Drift-diffusion simulation of the ephaptic effect in the triad synapse of the retina

Transcription

1 J Comput Neurosc (2015) 38: DOI /s Drft-dffuson smulaton of the ephaptc effect n the trad synapse of the retna Carl L. Gardner Jeremah R. Jones Steven M. Baer Sharon M. Crook Receved: 16 July 2013 / Revsed: 28 August 2014 / Accepted: 12 September 2014 / Publshed onlne: 28 September 2014 Sprnger Scence+Busness Meda New York 2014 Abstract Expermental evdence suggests the exstence of a negatve feedback pathway between horzontal cells and cone photoreceptors n the outer plexform layer of the retna that modulates the flow of calcum ons nto the synaptc termnals of cones. However, the underlyng mechansm for ths feedback s controversal and there are currently three competng hypotheses: the ephaptc hypothess, the ph hypothess, and the GABA hypothess. The goal of ths nvestgaton s to demonstrate the ephaptc hypothess by means of detaled numercal smulatons. The drft-dffuson (Posson-Nernst-Planck) model wth membrane boundary current equatons s appled to a realstc two-dmensonal cross-secton of the trad synapse n the goldfsh retna to verfy the exstence of strctly electrcal feedback, as predcted by the ephaptc hypothess. The effect on electrcal feedback from the behavor of the bpolar cell membrane potental s also explored. The computed steady-state cone calcum transmembrane current-voltage curves for several cases are presented and compared wth expermental data on goldfsh. The results provde convncng evdence that an ephaptc mechansm can produce the feedback effect seen n experments. The model and numercal methods presented here can be appled to any neuronal crcut where dendrtc spnes are nvagnated n presynaptc termnals or boutons. Keywords Synapse Retna Ephaptc effect Drft-dffuson model Acton Edtor: Mark van Rossum C. L. Gardner ( ) J. R. Jones S. M. Baer S. M. Crook School of Mathematcal & Statstcal Scences, Arzona State Unversty, Tempe AZ 85287, USA e-mal: carl.gardner@asu.edu 1 Introducton The amount of vsual processng that occurs n the retna before the sgnal reaches the vsual cortex s often underestmated. Varous feedback networks are known to exst wthn the retna, from the outer plexform layer to the neuropl of the nner plexform layer. To unravel how vsual processng takes place n the retna t s essental to understand how these feedback mechansms work. The most well-studed feedback s n the trad synapse n the outer plexform layer (OPL) where rods, cones, bpolar, and horzontal cells nteract. We have shown n a prevous study (Gardner et al. 2013) how the geometrc confguraton of these cells gves rse to an ephaptc feedback response, usng a vastly smplfed rectangular geometry for the ntersynaptc space. Here we demonstrate the ephaptc hypothess by means of detaled numercal smulatons n a realstc two-dmensonal crosssecton of the goldfsh trad synapse usng the drft-dffuson (Posson-Nernst-Planck) model plus membrane boundary current equatons. In the outer plexform layer of the retna, the synaptc termnals of rods and cones form synapses wth two other neuron types: horzontal cells and bpolar cells. The bpolar cells carry vsual nformaton to the nner plexform layer and the horzontal cells form a network connected through gap junctons that s confned to the OPL. Photoreceptors translate vsual nformaton nto electrc currents through changes n ther membrane potentals. Horzontal cells n turn respond to these changes, provdng the nput to bpolar cells. The horzontal cell feedback to the cones may be vewed as a sgnal processng mechansm that removes low spatal and temporal frequences to regulate release of glutamate by cones. In ths nvestgaton, we wll consder a partcular type of synapse n the OPL formed by a cone,

2 130 J Comput Neurosc (2015) 38: a horzontal cell, and a bpolar cell, referred to as a trad synapse. The synaptc termnal of a cone the cone pedcle forms a cavty-lke structure wth a hghly convoluted geometry. The pedcle s nvagnated by multple spnes extendng from the dendrtes of horzontal cells and bpolar cells. A trad synapse s a synapse n whch a bpolar cell, flanked by two or more horzontal cells, comes nto close proxmty wth the cone pedcle. In a typcal goldfsh cone pedcle, there are on average 5 16 trad synapses (Kamermans and Fahrenfort 2004). An dealzed dagram of a two dmensonal slce of a trad synapse s shown n Fg. 1. Note that we are modelng only calcum channels (along the thck red curve on the cone pedcle membrane) and hemchannels (along the thck dark blue curve on the horzontal cell membrane). The calcum transmembrane current flows nto the cone for the appled voltages of nterest here, whle the hemchannel transmembrane current can flow nto or out of the horzontal cell. A smple equvalent crcut model s gven n Fg. 3 of Kamermans and Fahrenfort (2004). Neurotransmsson n the trad synapse s modulated by the flow of calcum ons through the cone pedcle membrane. The area nsde the cone pedcle drectly across from the bpolar cell contans vescles that release the neurotransmtter glutamate. The rate at whch glutamate s beng released from the cone ncreases wth the cone s ntracellular calcum level. Fg. 1 Dagram of the trad synapse and physcal locatons of the calcum channels and hemchannels (CP = cone pedcle, HC = horzontal cell, and BC = bpolar cell). Lengths are n nm We apply our model to the experments performed by Verwejetal.(1996) on goldfsh retnas. In ther expermental setup, an solated goldfsh retna s saturated wth a65μm brght spot of red lght. The spot s a constant (non-flckerng) test regon n the presence and absence of a full-feld background llumnaton. The spot stmulates the cone and the background llumnaton stmulates the rods, whch have gap junctons wth the cones. The horzontal cells have synapses wth the cones (but not drectly wth the rods). The calcum current through the cone membrane s then measured wth and wthout background llumnaton usng patch clamp technques wth a Rnger s soluton desgned to block the currents contrbuted by other ons. In the absence of background llumnaton, cones depolarze, actvatng voltage-gated calcum channels n the cone andtherebyncreasngthe cone s ntracellular calcum levels. Ths ncrease n turn trggers the cone membrane to release more glutamate, whch dffuses across the synaptc cleft and bnds to receptor stes on the horzontal cell. The bndng of glutamate actvates caton-specfc channels n the horzontal cell, resultng n an nward current and depolarzaton of the postsynaptc membrane. On the other hand, when llumnated the cone hyperpolarzes, leadng to a reducton n glutamate release. Ths reducton causes fewer glutamate-gated channels n the horzontal cell to open and therefore less current enters the horzontal cell, leadng to a hyperpolarzaton of the horzontal cell membrane. It has been observed that when horzontal cells become hyperpolarzed the net result s an ncrease n ntracellular cone calcum levels, ncreasng glutamate release and brngng the horzontal cell back to ts restng state, mplyng a negatve feedback pathway from horzontal cells to cones (Kamermans and Fahrenfort 2004). The exstence of ths feedback pathway s rrefutable (Byzov and Shura-Bura 1986; Verwej et al. 1996) although the underlyng mechansm has been a subject of heated debate for over twenty years. There are three competng hypotheses regardng the feedback mechansms: the ephaptc hypothess, the ph hypothess, and the GABA hypothess. Expermental evdence exsts throughout the lterature that both supports and contradcts these three hypotheses, makng t dffcult to draw any sgnfcant conclusons. The most recent research n ths area suggests that the ephaptc and ph effects are both present but operate on dfferent tme scales (M. Kamermans, prvate communcaton). The goal of ths nvestgaton s to demonstrate the ephaptc hypothess through numercal smulatons of the mcroscopc drft-dffuson model wth membrane boundary current equatons. Ths model s appled to a realstc two-dmensonal cross-secton of the trad synapse

3 J Comput Neurosc (2015) 38: n the goldfsh retna to verfy the exstence of strctly electrcal feedback, as predcted by the ephaptc hypothess, reproducng the shft n the expermental background on/off cone calcum transmembrane current-voltage curve ( calcum IV curve for short) n Fg. 8 from a mcroscopc, electro-dffuson vewpont. The predcton of the shft n the calcum IV curve proves the ephaptc hypothess n the context of a mcroscopc model. Only the unshfted curve wthout background llumnaton s calbrated by adjustng the four parameters n the calcum channel model (13) wthn a physologcally relevant range. Then the drft-dffuson model, wth no further adjustment, gves the correctly shfted IV curve wth background llumnaton, provng the ephaptc hypothess. 1.1 The Ephaptc Hypothess The ephaptc hypothess was frst proposed n 1986 by Byzov and Shura-Bura (1986) and has snce been repeatedly tested and modfed. In short, the ephaptc hypothess clams that the negatve feedback pathway from horzontal cells to cones s electrcal n nature. The specalzed geometry of the trad synapse contans narrow extracellular regons between horzontal cells and the cone pedcle whch have a relatvely large resstvty. Ionc current passng from these hghresstance regons nto the horzontal cell va onc channels causes the extracellular potental n the cleft to become more negatve. The ncreased dfference n the cone membrane potental n turn actvates calcum channels. Horzontal cell hyperpolarzaton under background llumnaton actvates nward currents enhancng the cone membrane depolarzaton, whch ultmately leads to an ncrease n ntracellular calcum levels n the cone. In a voltage clamp experment, ths ncrease n cone calcum levels under background llumnaton s seen as a shft n the calcum IV curve to more negatve potentals. Ths mechansm depends on actve channels n the horzontal cell membrane that are responsble for the nward current. Byzov orgnally proposed the glutamate-gated channels at the tps of the horzontal cell as a canddate for ths mechansm (Byzov and Shura-Bura 1986). However, ths dea was dscarded when Kamermans and Spekrejse (1999) tested t n goldfsh usng dntroqunoxalne (DNQX), a glutamate antagonst, to block the transmembrane current through the glutamate-gated channels. The results showed no change n the shft n the calcum IV curve. Instead of abandonng the ephaptc hypothess, Kamermans and colleagues nstead modfed t by proposng that hemchannels n the tps of the horzontal cells were responsble for the nward current. Hemchannels are often consdered as one-way gap junctons, n the sense that they connect the nteror of a cell to the extracellular space wth no voltage- or lgand-gatng mechansm. Ths dea wa drven by physologcal studes that confrmed that such channels are ndeed located on the horzontal cell and are n close proxmty wth the calcum channels and glutamate release stes (Janssen-Benhold et al. 2001). The modfed hypothess has ganed momentum n recent years due to the successful experments desgned to test t. In one experment, Kamermans and colleagues performed dentcal voltage clamp experments on two groups of zebrafsh: a genetcally modfed group that lacked the codons necessary to specfy the hemchannel protens and an unmodfed control group (Klaassen et al. 2011). The results showed that the calcum IV curves n the modfed subjects were not shfted, whle the curves for the control group were shfted, a clear ndcaton that the feedback s ndeed dependent on hemchannels. Although the ephaptc hypothess has enjoyed some expermental success, t contnues to be controversal. Dmtrev and Mangel (2006) employed a crcut model to argue that the resstance of the extracellular cleft must be extremely large to nduce the observed feedback and that such an extreme resstance value s not physcally reasonable. However, the external resstvty s modeled successfully n Gardner et al. (2013), where t s shown that the drft-dffuson model creates the correct resstances for the ephaptc effect n the ntersynaptc space along the sdes of the horzontal cell spne. In addton, the other two hypotheses have receved some expermental support. 1.2 The ph Hypothess The ph hypothess proposes that feedback s modulated by changes n extracellular proton concentratons. Accordng to ths hypothess, hyperpolarzaton of horzontal cells alkalnzes the extracellular space whch serves to alter the gatng mechansm of ph-senstve calcum channels n the cone membrane (Barnes et al. 1993; Hrasawa and Kaneko 2003). However, the mechansm by whch horzontal cell polarzaton controls extracellular ph levels s stll unknown, although researchers have proposed many possble canddates (Kamermans and Fahrenfort 2004; Vessey et al. 2005; Bouver et al. 1992; Hrasawa and Kaneko 2003). The ph hypothess has a far amount of expermental support. It has been shown that extracellular ph levels can affect voltage-senstve calcum channels (DeVres 2001; Prodhom et al. 1987). Further, experments wth goldfsh, tger salamanders, and macaque monkeys have shown that nhbton of extracellular ph fluctuatons, nduced by

4 132 J Comput Neurosc (2015) 38: nsertng hgh concentratons of artfcal ph buffers, can greatly affect the feedback responses (Baba and Thoreson 2009; Cadett and Thoreson 2006; Davenport et al. 2008; Hrasawa and Kaneko 2003; Vessey et al. 2005). The valdty of ths hypothess has also been questoned. One study on the goldfsh retna showed that feedback responses were not altered n the presence of hgh concentraton of HEPES, an artfcal ph buffer (Mangel et al. 1985; Kamermans and Fahrenfort 2004). It has also been argued that the expermental technques used to test the hypothess can have unntended sde effects that would affect other feedback mechansms (Fahrenfort et al. 2009). For example, the nserton of ph buffers can cause acdfcaton of the ntracellular horzontal cell soluton, whch can nhbt hemchannel actvty. Snce the presence of ph affects most bologcal processes, on varyng tme scales from mllseconds to hours, t s dffcult to prove expermental support for a specfc ph effect. 1.3 The GABA Hypothess The GABA hypothess asserts that feedback s modulated by a chemcal neurotransmtter wth γ -amnobutyrc acd (GABA) beng the prmary canddate (Dunlap and Fschbach 1981; Gerschenfeld et al. 1980; Nelson et al. 1990; Pccolno 1995). The theory clams that horzontal cells constantly release GABA whch dffuses across the extracellular space, bndng to the cone membrane, nhbtng calcum channels. Under background llumnaton nduced hyperpolarzaton of the horzontal cell, the quantty of GABA released by the horzontal cell s reduced, allowng more calcum to flow nto the cone. The GABA hypothess has receved some expermental support. A GABA syntheszng enzyme, known as glutamc acd decarboxylase (GAD), has been found to exst n some horzontal cells of certan anmals (Chun and Wässle 1989; Guo et al. 2010; Johnson and Vard 1998; Lam et al. 1979; Vard et al. 1994). It has also been observed that GABA release stes on horzontal cells act n a manner consstent wth the hypothess,.e., they are nhbted by hyperpolarzaton (Ayoub and Lam 1985; Marcetal.1978; Schwartz 1982; 1987). Most mportantly, several pharmacologcal studes of the catfsh and carp retna have revealed that applcaton of GABA antagonsts does ndeed affect feedback under background llumnaton (Lam et al. 1978; Murakam et al. 1982a, b). Most opposton to the GABA hypothess stems from the fact that Kamermans experments were able to alter feedback responses n a GABA ndependent manner. It s most lkely that GABA does play some role n the overall process but only n certan nstances and for certan speces. However, t seems clear that n the goldfsh retna the feedback s not domnated by a GABA-ergc mechansm. 1.4 Summary of Scentfc Results Ths nvestgaton examnes the ephaptc hypothess by means of numercal smulatons of the goldfsh trad synapse at a mcroscopc level usng the drft-dffuson model wth membrane boundary current equatons. The drft-dffuson code s a general purpose membrane/onc bath smulator, and has already been appled to the potassum channel n Gardner and Jones (2011), and to membrane currents between heart muscle cells, n addton to the retna problem. The drft-dffuson model, as appled to the trad synapse, s calbrated n Gardner et al. (2013) to reproduce, n a smplfed geometry, the expermental calcum IV curves. The man result of ths nvestgaton s verfcaton of the ephaptc hypothess by reproducng the expermental shfted background on/off calcum IV curves from a mcroscopc, electro-dffuson smulaton wth only four parameters (all n the calcum channel current model n equaton (13)), wth the background llumnaton turned on and off by adjustng the ntracellular potental of the horzontal cell at the bottom of the computatonal regon. The expermental background on/off calcum IV curves can be reproduced n a smpler compartment model (Fahrenfort et al. 2009), but here we derve ths result from a local mcroscopc model. We also predct that there are 50 % ON and 50 % OFF type bpolar cells n the trad synapses (see Fg. 8). Related smulatons n a vastly smplfed rectangular geometry usng an extra parameter n the calcum channel current model to mplement turnng the background llumnaton on and off supportng the ephaptc mechansm are presented n Gardner et al. (2013); there we showed that the strength of the feedback response depends on the geometrc confguraton of the postsynaptc processes wthn the trad synapse. Baer et al. (to appear) dscuss the prmary mportance of the ephaptc effect vs. the effect of GABA on tmedependent smulatons of calcum current responses of the cat retna to steady and flckerng test stmul wth llumnated or unllumnated background, n the context of a mult-scale macroscopc two-dmensonal model. In vertebrate outer retna, changes n the membrane potental of horzontal cells affect the calcum nflux and glutamate release of cone photoreceptors va a negatve feedback mechansm. Ths feedback has a number of mportant physologcal consequences. One s called background-nduced flcker enhancement n whch the onset of dm background enhances the center flcker response of horzontal cells. Ths model, a partal dfferental equaton system, ncorporates both the GABA and ephaptc feedback mechansms on the scale of an ndvdual synapse and the scale of the receptve feld. Smulaton results, n comparson wth experments, ndcate that the ephaptc mechansm s domnant n

5 J Comput Neurosc (2015) 38: reproducng the major temporal dynamcs of backgroundnduced flcker enhancement. 2 Drft-Dffuson Equatons To model the potental and the onc currents n the trad synapse of the retna, and to compute the Ca 2+ currents nto the cone pedcle, we wll use the drft-dffuson model. The dscrete dstrbuton of ons s descrbed by contnuum on denstes n (x,t)for = Ca 2+, Na +, K +,andcl,and the postve and negatve ons flow n water n an electrc feld E(x,t). The drft-dffuson model s dervable from the Boltzmann transport equaton plus Posson s equaton; thus addtonal forces and flows can be ncorporated nto the model f expermentally observed. In the expermental setup, a voltage bas s appled between the cone and the horzontal cell by means of a patch clamp. Consder a regon such as that shown n the two dmensonal slce of the trad synapse n Fg. 1. Ths regon can be separated nto four compartments: cone nteror, horzontal cell (HC) nteror, bpolar cell (BC) nteror, and extracellular. Each compartment s assumed to be flled wth a salt soluton contanng the four common bologcal ons Ca 2+, Na +, K +,andcl, whch we treat as contnuous charge, rather than ndvdual ons. Ths contnuum model has been used successfully n many bologcal applcatons (Esenberg et al. 1995; Nonner and Esenberg 1995; Gardner et al. 2004; Gardner et al. 2013). The presence of dssocated ons n a salt bath nduces a potental feld, whch n turn affects the flow of ons. To model the evoluton of the on denstes and the electrc potental, we utlze a system of partal dfferental equatons known as the drft-dffuson or Posson-Nernst-Planck equatons that hold n the varous compartments. Treatment of the state varables on the membranes and boundares wll be dscussed below. We neglect water flow effects n ths nvestgaton; these effects (Esenberg et al. 2010; Mor et al. 2011) wll be ncluded n future work unless expermental work demonstrates that osmotcally nduced flows have no effect. By requrng conservaton of charge for each onc speces, we obtan the contnuty equaton n t + f = 0, (1) where =Ca 2+,Na +,K +,andcl, and where f s the flux of the th onc speces. Gauss Law relates the on denstes to the electrc potental φ: (ɛ φ) = ρ = q n, E = φ, (2) where ɛ s the delectrc coeffcent of water, ρ s the total charge densty, and q s the onc charge of speces. The onc flux has drft and dffuson terms f = z μ n E D n, (3) where z = q /e, D s the dffuson coeffcent, and μ the moblty coeffcent of onc speces. The dffuson and moblty coeffcents satsfy the Ensten relaton D = μ k B T/e where k B s the Boltzmann constant, T s the absolute temperature of the medum, and e>0sthe unt charge. For most bologcal applcatons, T 310 K, a typcal body temperature, so k B T 1/40 ev. The onc flux f for each onc speces can be converted nto electrc current denstes j va the smple relaton j = q f (4) and the total current densty j s j = j. (5) In general, the parameters D, μ,andɛ can be treated as functons of space. For our purposes, t s reasonable to assume that these parameters are constant n the physcal doman of the problem. The constant values used for the parameters are shown n Table 1. To summarze, the drftdffuson model reduces to the system n = D 2 n + z μ (n φ) (6) t 2 φ = 1 q n. (7) ɛ Ths model forms a nonlnear parabolc/ellptc system of N speces + 1 partal dfferental equatons where N speces s the number of onc speces ncluded n the model. The state varables of the model are n and φ, whch have Drchlet and/or Neumann boundary condtons. It s known expermentally that the on denstes n bologcal fluds reman at constant values when far away from cell membranes. The values of these constant on denstes Table 1 Physcal parameters used n the drft-dffuson equatons Parameter Value Descrpton D Ca 0.8 nm 2 /ns dffuson coeffcent of Ca 2+ D Cl 2nm 2 /ns dffuson coeffcent of Cl D Na 1.3 nm 2 /ns dffuson coeffcent of Na + D K 2nm 2 /ns dffuson coeffcent of K + μ Ca 32 nm 2 /(V ns) moblty coeffcent of Ca 2+ μ Cl 80 nm 2 /(V ns) moblty coeffcent of Cl μ Na 52 nm 2 /(V ns) moblty coeffcent of Na + μ K 80 nm 2 /(V ns) moblty coeffcent of K + ɛ 80 delectrc coeffcent of water

6 134 J Comput Neurosc (2015) 38: n b are referred to as the bath denstes and have been measured for several cases. For any gven onc speces, the bath denstes can be very dfferent dependng on whether the regon s nsde a cell or outsde a cell. For example, a typcal ntracellular bath densty for calcum n a mammalan organsm s n b,ca = 10 4 mm, whle a typcal extracellular bath densty s n b,ca = 2 mm. It s also known that bologcal fluds mantan charge neutralty away from membranes. To ensure ths, we must enforce the condton q n b = 0. (8) However, the expermentally measured values of the four common onc speces do not generally satsfy ths relaton snce there are a number of other charged molecules that contrbute. To get around ths, we use the typcal bath denstes for the postve ons and treat chlorde as the general negatve charge carrer, settng n b,cl = z n b. (9) =Cl The values for the bath denstes are shown n Table 2. Note that the values for n b,cl are not typcal and have been adjusted to ensure charge neutralty. The prmary flud dynamcs s determned n each ntracellular or extracellular regon by the domnant ons (see Table 2), wth a total densty n each case of approxmately 300 mm = ons/cm 3 = ons/μm 3 (1 mm = ons/cm 3 ), typcal of electron and hole denstes n semconductor devces, where the drft-dffuson model s known to gve excellent results for 1 μm devces. There may be some stochastc effects n the calcum currents, but the expermental data are not yet accurate enough to see such effects. n = 0, where n s the outward pontng unt normal vector to the boundary. The boundary condtons for φ are chosen n a way that attempts to mmc the voltage clamp experment. In such an experment, mcro-electrodes that are held at fxed potentals are nserted at specfc locatons, usually one nsde the cone and the other ground electrode far away from the cone. Along the top of the cone pedcle, U CP s set to V clamp,wherev clamp s the clamped potental, or holdng potental, wth respect to ground. Fgure 2 gves the boundary condtons on the electrostatc potental along the outer boundary of the computatonal doman. The specfcatons of the Drchlet boundary values for the holdng potental U CP and for the ntracellular potentals U BC and U HC are gven n Secton 3. U ref denotes a fxed reference potental, whch s not the absolute physcal ground φ = 0 snce that s far away from the computatonal regon, but s taken to be U ref = 40 mv, whch equals U off HC wth background llumnaton off. The appled voltage across the trad synapse s U CP U ref.a homogeneous Neumann boundary condton on the normal dervatve n φ = 0 s mposed on the remander of the outer boundary. CP HC 2.1 External Boundary Condtons On the external computatonal boundares we use a mxture of Drchlet and Neumann boundary condtons. The most natural boundary condton to use for the on denstes s the Drchlet condton n = n b, snce t s reasonable to assume the on denstes reman at ther bath values away from membranes. Along the y axs of symmetry (see Fg. 1), we use the homogeneous Neumann boundary condton n BC U CP U BC U HC U ref Table 2 Values of the ntracellular and extracellular bath denstes for the four common bologcal ons used n the smulatons Ion Intracellular Extracellular Ca mm 2 mm Cl 160 mm mm Na + 10 mm 140 mm K mm 2.5 mm Fg. 2 Boundary condtons on the electrostatc potental along the outer boundary of the computatonal doman. U CP,U BC,andU HC are specfed potentals, where CP = cone pedcle, BC = bpolar cell, and HC = horzontal cell. U ref denotes a fxed reference potental. The appled voltage across the trad synapse s U CP U ref. A homogeneous Neumann boundary condton on the potental s mposed on the remander of the outer boundary

7 J Comput Neurosc (2015) 38: The cone pedcle, horzontal cell, and bpolar cell are not sopotental (see Barclon et al. (1971) for nerve cells). The potental at the top boundary of the cone pedcle (see Fg. 2) sheldatafxedvoltageu CP (Drchlet boundary condton), whle the sdes of the cone pedcle obey a homogeneous Neumann boundary condton on the potental. We beleve that these are physcally relevant boundary condtons reflectng the fact that the electrode wthn the cone s very far above the computatonal regon, and that the voltage contours wll therefore be approxmately horzontal lnes near the top of the cone pedcle n the computatonal regon. The homogeneous Neumann boundary condtons at the sdes of the computatonal regon mathematcally represent a couplng to a reservor at ether sde, extendng the cone pedcle to the left and rght. In fact, the homogeneous Neumann boundary condton on the denstes as well as the potental at the left boundary makes t an axs of symmetry. At the rght boundary, the Drchlet boundary condtons on denstes represent a couplng to an nfnte reservor bath. There must be a voltage gradent wthn the cone pedcle, as well as wthn the horzontal cell and bpolar cell, as the voltage far above the cone termnal falls toward ground far below the computatonal regon. Large potental gradents do appear wthn the cone pedcle and horzontal cell, as a consequence of the potental dfferences appled across them n the boundary condtons, whch approxmate the voltage clamp expermental setup. The largest potental gradents actually occur across the capactatve cell membranes (see Fgs. 3, 4, 5 and 6), but wth an appled voltage of U CP U ref (see Fg. 2) n the range of [ 40, 50] mv across the vertcal doman of 0.9μm, large potental gradents must appear wthn the cone pedcle and horzontal cell except when the appled voltage s small ( U CP U ref 5mV). The 2D cross-secton that we have chosen s reflected about the vertcal axs at the left boundary by our boundary condtons (homogeneous Neumann boundary condtons on the denstes and the potental) to produce a good approxmaton to the 3D problem, whch has one bpolar cell and two horzontal cell spnes per trad synapse (the cone pedcle has on average 20 trad synapses). The addtonal 3D effects would generate only small correctons to the calcum IV curves, yet the computatonal tmes would be prohbtve for explorng the parameter space of the model. The calcum IV curves computed wth the 2D model agree wth the expermental calcum IV curves manly to wthn 10 %, and everywhere to wthn 20 %. Complete agreement s not to be expected because of the theoretcal argument presented by Klaassen et al. (2011) that the shft n the calcum IV curve under background llumnaton s a pure translaton, and that extraneous effects may have dstorted the expermental shft data n Verwej et al. (1996) (see further dscusson below n Secton 3). In future work we wll extend the trad synapse smulaton regon n Fg. 1 both vertcally and horzontally to nclude arrays of trad synapses. Ths extenson of the computatonal regon n the 2D cross-sectonal plane s much more mportant physcally than any 3D effects. 2.2 Membrane Boundary Condtons Bologcal fluds mantan charge neutralty away from membranes, though charge layers can accumulate on membranes, volatng the local neutralty. To resolve the charge Fg. 3 Steady-state potental n the synapse wth U CP = 0mV, U BC = 60 mv, U off HC = 40 mv, and UHC on = 60 mv. Lengths are n nm and the potental s n mv

8 136 J Comput Neurosc (2015) 38: Fg. 4 Steady-state potental n the synapse wth U CP = 20 mv, U BC = 60 mv, U off HC = 40 mv, and UHC on = 60 mv. Lengths are n nm and the potental s n mv layers, we must develop a model for the membrane surface charge denstes. Our approach to modelng the membrane follows that of Mor et al. (2007) and Mor and Peskn (2009). However, n ther treatment, they use asymptotc expansons wth ntermedate matchng to avod dealng wth charge layers, whle we actually resolve these layers. The man dea s to treat the membrane as a doublevalued sheet n three dmensons. We label the sdes of the membranes as + (ntracellular) and (extracellular). The membrane s modeled as a capactor wth zero thckness n whch ons can accumulate on and/or pass through ether sde, resultng n surface charge denstes σ ±,where ndexes the four onc speces and the ± superscrpt ndcates the sde of the membrane. The state varables of the drft-dffuson model, n and φ, are also defned on the membrane and are double-valued, denoted as n ± and φ ±, respectvely. To obtan the membrane boundary condtons for the on denstes, we relate the spatal charge denstes n ± to the surface charge denstes σ ± σ ± = q l D ± ( n ± n ± ) b, (10) where n ± s the on densty on the membrane, and where l D ± = ɛkb T (11) q2 n± b s the Debye length, whch s typcally around 1 nm for bologcal baths. Indeed, usng the parameter values from : Fg. 5 Steady-state potental n the synapse wth U CP = 40 mv, U BC = 60 mv, U off HC = 40 mv, and UHC on = 60 mv. Lengths are n nm and the potental s n mv

9 J Comput Neurosc (2015) 38: Fg. 6 Steady-state potental n the synapse wth U CP = 60 mv, U BC = 60 mv, U off HC = 40 mv, and UHC on = 60 mv. Lengths are n nm and the potental s n mv Tables 1 and 2, wehavethatl D nm and l D 0.79 nm. Assumng that the total charge on the membrane s overall neutral, the on denstes on the membranes must satsfy equaton (1). Usng ths fact and the defnton of electrc current densty n equaton (4), we have σ ± = l D ± t j± j m,, (12) where j m, s the transmembrane current. Note that we are usng the sgn conventon for j m, n whch current flowng nto a cell s negatve and current flowng out of a cell s postve. For the trad synapse, we model the two channel types n specfc locatons (see Fg. 1) on the membranes whch are mportant to ephaptc feedback: voltage-gated calcum channels n the cone pedcle membrane and hemchannels n the horzontal cell membrane (Kamermans et al. 2001; Kamermans and Fahrenfort 2004). We model the channel locatons on the membranes as a contnuum of channels wth a unform densty. The calcum channels n the cone pedcle (CP) have been expermentally shown to obey a nonlnear Ohm s law wth a voltage dependent conductance functon (Kamermans et al. 2001), whch we use n our model: g Ca,CP (V m E Ca,CP ) j m,ca = N s A m [1 + exp{(θ V m )/λ}], (13) where V m = φ + φ s the membrane potental, g Ca,CP s the maxmum calcum conductance, E Ca,CP s the reversal potental of calcum, N s s the average number of calcum channel stes n a cone pedcle, A m s the surface area of the secton of the cone pedcle contanng calcum channels, θ s the half-actvaton potental, and λ s a curve fttng parameter. The normalzaton factors N s and A m requre some explanaton. Equaton (13) s motvated by expermental data, whch measure actual currents nstead of current denstes. Further, the experments measure the total current through a gven cone pedcle, whch contans many trad synapses. On average, each pedcle has about N s = 20 calcum channel stes. In addton, the area A m of the regon of the cone pedcle contanng calcum channels s estmated to be about 0.1 μm 2. Thus dvdng the current by A m converts t nto a current densty and dvdng by N s gves the average current densty over a gven channel ste. The effects of the hyperpolarzaton of the horzontal cell on the cone calcum transmembrane current are modeled at a strctly local, mcroscopc level through the local values of the electrc potental, but the local values of the electrc potental change n response to a change n the boundary condton U HC under background llumnaton. The hemchannels n the horzontal cell are beleved to be non-specfc caton channels (Kamermans and Fahrenfort 2004) and thus we allow all catons to pass through them. The current-voltage relatonshp for hemchannels s expermentally observed to be lnear, wth an overall reversal potental of zero and a constant conductance of g hem (Klaassen et al. 2011). However, ths ncludes the current from all catons and does not gve any nformaton about ndvdual onc currents. A reasonable approach s then to model each current densty wth a lnear Ohm s law: j m, = g (V m E )/(N s A m ) (14) and mpose the constrants g = g hem (15)

10 138 J Comput Neurosc (2015) 38: Table 3 Hemchannel and other membrane parameters based on expermental estmatons Parameter Value Descrpton E Ca 50 mv reversal potental of Ca 2+ E Na 50 mv reversal potental of Na + E K 60 mv reversal potental of K + g Ca 1.5 ns conductance of Ca 2+ current g Na 1.5 ns conductance of Na + current g K 2.5 ns conductance of K + current g hem 5.5 ns total hemchannel conductance C m 1 μf/cm 2 capactance per area A m 0.1 μm 2 HC spne head area N s 20 number of HC spnes per synapse and g E = 0 (16) to guarantee consstency wth the expermental data. The hemchannel parameters are shown n Table 3. The calcum channel parameters are shown n Table 4. The locaton of the calcum channels n the cone membrane and the hemchannels on the horzontal cell membrane are not arbtrary and n fact are located n such a way as to enable ephaptc communcaton. Fgure 1 shows the regons n whch expermentalsts beleve the channels are located (Kamermans and Fahrenfort 2004), whch we also use n our model. In order to be able to compare our results to emprcal data, we must approxmate the current though an entre cone pedcle. To do ths, we frst compute the average current densty over the channel regon va the trapezodal rule for numercal ntegraton and then multply the result by the normalzaton factors N s and A m. Thus the calcum transmembrane current for an entre cone pedcle s approxmated as I Ca = N sa m j m,ca ds, (17) l C C where C s the segment of the membrane contanng the calcum channels and l C s the arc length of C. The membrane boundary condtons for the state varables on the membrane are determned n the followng way: Table 4 Calcum channel parameters for equaton (13) used n the smulatons Parameter Value Descrpton g Ca,CP 2.2 ns maxmum conductance of calcum channels E Ca,CP 50 mv reversal potental of calcum λ 3 mv knetc parameter θ 5 mv half-actvaton potental assumng we have solved equaton (12) at the gven tme, we can use equaton (10) to obtan the boundary condtons for n ±. Solvng for n± gves n ± = n ± b + σ ± q l D ±. (18) One of the boundary condtons for φ on the membranes can be determned by treatng the membrane as a capactor wth a surface charge densty σ and capactance per unt area C m, resultng n the jump condton [φ] φ + φ = σ C m, (19) where σ s defned as σ = σ + = σ. (20) Ths defnton assumes the membrane remans charge neutral, yeldng a second jump condton: [n m φ] n m φ + n m φ = 0, (21) where n m s the unt normal vector of the membrane pontng from the sde to the + sde. In other words, the normal component of the electrc feld s contnuous across the membrane. Lpd membranes and onc channels usually have permanent bult-n charges ndependent of the electrc potental, whch are functons of ph and calcum concentraton. These charge dstrbutons can be ncorporated (see Esenberg 1996) nto the model n future nvestgatons. 3 Smulaton Results In ths secton, we present the results of smulatons to steady state of the trad synapse. The TRBDF2 (Bank et al. 1985; Gardner et al. 2004; Gardner and Jones 2011; Gardner et al. 2013) (trapezodal rule-backward dfference formula second-order) method s appled to the transport equatons (6) and the ODEs (12). A parallelzed verson of the Chebyshev SOR method s used to solve Posson s equaton (7). The steady-state solutons n the fgures are computed by smulatng the tme-dependent equatons to steady state. The background llumnaton s smulated by adjustng the potental U HC nsde the horzontal cell (see Fg. 2). As mentoned n the ntroducton, background llumnaton hyperpolarzes the horzontal cell. Thus, n all smulatons, we use U off HC = 40 mv to smulate wthout background llumnaton and UHC on = 60mV to smulate wth background llumnaton. The bpolar cell ntracellular potental s set to U BC = 60 mv n Fgs The fxed reference potental s held at U ref = 40 mv for all smulatons. The unknown bologcal parameters used for the calcum

11 J Comput Neurosc (2015) 38: channel model (13)areshownnTable4. These parameters were tuned wthn a physologcally relevant range to get the best ft wth the expermental calcum IV curve for goldfsh (Fg. 11 of Verwej et al. (1996)) wth background llumnaton off. Then the computed shfted calcum IV curve wth an llumnated background s a predcton of our model. In Fgs. 3 6, we show color plots of the steady-state potental on a grd, wth and wthout background llumnaton, usng several dfferent values for U CP.The color scale for the potental vares over dfferent ranges n the fgures n order to best portray the detals of the potental varaton, from 60 mv to {0, 20, 40, 40} mv n Fgs. 3 6, respectvely. Note the hyperpolarzaton of the horzontal cell membrane potental and the depolarzaton of the cone pedcle membrane potental when the background s llumnated. To get a good vew of the charge layers, we zoom n on the regon contanng the channels and plot the steady-state charge densty as shown n Fg. 7. Ths fgure verfes that the baths reman charge neutral away from the membranes and the charge layers that accumulate on each sde are equal n magntude but opposte n sgn,.e., the membrane also mantans overall neutralty. We produced current-voltage curves n Fgs. 8 and 9 wth and wthout background llumnaton to observe how the model predcts the feedback response. The IV curves are generated by varyng the cone pedcle holdng potental U CP over the range [ 80, 10] mv and then computng the calcum current I Ca or hemchannel current I hem n steady state. Good qualtatve agreement s obtaned between the computed (Fg. 8) and expermental (Fg. 11 n Verwej et al. (1996)) IV curves. Klaassen et al. (2011) present both expermental data on zebrafsh and a theoretcal argument that the shft n the calcum IV curve under background llumnaton s a pure translaton, and that extraneous effects may have dstorted the shft n Verwej et al. (1996) our smulatons here ndcate a pure translaton shft. The hemchannel IV curves n Fg. 9 are just straght lnes, as mpled by the lnear Ohm s law (14). Also note there s no expermental data on the hemchannel currents wth whch to compare. The addton of the bpolar cell slghtly complcates thngs. Some bpolar cells are known to depolarze under background llumnaton (OFF bpolar cells) and others are known to hyperpolarze under background llumnaton (ON bpolar cells). To understand how the ntracellular potental of the bpolar cell affects the IV curves, we tred three dfferent cases, whch we label neutral, depolarzed, and hyperpolarzed. For all three cases, we use U off BC = 60mV wth no background llumnaton. When the background llumnaton s present, we use UBC on = 60 mv, 40 mv, and 80 mv for the neutral, depolarzed, and hyperpolarzed cases, respectvely. The bpolar cells are drven manly by glutamate release from cones (and possbly are nhbted by GABA release from horzontal cells), so U BC could n prncple be dervable from an extenson of the model. These effects wll be nvestgated n future work. Fgures 8 and 9 show the results for all three cases along wth the vertcal shfts n the calcum and hemchannel IV curves. Fgure 8 ndcates that the shft n the calcum IV curves s enhanced when the bpolar cell Fg. 7 A zoomed-n vew of the steady-state charge densty along the porton of the membranes contanng onc channels wth U CP = 20 mv, U HC = 40 mv, and U BC = 60 mv. Lengths are n nm and the charge densty s n e mm. Note that the charge layers on opposte sdes of the membranes are equal n magntude but opposte n sgn

12 140 J Comput Neurosc (2015) 38: Calcum Current (pa) Calcum Current (pa) Neutral Bpolar Cell HC/BC = 40/ 60 HC/BC = 60/ Holdng Potental (mv) Hyperpolarzed Bpolar Cell HC/BC = 40/ 60 HC/BC = 60/ Holdng Potental (mv) Fg. 8 Steady-state cone calcum transmembrane current-voltage curves (current or current shft n pa vs. cone pedcle holdng potental n mv) for dfferent bpolar cell ntracellular potentals. Top-left: Neutral bpolar cell. Top-rght: Depolarzed bpolar cell. Calcum Current (pa) Current Shft (pa) Depolarzed Bpolar Cell HC/BC = 40/ 60 HC/BC = 60/ Holdng Potental (mv) Shft Curves Neutral Depolarzed Hyperpolarzed 50 Holdng Potental (mv) Bottom-left: Hyperpolarzed bpolar cell. Bottom-rght: Vertcal dfference of calcum IV curves for all three cases. Compare wth Fg. 11 n Verwejetal.(1996) Hemchannel Current (pa) Hemchannel Current (pa) Neutral Bpolar Cell HC/BC = 40/ 60 HC/BC = 60/ Holdng Potental (mv) Hyperpolarzed Bpolar Cell HC/BC = 40/ 60 HC/BC = 60/ Holdng Potental (mv) Fg. 9 Steady-state horzontal cell hemchannel transmembrane current-voltage curves (current or current shft n pa vs. cone pedcle holdng potental n mv) for dfferent bpolar cell ntracellular potentals. Note that the hemchannel current can be ether postve Hemchannel Current (pa) Depolarzed Bpolar Cell HC/BC = 40/ 60 HC/BC = 60/ Holdng Potental (mv) Current Shft (pa) Shft Curves Neutral Depolarzed Hyperpolarzed 30 Holdng Potental (mv) or negatve, dependng on the holdng potental. Top-left: Neutral bpolar cell. Top-rght: Depolarzed bpolar cell. Bottom-left: Hyperpolarzed bpolar cell. Bottom-rght: Vertcal dfference of hemchannel IV curves for all three cases

13 J Comput Neurosc (2015) 38: s hyperpolarzed and reduced when the bpolar cell s depolarzed. 4 Concluson We have formulated a detaled spatal model of the trad synapse nvagnatng a cone pedcle n the outer plexform layer of the retna. Our goal was to demonstrate the valdty of the ephaptc hypothess as a feedback mechansm. The results of our smulatons have clearly verfed that the ephaptc mechansm correctly shfts the calcum IV curve n the trad synapse when background llumnaton s appled. Ths shft was produced smply by hyperpolarzng the horzontal cell ntracellular potental U HC. Note that the feedback mechansm can be obtaned by purely electrcal effects only f there are hgh resstance pathways that electrcally solate the ntersynaptc space. As mentoned above, we obtaned good qualtatve agreement wth the expermental calcum IV curves n Fg. 11 n Verwej et al. (1996). However our smulatons support the contenton of Klaassen et al. (2011) that both expermental data on zebrafsh and a theoretcal argument predct that the shft n the calcum IV curve under background llumnaton s a pure translaton, and that extraneous effects may have dstorted the shft n Verwej et al. (1996). The smulatons of the trad synapse also demonstrate the dependence of the feedback on the behavor of the bpolar cell under background llumnaton. The neutral case, shown n the upper left frame of Fg. 8 appears to be the closest match wth the expermental data from Verwej et al. (1996). Snce experments measure currents for an entre cone pedcle wth multple trad synapses, our results suggest that there are approxmately an equal number of ON and OFF bpolar cells n a gven pedcle, whch serve to balance out the shft effects. Ths result could be expermentally verfed by blockng the ON bpolar cells wth a chemcal nhbtor, whch would recover the contrbuton of the OFF bploar cells only. Prevous models of the feedback phenomena n the trad synapse have been compartment models n whch each cell and the extracellular space are treated as sopotental regons (Byzov and Shura-Bura 1986; Dmtrev and Mangel 2006; Usu et al. 1996; Smth 1995). As llustrated n Fgs. 3 6, our approach gves mportant nformaton on the spatal varaton n the potental wthn the cells of the synaptc crcut and the ntersynaptc space. Ths nformaton could facltate the formulaton of more detaled and accurate compartmental models of ths or other complex nvagnated synapses. The feedback mechansm addressed n ths paper s purely electrcal. In Secton 1.3 we dscussed how the nhbtory neurotransmtter GABA may also be a canddate mechansm for feedback. In a mult-scale computatonal approach, Baer et al. (to appear) propose that GABA plays a secondary or modulatory role n the feedback wth the ephaptc mechansm beng domnant. In a future study we wll test ths hypothess by nvestgatng the GABA mechansm at the mcroscopc onc level usng the drft-dffuson model approach presented here. Acknowledgments Research supported n part by the Natonal Scence Foundaton under grant DMS We thank Maarten Kamermans for valuable comments. Conflct of nterests of nterest. References The authors declare that they have no conflct Ayoub, G.S., & Lam, D.M.K. (1985). The content and release of endogenous GABA n solated horzontal cells of the goldfsh retna. Vson Research, 25(9), Baba, N., & Thoreson, W.B. (2009). Horzontal cell feedback regulates calcum currents and ntracellular calcum levels n rod photoreceptors of salamander and mouse retna. Journal of Physology, 587(10), Baer, S.M., Chang, S., Crook, S.M., Gardner, C.L., Jones, J.R., Nelson, R.F., Rnghofer, C., Zela, D. (to appear). A computatonal study of background-nduced flcker enhancement and feedback mechansms n the vertebrate outer retna: temporal propertes. Bank, R.E., Coughran, W.M., Fchtner, W., Grosse, E.H., Rose, D.J., Smth, R.K. (1985). Transent smulaton of slcon devces and crcuts. Barclon, V., Cole, J., Esenberg, R.S. (1971). A sngular perturbaton analyss of nduced electrc felds n nerve cells. SIAM Journal on Appled Mathematcs, 21, Barnes, S., Merchant, V., Mahmud, F. (1993). Modulaton of transmsson gan by protons at the photoreceptor output synapse. Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, 90, Bouver, M., Szatkowsk, M., Amato, A., Attwell, D. (1992). The glal cell glutamate uptake carrer countertransports ph-changng anons. Nature, 360, Byzov, A.L., & Shura-Bura, T.M. (1986). Electrcal feedback mechansm n the processng of sgnals n the outer plexform layer of the retna. Vson Research, 26(1), Cadett, L., & Thoreson, W.B. (2006). Feedback effects of horzontal cell membrane potental on cone calcum currents studed wth smultaneous recordngs. Journal of Neurophysology, 95, Chun, M., & Wässle, H. (1989). GABA-lke mmunoreactvty n the cat retna: Electron mcroscopy. Journal of Comparatve Neurology, 279(1), Davenport, C.M., Detwler, P.B., Dacey, D.M. (2008). Effects of ph bufferng on horzontal and ganglon cell lght responses n prmate retna: evdence for the proton hypothess of surround formaton. Journal of Neuroscence, 28, DeVres, S.H. (2001). Exocytosed protons feedback to suppress the CA 2+ current n mammalan cone photoreceptors. Nueronal, 32, Dmtrev, A.V., & Mangel, S.C. (2006). Electrcal feedback n the code pedcle: a computatonal analyss. Journal of Neurophysology, 95,

14 142 J Comput Neurosc (2015) 38: Dunlap, K., & Fschbach, G.D. (1981). Neurotransmtters decrease the calcum conductance actvated by depolarzaton of embryonc chck sensory neurons. Journal of Physology London, 317, Esenberg, R.S. (1996). Computng the feld n protens and channels. Journal of Membrane Bology, 150, Esenberg, R.S., Klosek, M.M., Schuss, Z. (1995). Dffuson as a chemcal reacton: stochastc trajectores between fxed concentratons. Journal of Chemcal Physcs, 102, Esenberg, B., Hyon, Y., Lu, C. (2010). Energy varatonal analyss EnVarA of ons n water and channels: feld theory for prmtve models of complex onc fluds. Journal of Chemcal Physcs, 133, Fahrenfort, I., Stejaert, M., Sjoerdsma, T., Vckers, E., Rpps, H., et al. (2009). Hemchannel-medated and ph-based feedback from horzontal cells to cones n the vertebrate retna. PLoS ONE, 4(6), e6090. Gardner, C.L., & Jones, J.R. (2011). Electrodffuson model smulaton of the potassum channel. Journal of Theoretcal Bology, 291, Gardner, C.L., Jones, J.R., Baer, S.M., Chang, S. (2013). Smulaton of the ephaptc effect n the cone-horzontal cell synapse of the retna. SIAM Journal on Appled Mathematcs, 73, Gardner, C.L., Nonner, W., Esenberg, R.S. (2004). Electrodffuson model smulaton of onc channels: 1D smulatons. Journal of Computatonal Electroncs, 3, Gerschenfeld, H.M., Pccolno, M., Neyton, J. (1980). Feed-back modulaton of cone synapses by L-horzontal cells of turtle retna. Journal of Expermental Bology, 89(0), Guo, C.Y., Hrano, A.A., Stella, S.L., Btzer, M., Brecha, N.C. (2010). Gunea pg horzontal cells express GABA, the GABAsyntheszng enzyme GAD(65) and the GABA vescular transporter. Journal of Comparatve Neurology, 518(1), Hrasawa, H., & Kaneko, A. (2003). ph changes n the nvagnatng synaptc cleft medate feedback from horzontal cells to cone photoreceptors by modulatng Ca 2+ channels. Journal of General Physology, 122, Janssen-Benhold, U., Schultz, K., Gellhaus, A., Schmdt, P., Ammermuller, J., Weler, R. (2001). Identfcaton and localzaton of connexn26 wthn the photoreceptor-horzontal cell synaptc complex. Vsual Neuroscence, 18, Johnson, M.A., & Vard, N. (1998). Regonal dfferences n GABA and GAD mmunoreactvty n rabbt horzontal cells. Vsual Neuroscence, 15, Kamermans, M., & Fahrenfort, I. (2004). Ephaptc nteractons wthn a chemcal synapse: hemchannel-medated ephaptc nhbton n the retna. Current Opnon n Neurobology, 14, Kamermans, M., Kraaj, D., Spekrejse, H. (2001). The dynamc characterstcs of the feedback sgnal from horzontal cells to cones n the goldfsh retna. Journal of Physology, 534(2), Kamermans, M., & Spekrejse, H. (1999). The feedback pathways from horzontal cells to cone: mn revew wth a look ahead. Vson Research, 39, Klaassen, L.J., Sun, Z., Stejaert, M.N., Bolte, P., Fahrenfort, I., et al. (2011). Synaptc transmsson from horzontal cells to cones s mpared by loss of connexn hemchannels. PLoS Bology, 9(7), e Lam, D.M.K., Lasater, E.M., Naka, K.-I. (1978). γ -amnobutyrc acd: a neurotransmtter canddate for cone horzontal cells of the catfsh retna. Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, 75(12), Lam, D.M.K., Su, Y.Y.T., Swan, L., Marc, R.E., Brandon, C., Wu, J.-Y. (1979). Immunocytochemcal localsaton of L-glutamc acd decarboxylase n the goldfsh retna. Nature, 278(5), Mangel, S.C., Arel, M., Dowlng, J.E. (1985). Effects of acdc amno acd antagonsts upon the spectral propertes of carp horzontal cells: crcutry of the outer retna. Journal Neuroscence, 5, Marc, R.E., Stell, W.K., Bok, D., Lam, D.M.K. (1978). GABAergc pathways n the goldfsh retna. Journal of Comparatve Neurology, 182(2), Mor, Y., Jerome, J.W., Peskn, C.S. (2007). A three-dmensonal model of cellular electrcal actvty. Bulletn of the Insttute of Mathematcs Academa Snca, 2, Mor, Y., & Peskn, C.S. (2009). A numercal method for cellular electrophysology based on the electrodffuson equatons wth nternal boundary condtons at the membrane. Communcatons n Appled Mathematcs and Computatonal Scences, 4, Mor, Y., Lu, C., Esenberg, R.S. (2011). A model of electrodffuson and osmotc water flow and ts energetc structure. Physca D: Nonlnear Phenomena, 240, Murakam, M., Shmoda, Y., Nakatan, K., Myach, E.I., Watanabe, S.I. (1982a). GABA-medated negatve feedback from horzontal cells to cones n carp retna. Japanese Journal of Physology, 32, Murakam, M., Shmoda, Y., Nakatan, K., Myach, E.I., Watanabe, S.I. (1982b). GABA-medated negatve feedback and color opponency n carp retna. Japanese Journal of Physology, 32, Nelson, R., Pflug, R., Baer, S.M. (1990). Background-nduced flcker enhancement n cat retnal horzontal cells. II. spatal propertes. Journal of Neurophysology, 64(2), Nonner, W., & Esenberg, R.S. (1995). Ion permeaton and glutamate resdues lnked by Posson-Nernst-Planck theory n L-type calcum channels. BoPhyscal Journal, 75, Pccolno, M. (1995). The feedback synapse from horzontal cells to cone photoreceptors n the vertebrate retna. Progress n Retnal and Eye Research, 14(1), Prodhom, B., Petrobon, D., Hess, P. (1987). Drect measurement of proton transfer rates to a group controllng the dhydropyrdnesenstve Ca 2+ channe. Nature, 329, Schwartz, E.A. (1982). Calcum-ndependent release of GABA from solated horzontal cells of the toad retna. Journal of Physology, 323, Schwartz, E.A. (1987). Depolarzaton wthout calcum can release gammaamnobutyrc acd from a retnal neuron. Scence, 238(4825), Smth, R.G. (1995). Smulaton of an anatomcally defned local crcut: the cone-horzontal cell network n cat retna. Vsual Neuroscence, 12, Usu, S., Kamjama, Y., Ish, H., Ikeno, H. (1996). Reconstructon of retnal horzontal cell responses by the onc current model. Vson Research, 36, Vard, N., Kaufman, D.L., Sterlng, P. (1994). Horzontal cells n cat and monkey retna express dfferent soforms of glutamc-acd decarboxylase. Vsual Neuroscence, 11(1), Verwej, J., Kamermans, M., Spekrejse, H. (1996). Horzontal cells feed back to cones by shftng the cone calcum-current actvaton range. Vson Res., 36(24), Vessey, J.P., Strats, A.K., Danels, B.A., Slva, N.D., Jonz, M.G., Lalonde, M.R., Baldrdge, W.H., Barnes, S. (2005). Protonmedated feedback nhbton of presynaptc calcum channels at the cone photoreceptor synapse. The Journal of Neuroscence, 25(16),