Abstract
Roska, Botond, Lubor Gaal, and Frank S. Werblin. Voltagedependent uptake is a major determinant of glutamate concentration at the cone synapse: an analytical study. J. Neurophysiol. 80: 1951–1960, 1998. It was suggested that glutamate concentration at the synaptic terminal of the cones was controlled primarily by a voltagedependent glutamate transporter and that diffusion played a less important role. The conclusion was based on the observation that the rate of glutamate concentration during the hyperpolarizing light response was dramatically slowed when the transporter was blocked with dihydrokainate although diffusion remained intact. To test the validity of this notion we constructed a model in which the balance among uptake, diffusion, and release determined the flow of glutamate into and out of the synaptic cleft. The control of glutamate concentration was assumed here to be determined by two relationships; 1) glutamate concentration is the integral over the synaptic volume of the rates of release, uptake, and diffusion, and 2) membrane potential is the integral over the membrane capacitance of the dark, leak, and transportergated chloride current. These relationships are interdependent because glutamate uptake via the transporter is voltage dependent and because the transportergated current is concentration dependent. The voltage and concentration dependence of release and uptake, as well as the lightelicited, transportergated, and leak currents were measured in other studies. All of these measurements were incorporated into our predictive model of glutamate uptake. Our results show a good quantitative fit between the predicted and the measured magnitudes and rates of change of glutamate concentration, derived from the two interdependent relationships. This close fit supports the validity of these two relationships as descriptors of the mechanisms underlying the control of glutamate concentration, it verifies the accuracy of the experimental data from which the functions used in these relationships were derived, and it lends further support to the notion that glutamate concentration is controlled primarily by uptake at the transporter.
INTRODUCTION
An earlier study (Gaal et al. 1998) suggested that voltage and concentrationdependent uptake by the glutamate transporter at the cone synaptic terminal provided the essential link between cone membrane potential and glutamate concentration. The main evidence for this was the observation that the rate of glutamate removal during a light flash, measured by the rate of horizontal cell hyperpolarization, was dramatically slowed by dihydrokainate (DHK), a glutamate transporter blocker that acted specifically at the cones but not the Mueller cells. Glutamate concentration is thought to be controlled by the integral of the rates of uptake release and diffusion. This study defines the relative rates of each of these quantities and the changes that take place during the light response.
The overall system relating glutamate concentration to light intensity can be described by two relationships. One states that the rate of change of membrane potential is determined by the integral of the lightdependent current, the leak current, and the glutamateelicited, transportergated chloride current over the membrane capacitance. The other states that the rate of change of glutamate concentration is determined by the integral of uptake, diffusion, and vesicular release over the volume of the space at the synapse.
The underlying functions relating currents and glutamate uptake to voltage, concentration, and light intensity were measured in earlier studies. Transport rate and the associated chloride current are known as a function of glutamate concentration and membrane voltage (Eliasof and Werblin 1993; Gaal et al. 1998; Picaud et al. 1995; Wadiche et al. 1995). A part of vesicular release depends on membrane potential (Copenhagen and Jahr 1989), but part appears to be potential independent (Rieke and Schwartz 1994). The lightelicited current was measured as a function of membrane voltage and light intensity (Attwell et al. 1982; Haynes and Yau 1985). The relationship between glutamate concentration and horizontal cell potential was also measured (Gaal et al. 1998).
The two relationships above are sufficiently complete to allow us to predict the time course of the light response in horizontal cells under normal and transporterblocked conditions. We can also predict quantities that are unmeasurable such as the time course of glutamate concentration change and rates of diffusion, release, and transport after a light flash. We have no independent measure of the volume of the synaptic space, so we have left all results scalable to this value.
We found that the previous measurements of voltage and glutamatedependent uptake, voltagedependent release, and glutamategated chloride current, when incorporated into the two relationships, predict time courses of glutamate removal that are very close to those actually measured. This good fit suggests that the two interdependent relationships outlined previously provide a reasonable approximation to the underlying mechanism that controls glutamate concentration as a function of cone membrane potential. The fit also supports the notion that the transporter is a significant mechanism along with vesicular release that links glutamate concentration to cone membrane potential.
METHODS
Electrical recording, solutions, and drugs
Briefly, horizontal cells and cones were patch recorded in tiger salamander retinal slices, and solutions and drugs were applied as described by Gaal et al. (1998).
Equivalent circuit of the cone output synapse: the resistive two port model
The proposed equivalent circuit of the cone output synapse is shown in Fig. 1. The two differential equations, describing the charging of the cone membrane capacitance and the filling of the synaptic cleft with neurotransmitter, are represented by two firstorder circuits. The two circuits are joined together with a resistive twoport, which represents voltage and glutamate concentrationdependent uptake. The voltage of the first circuit (V _{m}) represents cone voltage across the membrane capacitance (C _{m}), which is modulated by a lightcontrolled current (i _{dark}[I, V _{m}]) as well as leak current (i _{leak}[V _{m}]) and chloride current (i _{chloride}[V _{m}, V _{g}]). The voltage of the second circuit (V _{g}) represents glutamate concentration (G) in the volume of the synaptic cleft (C _{g}). Release, which increases glutamate concentration in the cleft, is modeled with a current source (i _{release}[V _{m}]) controlled by the state of the first circuit (V _{m}). Diffusion (i _{diffusion}[V _{g}]) is modeled by a simple resistor in series with a voltage source that represents glutamate concentration outside the synapse. Finally uptake (i _{uptake}[V _{m}, V _{g}]) is represented by a current flowing through a resistive twoport and controlled by both V _{m} and V _{g}.
Mathematical model
The formal relationship between glutamate concentration and cone membrane potential can be described by two differential equations. The first states that the chloride, dark, and leak currents determine the electrical charging of the membrane capacitance
The second relationship states that the flows of glutamate into and out of the diffusionlimited extracellular synaptic space, caused by release, uptake, and diffusion, determine the rate of “charging” of the glutamate concentration in the synaptic region
For Eq. 1 we make the assumption that the cone is isopotential. This is reasonable because the electrotonic distances are small. For Eq. 2 we assume that the diffusionlimited compartment containing the glutamate concentration is compact; in other words, glutamate concentration throughout the diffusionlimited volume is uniform.
Dark current
The dark current (Fig. 2
A) depends on I and V
_{m} independently according to the following
The IV
_{m} curve of the dark current was fitted by Haynes and Yau (1985) by the sum of two exponentials
Chloride current
The chloride current (Fig. 2
B) is gated by glutamate (Eliasof and Werblin 1993; Picaud et al. 1995) and therefore depends on both V
_{m} and G. The dependence on glutamate concentration can be approximated by a scaled hyperbolic function with onehalf saturating concentration K
_{Clm} = 12 μM (Eliasof and Werblin 1993)
We fitted by the least square fit method the experimental IV
_{m} relationship at saturating glutamate concentrations measured by Picaud et al. (1995) with a sum of two exponentials that can be interpreted as a single energy barrier placed at γ = 0.91 fractional distance from the intracellular boundary of the membrane (Hille 1992)
The independence of V
_{m} and G (Barbour et al. 1991) again allows us to take the product of Eqs. 6
and 7 for the overall expression
Leak current
Leak current (Fig. 2
C) refers to all other currents and is assumed to be ohmic, although it is known that a potassium current at the inner segment is outward rectifying and time dependent; a calcium current and a calcium dependent potassium and chloride current also exist. However, in the physiological operating range the leak current can be approximated by a linear curve (Attwell et al. 1982) with slope and offset adjusted to create a typical cone light response
Release
Here we describe both voltagedependent and voltageindependent components (Rieke and Schwartz 1994) of release (Fig. 2
D). This curve shows a transition near −40 mV, the activation point of Ltype Ca^{2+} channels present in salamander cone terminals. As V
_{m} tends from −40 to −∞ infinity, release approximates a constant (the voltageindependent component). Positive to −40 mV, release increases monotonically according to the activation curve for Ca^{2+}. The relationship is described by the following equation
N _{1} is proportional to the number of release sites, number of vesicles per release site, and the number of transmitter molecules per vesicle. It determines the magnitude of release at every membrane voltage.
Range is defined as the release at −35 mV divided by the voltage independent release
Transport
Transport (Fig. 2
E), as the chloride current, is affected by both voltage and glutamate concentration independently according to the following relationship (Wadiche et al. 1995)
Diffusion
Diffusion (Fig. 2
F) from the synaptic region is assumed to be linear with the concentration gradient. Glutamate concentration outside the synapse is assumed to be quite low and therefore modeled to be zero
The horizontal cell as a glutamate electrode
Throughout the measurements the horizontal cell (which is postsynaptic to the cone) membrane potential was used to indicate glutamate concentration in the synapse. To compare the predictions of the model with the measured light response, we calibrated the horizontal cell membrane potential in the presence of different concentrations of glutamate (Gaal et al. 1998). The data can be fit by the following relationship
Determination of N_{1}, N_{2}, N_{3}, K_{m}, m, range, and S parameters
Our strategy to find these parameters was as follows. The ordinate values of the normal light response of a horizontal cell (Fig. 3 A, control) and the light response when glutamate uptake was blocked with DHK (Fig. 3 A, uptake blocked) were converted to glutamate concentration units (Fig. 3 B) with the calibration curve described by Eq. 13 . The shaded bar in Fig. 3 B indicates the time frame when cone voltage remained close to −50 mV. The change of glutamate concentration over time (−dG/dt) is plotted against glutamate concentration in Fig. 3 C in the time frame indicated by the shaded bar in Fig. 3 B. When uptake is blocked the ordinate values represent −i _{diffusion} [G] − i _{release} [−50]. The slope of the fitted linear curve determines the diffusion constant, N _{3} (4.2 s^{−1}), and the offset determines the rate of voltage independent release, N _{1}/range. When uptake is intact the data points represent −i _{uptake} [−50,G] − i _{diffusion} [G] − i _{release} [−50]. Because both i _{diffusion} [G] and i _{release} [−50] were already determined, fitting a function of the form of i _{uptake} [−55, G, Km, N _{2}, m] + i _{diffusion} [G] + i _{release} [−55] determines K _{m} (3.96 μM) and sets the value of N _{2} exp[−55/m]. i _{diffusion} [G] + i _{release} [−55] and i _{uptake} [−55, G] are plotted as solid curves in Fig. 3 D. Their intersection sets the glutamate concentration in light when cones are hyperpolarized to −50 mV. In dark, when cones are depolarized to −35 mV, the i _{diffusion} + i _{release} curve is shifted to right (dashed curve) and intersects the abscissa at the dark concentration of glutamate when uptake is blocked (Fig. 3 B). The offset of this shifted curve determines i _{release} [−35], which equals N _{1} (370 μM s^{−1}). From N _{1} and the voltageindependent release, N _{1} /range, range can be calculated (1.99). In dark this shifted linear curve intersects i _{uptake} [−35, G] at the normal dark concentration of glutamate (Fig. 3 B), which together with the known value of N _{2} exp[−55/m] sets N _{2} (3.87 μM s^{−1}) and m (11.32 mV). Because we had no experimental way to determine the volume of the synaptic cleft, S, it was set to unity. The values of N _{1}, N _{2}, and N _{3} are relative to S, so the true diffusion constant and number of transporters are scaled to the (unknown) volume of the synaptic cleft. We note that horizontal cells are not ideal glutamate electrodes, so the delay caused by their capacitance is also included in S.
RESULTS
Our goal was to generate simulated light responses by using the equations and functions outlined in methods. We then compared the simulated responses with the actual measurements under different pharmacological conditions (Gaal et al. 1998), when either release or uptake were blocked. We considered the following three experimental situations: 1) normal light response, 2) light response in the presence of different doses of DHK to block transport, and 3) light response in the presence of different doses of magnesium (Mg^{2+}) to block release.
Normal light response
Figure 4 shows the response of the model to a light flash eliciting maximal response in cones under control conditions. The shape and range of the cone (Fig. 4 A) and horizontal cell (Fig. 4 B) membrane potential responses are similar to those in the living system (Fig. 4, C and D). The characteristic initial peak hyperpolarization and depolarization of the cone at light onset and offset, respectively, are illustrated on Fig. 4 A.
The model can predict the change in glutamate concentration as well as the different components of the glutamate flow (release, uptake, and diffusion) during a light flash. Figure 5 shows these “hidden” events. The shape of the diffusion curve (Fig. 5 D) is similar to the G response (Fig. 5 A) because the diffusion depends only on G. Although glutamate release is a function of only V _{m}, the release curve (Fig. 5 B) is different from the cone V _{m} response. This discrepancy is due to the fact that release is insensitive to voltages more negative than −40 mV but becomes strongly dependent on V _{m} around −35 mV (Fig. 2 D). This asymmetry is clearly demonstrated on Fig. 5 B. The initial peak hyperpolarization of the cone V _{m} at light onset has no effect on release, but the peak depolarization at light offset causes a large release peak.
The transport curve (Fig. 5 C) reveals the dependency of uptake on G and also on V _{m}. The sudden increase and decrease of uptake rate at the light onset and offset is a consequence of the fast hyperpolarization and depolarization, respectively, of the cone terminal membrane.
Light response in the presence of different doses of DHK
In the model, DHK, a competitive inhibitor of the glutamate transporter (Arriza et al. 1994; Barbour et al. 1991; Eliasof and Werblin 1993; Picaud et al. 1995), changes the K _{m} of the transporter channel to K _{m} (1 + i/K _{i}), where i is the concentration of DHK and K _{i} is its dissociation constant (Stryer 1990).
According to the experiments of Gaal et al. (1998), there are three important consequences of DHK to the light response: 1) cones are depolarized in dark, 2) horizontal cells are depolarized in dark and light, and 3) the rate of hyperpolarization of horizontal cells decreases at light onset.
The effect of DHK on the cone light response is shown in Fig. 6, A (model) and B (measurements). After introducing DHK (t = 0 s) in the model, there is a significant depolarization of cones in dark that increases the operating range of cones (the measurements reflect only the steady state). The depolarization of horizontal cells in dark and light and the slow down of the light response at on are shown in Fig. 6, C (model) and D (measurement).
Figure 6, E and F, depict the results of analysis of the effect of DHK on the kinetics of the horizontal cell light response. The initial rate is defined as the average slope between t = 0.5 s and t = 0.6 s for onset and t = 1.5 s and t = 1.6 s for offset. In Fig. 6, E (model) and F (measurements), the normalized initial rate (defined as the initial rate in the presence of DHK/control initial rate × 100) is plotted against DHK concentration.
Light response in the presence of different doses of Mg^{2+}
Magnesium decreases the calciumdependent release of glutamate (Dowling and Ripps 1973). We can introduce this blocking effect into the model by scaling the N _{1} variable of release by a Boltzman function fitted to the normalized horizontal cell darkvoltage versus Mg^{2+} concentration experimental curve: 1/{exp[([Mg^{2+}] − 3)/0.6] + 1}. Increasing Mg^{2+} therefore decreases N _{1} with halfmaximal concentration of 3 mM.
The characteristic changes in the light response of horizontal cells caused by Mg^{2+} were 1) hyperpolarization of horizontal cells in dark and light and 2) a decrease in rate of depolarization at the light offset and change of the onset kinetics. Our model displays both effects. Figure 7 shows the effect of Mg^{2+} on the light response of horizontal cells in the model (Fig. 7 A) and in the measurement (Fig. 7 B). The change in the kinetics of the horizontal cell light response with Mg^{2+} is rather complex (Fig. 7, C and D). The normalized initial rate at light off is monotonically decreasing with increasing Mg^{2+} concentrations (Fig. 7, C, dashed line for model, and D, dashed line for measurement). At light on the normalized initial rate shows a bellshape dependence on Mg^{2+} (Fig. 7, C, solid line for model, and D, solid line for measurement). This can be explained as follows. The rate of removal of glutamate from the synaptic cleft depends on the balance between inflow (release) and outflow (transport and diffusion), so the expected effect of Mg^{2+}, a release blocker, is an increase in rate of the glutamate depletion. However Mg^{2+} also lowers the glutamate concentration of the cleft in dark, which decreases the rate of transport and diffusion. This decrease in removal counteracts the effect on release and slows down the onset response. At lower concentrations of Mg^{2+} the direct effect on release is stronger so the initial rate increases with increasing Mg^{2+}. At higher Mg^{2+} concentrations the indirect effect on transport and diffusion becomes increasingly stronger, counteracting the direct effect and causing the slope of the initial rate versus Mg^{2+} curve to become zero then negative later. This could account for the bellshape dependence of the initial rate curve on Mg^{2+}.
DISCUSSION
At the output synapse of cones in the tiger salamander retina the glutamate concentration is determined by the integral of three rates: release, uptake, and diffusion. The interactions that set concentration are complex because uptake is both voltage and glutamate concentration dependent. Further, the transporter appears to gate a chloride channel, generating a negative feedback signal that alters cone membrane potential. This potential controls both release and uptake.
The study of Gaal et al. (1998) proposed that the transporter is mainly responsible for setting concentration, but that study raised the more general question as to the relative contributions of each of these rates to the control of glutamate concentration. In this study we attempted to evaluate relative contribution of each of these rates in setting glutamate concentration. We used a simple pair of interrelated equations to describe these complex interactions. These equations were fitted with previously measured functions relating dark current, chloride current, leak current, release, and uptake to membrane voltage and glutamate concentration. The solution of these equations with previously measured functions generated rates of glutamate concentration change that were quite close to the quantities actually measured by Gaal et al. (1998), suggesting that the model might be a good approximation to the mechanism underlying glutamate concentration control.
Relative contributions of uptake and release are functions of membrane potential
Figure 8 shows how uptake, release, and diffusion interact to set glutamate concentration as uptake and release change with lightelicited variations in cone membrane potential. The i _{release} and i _{uptake} + i _{diffusion} curves are shown as functions of glutamate concentration. The intersection of the two curves sets the steadystate glutamate concentration in the synaptic cleft at a given cone membrane potential, which in turn is a function of light intensity. In the dark, release balanced by uptake and diffusion sets the glutamate concentration to 67 μM (Fig. 8 A). When cones are illuminated they respond with hyperpolarization, which decreases release over the potential range from −35 to −40 mV and increases uptake. A downward shift in the release curve moves the intersection of the curves to the left, decreasing glutamate concentration. When the cone voltage reaches −40 mV (Fig. 8 B) release is no longer voltage dependent (Rieke and Schwartz 1994), but uptake continues to increase in magnitude with hyperpolarization, steepening the uptake plus diffusion curve and pushing the intersection toward lower glutamate concentrations. When cone voltage reaches its maximum hyperpolarization of −50 mV, glutamate uptake increased to the level where the intersection of the curves moves the glutamate concentration to 5 μM (Fig. 8 C).
Except for a narrow part of the response range where vesicular release is voltage dependent, cones appear to use a strategy different from other neurons to control transmitter concentration over most of the cone response range; cone voltage controls the rate of uptake of neurotransmitter from the synaptic cleft.
Chloride current may generate a significant negative feedback at the cone synapse
The voltage and glutamate concentrationdependent transporter not only locks glutamate concentration to cone voltage but also provides a feedback signal from glutamate concentration to cone voltage. The feedback signal is provided by a chloride channel, which is incorporated into the transporter (Picaud et al. 1995; Wadiche et al. 1995). The feedback is negative because the chloride equilibrium potential (E _{Cl}) lies negative to the cone operating range (−35 to −50 mV) (Werblin and Dowling 1969).
This negative feedback could act to accelerate the light response and help to prevent perturbations of glutamate concentration brought about by variations in rate of vesicular release. Such perturbations can be caused by change in temperature or pH, both of them shown to modulate release. (Barnes and Bui 1991; Barnes et al. 1993)
Input–output relationship of the conehorizontal cell synapse
The rate of glutamate concentration change depends on cone voltage and glutamate concentration in the synaptic cleft. Plotting uptake, diffusion, or release as a function of cone voltage and glutamate concentration defines a threedimensional surface. In Fig. 9 A we plotted the release and uptake + diffusion surfaces. If we project the intersection of the two curves to the cone voltageglutamate concentration plane (Fig. 9 B), the resulting curve is the steadystate input–output relationship of the cone output synapse. Converting glutamate concentration units to horizontal cell voltages with the calibration curve described by Eq. 13 leads to the input–output relationship from cone voltage to horizontal cell voltage (Fig. 9 C). If uptake is blocked, the steadystate input–output relationships (Fig. 9, E and F) can be obtained by projecting the intersection of the release and diffusion surfaces to the cone voltageglutamate concentration plane. In the absence of uptake the input–output curve is highly nonlinear compared with the almost linear relationship if uptake is present (cf. Fig. 9, C and F).
Footnotes

Address for reprint requests: F. Werblin, 145 Life Sciences Addition, University of California, Berkeley, CA 94720.