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Modeling Starburst Cells' GABA_B Receptors and Their Putative Role in Motion Sensitivity

Norberto M. Grzywacz ^* and Charles L. Zucker 

^* Department of Biomedical Engineering, Neuroscience Graduate Program, and Center For Visual Science and Technology, University of Southern California, Los Angeles, California; and Department of Anatomy and Neurobiology, Boston University School of Medicine, Boston, Massachusetts

Correspondence: Address reprint requests to Norberto M. Grzywacz, Tel.: 213-821-1150; E-mail: nmg{at}bmsr.usc.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
Neal and Cunningham (Neal, M. J., and J. R. Cunningham. 1995. J. Physiol. (Lond.). 482:363–372) showed that GABA_B agonists and glycinergic antagonists enhance the light-evoked release of retinal acetylcholine. They proposed that glycinergic cells inhibit the cholinergic Starburst amacrine cells and are in turn inhibited by GABA through GABA_B receptors. However, as recently shown, glycinergic cells do not appear to have GABA_B receptors. In contrast, the Starburst amacrine cell has GABA_B receptors in a subpopulation of its varicosities. We thus propose an alternate model in which GABA_B-receptor activation reduces the release of ACh from some dendritic compartments onto a glycinergic cell, which then feeds back and inhibits the Starburst cell. In this model, the GABA necessary to make these receptors active comes from the Starburst cell itself, making them autoreceptors. Computer simulations of this model show that it accounts quantitatively for the Neal and Cunningham data. We also argue that GABA_B receptors could work to increase the sensitivity to motion over other stimuli.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
One of the most important unanswered questions in retinal neurobiology is why the Starburst cholinergic amacrine cells have two neurotransmitters (1–4). These cells produce both acetylcholine (ACh) and -aminobutyric acid (GABA), releasing them upon light stimulation (5–9). The released ACh has many roles in the retina, including the enhancement of motion sensitivity (10–12) and the establishment of directional selectivity for some types of stimuli (13–20). In turn, recent evidence shows that GABA is the main transmitter involved in directional selectivity (6,7,21). In this article, we discuss another possible role suggested by the cholinergic amacrine cells themselves containing GABA_B receptors (22). Because these receptors often work as autoreceptors in the brain (23–25), this raises the possibility that GABA from these cells feeds back onto them to control their release of ACh. This control could be by hyperpolarization (26,27), by reducing a Ca²⁺-dependent current (28–30) through a G-protein mechanism (30–32), or by facilitating a L-type Ca²⁺ channel (33). The results of Neal and Cunningham (5) coupled to the results of Zucker et al. (22) lend some support to such an autoreceptor control (see also (34)). Neal and Cunningham showed that the GABA_B agonist baclofen and the glycinergic antagonist strychnine enhance the light-evoked release of retinal ACh. Considering these results, they proposed that glycinergic cells inhibit the Starburst cells and are in turn inhibited by GABA through GABA_B receptors. However, as shown by Zucker et al. (22), glycinergic cells do not appear to have GABA_B receptors. Consequently, one must search an alternate hypothesis for the role of these receptors. The simplest alternative given the available data is that GABA_B agonists enhance the release of ACh by acting on the Starburst cells themselves. These cells may synapse onto glycinergic cells (which probably include the cholinoreceptive DAPI-3 cell (22,35,36) through muscarinic receptors. (The receptors may be chemically ephaptic, that is, ACh may diffuse to targets far from the presynaptic site; this would help to explain the apparent dearth of conventional synapses made by Starburst cells onto noncholinergic amacrine cells (37)). In addition, retinal cholinergic receptors are often far away from the site of cholinergic release (38)—the glycinergic cells of ACh may provide contact back onto Starburst cells (5,39). Hence, the activation of GABA_B receptors may result in disinhibition.

In this article, we use a biophysical model to test the feasibility of the GABA_B-autoreceptor hypothesis for Starburst cells. To know whether this hypothesis will work is not so easy. One difficulty is to know how GABA_B agonists reduce the muscarinic input to the glycinergic cell at the same time that they increase the overall release of ACh. Perhaps the answer lies in the recent surprising finding that only 25% of Starburst-cell varicosities contain GABA_B receptors (22). If the input to glycinergic cells came only from these varicosities, then GABA_B agonists might affect these cells without reducing ACh release from other varicosities. However, the model must solve another problem with GABAergic action on Starburst cells. The release of ACh from Starburst cells may be also inhibited by GABA through GABA_A receptors (8,16,39,40). How is it that the GABA that putatively feeds back to the GABA_B autoreceptor does not inhibit the ACh release through the GABA_A heteroreceptor? The model provides answers to these questions and fits the Neal and Cunningham data well. An abstract version of the model appeared elsewhere (41).

The next section of this article will provide the model assumptions and their justifications, along with a physical description of the model. That section will include no equations to facilitate the comprehension of the ideas. The model equations and the parameters used in the simulations will appear in Appendices A and B, respectively.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
Fig. 1 shows the architecture of the model. We do not intend it to capture all the known details of the cells. Rather, we want it to be the simplest model that accounts for the Neal and Cunningham article (5) and be generally consistent with the knowledge in the field. To start, the model simulations used only the ON-pathway (42–45), assuming the OFF-pathway to be similar. Moreover, all model cells except for the Starburst cell had just one compartment. In turn, the Starburst cell had five compartments, a soma, and two proximal and two distal dendritic compartments. The reason for having two of each of these compartments was to enforce morphological symmetry about the soma (44,46).

View larger version (22K): [in this window] [in a new window]   FIGURE 1  New model for the function of GABAB receptors in the control of retinal cholinergic release. In this figure, circles, and rounded rectangles represent somas and dendrites, respectively. The T terminations represent synapses onto somas and onto dendrites, with the plus symbol (+) indicating excitatory synapses and the minus symbol (–) indicating inhibitory synapses. The main cell in the model is the Starburst cell, whose soma is labeled SA and which, for simplicity, contains four dendritic compartments. Other cells include the bipolar cells (labeled B) and two types of amacrine cells (labeled A), one GABAergic and one glycinergic (labeled Gly). The direct pathway to the Starburst cell appears textured, whereas the inhibitory feedback pathway appears in black. Bipolar cells making glutamatergic synapses (labeled Glu) are the beginning of the excitatory pathway. They feed the Starburst cell (synapses labeled 1) and the GABAergic cells (synapses labeled 2). In turn, the GABAergic cells feed the bipolar inputs (synapses labeled 3) and the distal dendrites of the Starburst cells (synapses labeled 4). The proximal dendrites release ACh, which excites the glycinergic cell through muscarinic receptors (labeled ACh M and synapses labeled 5). This ACh release that activates muscarinic receptors is controlled by GABA, which is released from the proximal dendrite and feeds back to it through GABAB receptors (synapses labeled 6). Activation of the muscarinic receptors on the glycinergic cell causes it to inhibit the Starburst cell (synapses labeled 7). As a result, there is a reduction of cholinergic release from its distal dendrites. (These dendrites normally feed ganglion cells through nicotinic receptors—labeled ACh N and synapses labeled 8.) If the distal cholinergic release is much larger than the proximal one, then the model can account for the results of Neal and Cunningham (5). (Evidence suggests that different dendritic varicosities of the Starburst cell may have different relative amounts of GABAA and GABAB receptors (see Model, in text, for arguments. The division into proximal and distal dendritic compartments is only for simulation simplicity.)

Because GABA_B receptors are proximal in the model, the synapse to the glycinergic cell must also be proximal. From the data of Neal and Cunningham (5) and Zucker et al. (22), we assume that this synapse is muscarinic. (One could also use nicotinic and GABA_A synapses, but this would make the model unnecessarily complex.) Furthermore, it is postulated that GABA is released in a Ca²⁺-independent manner (48–50) by the proximal dendritic compartment and feeds back onto it. This assumption is not completely realistic, because some GABA release from Starburst cells is Ca²⁺ dependent (51). However, in varicosities with GABA_B receptors, their action suppresses Ca²⁺-dependent release of GABA. Consequently, such a release would only occur early in the response, that is, before GABA_B action begins. Because the stimulus' period (333 ms; see Appendix B) is much slower than most synapses, any Ca²⁺-dependent release would occur during a negligible portion of the response in the proximal compartment of the model. Nevertheless, such a release could happen in the distal compartment, making the model still compatible with the data of Zheng et al. (51). Thus, for simplicity, we only included Ca²⁺-independent release of GABA in the proximal compartment.

The activation of the GABA_B autoreceptor would reduce the gain of the muscarinic synapse by lowering the influx of Ca²⁺ into the presynaptic site (28–30). Hence, the glycinergic cell and its synapse would be less active and thus inhibit less the Starburst cell (Fig. 1). (Alternatively, the glycinergic cell could inhibit the bipolar-cell input onto the Starburst amacrine cell (37).) This disinhibition would increase the release of ACh. How is it possible for the ACh release to increase at the same time that the muscarinic synapse is becoming weaker? Again, since GABA_B receptors exist only proximally in the model, one can answer this question if one postulates that there is distal cholinergic release. What one needs is for the distal release to dominate the overall release. Therefore, even though the ACh release that activates muscarinic receptors falls, the total release may increase. Fortunately, there is evidence that Starburst dendrites normally release ACh onto ganglion-cells' nicotinic receptors (7,52–56).

Bipolar cells making glutamatergic synapses are the beginning of the model's excitatory pathway. These cells receive the visual input (not shown in Fig. 1), and synapse onto the Starburst and the GABAergic amacrine cells. In turn, the GABAergic cells feed the bipolar inputs and the distal dendrites of the Starburst cells. There is evidence for a GABAergic feedback onto the bipolar cells feeding the Starburst cell (57). The GABAergic synapse onto the bipolar cells could be through GABA_A (58–61) or GABA_C (59,62–64) receptors. In this article, we will assume for simplicity that the GABAergic synapses onto bipolar and distal Starburst dendrites are identical.

	METHODS

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
With one exception, the simulations of the model included only one type of stimulus: a full-field square-wave stimulation whose definition and parameters appear in the Appendices. (The exception appears in the Discussion and we describe it there.) The computer simulated the response by solving numerically the system of differential equations representing the model (Appendix A). This solution used the Dormand-Prince method with adaptive step size (65) and was implemented in MatLab using Simulink (The MathWorks, Natick, MA).

All the solutions of the equations used the same parameters, which appear in Appendix B. With one exception (explained below), we did not attempt to optimize parameters finely, but only to capture the data qualitatively. Several parameters were around values estimated from the literature. We used parameters as close as possible to the cells of interest, but sometimes, the only ones we could find were not even for the retina. We used them anyway, as they at least constrained our simulations to realistic values.

Parameters estimated from the literature were as follows: We used a specific membrane-capacitance value of 1 µF/cm². This was close to the value measured for bipolar, amacrine, and ganglion cells of mudpuppy and tiger-salamander retinas (66). In turn, for membrane time constants, we used characteristic vertebrate-neuron values of 10–20 ms (but see (66)). These values came, for example, from the Guinea-pig's hippocampal neurons (67), and from the cat's spinal motoneurons (68) and sensory-motor-cortex cells (69). Values for resting potentials were –50 mV, as recorded in the turtle's amacrine cells (70). Amacrine-cell soma diameters were between 5 and 10 µm, as in the cat's AII amacrine cells (71), the primate's Starburst amacrine cells (72), and the pigeon's amacrine cells (73). GABA_A and glycinergic reversal potentials were typical for the retina by being between –60 and –50 mV. For instance, studies on GABA_A and glycinergic synapses were performed respectively with the turtle's cone photoreceptors (74) and the tiger salamander's amacrine cells (75). In turn, muscarinic receptors reversed at 0 mV in most vertebrate tissues. The information on these receptors came, for example, from the bullfrog's sympathetic ganglia (76), the Guinea-pig's smooth muscles (77), and the bovine ciliary muscle cells (78). Finally, we set glutamatergic receptors to reverse at 20 mV, as for the Guinea-pig's laterodorsal tegmental neurons (79).

Different from these parameters, we had no good experimental basis to select other synaptic parameters. For instance, maximal synaptic conductances for our simulations would depend on light-stimulus properties, such as intensity. Such conductances had not been measured in conditions similar to those that we were trying to simulate. Hence, we selected these conductances based on the resting conductances of the membrane. The maximal glutamatergic conductance was the same as the resting conductance of the cellular compartments. Our rationale was that much smaller glutamatergic conductances would depolarize the cell too little, whereas much larger ones would saturate it. Similar considerations were used for the muscarinic conductance. In contrast, the glycinergic conductance was four times stronger than the resting conductance to produce shunting inhibition (with a similar rationale applying to the GABA_A synapse). We tested the robustness of our results with these conductance choices by varying each of them individually by 25% up or down. Our fits remained inside the range of the experimental values (Figs. 6 and 8) despite these variations.

View larger version (10K): [in this window] [in a new window]   FIGURE 6  Comparison between experiments and model on the effects of baclofen and strychnine. The experimental data are total ACh releases taken from Neal and Cunningham (5). These data appear either as standard-error bars (if provided by the authors), or as ranges (given by the lower and higher white squares) and medians (given by the middle square). The categories in the data's horizontal axes are as follows: Light/Back is the ratio between light-evoked and spontaneous releases in control condition. Bacl Back and Stryc Back are the post- to pre-drug ratios of spontaneous release for baclofen and strychnine, respectively. Bacl Light and Stryc Light are the post- to pre-drug ratios of light-evoked release for baclofen and strychnine, respectively. B+S Light is the same, but for using baclofen and strychnine simultaneously. The model values fall within the experimental ranges and error estimates for all control, baclofen, and strychnine conditions.

View larger version (7K): [in this window] [in a new window]   FIGURE 8  Comparison between experiments and model on the effects of bicuculline and baclofen. The data's source and figure's conventions are as in Fig. 6. The categories in the data's horizontal axes are as follows: Bic Back and Bic Light are the post- to pre-drug ratios of spontaneous and light-evoked releases, respectively, for bicuculline. B+B Light is the post- to pre-drug ratio of light-evoked releases for baclofen starting from a bicuculline condition. Again, the model values fall within the experimental ranges and error estimates for all bicuculline and baclofen conditions.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
The simulations in this section are organized according to the experimental conditions used, first considering the control condition, and then the effects of baclofen, strychnine, and bicuculline.

Control
The first things to consider, in understanding the model's behavior, are the potentials in control conditions in the various compartments. The most-central compartments for this understanding are those of the Starburst amacrine cell. In Fig. 2, we show the potentials in this cell's soma and distal dendritic compartment (Fig. 1) in response to a full-field square-wave modulation. Appropriately, the response is periodic. As seen in Fig. 2 A, the response reaches equilibrium essentially during the first cycle. Fig. 2 B shows the voltage's details in one of the cycles. Not surprisingly, there is much voltage attenuation from the distal dendrite to the soma. In addition, the voltage waveforms are complex, showing two peaks during the cycle. The reason for these peaks will be understood when we consider the glycinergic input to the Starburst cell.

View larger version (12K): [in this window] [in a new window]   FIGURE 2  Somatic and dendritic voltages of the model's Starburst cell in control condition. The recorded dendrite is the distal one (top) in Fig. 1. The stimulus giving rise to these recordings is a 3 Hz full-field square-wave modulation over 2 s with 25% duty-cycle. As seen in panel A, model cells reach equilibrium essentially during the first cycle. Panel B is for the second cycle of the top graph, but with an expanded timescale. One can see that the depolarization is attenuated from the dendrite to the soma and that the waveforms are complex, having two peaks during the cycle.

View larger version (11K): [in this window] [in a new window]   FIGURE 3  Cholinergic release in control condition in the model. The release is shown for the synapses 5 (here labeled muscarinic) and 8 (here labeled nicotinic) of Fig. 1. We are normalizing the individual releases by the maximum of their sum and the period of the release is the same as in Fig. 2 B. The ACh release that activates nicotinic receptors is much larger than its muscarinic counterpart. The nonzero release between 0.46 and 0.66 s reflects the tonic cholinergic release without light stimuli.

View larger version (16K): [in this window] [in a new window]   FIGURE 4  Effect of baclofen in the model. This effect is shown for the voltages at the glycinergic cell and at the distal dendrite of the Starburst cell (A and C). Moreover, the figure displays the effect for transmitter release at the Starburst cell's muscarinic (synapses labeled 5 in Fig. 1) and GABAB (synapses labeled 6 in Fig. 1) synapses (B and D). We show these voltages and releases over the same period as in Fig. 2 B. At each synapse, the plots normalize release to the maximum release in control condition. Panel A shows that the two control-voltage peaks result from the rise of glycinergic activity exactly at the minimum between the peaks. Baclofen eliminates the glycinergic activity, destroying the two peaks and increasing depolarization in the Starburst cell (C). This elimination of glycinergic activity is due to the blockade of ACh release that activates muscarinic receptors by baclofen (D). Baclofen's induced increase in depolarization causes several processes to increase in amplitude, including the release at the GABAB synapse. This synapse's role can be appreciated in control conditions when its maximal release coincides with a small local minimum at the ACh release that activates muscarinic receptors (B).

View larger version (18K): [in this window] [in a new window]   FIGURE 5  Effect of strychnine in the model. This effect is shown for the same variables as in Fig. 4, except that the releases are of ACh (sum of synapses labeled 5 and 8 in Fig. 1) and glycine (synapses labeled 7 in Fig. 1). (A–D) Plotting conventions, and periods are also as in Fig. 4. This figure confirms and extends the conclusion of Fig. 4 that the two control-voltage peaks result from the rise of glycinergic activity exactly at the minimum between the peaks (A). Strychnine does not eliminate glycinergic release (D), but eliminates its postsynaptic action. As a result, strychnine destroys the two peaks and increases the depolarization in the Starburst and glycinergic cells (C). Therefore, ACh and glycine releases also increase (B and D).

Bicuculline
Neal and Cunningham also used bicuculline, a GABA_A antagonist, in addition to baclofen and strychnine. Fig. 7 shows how the model reacts to bicuculline and to its combination with baclofen. Bicuculline causes a relatively small increase in the Starburst potential without eliminating its two peaks (Fig. 7 A). This increase follows an increase in the bipolar input to the Starburst cell. Furthermore, the increase in Starburst potential follows a reduction of the inhibition in this cell's distal dendritic compartment (Fig. 1). As a result of this increase in potential, the cholinergic release also undergoes a small augmentation (Fig. 7 B). The increase in potential is much more dramatic when one adds baclofen to the bicuculline (Fig. 7 A). Comparison of Figs. 4 and 7 reveals that the combination of bicuculline and baclofen causes a larger increase in potential than with each drug in isolation. This dramatically increased potential leads to a boosted release of ACh (though the boost seems to be in the duration of the release and not in its total amplitude).

View larger version (11K): [in this window] [in a new window]   FIGURE 7  Effect of bicuculline and baclofen in the model. To illustrate this effect, we use the voltage at the distal dendrite of the Starburst cell (A). Furthermore, the figure displays the effect for cholinergic release from this cell (sum of Synapses 5 and 8 in Fig. 1). (B) The display conventions and periods are as in Fig. 4. In this figure, the curve Bic+Bac corresponds to the condition where bicuculline and baclofen are both applied. Bicuculline increases the depolarization of the Starburst cell and thus the cholinergic release. Adding baclofen to bicuculline causes further depolarization and cholinergic release, though the latter rises mostly due to the prolongation of the response.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
We model the circuit linking Starburst and glycinergic amacrine cells, and account for a host of pharmacological conditions (5). The model postulates that GABA_B receptors on the Starburst cell function as autoreceptors, whose role is to control the cholinergic contact onto glycinergic cells. In turn, these cells feed-back onto the Starburst cell, inhibiting it with glycine. This feedback loop is sufficient to account quantitatively for the increased release of ACh under baclofen and strychnine (Figs. 4–6). Moreover, this loop can explain the reduced glycinergic release by baclofen (Fig. 4) and the effect of muscarinic drugs on this release (Fig. 1). Effects involving GABA_A antagonists are accounted for, too (Figs. 7 and 8).

Limitations of the model
The main limitations of the model are due to the simplicity of its components. Perhaps its main simplifying feature is its segregation of GABAergic receptors, and of ACh release that activates nicotinic and muscarinic receptors into different dendritic compartments of the Starburst cell. We already discussed the rationale and limitations of this segregation assumption in Model. In Experimental Predictions, we will address some of the consequences of segregation.

To make the model more realistic, one would have to include better morphological structure for all cells (Fig. 1). In addition, one would have to use synaptic mechanisms and inputs to bipolar cells that are more realistic (Appendix A). A benefit of using realistic morphology for the Starburst cell would have been an assessment of the spatial-separation requirements for this cell's synaptic inputs and outputs. The details of this cell's response dynamics would also change, but we do not believe that the main conclusions would be modified (Model). The synapses are modeled without dynamics—that is, assuming that they are sufficiently fast to be in quasi steady state. This is probably true, since the stimulus' period (333 ms; see Appendix B) is much slower than most synapses. However, this may not apply to the GABA_B synapse, since it may depend on a G-Protein second-messenger pathway (30–32). If we were to slow down this synapse in the model, the dynamics of the responses would change and the autoreceptor control may lose its effectiveness at high temporal frequencies. The stimulus' slowness is also our justification for disregarding the dynamics of the input to the bipolar cell (Appendix A). However, it would have been better not to neglect these dynamics if we wanted to predict the results of faster experiments. Moreover, the model input to the bipolar cell neglected surround-inhibition from horizontal cells. This hampers the ability to make predictions on the effect of spatial frequency. If spatial frequency were of interest, then one would have to include a richer spatial representation of the inputs. Because the stimuli are full-field square-waves (Methods), there is no need to worry about spatial frequency. For these stimuli, surround-inhibition only reduces the bipolar-cell gain.

Functional roles
The roles of several components of the model have been discussed elsewhere. For instance, the role of the GABAergic input to the bipolar cells may be to give the retina a degree of transience (81–83). In turn, one role of the GABAergic input to the distal dendrites of the Starburst cell may be to contribute to directional selectivity ((16,84); but see (85)). What could be the role of the GABA_B-controlled glycinergic feedback loop to the Starburst cell? We propose that this loop could work to make the sensitivity of ACh release in response to motion larger than those in response to other stimuli. Acetylcholine has been proposed to be involved in the enhancement of retinal motion sensitivity (10–12) and in preferred-direction facilitation in directionally selective cells (10,86,87). Here is how we think of the GABA_B-receptor involvement in these motion functions. If one suddenly delivers a no-motion stimulus to the Starburst cell, then the GABA_B action on the muscarinic synapse may not have time to react. This is because a slow G-protein second-messenger system may mediate this action (30–32). Without this action, muscarinic activation of the glycinergic cell can occur and thus, there is inhibitory feedback onto the Starburst cell (Fig. 1). If instead of sudden stimuli, a sufficiently slow motion sweeps through the Starburst cell, then a different set of events take place. Consider a motion sweeping from left to right in Fig. 1. The left bipolar cells will be the first to be made active, depolarizing the Starburst cell. This depolarization would start the GABA_B loops (synapses labeled 6 in Fig. 1) even before the motion reaches the right bipolar cells. In this case, when the motion finally reaches them, the ACh release that activates muscarinic receptors to the glycinergic cell is truncated, eliminating the glycinergic feedback onto the Starburst cell.

To illustrate these predictions of the model, we added an artificial constant delay to the GABA_B synapse (Eq. A4 in Appendix A). Furthermore, we activated the bipolar cells sequentially to simulate motion. These cells were still activated by a 3-Hz square-wave (Eq. A1). However, the cycle of any given bipolar cell started after that of the neighbor to its left by a preset delay. For example, in Fig. 9 A, this delay was 100 ms, implying an edge motion sweeping the Starburst cell in 400 ms from left to right. The figure shows the response as recorded on the right distal dendritic compartment. As predicted, the response to the motion was larger than the response to the full-field stimulation (Fig. 9 A). With this delay and this motion, the facilitation of the amplitude of response above resting potential was 84.1%. Without the GABA_B delay, facilitation disappeared (only 7.6% for this motion and negative for other motions). However, although the GABA_B delay had effects on the details of the responses, it did not affect the qualitative effects of the various pharmacological drugs described earlier in the article. Even with this delay, baclofen and strychnine still increased the response (compare Fig. 9 B with the solid line of Fig. 9 A). Bicuculline alone did not have a major effect on the response (compare Fig. 9, A and B). But bicuculline augmented the effect of baclofen (Fig. 9 B). These results with baclofen, strychnine, and bicuculline were similar to those in Figs. 4–8.

View larger version (11K): [in this window] [in a new window]   FIGURE 9  Effect of adding a constant delay to the GABAB synapse. We illustrate this effect by showing voltage at the right distal dendrite as in Fig. 2 B. Here, we show the results for a 50-ms delay. In panel A, we activate the bipolar cells sequentially to simulate motion. These cells are still activated by a 3-Hz square-wave (Eq. A1). However, the cycle of any given bipolar cell starts after that of the neighbor to its left by a preset delay. In this figure, this delay is 100 ms, implying an edge motion sweeping the Starburst cell in 400 ms from left to right. As predicted, the response to the motion is larger than the response to the full-field stimulation. In contrast, without the GABAB delay, motion facilitation over the full-field responses disappeared (see text for details). In panel B, we use full-field stimulation, with different combinations of drugs. The curves for baclofen and strychnine superimpose. Comparison of these curves with those in panel A reveals that even with the delay, these drugs augment the response. In turn, bicuculline on its own has little effect, but increases the effect of baclofen on the response (Bic+Bac), as in Figs. 7 and 8.

Experimental predictions
One novel aspect of the model is the proposal that GABA_B receptors work as autoreceptors in the Starburst cell. There are many experiments in the literature trying to prove that particular receptors work as autoreceptors (23–25). For the most part, these experiments try to investigate whether a cell containing a particular transmitter responds to that transmitter. It is known that Starburst cells respond to GABA, but no current or voltage responses to baclofen have been observed (39). However, we propose that baclofen has an effect on the muscarinic synapse without affecting particular conductances. An ideal experiment would measure baclofen's effect on ACh's release from Starburst cells synaptically isolated from the rest of the retina. One could attempt to isolate the Starburst cells synaptically by using low-Ca²⁺ media. Unfortunately, this condition would also shut down these cells' synapses, eliminating ACh release. Alternatively, one could use a cell culture to try to isolate the cholinergic Starburst cell. An experiment that is more feasible is to study whether the effect of baclofen on cholinergic and glycinergic releases is interrupted by drugs that interfere with G-protein pathways. This would not prove the existence of autoreceptors, but would provide circumstantial evidence for them as they tend to work through such proteins (30–32).

Another novel aspect of the model is that different types of GABAergic receptors may have different distributions across the Starburst cell's dendritic tree. Indirect evidence is now available for such different distributions of receptors (see Model for detailed arguments). The same differences should apply for cholinergic synapses. A test for such different distributions would be to use antibodies against these receptors and synapses in an identified Starburst cell. One could then check microscopically whether there are different distributions of immunoreactivity with the different antibodies. If such differences exist, then they would raise the possibility that the Starburst dendrite has more functionality than previously thought. Some studies propose that this cell's dendritic branches may function as independent electrotonic subunits (16,84,85,88). If different receptors and synapses have different distributions in the dendritic tree, then this may allow different portions of a dendritic branch to have different roles in information processing.

	APPENDIX A: EQUATIONS

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
This Appendix presents all the equations used to simulate the model in Fig. 1.

Light stimuli
With the exception of the motion described in the Discussion, the light stimuli used in the simulations paralleled those used by Neal and Cunningham (5). Here, we will discuss the model only in the context of their stimuli. We used full-field square-wave stimulation,

(1)

where a, f, and d are the amplitude, frequency, and duty-cycle of the wave, respectively.

Biophysical notation
Because the stimulus is full field, all the bipolar compartments in Fig. 1 respond alike. The same holds for the GABAergic compartments, glycinergic compartments, the two proximal dendritic compartments of the Starburst cell, and its two distal compartments. In this Appendix, we will denote a compartment by the subindex c. This subindex can attain the values of b, , g, s, p, and d for the bipolar, GABAergic, and glycinergic cells, soma of the Starburst cell, and its proximal and distal compartments, respectively. (Although the simulations do not include the ganglion cell, its subindex will be gc for simplifying the notation below.) Besides cells and compartments, release of neurotransmitters is identical for several different synapses containing the same transmitter. We will denote a neurotransmitter by the subindex n. This subindex can attain the values of gu for glutamate, a for GABA released from GABAergic cells toward GABA_A receptors, gy for glycine, b for GABA released onto GABA_B autoreceptors, and mu and ni for ACh released onto muscarinic and nicotinic receptors, respectively.

This article models the cellular compartments through standard cable-theory assumptions (89). The most important variable in this model is the compartment's potential (voltage), V_c(t). A compartment's potential causes transmitter release, which is denoted by with c₁ and c₂ being the pre- and postsynaptic compartments, respectively. For instance, R_{p, mu, g} is the ACh release that activates muscarinic receptors from the proximal dendritic compartment of the Starburst cell onto the glycinergic cell. In turn, neurotransmitter release causes the activation of a membranous postsynaptic conductance, whose symbol is Besides synaptic conductances, between any two neighbor compartments of the Starburst cell, there is an axial conductance And all the model's compartments also have resting conductances, which we denote g_{c, r, c}.

We denote by I_x,y,c the current flowing through the conductance g_x,y,c into compartment c. To calculate this current, one needs the compartment's voltage (V_c(t)) and the conductance's reversal potential, E_y,c. The axial-conductance current flowing from compartment c₁ to compartment c is From the axial and membranous currents, one can calculate the temporal evolution of the voltage at the compartment if one has its capacitance C_c. This capacitance and resting conductance were computed from the multiplication of the compartment's membranous area (A_c) by their specific values, denoted and respectively.

Synapses
All synapses in the model, except for the GABA_B synapse, respond conventionally to presynaptic voltage. We model conventional synapses as if they are not the temporal bottlenecks of the system. The sigmoidal pre-synaptic-to-transmitter relationship in the model conventional synapses follows the steady-state input-output curves found experimentally (90,91). This relationship is

(2)

where is the synapse's gain, _n > 0 is the slope of the synapse's input-output relationship, and _n is the synaptic threshold. Without loss of generality, one can set the post-synaptic conductance equals to release, since could be made to have units of conductance:

(3)

In Appendix B we express with such units.

The GABA_B synapse responds differently to presynaptic voltage than conventional synapses, since this synapse does not induce, directly, a conductance change (see Model). Instead, this synapse reduces the presynaptic Ca²⁺ conductance in the muscarinic synapse to the glycinergic cell. As a result, this synapse's gain falls. We model this fall by letting ß_{mu, g} depend on the locally released GABA,

(4)

where is the gain of the muscarinic synapse when the GABA_B receptors are not activated. (This equation assumes that the Starburst-cell release of GABA has the same dependence on presynaptic voltage as other releases (i.e., Eq. 2). This assumption cannot be, strictly speaking, correct. This is because we assume that this release is Ca²⁺ independent and thus not affected by the activation of the autoreceptors. The use of Eq. A2 is only justified for the sake of model simplicity, since there are no good models of Ca²⁺-independent release.) Such a model must be taken as a simple abstraction of the complicated G-protein-dependent mechanism by which the GABA_B autoreceptors work (30–32). A complete model would have to include the complete G-protein pathway.

Currents and potentials
This section describes how we calculate the voltages across the membranes of the different cellular compartments. This calculation is identical in all compartments except for the bipolar cells. The currents flowing through synaptic or resting conductances depend on the voltage at nonbipolar compartments and on the conductances' reversal potentials. These currents are

(5)

The current flowing into compartment c from a compartment c₁ through an axial conductance is

(6)

From these currents, the potential in the nonbipolar compartment follows

(7)

where N_c is the number of types of currents flowing into compartment c.

The bipolar cells are special cases among all compartments, because one of its inputs does not come from synapses but directly from light. We model this input as a current being injected onto the bipolar compartment. In addition, this compartment receives a GABAergic current and has a resting-conductance current (Fig. 1), making N_b = 3 in Eq. A7. For simplicity, the stimulus current is exactly S(t) (Eq. A1). This means that the simulations neglect the temporal properties of the photoreceptors feeding the bipolar cells, that is, they consider the input's dynamic sufficiently slow. For the same reason, the simulations assume that the bipolar compartment has zero capacitance, that is, C_b = 0. With these assumptions, one can set the left-hand side of Eq. A7 to 0 and use Eq. A5 to get

where g_{, a, b}(t) obeys Eq. 3. This voltage is then fed to the bipolar synapse, whose glutamatergic release is governed by Eq. 2.

	APPENDIX B: PARAMETERS

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
This Appendix presents the simulation parameters, with the notation of Appendix A, and organized in four sets: stimulus, synaptic, cellular, and pharmacology.

Stimulus parameters
The three parameters of the stimulus (Eq. 1) were a = 8 pA, f = 3 Hz, and d = 0.25.

Synaptic parameters
We present the synaptic parameters (Eqs. 2–5) in six subsets: glutamatergic, GABA_A, glycinergic, GABA_B, muscarinic, and nicotinic synapses.

Glutamatergic synapses

GABA_A synapses

Glycinergic synapses

GABA_B synapses

Muscarinic synapses

Nicotinic synapses

The only exception is that pS for the central glycinergic cell, so its total muscarinic input is equal to the other glycinergic cells in the full-field experiments.

Cellular parameters
Three cellular parameters (Eqs. 5–7) were identical for all compartments: the specific membrane capacitance (=1 µF/cm²), the resting potential (E_{r, c} = –50 mV), and the axial conductance ( pS). The rest of parameters are organized in two sets: compartmental membranous areas and specific resting conductances.

Compartmental membranous areas

Specific resting conductances

Pharmacology parameters
The simulations of the pharmacology were performed by setting the appropriate parameters to zero when a particular drug was used. We present the pharmacology parameters in three subsets: baclofen, strychnine, and bicuculline.

Baclofen

Strychnine

Bicuculline

	ACKNOWLEDGEMENTS

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
We thank Susmita Chatterjee, Eun-Jin Lee, David Merwine, Mónica Padilla, Joaquín Rapela, and Jeff Wurfel for discussions during the performance of this project. We also thank Ms. Denise Steiner for administrative help.

This work was supported by National Eye Institute grants No. EY08921 and No. EY11170 to N.M.G., and by National Eye Institute grant No. EY07552 to C.L.Z.

Submitted on August 8, 2005; accepted for publication April 10, 2006.

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TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION APPENDIX A: EQUATIONS APPENDIX B: PARAMETERS ACKNOWLEDGEMENTS REFERENCES

 
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