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Ionic current model of rabbit retinal horizontal cell
Neuroscience Research 37 (2000) 141 – 151 www.elsevier.com/locate/neures
Ionic current model of rabbit retinal horizontal cell Toshihiro Aoyama a, Yoshimi Kamiyama b, Shiro Usui a,*, Roman Blanco c, Cecilia F. Vaquero c, Pedro de la Villa c a
Department of Information and Computer Sciences, Toyohashi Uni6ersity of Technology, Toyohashi, Aichi, 441 -8580, Japan b Faculty of Information Science and Technology, Aichi Prefectural Uni6ersity, Nagakute, Aichi, 480 -1198, Japan c Department of Physiology and Pharmacology, School of Medicine, Uni6ersity of Alcala de Henares, Madrid, 28871, Spain Received 30 August 1999; accepted 28 February 2000
Abstract We propose a mathematical model of rabbit retinal horizontal cell based on the ionic current mechanisms. Five types of ionic currents in rabbit retinal horizontal cell, INa, ICa, IKv, IA and IKa, are described by Hodgkin – Huxley type equations based on voltage clamp measurements. In simulation the model reproduced similar responses to voltage and current clamp experiments. Under the current clamp experiment a repetitive action potential was found on A-type rabbit horizontal cells. Our result suggests that the repetitive action potential is generated by an interaction Of ICa and IKv. © 2000 Elsevier Science Ireland Ltd and the Japan Neuroscience Society. All rights reserved. Keywords: Rabbit; Retinal horizontal cell; Ionic current model; Simulation; Reptitive; Action potential
1. Introduction Retinal horizontal cells have been studied in a variety of animals for the last 40 years. In lower vertebrate retinas, horizontal cells play a fundamental role in forming center-surround receptive fields and in generating color opponent signals (reviewed in Kaneko (1987)). On the other hand, anatomical and physiological studies suggest that there are structural and functional differences between lower vertebrate and mammalian horizontal cells. Mammalian horizontal cells are morphologically classified into A- and B-type cells. The A-type cell has large, thin dendrites which exclusively connect with cones, and the B-type cell has dendrites which connect with cones and an axon and an axon terminal which contacts only with rods (Kolb, 1974; Boycott et al., 1978; Bloomfield and Miller, 1982). In the primate retina all the horizontal cells are functionally classified as luminosity-type, these cells do not perform a color opponent transformation (Dacey et al., 1996). Lankheet et al. (1990) have performed intracellular recordings from cat horizontal cells and found * Corresponding author. Tel.: +81-532-446764; fax: + 81-532467806.
that the spatial properties of response have non-passive electrical spread and nonlinear summation of potential. A nonlinear relationship between voltage response and light intensity has been found (Lankheet et al., 1991a,b). Some of these nonlinear properties are thought to be caused by the membrane properties of the horizontal cells. Membrane currents which characterize the electrical properties of horizontal cells have been analyzed using isolated cell preparations. Blanco et al. (1996a) recently found that the rabbit A-type horizontal cells exhibit repetitive action potentials which have not been observed in distal neurons of the vertebrate retina previously. Although the horizontal cells normally respond to light with a graded potential, it is of interest to elucidate how the cells generate the spike trains and to understand their functional role in vision. However, it is difficult to analyze the mechanism underlying repetitive action potentials solely using traditional electrophysiology. We have developed a quantitative model of horizontal cells to investigate the mechanism of the action potential. Recently, mathematical models based on ionic currents have been developed for lower vertebrate photoreceptors (Kamiyama et al., 1996), horizontal cells (Usui
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et al., 1996a) and bipolar cells (Usui et al., 1996b). These models have been used to elucidate complicated electrical behavior in these retinal cells. In this paper we developed a mathematical model of the rabbit retinal horizontal cell based on membrane ionic current mechanisms. We confirmed the validity of the model through current clamp simulation, and analyzed the mechanism of repetitive spike generation found in rabbit horizontal cells.
(Kamiyama et al., 1996), horizontal cells (Usui et al., 1996a) and bipolar cells (Usui et al., 1996b). In the present study, the system of differential equations is numerically solved by using a Runge–Kutta– Fehlberg method (Ralston and Rabinowits, 1978) with an adaptive step size. All simulations were performed on a UNIX work station (DEC Personal Work Station 500au).
2.2. Electrophysiology
where m and h are the open and close gating variables, M and H are constants which determine the rising and falling phases of the conductance and gi is the maximum conductance. These gating variables (x = m, h) depend on the membrane voltage and time, and are described as:
The detail of the experimental method has been reported previously (Blanco et al., 1996a) and will be summarized briefly here. Horizontal cells were dissociated enzymatically from the rabbit retina. For enzymatic dissociation, the retina was incubated in oxygenated standard solution containing 75 mg/ml papain (Carl Roth, Kahlsruhe, Germany) and 0.1 mg/ml DL-eysteine (Sigma, St Louis, MO) for 20–30 min at 30°C and rinsed with a standard solution containing 0.01% BSA. Standard solution contained 135 mM NaCl, 5 mM KCl, 1 mM MgCl2, 2 mM CaCl2, 10 mM glucose and 10 mM Hepes (pH adjusted to 7.4 with NaOH). The cells were maintained at 5°C for 1–5 h before use in a plastic culture dish containing 2 ml standard solution with BSA. Membrane voltage and current were recorded by patch pipette in the whole-cell configuration. Pipette solutions contained 120 mM KCl, 1 mM MgCl2, 5 mM EGTA, 0.5 mM CaCl2 and 10 mM Hepes (pH adjusted to 7.2 with KOH). Patch pipettes (Clark Electromedical: 1.0 mm o.d.) were pulled by the Sutter pipette puller (model P-90). The resistance of the pipette as measured in the bath was usually about 3–8 MV. Electrode series resistance ranged from 10–20 MV. In all experiments the drugs were dissolved in the standard solution and applied to the cell by a Y-tube system.
dx = ax (1− x)−bxx, dt
3. Results
2. Methods
2.1. Modeling and simulation The electrical property of a cell membrane is represented by a parallel conductance model (Johnston and Wu, 1995). The membrane current I is described as: I=C
dV dV +%gi(V − Ei) +%Ii = C dt dt
(1)
where C is the membrane capacitance (nF), V is the membrane potential (mV), Ii is the ionic current (pA), gi is the conductance of the ionic current (nS) and Ei is the reversal potential (mV) of the current. In the Hodgkin and Huxley formulation the conductance of an ionic current is written as the product of gating variables and the maximum conductance (Hodgkin and Huxley, 1952): gi = m Mh Hgi,
(2)
(3)
where ax, bx are expressed as: C1 exp(C2(V +C3) + C4(V + C5)) . ax, bx = exp(C6(V + C3) +C7)
(4)
The parameters Cl – C7 are calculated from following equations: ax =
x tx
(5)
bx =
1−x tx
(6)
x and tx are stable gating variables (x) and time constant of gating variable (x) calculated from experimental voltage clamp data. The method has been applied to developing models of vertebrate photoreceptors
3.1. Summary of ionic current of mammalian horizontal cells In mammalian horizontal cells ionic currents have been measured in the cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994). These currents are summarized in Table 1. Five types of voltage dependent ionic currents have been found in the cell body: a sodium current (INa), a calcium current (ICa), a delayed rectifying potassium current (IKv,), a transient outward potassium current (IA) and an anomalous rectifying potassium current (IKa). These voltage dependent currents also have been identified in lower vertebrate horizontal cells, goldfish (Tachibana, 1983), catfish (Shingai and Christensen, 1983) and white bass (Sullivan and
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Table 1 Summary of the ionic current of rabbit horizontal cell Current
Reversal potential (mV)
Activate potential (mV)
I–V property
ICa INa IKv IA IKa
54 55 −80 −80 −80
Over −30 Over −35 Over −50 Over −40 Below −80
Bell Bell Outward rectifying Outward rectifying Inward rectifying
Lasater, 1990). One of the differences in the ionic currents between lower vertebrate and mammalian horizontal cells is the time course of the calcium current. ICa is inactivated by the accumulation of intracellular calcium ions in goldfish horizontal cells (Tachibana, 1983); however, this inactivation has never been observed in the cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994) horizontal cells. The calcium dependent potassium current (IKc) is found in rabbit B-type horizontal cells (Lo¨hrke and Hofmann, 1994), but has never been observed in lower vertebrates and cat.
al., 1992) and rabbit horizontal cells (Lo¨hrke and Hofmann, 1994). These results show that the L-type current is not suppressed by an accumulation of intracellular calcium as in the goldfish (Tachibana, 1983). Lo¨hrke and Hofmann (1994) reported that there is a transitory T-type calcium current, which activates at more than − 50 mV, in rabbit horizontal cells. Since our data recorded from rabbit A-type horizontal cells did not show clear inactivation, we only modeled the sustained L-type calcium current. All parameters are estimated from voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
3.2. Description of ionic currents in the horizontal cells
3.5. Delayed rectifying potassium current (IK6)
We modeled these five types of voltage dependent ionic currents identified in rabbit horizontal cells. In the present model we omitted the calcium dependent potassium current (IKc) since the conductance of that current is far smaller than other voltage dependent currents. The membrane potential of the model (Fig. 1) for the rabbit horizontal cells is given by the following ordinary differential equation:
IKv is found in cat and rabbit A- and B-type horizontal cells (Ueda et al., 1992; Lo¨hrke and Hofmann, 1994). IKv slowly activates on depolarization by more than −35 mV, is blocked by tetraethylammonium (TEA), and is inactivated by depolarizing holding potential. Both parameters were estimated from voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
C
dV = − (INa +ICa +IKv +IA +Il). dt
(7)
where Il is leakage current. The description of Il is also added in Appendix A.
3.3. Sodium current (INa) INa is found in cat A-type (Ueda et al., 1992) and rabbit A- and B-type horizontal cells (Lo¨hrke and Hofmann, 1994). The current activates around − 50 mV, and reaches a peak at about − 10 mV, and is decreased by depolarization greater than − 10 mV. The current activates within a few milliseconds, and completely inactivates in less than 10 ms. This fast inward current is blocked by tetrodotoxin (TTX). All parameters of the INa equations are estimated from voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
3.6. Transient outward potassium current (IA) The transient outward potassium current, IA, is found in cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994) horizontal cells. The current activates at depolarization of more than − 40 mV, reaches a peak within a few milliseconds and inactivates in less than a few hundred milliseconds. The time constant of activation and the amplitude of the current increases as
3.4. Calcium current (ICa) The L-type calcium current is found in cat (Ueda et
Fig. 1. Parallel conductance model of rabbit horizontal cells
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Fig. 2. Current responses to the horizontal cell model under the voltage clamp condition. Left panel shows the time courses and right panel shows the I–V characteristics. Vh is − 80 mV and Vc is − 120–60 mV (20 mV steps). The amplitude of the currents was measured at the peak time of a response (Vc =60 mV).
the depolarization increases. This current is inactivated by 4-aminopyridine (4-AP). We estimated the parameters of the current from our voltage clamp recordings (Blanco et al., unpublished data).
estimated from the voltage clamp data, simulations of current clamp data are necessary to confirm the physiological validity of the proposed model. Under current clamp conditions, the membrane potential (V) of the horizontal cell is described as:
3.7. Anomalous rectifying potassium current (IKa) The anomalous rectifying potassium current IKa is found in cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994) horizontal cells. IKa is activated by hyperpolarization below − 80 mV and shows anomalous (inward) rectification near the resting potential. The current is blocked by a solution with Cs or Ba, and is found in both A- and B-type cells. The parameters are estimated from our voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
3.8. Voltage clamp simulations Under voltage clamp conditions, total membrane current (I) in a horizontal cell is described as: I = INa +ICa +IKv + IA +IKa +Il.
(8)
Fig. 2 shows the simulated membrane current response (I) elicited by voltage steps from the holding potential (Vh = − 80 mV) to various command voltages (Vc = − 120– − 60 mV in 20 mV steps). Fig. 3 shows the simulated ionic current responses. The experimental properties of each current (Ueda et al., 1992; Lo¨hrke and Hofmann, 1994) are reproduced well.
3.9. Current clamp simulations Since the parameters of the horizontal cell model are
V=
&
1 {I−(INa + ICa + IKv + IKa + IA + Il)} dt+V0. C (9)
Fig. 4 shows the experimental responses to current injection of rabbit horizontal cells. In control conditions, a current injection of more than 40 pA induces a long lasting depolarization (Fig. 4A and C). Since this depolarization is eliminated by replacing the extracellular calcium with cobalt (Fig. 4B, D and E), the long lasting depolarization results from an influx of Ca2 + . The depolarized membrane potential is returned to the resting potential by a hyperpolarizing current injection (Fig. 4A arrow). A large current pulse accelerates the rising phase of depolarization (Fig. 4C). As shown in Fig. 4F, application of TTX extracellularly slows the rising phase of the depolarizing response. The result suggests that INa contributes to the depolarization. These experimental findings are well reproduced by the proposed model as shown in Fig. 5. Under control conditions (all five types of ionic currents included), the model reproduces the long lasting depolarization caused by the activation of the calcium current (Fig. 5A and C). The membrane potential is returned to the resting potential by a hyperpolarizing current injection of 150 pA (Fig. 5A) and the long lasting depolarization disappears in a model without ICa (Fig. 5B, D and E). The dynamic properties of the
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rising phase are also well simulated by the model, i.e. the rising phase of depolarization becomes faster as the current stimulus is increased (Fig. 5C) and is slower without INa (Fig. 5F). The similarity between measured and modeled voltage responses suggests the model is valid.
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3.10. Analysis of repetiti6e action potential As stated in the introduction, repetitive action potentials have been observed in isolated rabbit A-type horizontal cells (Blanco et al., 1996a). In response to a large amplitude current injection, an action potential or
Fig. 3. Each ionic current response of the horizontal cell model under the voltage clamp condition. Each panel shows the time courses (left) and the I– V characteristics (right) for each ionic current. INa: Vh is − 80 mV and Vc is − 70 – 40 mV (10 mV steps). ICa: Vh is − 80 mV and Vc is − 50– 50 mV (10 mV steps). IA: Vh is − 80 mV and Vc is −60 – 60 mV (20 mV steps). IKv: Vh is − 80 mV and Vc is − 70 – 110 mV (20 mV steps). IKa: Vh is −40 mV and Vc is − 140–20 mV (20 mV steps).
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Fig. 4. Experimental results of current injection. Action potentials recorded from horizontal cells dissociated from the rabbit retina. (A) Voltage responses to 20, 40 and 60 pA current pulses in control solution. For a current pulse over 20 pA, a long lasting depolarization was observed. This depolarization is returned to the resting potential by a short hyperpolarizing inward current (shown by arrow). (B) Voltage responses in 4 mM Co2 + external solution. (C) Voltage records from (A) were superimposed in an expanded time scale. (D) Voltage records from (B) were superimposed in an expanded time scale. (E) Superimposed traces of voltage responses to 40 pA current pulses from (C) and (D). (F) Voltage responses in the presence of 1 mM TTX with the response in control solution.
repetitive action potentials were observed in seven of 17 horizontal cells. In the rest of the cells the long-lasting depolarization, which is similar to the response to current clamp in Fig. 4C, was observed. The underlying mechanism of spike generation was not clarified at that time. Therefore, it is of interest to determine how the horizontal cell produces repetitive spikes. We analyze the mechanism using the model described above. Fig. 6 shows simulated voltage responses to current injection under control conditions. The membrane potential is kept at − 80 mV, and currents of 10, 25, 40 and 70 pA are injected. Fig. 6B shows the ionic current responses during current injection. In control conditions, repetitive action potentials are not observed, and only a long-lasting depolarization similar to Fig. 5A and C is observed. The membrane depolarization in-
duced by current injection activates ICa. and the activated ICa depolarizes the membrane. IKv is activated slower than ICa. When the current injection is terminated, the potential stabilizes at a depolarized level in which ICa is balanced by IKv. This simulated result also corresponds to the long-lasting Co2 + -sensitive action potential observed in the horizontal cells of the cat (Ueda et al., 1992) and goldfish (Tachibana, 1981). Repetitive spikes require not only a calcium current but also ionic currents which hyperpolarize the membrane. In order to determine which current is involved in repetitive action potentials, we analyzed individual ionic currents in the rabbit horizontal cell. The current hyperpolarizing the membrane should flow outward and should activate slower than the calcium current. The delayed rectifying potassium current (IKv) is the most likely candidate since this current is an outward current activated by depolarization. We simulated the response to current injection in models with different values of the maximum conductance of IKv. When the conductance is about eight times larger than the control condition, repetitive action potentials are generated (Fig. 7). Fig. 7A shows the voltage responses to current injection in the model with a large conductance of IKv and Fig. 7B shows the ionic current responses during current injection when IKv conductance is large. When a current pulse of amplitude over 25 pA is applied, an action potential or repetitive action potentials are produced by the model. These responses are highly similar to those measured experimentally (Fig. 2 in Blanco et al. (1996a)). This suggests the following mechanism for generating repetitive spikes. A current injection depolarizes the membrane. The depolarization activates ICa and an increase of ICa causes more depolarization. IKv is activated by the depolarization more slowly than is ICa. This sequence of events results in the membrane potential illustrated in Fig. 5C. In contrast, when IKv has a large conductance, it hyperpolarizes the membrane which results in a transient depolarizing response. After hyperpolarization, since the steady stimulus current induced the membrane depolarization, ICa is reactivated. ICa and IKv repeat this activation and inactivation during current injection, which generates repetitive spikes (ICa and IKv in Fig. 7B). These results show that the delayed rectifying potassium current (IKv) and the calcium current (ICa) are responsible for the repetitive action potentials. The another outward potassium current, IA is known to regulate inter-spike interval time (Hille, 1992). In response to current injection, a model with a reduced IA conductance produces a larger number of spikes than the model with the control value of IA conductance.
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Fig. 8 shows the voltage responses under both small IA and large IKv condition (solid line), only large IKv (broken line), only small IA (chain line) and control condition (dotted line). IA is activated rapidly when the potential reaches the threshold (− 40 mV). The current
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tends to hyperpolarize the membrane and prevents or delays the activation of ICa. These results lead to the conclusion that IA is not a direct candidate for generating the spikes, but IA is a candidate for modulating the spikes.
Fig. 5. Simulated responses of the horizontal model under current clamp conditions. (A) Simulated responses to 20, 40 and 60 pA current pulses in control condition. The current injections of 40 and 60 pA induced a long lasting depolarization, which is returned to the resting potential by a short hyperpolarizing inward current (150 pA). (B) Simulated responses on the model without ICa. (C) Voltage records from (A) were superimposed in an expanded time scale. (D) Voltage records from (B) were superimposed in an expanded time scale. (E) Superimposed traces of voltage responses to 40 pA current pulses from (C) and (D). (F) Simulated responses on the model without INa.
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complicated membrane responses, such as repetitive action potentials. Simulated responses suggest that repetitive spikes are caused by ICa and a large conductance of IKv. There is room for argument on the large conductance IKv. One possibility is a difference of membrane property among the cells. Unfortunately, we do not have enough data to compare quantitatively the conductance of IKv in voltage clamp experiment, but several studies suggest that the conductance of ionic current varies among the
Fig. 6. Current clamp simulations under the control condition (gKv = 4.5 nS). (A) Membrane responses to 10, 25, 40 and 70 pA current pulses. (B) Each ionic current response during current clamp condition.
4. Discussion We constructed a model of rabbit horizontal cells based on ionic current mechanisms. The model is able to reproduce voltage- and current-clamp responses, and repetitive spikes. Using traditional biophysical techniques, it is not technically feasible to analyze how individual ionic currents participate in generating membrane responses. The mathematical model proposed here allows us to analyze the mechanism underlying
Fig. 7. Current clamp simulations under large conductance if IKv (gKv =34 nS). (A) Membrane responses to 10, 25, 40 and 70 pA current pulses. (B) Each ionic current response during current clamp condition.
T. Aoyama et al. / Neuroscience Research 37 (2000) 141–151
Fig. 8. Current clamp simulations under large IKv (gKv = 34 nS) and small IA (gA = 0.5 nS). Dotted line: control condition, broken line: large IKv condition (gKv = 34 ns), solid line: large IKv and small IA condition (gA = 0.5 ns, gKv = 34 ns), chain line: small IA condition (gA =0.5 ns).
cells, for example IKv (0.06 90.04 pA/mm2) (Malchow et al., 1990), IA (2777 91441 pA) (Sullivan and Lasater, 1990) and ICa (0.57 90.23 pA/mm2) (Kaneda and Kaneko, 1991). These observations may suggest that a conductance of ionic current varies among the cells. In addition, IKv conductance was estimated from one clear series of voltage clamp data of a cell, not from average data of cells. In current clamp condition, the horizontal cells which occurred the repetitive spikes are seven of 17 cells and the cells which did not do were also found (Blanco et al., 1996a). This suggests that both cells with comparatively small IKv and cells with comparatively large IKv were recorded. Our model reproduced well similar non repetitive spiking response of a cell without any change of IKv conductance. This suggests that the cell which was used for estimation of IKv conductance had comparatively small IKv conductance, and the parameters were sufficient to reproduce the non-repetitive spiking responses observed in ten of 17 cells. However, to reproduce the repetitive spike response by the model, we had to set IKv conductance to eight times larger than that of the control condition. We also found that the repetitive response can be reproduced with four times larger IKv, half of control IA and half of control ICa. The smaller IA, the larger the number of repetitive responses. The result of simulation predicts that it depends mainly on IKv conductance whether the cells produce repetitive spikes or not. Further experimental study should be carried out to demonstrate these simulated results. In the previous studies (Maricq and Korenbrot, 1988; Blanco et al., 1996b) two types of chloride currents have been found in retinal cells, i.e. GABA-induced
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chloride current through GABAA receptors and calcium-induced chloride current. Blanco et al. (1996b) studied the effects of GABA on rabbit horizontal cells and found GABA-induced chloride current in the horizontal cells. Chloride current which is induced by intracellular calcium ion has never been found in the retinal horizontal cells. Therefore, if we model the chloride current, the current should be introduced as the GABA-induced current. However, GABA was not applied in the experimental condition in which the repetitive spikes were observed, it is reasonable to omit modeling the GABA-induced chloride current. Horizontal cells are depolarized (around − 35 mV) in the dark by the glutamate release from photoreceptors (reviewed in Massey (1990)). Horizontal cells normally respond with graded hyperpolarization to light. Therefore, it is unlikely that the action potential is generated during the light response. In excitable cells it is known that an action potential can be observed at the offset of a hyperpolarizing current step, called anode break (Johnston and Wu, 1995). Since the light stimulus behaves as a hyperpolarizing current to the horizontal cells a similar phenomenon to the anode break may occur at the light offset. In the future, in order to evaluate this possibility, we plan to add a model of the glutamate receptor to the present model and simulate the responses to light.
Acknowledgements Dr Kim Blackwell’s insightful assistance with the preparation of the manuscript is gratefully acknowledged. This work was supported in part by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research on Priority Areas (No. 08650486) and for Scientific Research (C) (No. 10650407), and the Ministry of Education and Culture of Spain (Ref. No. SAF98-098-CO2-01 to PV).
Appendix A. Description of the ionic current model Sodium current (INa)
200·(38− V) 38−V exp −1 25 55+ V bmNa = 2000·exp − 18 amNa =
dmNa = amNa·(1−mNa)− bmNa·mNa dt
ahNa = 1000·exp −
80+ V 8
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bhNa =
800 80− V exp +1 75
bhA =
20 V+40 exp − +1 5
dhNa =ahNa·(1− hNa) − bhNa·hNa dt
dhA = ahA·(1−hA)− bhA·hA dt
INa =gNa·m 3Na·hNa·(V −ENa)
IA = gA·m 3A·hA·(V −EK)
Calcium current (ICa)
240·(68−V) amCa = 68−V exp −1 21 800 bmCa = 55+V exp +1 55 dmCa = amCa · (1− mCa) −bmCa · mCa dt ECa =12.9·log
[Ca]o [Ca]d
Delayed rectifying potassium current (IKv)
0.40·(65−V) 65− V −1 exp 50 45−V bmKv =4.8·exp 85
dmKv = amKv·(1−mKv) − bmKv·mKv dt
1500 V+ 92 +1 exp 7 80 bhKv = +0.02 V+100 exp +1 15 dhKv =ahKv·(1− hKv) − bhKv·hKv dt ahKv =
mKa =
1 V+ 60 1+exp 12
IKa = gKa·m 5Ka·(V −EK) Leakage current (Il) Il = gl·(V− El) Parameter of the ionic current model in control condition
ICa =gCa·m 4Ca·(V − ECa)
amKv =
Anomalous rectifying potassium current (IKa)
IKv =gKv·m 4Kv·hKv·(V −EK)
gA gKa gKv gCa gNa gl EK ENa El [Ca]o [Ca]d C V0 mA0 hA0 mNa0 hNa0 mKv0 hKv0 mCa0
maximum conductance of IA (15.0 ns) maximum conductance of IKa (4.5 ns) maximum conductance of IKv (4.5 ns) maximum conductance of ICa (9.0 ns) maximum conductance of INa (2.4 ns) maximum conductance of Il (0.5 ns) reversal potential of potassium current (−80 mV) reversal potential of sodium current (55 mV) reversal potential of leak current (−80 mV) extracellular Ca2+ concentration (2.0 mM) domain Ca2+ concentration (30 mM) membrane capacitance (0.106 nF) initial membrane potential (−80 mV) initial value Of mA (0.030) initial value of hA (0.998) initial value Of mNa (0.026) initial value of hNa (0.922) initial value Of mKv (0.139) initial value of hKv (0.932) initial value of mCa (0.059)
Transient outward potassium current (IA)
2400 V− 50 1+ exp − 28 −V bmA =80·exp 36 amA =
dmA =amA·(1− mA) −bmA·mA dt
ahA = 1·exp −
V 60
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Lo¨hrke, S., Hofmann, H.D., 1994. Voltage-gated currents of rabbit aand b-type horizontal cells in retinal monolayer cultures. Vis. Neurosci. 11, 369 – 378. Malchow, R.P., Qian, H., Ripps, H., Dowling, J.E., 1990. Structual and functional properties of two types of horizontal cell in the skate retina. J. Gen. Physiol. 95, 177 – 198. Maricq, A.V., Korenbrot, J.I., 1988. Calcium-dependent chloride currents generate action potentials in solitaly cone photoreceptors. Neuron 1, 503 – 515. Massey, S.C., 1990. Cell types using glutamate as a neurotransmitter in the vertebrate retina. In: Osborne, N., Chader, G. (Eds.), Progress in Retinal Research, vol. 9. Pergamon Press, Oxford, pp. 399 – 425. Ralston, A., Rabinowits, P., 1978. A First Course in Numerical Analysis, 2nd edn. McGraw-Hill, NewYork, pp. 164 – 246. Shingai, R., Christensen, B.N., 1983. Sodium and calcium current measured in isolated catfish horizontal cell under voltage clamp. Neuroscience 10, 893 – 897. Sullivan, J.M., Lasater, E.M., 1990. Sustained and transient potassium currents of cultured horizontal cells isolated from white bass retina. J. Neurophysiol. 64, 1758 – 1766. Tachibana, M., 1981. Membrane properties of solitary horizontal cells isolated from goldfish retina. J. Physiol. 321, 141 – 161. Tachibana, M., 1983. Ionic currents of solitary horizontal cells isolated from goldfish retina. J. Physiol. 345, 329 – 351. Ueda, Y., Kaneko, A., Kaneda, M., 1992. Voltage-dependent ionic currents in solitary horizontal cells isolated from cat retina. J. Neurophysiol. 68, 1143 – 1150. Usui, S., Kamiyama, Y., Ishii, H., Ikeno, H., 1996a. Reconstruction of retinal horizontal cell responses by the ionic current model. Vision Res. 36, 1711 – 1719. Usui, S., Ishihara, A., Kamiyama, Y., Ishii, H., 1996b. Ionic current model of bipolar cells in the lower vertebrate retina. Vision Res. 36, 4069 – 4076.
Ionic current model of rabbit retinal horizontal cell Toshihiro Aoyama a, Yoshimi Kamiyama b, Shiro Usui a,*, Roman Blanco c, Cecilia F. Vaquero c, Pedro de la Villa c a
Department of Information and Computer Sciences, Toyohashi Uni6ersity of Technology, Toyohashi, Aichi, 441 -8580, Japan b Faculty of Information Science and Technology, Aichi Prefectural Uni6ersity, Nagakute, Aichi, 480 -1198, Japan c Department of Physiology and Pharmacology, School of Medicine, Uni6ersity of Alcala de Henares, Madrid, 28871, Spain Received 30 August 1999; accepted 28 February 2000
Abstract We propose a mathematical model of rabbit retinal horizontal cell based on the ionic current mechanisms. Five types of ionic currents in rabbit retinal horizontal cell, INa, ICa, IKv, IA and IKa, are described by Hodgkin – Huxley type equations based on voltage clamp measurements. In simulation the model reproduced similar responses to voltage and current clamp experiments. Under the current clamp experiment a repetitive action potential was found on A-type rabbit horizontal cells. Our result suggests that the repetitive action potential is generated by an interaction Of ICa and IKv. © 2000 Elsevier Science Ireland Ltd and the Japan Neuroscience Society. All rights reserved. Keywords: Rabbit; Retinal horizontal cell; Ionic current model; Simulation; Reptitive; Action potential
1. Introduction Retinal horizontal cells have been studied in a variety of animals for the last 40 years. In lower vertebrate retinas, horizontal cells play a fundamental role in forming center-surround receptive fields and in generating color opponent signals (reviewed in Kaneko (1987)). On the other hand, anatomical and physiological studies suggest that there are structural and functional differences between lower vertebrate and mammalian horizontal cells. Mammalian horizontal cells are morphologically classified into A- and B-type cells. The A-type cell has large, thin dendrites which exclusively connect with cones, and the B-type cell has dendrites which connect with cones and an axon and an axon terminal which contacts only with rods (Kolb, 1974; Boycott et al., 1978; Bloomfield and Miller, 1982). In the primate retina all the horizontal cells are functionally classified as luminosity-type, these cells do not perform a color opponent transformation (Dacey et al., 1996). Lankheet et al. (1990) have performed intracellular recordings from cat horizontal cells and found * Corresponding author. Tel.: +81-532-446764; fax: + 81-532467806.
that the spatial properties of response have non-passive electrical spread and nonlinear summation of potential. A nonlinear relationship between voltage response and light intensity has been found (Lankheet et al., 1991a,b). Some of these nonlinear properties are thought to be caused by the membrane properties of the horizontal cells. Membrane currents which characterize the electrical properties of horizontal cells have been analyzed using isolated cell preparations. Blanco et al. (1996a) recently found that the rabbit A-type horizontal cells exhibit repetitive action potentials which have not been observed in distal neurons of the vertebrate retina previously. Although the horizontal cells normally respond to light with a graded potential, it is of interest to elucidate how the cells generate the spike trains and to understand their functional role in vision. However, it is difficult to analyze the mechanism underlying repetitive action potentials solely using traditional electrophysiology. We have developed a quantitative model of horizontal cells to investigate the mechanism of the action potential. Recently, mathematical models based on ionic currents have been developed for lower vertebrate photoreceptors (Kamiyama et al., 1996), horizontal cells (Usui
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et al., 1996a) and bipolar cells (Usui et al., 1996b). These models have been used to elucidate complicated electrical behavior in these retinal cells. In this paper we developed a mathematical model of the rabbit retinal horizontal cell based on membrane ionic current mechanisms. We confirmed the validity of the model through current clamp simulation, and analyzed the mechanism of repetitive spike generation found in rabbit horizontal cells.
(Kamiyama et al., 1996), horizontal cells (Usui et al., 1996a) and bipolar cells (Usui et al., 1996b). In the present study, the system of differential equations is numerically solved by using a Runge–Kutta– Fehlberg method (Ralston and Rabinowits, 1978) with an adaptive step size. All simulations were performed on a UNIX work station (DEC Personal Work Station 500au).
2.2. Electrophysiology
where m and h are the open and close gating variables, M and H are constants which determine the rising and falling phases of the conductance and gi is the maximum conductance. These gating variables (x = m, h) depend on the membrane voltage and time, and are described as:
The detail of the experimental method has been reported previously (Blanco et al., 1996a) and will be summarized briefly here. Horizontal cells were dissociated enzymatically from the rabbit retina. For enzymatic dissociation, the retina was incubated in oxygenated standard solution containing 75 mg/ml papain (Carl Roth, Kahlsruhe, Germany) and 0.1 mg/ml DL-eysteine (Sigma, St Louis, MO) for 20–30 min at 30°C and rinsed with a standard solution containing 0.01% BSA. Standard solution contained 135 mM NaCl, 5 mM KCl, 1 mM MgCl2, 2 mM CaCl2, 10 mM glucose and 10 mM Hepes (pH adjusted to 7.4 with NaOH). The cells were maintained at 5°C for 1–5 h before use in a plastic culture dish containing 2 ml standard solution with BSA. Membrane voltage and current were recorded by patch pipette in the whole-cell configuration. Pipette solutions contained 120 mM KCl, 1 mM MgCl2, 5 mM EGTA, 0.5 mM CaCl2 and 10 mM Hepes (pH adjusted to 7.2 with KOH). Patch pipettes (Clark Electromedical: 1.0 mm o.d.) were pulled by the Sutter pipette puller (model P-90). The resistance of the pipette as measured in the bath was usually about 3–8 MV. Electrode series resistance ranged from 10–20 MV. In all experiments the drugs were dissolved in the standard solution and applied to the cell by a Y-tube system.
dx = ax (1− x)−bxx, dt
3. Results
2. Methods
2.1. Modeling and simulation The electrical property of a cell membrane is represented by a parallel conductance model (Johnston and Wu, 1995). The membrane current I is described as: I=C
dV dV +%gi(V − Ei) +%Ii = C dt dt
(1)
where C is the membrane capacitance (nF), V is the membrane potential (mV), Ii is the ionic current (pA), gi is the conductance of the ionic current (nS) and Ei is the reversal potential (mV) of the current. In the Hodgkin and Huxley formulation the conductance of an ionic current is written as the product of gating variables and the maximum conductance (Hodgkin and Huxley, 1952): gi = m Mh Hgi,
(2)
(3)
where ax, bx are expressed as: C1 exp(C2(V +C3) + C4(V + C5)) . ax, bx = exp(C6(V + C3) +C7)
(4)
The parameters Cl – C7 are calculated from following equations: ax =
x tx
(5)
bx =
1−x tx
(6)
x and tx are stable gating variables (x) and time constant of gating variable (x) calculated from experimental voltage clamp data. The method has been applied to developing models of vertebrate photoreceptors
3.1. Summary of ionic current of mammalian horizontal cells In mammalian horizontal cells ionic currents have been measured in the cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994). These currents are summarized in Table 1. Five types of voltage dependent ionic currents have been found in the cell body: a sodium current (INa), a calcium current (ICa), a delayed rectifying potassium current (IKv,), a transient outward potassium current (IA) and an anomalous rectifying potassium current (IKa). These voltage dependent currents also have been identified in lower vertebrate horizontal cells, goldfish (Tachibana, 1983), catfish (Shingai and Christensen, 1983) and white bass (Sullivan and
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Table 1 Summary of the ionic current of rabbit horizontal cell Current
Reversal potential (mV)
Activate potential (mV)
I–V property
ICa INa IKv IA IKa
54 55 −80 −80 −80
Over −30 Over −35 Over −50 Over −40 Below −80
Bell Bell Outward rectifying Outward rectifying Inward rectifying
Lasater, 1990). One of the differences in the ionic currents between lower vertebrate and mammalian horizontal cells is the time course of the calcium current. ICa is inactivated by the accumulation of intracellular calcium ions in goldfish horizontal cells (Tachibana, 1983); however, this inactivation has never been observed in the cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994) horizontal cells. The calcium dependent potassium current (IKc) is found in rabbit B-type horizontal cells (Lo¨hrke and Hofmann, 1994), but has never been observed in lower vertebrates and cat.
al., 1992) and rabbit horizontal cells (Lo¨hrke and Hofmann, 1994). These results show that the L-type current is not suppressed by an accumulation of intracellular calcium as in the goldfish (Tachibana, 1983). Lo¨hrke and Hofmann (1994) reported that there is a transitory T-type calcium current, which activates at more than − 50 mV, in rabbit horizontal cells. Since our data recorded from rabbit A-type horizontal cells did not show clear inactivation, we only modeled the sustained L-type calcium current. All parameters are estimated from voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
3.2. Description of ionic currents in the horizontal cells
3.5. Delayed rectifying potassium current (IK6)
We modeled these five types of voltage dependent ionic currents identified in rabbit horizontal cells. In the present model we omitted the calcium dependent potassium current (IKc) since the conductance of that current is far smaller than other voltage dependent currents. The membrane potential of the model (Fig. 1) for the rabbit horizontal cells is given by the following ordinary differential equation:
IKv is found in cat and rabbit A- and B-type horizontal cells (Ueda et al., 1992; Lo¨hrke and Hofmann, 1994). IKv slowly activates on depolarization by more than −35 mV, is blocked by tetraethylammonium (TEA), and is inactivated by depolarizing holding potential. Both parameters were estimated from voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
C
dV = − (INa +ICa +IKv +IA +Il). dt
(7)
where Il is leakage current. The description of Il is also added in Appendix A.
3.3. Sodium current (INa) INa is found in cat A-type (Ueda et al., 1992) and rabbit A- and B-type horizontal cells (Lo¨hrke and Hofmann, 1994). The current activates around − 50 mV, and reaches a peak at about − 10 mV, and is decreased by depolarization greater than − 10 mV. The current activates within a few milliseconds, and completely inactivates in less than 10 ms. This fast inward current is blocked by tetrodotoxin (TTX). All parameters of the INa equations are estimated from voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
3.6. Transient outward potassium current (IA) The transient outward potassium current, IA, is found in cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994) horizontal cells. The current activates at depolarization of more than − 40 mV, reaches a peak within a few milliseconds and inactivates in less than a few hundred milliseconds. The time constant of activation and the amplitude of the current increases as
3.4. Calcium current (ICa) The L-type calcium current is found in cat (Ueda et
Fig. 1. Parallel conductance model of rabbit horizontal cells
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Fig. 2. Current responses to the horizontal cell model under the voltage clamp condition. Left panel shows the time courses and right panel shows the I–V characteristics. Vh is − 80 mV and Vc is − 120–60 mV (20 mV steps). The amplitude of the currents was measured at the peak time of a response (Vc =60 mV).
the depolarization increases. This current is inactivated by 4-aminopyridine (4-AP). We estimated the parameters of the current from our voltage clamp recordings (Blanco et al., unpublished data).
estimated from the voltage clamp data, simulations of current clamp data are necessary to confirm the physiological validity of the proposed model. Under current clamp conditions, the membrane potential (V) of the horizontal cell is described as:
3.7. Anomalous rectifying potassium current (IKa) The anomalous rectifying potassium current IKa is found in cat (Ueda et al., 1992) and rabbit (Lo¨hrke and Hofmann, 1994) horizontal cells. IKa is activated by hyperpolarization below − 80 mV and shows anomalous (inward) rectification near the resting potential. The current is blocked by a solution with Cs or Ba, and is found in both A- and B-type cells. The parameters are estimated from our voltage clamp recordings of rabbit A-type horizontal cells (Blanco et al., unpublished data).
3.8. Voltage clamp simulations Under voltage clamp conditions, total membrane current (I) in a horizontal cell is described as: I = INa +ICa +IKv + IA +IKa +Il.
(8)
Fig. 2 shows the simulated membrane current response (I) elicited by voltage steps from the holding potential (Vh = − 80 mV) to various command voltages (Vc = − 120– − 60 mV in 20 mV steps). Fig. 3 shows the simulated ionic current responses. The experimental properties of each current (Ueda et al., 1992; Lo¨hrke and Hofmann, 1994) are reproduced well.
3.9. Current clamp simulations Since the parameters of the horizontal cell model are
V=
&
1 {I−(INa + ICa + IKv + IKa + IA + Il)} dt+V0. C (9)
Fig. 4 shows the experimental responses to current injection of rabbit horizontal cells. In control conditions, a current injection of more than 40 pA induces a long lasting depolarization (Fig. 4A and C). Since this depolarization is eliminated by replacing the extracellular calcium with cobalt (Fig. 4B, D and E), the long lasting depolarization results from an influx of Ca2 + . The depolarized membrane potential is returned to the resting potential by a hyperpolarizing current injection (Fig. 4A arrow). A large current pulse accelerates the rising phase of depolarization (Fig. 4C). As shown in Fig. 4F, application of TTX extracellularly slows the rising phase of the depolarizing response. The result suggests that INa contributes to the depolarization. These experimental findings are well reproduced by the proposed model as shown in Fig. 5. Under control conditions (all five types of ionic currents included), the model reproduces the long lasting depolarization caused by the activation of the calcium current (Fig. 5A and C). The membrane potential is returned to the resting potential by a hyperpolarizing current injection of 150 pA (Fig. 5A) and the long lasting depolarization disappears in a model without ICa (Fig. 5B, D and E). The dynamic properties of the
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rising phase are also well simulated by the model, i.e. the rising phase of depolarization becomes faster as the current stimulus is increased (Fig. 5C) and is slower without INa (Fig. 5F). The similarity between measured and modeled voltage responses suggests the model is valid.
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3.10. Analysis of repetiti6e action potential As stated in the introduction, repetitive action potentials have been observed in isolated rabbit A-type horizontal cells (Blanco et al., 1996a). In response to a large amplitude current injection, an action potential or
Fig. 3. Each ionic current response of the horizontal cell model under the voltage clamp condition. Each panel shows the time courses (left) and the I– V characteristics (right) for each ionic current. INa: Vh is − 80 mV and Vc is − 70 – 40 mV (10 mV steps). ICa: Vh is − 80 mV and Vc is − 50– 50 mV (10 mV steps). IA: Vh is − 80 mV and Vc is −60 – 60 mV (20 mV steps). IKv: Vh is − 80 mV and Vc is − 70 – 110 mV (20 mV steps). IKa: Vh is −40 mV and Vc is − 140–20 mV (20 mV steps).
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Fig. 4. Experimental results of current injection. Action potentials recorded from horizontal cells dissociated from the rabbit retina. (A) Voltage responses to 20, 40 and 60 pA current pulses in control solution. For a current pulse over 20 pA, a long lasting depolarization was observed. This depolarization is returned to the resting potential by a short hyperpolarizing inward current (shown by arrow). (B) Voltage responses in 4 mM Co2 + external solution. (C) Voltage records from (A) were superimposed in an expanded time scale. (D) Voltage records from (B) were superimposed in an expanded time scale. (E) Superimposed traces of voltage responses to 40 pA current pulses from (C) and (D). (F) Voltage responses in the presence of 1 mM TTX with the response in control solution.
repetitive action potentials were observed in seven of 17 horizontal cells. In the rest of the cells the long-lasting depolarization, which is similar to the response to current clamp in Fig. 4C, was observed. The underlying mechanism of spike generation was not clarified at that time. Therefore, it is of interest to determine how the horizontal cell produces repetitive spikes. We analyze the mechanism using the model described above. Fig. 6 shows simulated voltage responses to current injection under control conditions. The membrane potential is kept at − 80 mV, and currents of 10, 25, 40 and 70 pA are injected. Fig. 6B shows the ionic current responses during current injection. In control conditions, repetitive action potentials are not observed, and only a long-lasting depolarization similar to Fig. 5A and C is observed. The membrane depolarization in-
duced by current injection activates ICa. and the activated ICa depolarizes the membrane. IKv is activated slower than ICa. When the current injection is terminated, the potential stabilizes at a depolarized level in which ICa is balanced by IKv. This simulated result also corresponds to the long-lasting Co2 + -sensitive action potential observed in the horizontal cells of the cat (Ueda et al., 1992) and goldfish (Tachibana, 1981). Repetitive spikes require not only a calcium current but also ionic currents which hyperpolarize the membrane. In order to determine which current is involved in repetitive action potentials, we analyzed individual ionic currents in the rabbit horizontal cell. The current hyperpolarizing the membrane should flow outward and should activate slower than the calcium current. The delayed rectifying potassium current (IKv) is the most likely candidate since this current is an outward current activated by depolarization. We simulated the response to current injection in models with different values of the maximum conductance of IKv. When the conductance is about eight times larger than the control condition, repetitive action potentials are generated (Fig. 7). Fig. 7A shows the voltage responses to current injection in the model with a large conductance of IKv and Fig. 7B shows the ionic current responses during current injection when IKv conductance is large. When a current pulse of amplitude over 25 pA is applied, an action potential or repetitive action potentials are produced by the model. These responses are highly similar to those measured experimentally (Fig. 2 in Blanco et al. (1996a)). This suggests the following mechanism for generating repetitive spikes. A current injection depolarizes the membrane. The depolarization activates ICa and an increase of ICa causes more depolarization. IKv is activated by the depolarization more slowly than is ICa. This sequence of events results in the membrane potential illustrated in Fig. 5C. In contrast, when IKv has a large conductance, it hyperpolarizes the membrane which results in a transient depolarizing response. After hyperpolarization, since the steady stimulus current induced the membrane depolarization, ICa is reactivated. ICa and IKv repeat this activation and inactivation during current injection, which generates repetitive spikes (ICa and IKv in Fig. 7B). These results show that the delayed rectifying potassium current (IKv) and the calcium current (ICa) are responsible for the repetitive action potentials. The another outward potassium current, IA is known to regulate inter-spike interval time (Hille, 1992). In response to current injection, a model with a reduced IA conductance produces a larger number of spikes than the model with the control value of IA conductance.
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Fig. 8 shows the voltage responses under both small IA and large IKv condition (solid line), only large IKv (broken line), only small IA (chain line) and control condition (dotted line). IA is activated rapidly when the potential reaches the threshold (− 40 mV). The current
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tends to hyperpolarize the membrane and prevents or delays the activation of ICa. These results lead to the conclusion that IA is not a direct candidate for generating the spikes, but IA is a candidate for modulating the spikes.
Fig. 5. Simulated responses of the horizontal model under current clamp conditions. (A) Simulated responses to 20, 40 and 60 pA current pulses in control condition. The current injections of 40 and 60 pA induced a long lasting depolarization, which is returned to the resting potential by a short hyperpolarizing inward current (150 pA). (B) Simulated responses on the model without ICa. (C) Voltage records from (A) were superimposed in an expanded time scale. (D) Voltage records from (B) were superimposed in an expanded time scale. (E) Superimposed traces of voltage responses to 40 pA current pulses from (C) and (D). (F) Simulated responses on the model without INa.
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complicated membrane responses, such as repetitive action potentials. Simulated responses suggest that repetitive spikes are caused by ICa and a large conductance of IKv. There is room for argument on the large conductance IKv. One possibility is a difference of membrane property among the cells. Unfortunately, we do not have enough data to compare quantitatively the conductance of IKv in voltage clamp experiment, but several studies suggest that the conductance of ionic current varies among the
Fig. 6. Current clamp simulations under the control condition (gKv = 4.5 nS). (A) Membrane responses to 10, 25, 40 and 70 pA current pulses. (B) Each ionic current response during current clamp condition.
4. Discussion We constructed a model of rabbit horizontal cells based on ionic current mechanisms. The model is able to reproduce voltage- and current-clamp responses, and repetitive spikes. Using traditional biophysical techniques, it is not technically feasible to analyze how individual ionic currents participate in generating membrane responses. The mathematical model proposed here allows us to analyze the mechanism underlying
Fig. 7. Current clamp simulations under large conductance if IKv (gKv =34 nS). (A) Membrane responses to 10, 25, 40 and 70 pA current pulses. (B) Each ionic current response during current clamp condition.
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Fig. 8. Current clamp simulations under large IKv (gKv = 34 nS) and small IA (gA = 0.5 nS). Dotted line: control condition, broken line: large IKv condition (gKv = 34 ns), solid line: large IKv and small IA condition (gA = 0.5 ns, gKv = 34 ns), chain line: small IA condition (gA =0.5 ns).
cells, for example IKv (0.06 90.04 pA/mm2) (Malchow et al., 1990), IA (2777 91441 pA) (Sullivan and Lasater, 1990) and ICa (0.57 90.23 pA/mm2) (Kaneda and Kaneko, 1991). These observations may suggest that a conductance of ionic current varies among the cells. In addition, IKv conductance was estimated from one clear series of voltage clamp data of a cell, not from average data of cells. In current clamp condition, the horizontal cells which occurred the repetitive spikes are seven of 17 cells and the cells which did not do were also found (Blanco et al., 1996a). This suggests that both cells with comparatively small IKv and cells with comparatively large IKv were recorded. Our model reproduced well similar non repetitive spiking response of a cell without any change of IKv conductance. This suggests that the cell which was used for estimation of IKv conductance had comparatively small IKv conductance, and the parameters were sufficient to reproduce the non-repetitive spiking responses observed in ten of 17 cells. However, to reproduce the repetitive spike response by the model, we had to set IKv conductance to eight times larger than that of the control condition. We also found that the repetitive response can be reproduced with four times larger IKv, half of control IA and half of control ICa. The smaller IA, the larger the number of repetitive responses. The result of simulation predicts that it depends mainly on IKv conductance whether the cells produce repetitive spikes or not. Further experimental study should be carried out to demonstrate these simulated results. In the previous studies (Maricq and Korenbrot, 1988; Blanco et al., 1996b) two types of chloride currents have been found in retinal cells, i.e. GABA-induced
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chloride current through GABAA receptors and calcium-induced chloride current. Blanco et al. (1996b) studied the effects of GABA on rabbit horizontal cells and found GABA-induced chloride current in the horizontal cells. Chloride current which is induced by intracellular calcium ion has never been found in the retinal horizontal cells. Therefore, if we model the chloride current, the current should be introduced as the GABA-induced current. However, GABA was not applied in the experimental condition in which the repetitive spikes were observed, it is reasonable to omit modeling the GABA-induced chloride current. Horizontal cells are depolarized (around − 35 mV) in the dark by the glutamate release from photoreceptors (reviewed in Massey (1990)). Horizontal cells normally respond with graded hyperpolarization to light. Therefore, it is unlikely that the action potential is generated during the light response. In excitable cells it is known that an action potential can be observed at the offset of a hyperpolarizing current step, called anode break (Johnston and Wu, 1995). Since the light stimulus behaves as a hyperpolarizing current to the horizontal cells a similar phenomenon to the anode break may occur at the light offset. In the future, in order to evaluate this possibility, we plan to add a model of the glutamate receptor to the present model and simulate the responses to light.
Acknowledgements Dr Kim Blackwell’s insightful assistance with the preparation of the manuscript is gratefully acknowledged. This work was supported in part by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research on Priority Areas (No. 08650486) and for Scientific Research (C) (No. 10650407), and the Ministry of Education and Culture of Spain (Ref. No. SAF98-098-CO2-01 to PV).
Appendix A. Description of the ionic current model Sodium current (INa)
200·(38− V) 38−V exp −1 25 55+ V bmNa = 2000·exp − 18 amNa =
dmNa = amNa·(1−mNa)− bmNa·mNa dt
ahNa = 1000·exp −
80+ V 8
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bhNa =
800 80− V exp +1 75
bhA =
20 V+40 exp − +1 5
dhNa =ahNa·(1− hNa) − bhNa·hNa dt
dhA = ahA·(1−hA)− bhA·hA dt
INa =gNa·m 3Na·hNa·(V −ENa)
IA = gA·m 3A·hA·(V −EK)
Calcium current (ICa)
240·(68−V) amCa = 68−V exp −1 21 800 bmCa = 55+V exp +1 55 dmCa = amCa · (1− mCa) −bmCa · mCa dt ECa =12.9·log
[Ca]o [Ca]d
Delayed rectifying potassium current (IKv)
0.40·(65−V) 65− V −1 exp 50 45−V bmKv =4.8·exp 85
dmKv = amKv·(1−mKv) − bmKv·mKv dt
1500 V+ 92 +1 exp 7 80 bhKv = +0.02 V+100 exp +1 15 dhKv =ahKv·(1− hKv) − bhKv·hKv dt ahKv =
mKa =
1 V+ 60 1+exp 12
IKa = gKa·m 5Ka·(V −EK) Leakage current (Il) Il = gl·(V− El) Parameter of the ionic current model in control condition
ICa =gCa·m 4Ca·(V − ECa)
amKv =
Anomalous rectifying potassium current (IKa)
IKv =gKv·m 4Kv·hKv·(V −EK)
gA gKa gKv gCa gNa gl EK ENa El [Ca]o [Ca]d C V0 mA0 hA0 mNa0 hNa0 mKv0 hKv0 mCa0
maximum conductance of IA (15.0 ns) maximum conductance of IKa (4.5 ns) maximum conductance of IKv (4.5 ns) maximum conductance of ICa (9.0 ns) maximum conductance of INa (2.4 ns) maximum conductance of Il (0.5 ns) reversal potential of potassium current (−80 mV) reversal potential of sodium current (55 mV) reversal potential of leak current (−80 mV) extracellular Ca2+ concentration (2.0 mM) domain Ca2+ concentration (30 mM) membrane capacitance (0.106 nF) initial membrane potential (−80 mV) initial value Of mA (0.030) initial value of hA (0.998) initial value Of mNa (0.026) initial value of hNa (0.922) initial value Of mKv (0.139) initial value of hKv (0.932) initial value of mCa (0.059)
Transient outward potassium current (IA)
2400 V− 50 1+ exp − 28 −V bmA =80·exp 36 amA =
dmA =amA·(1− mA) −bmA·mA dt
ahA = 1·exp −
V 60
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