated receptor channel with two bound states

(2)

(Bier et al. 1996) where R_A is the unbound form, GABA-R_A is the bound and closed form, and GABA-R^*_A is the open and conducting form of the GABA_A channel. The GABA_A channel transitions from the closed state to the bound state with rate constants k₁ = 0.13e-3 µM¹-ms¹ and k₂ = 0.16 ms¹. A second voltage-independent transition from the bound state to the open state occurs with rate constants, k₃ = 0.019 ms¹ and k₄ = 0.009 ms¹. The open state can return to the closed state either through the bound state or directly with rate constants k₅ = 0.031e-3 µM¹-ms¹ and k₆ = 0.165 ms¹. 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 GABA_A 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 GABA_A equations produce a small fraction of channels in the open state. The observed GABA_A conductance is <14 nS, which is close to the value observed by Rogers et al. (1994).

When the GABA_B metabotropic receptor binds to GABA, it catalyzes the activation of G protein

(3)

The rate constants describing GABA binding to the GABA_B receptor are g₁ = 0.06e-3 µM¹-ms¹ and g₂ = 0.5 ms¹. The bound and active GABA_B receptor binds to the inactive G protein (composed of G and G subunits) with rate constants g₃ = 2.0 µM¹-ms¹ and g₄ = 0.5 ms¹; 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 GABA_B postsynaptic current generated (Tempia et al. 1998). The active G subunit, G-GTP, is produced with rate constant g₅ = 0.5 ms¹ (Mukhopadhyay and Ross 1999); degradation of G-GTP (hydrolysis of the bound GTP) occurs with rate constant g₆ = 0.02 ms¹ (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 GABA_B potassium permeable channel

(4)

where K_B is the unbound form, G-K_B is the bound and closed form, and G-K^*_B is the open and conducting form of the GABA_B channel. The GABA_B channel transitions from the closed state to the bound state with rate constants  = 0.018 µM¹-ms¹ and  = 0.05 ms¹. A second voltage-independent transition from the bound state to the open state occurs with rate constants,  = 0.01 ms¹ and  = 0.002 ms¹. 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 GABA_B 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 GABA_A and GABA_B 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 GABA_A and GABA_B 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 GABA_B 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 µM¹-ms¹, and  = 0.5 ms¹. The production of IP₃ from phosphatidyl inositol bisphosphate (PIP₂) by active PLC (G-PLC*) is governed by Michaelis-Menten dynamics

(6)

V_max(PLC) for production of IP₃ ranges from 0.01 to 0.1 ms¹ (µmol IP₃ · ms¹ · µmol PLC¹) (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 (K_M) 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 (D_IP3) 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 µM¹-ms¹, M₂ = 0.8e-3 µM¹-ms¹, L₁ = 7.6e-3 ms¹, and L₂ = 0.84e-3 ms¹. 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, F_max(RyR), is 0.08 ms¹ 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 K_D(SERCA) is 0.1 µM (Li and Rinzel 1994), and, unless otherwise indicated, V_max(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; J_L-S has units of ms¹, 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 K_D(PMCA) is 1.0 µM (Enyedi et al. 1994), area = surface area of the cell membrane, vol = volume of the cytosolic compartment, and J_L-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, V_max(PMCA) is 0 µMol/ms/cm², but this equation is included because one set of simulations evaluates the effect of a nonzero V_max(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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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 GABA_B 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.

View larger version (28K): [in this window] [in a new window]   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.

View larger version (47K): [in this window] [in a new window]   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 ms1, Vmax(SERCA) = 0.6 µM/ms. Right: calcium concentration for Fmax(RyR) = 0 ms1, Vmax(SERCA) = 0.3 µM/ms. A: Vmax(PLC) = 0.1 ms1. 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 ms1. C: Vmax(PLC) = 0.01 ms1. 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.

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 V_max(PLC) between 0.01 and 0.1 ms¹, 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 F_max(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 V_max(PLC) = 0.1 ms¹, IP₃ concentration reaches 0.2 µM (the threshold for IICR) as far as 54 µm from the distal end of the neurite, whereas for V_max(PLC) = 0.01 ms¹, IP₃ concentration reaches 0.2 µM only as far as 8 µm. In all cases for F_max(RyR) = 0, the calcium wave propagates to the distance at which IP₃ reaches 0.2 µM. In contrast, for F_max(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 V_max(PLC) = 0.01 ms¹.

View larger version (21K): [in this window] [in a new window]   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 ms1, Vmax(SERCA) = 0.6 µM/ms, Vmax(PLC) = 0.01 ms1. 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 ms1, 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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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 V_max(SERCA). All changes in V_max(SERCA) are accompanied by compensatory changes in the J_L-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 V_max(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.

View larger version (46K): [in this window] [in a new window]   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 ms1, Vmax(PLC) = 0.01 ms1 (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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View larger version (38K): [in this window] [in a new window]   Fig. 6. Calcium concentration as a function of time and distance along the neurite. A: Fmax(RyR) = 0.16 ms1, Vmax(PLC) = 0.01 ms1, 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 ms1, Vmax(PLC) = 0.03 ms1, 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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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 V_max(SERCA) leads to faster wave propagation.

Cause of late depolarization

The second issue addressed by this study is the origin of the late depolarization and increase in R_N 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 R_N using the formula %R_N = 100 * (V_post  V_pre)/V_pre. Figure 7A illustrates that both a late depolarization and an increase in R_N 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 R_N; 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 V_max(PLC) is 0.1 ms¹, 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 R_N 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.

View larger version (33K): [in this window] [in a new window]   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.

The GABA_A and GABA_B conductance underlying the hyperpolarization are illustrated in Fig. 7C. The GABA_A conductance (offset by 7 nS in the figure) consists of multiple brief channel activations; in contrast, the GABA_B conductance shows the slow and prolonged time course characteristic of G-protein-gated channels. The large fluctuations in membrane potential caused by the fast GABA_A 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), R_N 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 R_N (Fig. 8, right). This is shown for V_max(PMCA) (A), V_max(PLC) (B), V_max(SERCA) (C), for F_max(RyR) = 0.08 ms¹, and F_max(RyR) (D), for the ratio of F_max(RyR)/V_max(SERCA) = 0.133 µM¹. 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 R_N following GABA stimulation.

View larger version (41K): [in this window] [in a new window]   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 ms1, and Fmax(RyR) (D) for ratio Fmax(RyR)/Vmax(SERCA) = 0.133 µM1. 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 ms1 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 ms1 for A, C, and D.

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, I_CaT, has a half activation of 40 mV and half inactivation of 48 mV, and the hyperpolarization-activated current, I_H, is active at 60 mV. The decrease in I_CaT inactivation caused by hyperpolarization may allow an increase in I_CaT following membrane repolarization. This in turn may cause a small depolarization and consequent inactivation of I_H, causing an increase in R_N. This possibility was investigated with one additional set of simulations using the following modifications to the model: both I_H and I_CaT 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 I_H = 833 nS/cm² and maximal permeability of I_CaT = 4e-7 cm/s. Activation of PLC by G-GTP was eliminated to determine if calcium influx through I_CaT 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 I_CaT. Furthermore, because I_H has a reversal potential of 30 mV, any inactivation of I_H sufficient to cause an increase in R_N 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 R_N was observed due to potassium leak channel reduction. Calcium influx through I_CaT 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 GABA_B current in the symmetric case is very close to that in the high-density asymmetric case (Fig. 9B, top). However, the GABA_A 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 GABA_A reversal potential and reverses the driving potential for the GABA_A current for several hundred milliseconds (Fig. 9B, bottom).

View larger version (30K): [in this window] [in a new window]   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 ms1, Vmax(SERCA) = 0.6 µM/ms, Vmax(PMCA) = 0, and Vmax(PLC) = 0.01 ms1. 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. <