Cellular Inhibitory Behavior Underlying the Formation of Retinal Direction Selectivity in the...

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Journal of Integrative Neuroscience, Vol. 9, No. 3 (2010) 299– 335c Imperial College Press

DOI: 10.1142/S0219635210002457

CELLULAR INHIBITORY BEHAVIOR UNDERLYINGTHE FORMATION OF RETINAL DIRECTION SELECTIVITY

IN THE STARBURST NETWORK

R. R. POZNANSKINeural Networks Research Group

Faculty of Computer Science & Information Technology University of Malaya, 50603 Kuala Lumpur, Malaysia

[email protected]

Received 10 August 2010Accepted 26 August 2010

Optical imaging of dendritic calcium signals provided evidence of starburst amacrine cellsexhibiting calcium bias to somatofugal motion. In contrast, it has been impractical to usea dual-patch clamp technique to record membrane potentials from both proximal den-drites and distal varicosities of starburst amacrine cells in order to unequivocally provethat they are directionally sensitive to voltage, as was rst suggested almost two decadesago. This paper aims to extend the passive cable model to an active cable model of a star-burst amacrine cell that is intrinsically dependent on the electrical properties of starburstamacrine cells, whose various macroscopic currents are described quantitatively. The cou-pling between voltage and calcium just below the membrane results in a voltage-calciumsystem of coupled nonlinear Volterra integral equations whose solutions must be integratedinto a prescribed model for example, for a synaptic couplet of starburst amacrine cells.Networks of starburst amacrine cells play a fundamental role in the retinal circuitry under-lying directional selectivity. It is suggested that the dendritic plexus of starburst amacrinecells provides the substrate for the property of directional selectivity, while directionalselectivity is a property of the exclusive layerings and connement of their interconnec-tions within the sublaminae of the inner plexiform layer involving cone bipolar cells anddirectionally selective ganglion cells.

Keywords : Motion perception; physiological network; intact retina; rabbit; primate;starburst amacrine cell; calcium; reciprocal synapse; co-transmission.

1. Introduction

The anatomical and physiological substrates of direction selectivity in retinal gan-glion cells have been studied for many years, guided by models that were proventhrough increasingly sophisticated experimental work to be incomplete. This earlierwork has established many of the biophysical and pharmacologic bases of gan-glion cell direction selectivity. Further, it has demonstrated that direction selectiv-ity originates in complex subcellular interactions between the dendrites of starburstamacrine cells (SACs) arrayed in a laminar fashion around the directionally selective

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http://dx.doi.org/10.1142/S0219635210002457

http://dx.doi.org/10.1142/S0219635210002457



300 Poznanski

ganglion cells (DSGCs). The design of the underlying retinal circuitry is understoodin broad terms [124, 125], but it is still not clearly understood how the directionallyselective signal is actually computed [33, 48].

Cholinergic amacrine cells are found in the retinae of all vertebrate species includ-ing humans and mammals. These small-diameter sensory interneurons are radiallysymmetric with a soma of approximately 10 µm in diameter and a dendritic eldof approximately 300–400 µm (see Fig. 1). They exhibit a unique neuroanatomywith ne proximal dendrites 0.28–1.46 µm [88] and varicosities (∼3 µm) exclusivelylocated in the distal regions resembling a “starburst” [108]. Primate SACs are smallerin comparison to the rabbit retina with dendritic eld diameters of 100 µm near thefovea and 350 µm in the peripheral retina [94]. The inputs and outputs of such SACswere shown by Famiglietti [37] to have asymmetrical distributions of input and out-put with inputs from bipolar cells and other SACs being distributed throughoutthe entire dendritic tree, while the output to DSGCs was restricted to the varicosedistal arbors (see also [38]). However, cofasciculation of SACs dendrites raises theintriguing possibility of asymmetrical bipolar input impinging on SACs within theplexus of the network, which could play a functional role in directional selectiv-ity. There are two paramorphic types of SACs: the ON SACs that stratify withinsublamina b of the inner plexiform layer (also known as cb SACs) and OFF SACswhich stratify within sublamina a of the inner plexiform layer [14]. ON-OFF DSGCshave bistratied morphology entwined with dendrites from both ON and OFF SACs

[4, 34, 35, 39].The cellular inhibitory behavior manifesting in dendritic processes of individualSACs as described in this paper most likely underlies the formation of directionalselectivity. Moreover, it is robust through morphological constraints involving asym-metrical excitatory connectivity with cone bipolar cells and reciprocal synapsesbetween cone bipolar cells and SACs. Recent electrophysiological studies have iden-tied a reportoire of different types of ionic currents (and ionic uxes converted to

Fig. 1. A uorescence image of a Lucifer yellow-lled single starburst amacrine cell in the rabbitretina. Scale bar = 50 µm.



Cellular Inhibitory Behavior Underlying the Formation of Retinal Direction Selectivity 301

currents) in SACs mediating γ -aminobutyric acid (GABA)-dependent directionallyselective responses. An overview of the literature leading to the presynaptic co-transmission model of retinal direction selectivity is described (see Section 2), basedupon light-independent transporter-carried release of GABA, and light-dependentrelease of acetylcholine (ACh) from SACs. Calcium (Ca 2+ ) bias to somatofugalmotion in SACs results from a dendritic-autonomous mechanism of the synaptology,which included co-transmission in terms of a conjoint release of excitation/inhibition(i.e., a bi-synaptic relay of exogenous ACh and GABA) in distal varicosities of indi-vidual SACs. The transmission is non-diphasic since ACh release is preferentiallysuppressed by synaptically and extrasynaptically positioned GABA A receptors vialigand-gated chloride (Cl − ) channels. Tonically mediated inhibition activated byGABA B receptors reduces [Ca 2+ ]i levels in the distal dendrites of SACs (see Sec-tion 3). These inhibitory mechanisms emanating from the starburst network plexusin the inner plexiform layer of the rabbit’s retina are directly responsible for theformation of directionally selective responses of ganglion cells. In addition, the spa-tial asymmetry imposed by the cofasciculation of SACs with glutamatergic conebipolar cells; and the laterally offset inhibition from SACs negating preferentially,direct glutamatergic cone bipolar inputs onto ON–OFF DSGCs through reciprocalsynapses between bipolar cell axon terminals and SACs, underlie the retinal circuitryof directional selectivity in the primate retina (see Section 4).

Few attempts have been made at quantifying the new experimental data in

terms of mathematical equations that would elucidate the underlying mechanismsof directional behavior in the network of SACs. The development of network modelsof the dendritic plexus of SACs without recourse to arbitrary discretization protocolsinherent in compartmental models thus far attempted (see e.g., [31 , 77]) have notbeen imminent. In order to model biologically realistic networks of SACs, both thespatial structure of single dendrites and the asymmetrical transmitter release that isdependent on the subcellular computation in the terminals of the SACs have to bedeveloped (see Section 5). Such models utilize a rich repertoire of voltage-dependention channels present in the dendrites of SACs. In this paper, a formulation of the

equations underlying directional behavior in the network of SACs is proposed. Itis clear that commonly used models that use compartments to model SACs areinappropriate, since spatiotemporal integration is pivotal in understanding the cel-lular inhibitory behavior underlying the formation of retinal direction selectivity.This paper presents a model of a network of SACs and the circuitry underlying theformation of directional selectivity in the mammalian retina.

2. The Road to the Presynaptic Model Underlying DirectionSelectivity in the Mammalian Retina

The classic electrophysiological study by Barlow et al. [9] rst characterized thereceptive eld properties of directionally selective units intraocularly recorded inresponse to stimulus movement across the visual streak of anesthetized rabbits.



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When a stimulus in the visual eld was moved along the preferred axis, spikes wereobserved, but almost none or very few spikes occurred in response to a stimulus mov-ing in the opposite or null direction. The receptive eld map was shown to containmultiple such directionally selective units, all with the same preferred null axis. Thissuggested that the mechanism for its operation was localized within a small areaof the receptive eld. Barlow and Levick [11] proposed two different “black box”mechanisms of directional selectivity: an excitatory mechanism and an inhibitorymechanism. They favored a mechanism mediated primarily by inhibition, activatedby null-direction movement. A further physiological property of retinal directionselectivity was observed by showing that the receptive eld of a directionally selec-tive unit has a non-discriminating zone, which is a laterally displaced inhibitionresembling a cardioid shape [117].

The rst intracellular recordings by Dacheux [26] from the soma of a ON–OFFDSGC in the rabbit’s retina provided evidence of hyperpolarization, and there-fore physiologically identied an inhibitory mechanism. Strong support for suchan inhibitory model was further provided by pharmacological experiments, whichshowed that GABA receptor antagonists abolished directional selectivity [6 , 20]. Butbecause retinal cell types other than DSGCs expressed GABA receptors, it was notclear if the inhibition occurred pre- or postsynaptically to the DSGC. Therefore, twocellular mechanisms were postulated: (1) a presynaptic scheme, where the inhibitionimpinges directly on bipolar cells [10 , 98] or cholinergic amacrine cells [69]; and (2)a postsynaptic scheme, where the inhibition impinges on the dendrites of DSGCs[106]. However, without a clear understanding of the synaptology and morphology of the circuitry involved, the cellular mechanism underlying retinal direction selectivityremained open to further research.

The rst support for a presynaptic scheme came from Dowling [29], who proposedthat a serial synapse involving an amacrine cell is centrally involved in the direction-ally selective mechanism in the mudpuppy ( Necturus maculosus ). Further, intracel-lular recordings in the turtle retina made by Criswell [25] and later Borg-Graham[15] suggested that directionally selective bipolar and amacrine cells contributed to

the directional responses produced by DSGCs. On the other hand, Torre and Poggio[106] hypothesized a postsynaptic scheme based on shunting inhibition that couldmediate the directionally selective mechanism. In support of the Torre-Poggio model,Koch [61] speculated that the locus of retinal direction selectivity might involve local-ized interactions between inhibitory inputs that were delayed relative to excitatoryinputs, as a consequence of their spatial juxtaposition on a DSGC. This meant that,in the preferred direction, the spatially offset inhibition is delayed and arrived toolate to negate excitation, while in the null direction, the delayed inhibitory synapsematched excitation because it is spatially offset relative to the excitatory input. For

such an arrangement to satisfy Barlow and Levick’s observation that inhibition actslocally within the receptive eld of the DSGC, Koch-Poggio-Torre proposed an “on-path” rule where an inhibitory synapse is always located proximally to the soma




with respect to an excitatory synapse. However, the on-path rule required nonlinearinteractions to be located on ne distal dendrites in order for the veto mechanism towork, leaving a “hole” in the receptive eld; and the postsynaptic scheme requiredlong-range inhibitory interactions not seen in Barlow and Levick’s experiments.

Taylor et al. [102] measured the postsynaptic currents using a whole-cell patchclamp on the soma of a DSGC. They depolarized the membrane potential to − 30mVand measured the net outward current as a result of a moving light stimulus, reveal-ing that inhibition was postsynaptic. The problem with such an experimental set-upis that it did not reveal synaptic currents elicited in the ne distal dendrites wherethe postsynaptic scheme was believed to take place (see Koch [61]). The experi-ments did show that postsynaptic inhibition impinges on the soma in generatinga directionally selective response, but the experiments did not rule out the possi-bility that ne distal dendrites (as opposed to the soma) could in fact be drivenby a presynaptic mechanism. The experimental studies only reected the net out-ward current as an average of the whole neuron measured at the soma. If the nedistal dendrites are isolated from the cell body, then the somatic whole-cell patchclamp will not measure the excitatory postsynaptic currents generated in the nedistal dendrites because of a poor space-clamp. Abolishing the directionally selec-tive response by lling the ganglion cell with a solution of Cl − ions only indicatedthat the outward current is abolished, but does not prove whether the directionallyselective excitability arose pre- or postsynaptically. In other words, the experimentperformed by Taylor et al. [102] did not reveal the locus of the directionally selectivemechanism, but instead indicated the presence of GABAergic-activated Cl − chan-nels on the somata of DSGCs. Such non-local directionally selective responses canbe interpreted as long-range interactions not seen in the classical Barlow–Levickexperiments.

Since then the morphology of DSGCs has been described by Amthor et al. [5]and Oyster et al. [83], revealing a convoluted branching pattern. Such convolutedbranching patterns would require each localized unit to show identical null-preferreddirections, necessitating a highly hard-wired presynaptic circuitry being imposed on

such a postsynaptic scheme of retinal direction selectivity (cf. [61 , 74]). To elucidatewhether inhibition occurs postsynaptically, various receptors or receptor subtypesneed to be mapped onto morphologically and physiologically identied DSGCs,using a combination of electrophysiological recording, intracellular injection andreceptor immunohistochemistry. Jeon and colleagues [54 , 55, 65] accomplished suchexperiments by using a triple-labeling technique. They provided conclusive micro-pharmacological evidence of random distribution of both nicotinic cholinergic andglutamatergic receptors on dendrites of DSGCs. Similar conclusions were generatedby Dacheux et al. [27], who used electron microscopy to reconstruct the contacts on

individual segments of the dendritic arbor of a DSGC. This kind of experimental evi-dence helped to argue against the notion that direction selectivity can be computedby excitatory inputs shunted by on-path inhibition on the dendrites of DSGCs. As



304 Poznanski

a result, the idea of a presynaptic scheme, where cholinergic amacrine cells play anactive role, became a viable possibility.

Vaney et al. [114] proposed that the proximo-distal segregation of the inputand output synapses could provide the spatial asymmetry that is necessary for thegeneration of directional selectivity. This could be done if the dendrites on differentsides of the SACs provided selective output to the DSGCs with different directions.Given that the cholinergic SACs were found to synthesize and store GABA [115], aco-transmission model was postulated, where SACs released either excitatory (ACh)or inhibitory (GABA) neural transmitters, depending on whether the somatofugaldirection fell on the preferred or null direction, respectively (see also [75 , 111, 114]).A SAC pointing in one radial direction will both selectively excite a DSGC with amatching preferred direction, as well as selectively inhibit an overlapping DSGC withan opposite preferred direction (see also [109]). Although each varicosity was capableof mediating conjoint excitation and inhibition (i.e., co-transmission), the response isnever diphasic, if the dendrites of SACs are electrotonically isolated from each other,as shown with passive cable modeling [76]. The co-transmission model, however, didnot explicitly show how both GABAergic and cholinergic inputs asymmetricallyinteract onto a DSGC. In other words, the retinal circuitry was not delineated withprecision, and since the notion of co-release undermines Dale’s law, which positedthat one transmitter is released from a synapse, therefore a presynaptic scheme withtwo distinct SACs rather than a co-transmission model pre-dominated the literature(see [30, 47, 53, 113]).

In a modeling study undertaken by Poznanski [88] (see also [87]), a presynapticco-transmission model was developed, where the “directional locus” correspondedto the length of dendritic segments in the distal dendritic zone of SACs requir-ing lateral inhibition for directional selectivity (see also [66]). In this presynapticco-transmission model, the asymmetric cholinergic pathway is different from theBorg-Graham and Grzywacz model [16], in that the GABAergic input is distributeddistally and not throughout the entire dendritic tree of a SAC. The presynapticco-transmission model required each SAC to show a signicant reduction of the

membrane potential in the somatofugal, but not the somatopetal direction as aresult of direct GABAergic contact onto the distal dendrites of SACs. Poznanski[88] was the rst to show this behavior of SAC through the use of passive cablemodeling, although it remained unclear how the results would differ if active mem-brane properties were to be included in the model [48].

Bloomeld [12, 13] recorded from the cell bodies of mature SACs to show thatthey were capable of generating Na + spikes and Zhou and Fain [123] recorded Na +

spikes from immature SACs. Hausselt et al. [48] recorded spike-like [Ca2+ ]i tran-sients from mature SACs and Zheng et al. [121] recorded Ca2+ spikes mediated

by Ca2+

-dependent potassium currents in immature SACs. Thus, synaptic currentscannot be propagated passively, as in a leaky cable, but must be amplied by avoltage-dependent regenerative process. This brings into question the possibility that




SACs dendrites are isolated as required in the presynaptic co-transmission model.Hence, the isolation criteria for asymmetrical activity must be reconciled with theactive membrane properties of SACs. If SACs show regenerative membrane prop-erties then it would be difficult to justify the existence of electrotonically isolateddendritic compartments within SACs (cf. [116]). The earlier models of SACs incor-porating active membrane properties were based on inadequate assumptions (seefor example, [107, 116]). For instance, Velte and Miller [116] developed a compart-mental model of a single SAC with active ionic currents found in ganglion cellsfrom intact amphibian retina. Such modeling should be treated with skepticism, asthere is a serious hazard in generalizing across species. While Tukker et al. [107]included only Ca 2+ channels in a compartmental model of SAC dispensing with theneed for inhibition on the grounds that directionally selective dendritic Ca 2+ signalin SACs arises from their distinctive morphology. The validity of this assumptionseems unlikely to hold because it imposes a high degree of specicity in the connec-tions between the DSGCs and SACs. Such unnecessary complexity is avoided in theparsimonious presynaptic co-transmission model described here.

Euler et al. [32] showed that the dendritic tips of a single SAC displayed direction-ally selective cytosolic Ca 2+ by recording optically the light-evoked Ca 2+ responses.These tended to be greater for somatofugal than somatopetal stimuli, and were alsoindifferent to the presence of GABA A receptor blockers (e.g., picrotoxin), implyingthat inhibition is not necessary to generate somatofugal bias in individual SACsand therefore suggesting that the bias to somatofugal motion is driven by intrinsicelectrical properties of SACs, such as cytosolic [Ca 2+ ]i reduction via a Ca 2+ − Na+

exchanger current [50] and a Ca 2+ transporter current. Although these ndings ruledout an early hypothesis that the electrotonic generation of directional selectivity inSACs is augmented by on-the-path inhibition between the soma and the termi-nals (cf. [16]), these ndings do not diminish the role of GABA A receptors in AChsuppression as a pivotal requirement for the mechanism of directional selectivity,since at the network level, the presynaptic inhibition caused by GABA A autore-ceptors responsible for a prolonged negative feedback preventing any increase to

Ca2+

-dependent responses in the somatopetal direction would not be affected bythe GABA A receptor blockers infused into the single SAC recorded by Euler et al.[32], which may have no effect upon extrasynaptically coupled SACs within the net-work for the operation of directional selectivity. More recently it was suggested thatsomatofugally biased response to motion of SACs accounted for directional asymme-try without the need for synaptic inhibition [48]. However, this criteria alone wouldbe insufficient to account for directional signals observed in SACs, especially whenthe stimulus moved somatopetally from outside the receptive eld of the SACs [124].Indeed, localization of dendritic Ca 2+ signals in SAC is insufficient for the operation

of directional selectivity which is known to depend on mostly electrogenic synapses,and GABAergic [120] and cholinergic connectivity [73].



306 Poznanski

Fried et al. [42] used a dual-patch clamp technique by clamping the DSGC’svoltage to directly record excitatory and inhibitory currents and therefore infer onthe nature of the inputs. By electrically stimulating through light stimulus, a SACwhose dendrites overlapped a DSGC on its null side recorded an inhibitory cur-rent in the DSGC, but when the SAC was on the DSGC’s preferred side, electricalstimulation did not affect the DSGC. They hypothesized that each SAC dendritepointing in different directions would provide the GABA inhibition somatofugally toDSGCs with different preferred directions (i.e., the processes of SACs are connectedasymmetrically to DSGCs). In their model, SAC processes on either side of thesoma contact DSGCs with opposite preferred null direction, with directional selec-tivity being formed by laterally offsetting inhibition. Similar studies have shown thatinhibitory inputs to DSGCs are themselves directionally selective; implying that reti-nal direction selectivity occurs predominantly because of presynaptic mechanisms(excitatory and inhibitory) to DSGCs [43].

Taylor and Vaney [103] showed that for “robust” directionality there is a need forboth presynaptic and postsynaptic scheme in the rabbit retina. The precise wiringleading to a postsynaptic scheme of directional behavior involving non-robust direc-tional selectivity was proposed by Schacter et al. [95] requiring the “ on-path-rule ” forlong-range inhibitory interactions as well as electrotonically isolated patches inter-mingled with dendritic “ hot-spots ” in DSGCs for local excitatory and inhibitoryinputs to initiate dendritic spikes. Both long-range inhibitory interactions and “ hot-

spots ” of excitability are controversial; their conclusions predicated that the rationalefor a presynaptic scheme was to merely assist a postsynaptic scheme in deliver-ing robust directionality because the latter scheme’s inuence would always decayalmost to zero near the soma of a DSGC. However, their predications are awedbecause they include active channels in a compartmental model of a DSGC withmetrics used for dendritic isolation based on passive theory. For instance, they mea-sure how directionality is affected by changes in the space-constant in a passive in-nite cable. One way to circumvent such problems is to superimpose active channelson the passive membrane (see Appendix), assuming a sparse density distribution of

active channels for dendritic spike propagation as opposed to the Hodgkin-Huxleyapproach for action potentials based on a continuous distribution of active channelsin unmyelinated axons. Preliminary work in this direction has revealed no dendriticisolation [90, 92].

3. The Synaptology by which the Starburst Amacrine Cells Cometo Favor Somatofugal Motion

The purpose of this section is to provide the precise synaptology in the design of the underlying retinal circuit which is understood to be involved in the mehanismof directional selectivity.

As shown by Euler et al. [32], dendritic Ca 2+ responses to light stimuli arelocal and independent of each dendritic branch of a SAC. The decrease of cytosolic




[Ca2+ ]i in the somatopetal direction was not blocked by GABA A receptor antago-nists, implying that this decrease could be mediated by either intrinsic properties of SACs, such as P-type Ca 2+ channels [32], or by the suppression of tonic input frombipolar cell terminals by GABA

Creceptor-mediated presynaptic inhibition [70], or

the spatial asymmetry in distribution of Cl − transporters along the SAC dendrites[44]. Input from bipolar cells onto DSGCs and SACs is tonic, and the tonic inputfrom bipolar cells is suppressed by SACs [32]. However, the small dendritic diame-ter of SACs with a small volume-to-surface ratio would favor metabolic pump cur-rents and exchanger currents in maintaining ionic uxes, thereby effecting cytosolic[Ca2+ ]i levels. To resolve this dilemma, Gavrikov et al. [44] postulated that the bio-physical substrate underlying somatofugal motion bias must include an asymmetricdistribution of two co-transporters along the SAC dendrites. However, this asymme-

try depends on the distributed properties of SACs with Na+

–K+

–Cl−

(transportsCl− into cells) located proximally to the soma which depolarizes the dendrites andK+ –Cl− (extrudes Cl − from neurons) distally located which hyperpolarizes the distaltip of SACs.

Stimulus movement in the somatopetal direction evokes hyperpolarization in theSACs, which is K + –Cl− co-transporter-dependent and likely mediates the responseof SACs to stimulus in the somatopetal direction. Gavrikov et al. [44] also suggestedthat the asymmetry could take place because GABA could act as either an excita-tory or inhibitory neural transmitter depending on the activity of Cl − transporters:

(1) Na+

–K+

–Cl−

(transports Cl−

into the cell) which is proximally located anddepolarized the dendrite because E Cl > E rest ; (2) K+ –Cl− (takes out Cl − ) distallylocated and hyperpolarizes the dendrite because E Cl < E rest . Such a neural circuitryis capable of generating a hyperpolarization in the null direction, although it doesnot explain the preferential release of GABA, in the context of a co-transmissionmodel, at the locus where the output of SACs directly applies. Furthermore, it isunclear if the transporter activity can signicantly alter Cl − concentration.

Intracellular Ca 2+ levels do not produce directional selective release of GABA;it is some other mechanism that produces the asymmetry in [Ca 2+ ]i level increases

in the rst place. The GABA transport system in SACs has multiple transmissionmodes which could explain this asymmetry. Most of the synaptic GABA releaseis Ca2+ -dependent although transporter-mediated release also exists under cer-tain conditions. There are several mechanisms experimentally observed in SACs.The rst mechanism involves non-vesicular and Ca 2+ -independent carrier-mediatedexogenous GABA release. This mechanism is light-independent and dependent ondepolarization rather than on Ca 2+ inux for fusion of vesicles with the plasmamembrane [82]. Depolarization causes I GABATransporter to move GABA outward of the SAC. As GABA carrier is electrogenic the net direction of GABA ow would

depend on the GABA concentration on each side of the SAC membrane. The secondmechanism involves tonically mediated inhibition of SACs through GABA B recep-tors [126] which is light-independent vesicular release that is Ca 2+ -dependent and



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uptake is by I GABA into the SAC. The third mechanism involves light-dependent viaCa 2+ -dependent vesicular release where both synaptic and extrasynaptic GABA Areceptors mediate the negative feedback current I Cl through Cl − channels hyperpo-larizing the SAC [120, 122].

How do SACs co-release GABA and ACh? Light-evoked vesicular release of AChis Ca2+ -dependent [82], and GABA has been shown to suppress the release of AChfrom SACs in the rabbit retina [71]. In addition, extrasynaptic autoreceptors (i.e.,GABA A receptors) cause a suppression of ACh release, a conjecture rst proposedby Zhou and Fain [122] after GABA A receptors were found to be expressed on SACs.Removing inhibition by picrotoxin would increase the overall release [81]. However,pharmacological evidence by Linn and Massey [67] argues against this mechanismbecause when kainate (a glutamate receptor agonist) was used to induce ACh release(bypassing the bipolar cells), the release was not decreased by muscimol (a GABAagonist). Based on measurements of ACh release, they proposed that GABA actsindirectly on SACs via GABAergic feedback synapses on bipolar cell terminals.

GABA receptors are an essential requirement in the hypothesis that the SACnetwork provide a direct substrate for the asymmetric GABAergic inhibition. Zhenget al. [120] used dual-patch clamp to show those GABA A receptors are on SACs.Firth et al. [41] have used a combination of pharmacological methods and immuno-labeling to show AMPA receptors are present on SACs. This is important becauseAMPA receptor would allow SACs to respond more effectively to rapidly changingstimuli than AMPA-KA receptors as was previously suggested by Poznanski [91].Starburst amacrine cells receive synaptic input from cone bipolar cells and otherSACs [17, 38, 73]. Cholinergic–cholinergic synapses in the rabbit retina are possi-ble because SACs express nicotinic cholinergic receptors, and also SACs receivedirect GABAergic, cholinergic and glutamatergic inputs [58]. SACs could inhibiteach other via GABAergic synapses, since these cells synapse onto each other. Star-burst amacrine cells using GABA to communicate with each other is a componentin the network responsible for directional selectivity [120].

Zheng et al. [120] showed experimental evidence of GABA A receptors on SACs,

thus providing convincing evidence for SACs being a direct site of action for GABAto suppress ACh release. Using dual-patch clamp, SACs were shown to synapse ontoother SACs via reciprocal GABA A synapses that mediated light-evoked co-releaseof ACh and GABA. Furthermore, blocking GABA A receptors did not cause lossin directional selectivity while increasing the light-stimulated release of ACh [81].This provides indirect evidence that GABA release to neighboring SACs is throughI GABATransporter and that the GABA A receptors on the same SAC block the AChresponse, leading to preferential GABA release. Also carrier-mediated release of GABA was shown to be light-independent [82]. GABA release is carrier-mediated,

meaning that it depends on the elevated level of cytosolic [Ca2+

]i . When the pro-longed wave of excitation travels in the somatofugal direction and reaches the distalvaricosities containing Ca 2+ channels, the depolarization cause a Ca 2+ inux thatpromotes an increase in cytosolic [Ca 2+ ]i . This in turn promotes an efflux of K +




through Ca 2+ -dependent K + channels and carrier-mediated efflux (outward move-ment) of GABA is induced by decreases in the cytosolic [Ca 2+ ]i levels in SACs.Evoked hyperpolarization is K + –Cl− co-transporter-dependent and mediates theSACs response to stimulus movement in the somatopetal direction by localizing thespread of cytosolic Ca 2+ in the somatopetal direction. In the null direction, the SACdendrites are hyperpolarized because of the activity of the K + –Cl− co-transporteror possibly by the inux of K + through an inward rectifying K + current. As a result,GABA release from distal tip of SAC will be minimal.

Cohen [23] found N (CaV2.2) and P/Q-type (CaV 2.1) Ca 2+ channels present onSACs. These channels are high voltage activated Ca 2+ channels. However, N-typeCa2+ channels were located on the soma of SACs and are therefore not involvedin synaptic release. Calcium channels can shape the light response of neurons by

activating the Ca2+

-dependent K+

current. The bipolar synaptic release of gluta-mate was being mediated by T- and L-type Ca 2+ channels [57]. Jensen [52] foundthat P/Q-type Ca 2+ channel antagonists abolished directional selectivity. Activa-tion of a sustained depolarization can activate Ca 2+ channels (such as P/Q types).Hausselt et al. [48] observed large spike-like Ca 2+ transients riding on top of depo-larizating responses. If the dendrites of SACs favor somatofugal motion, a pro-longed wave of excitation propagating somatofugally may play a role in somatofugalbias to motion by depolarizing distally located varicosities. This elevates cytoso-lic Ca2+ to activate a K + current and signal transduction pathways, resulting in

GABA release through a GABA transporter. The release of exogeneous GABA inthe extracellular uid binds to GABA A receptors, which in turn causes a hyper-polarization through Cl − channels on the postsynaptic SAC, as well as on thesame SAC. The latter is referred to as GABA-induced extrasynaptically positionedautoreceptors.

Patch-clamp experimental data revealed sodium (Na + ) to be a major voltage-dependent ion channel responsible for a regenerative active mechanism of conduc-tion in the soma of SACs [23]. In current-clamp recordings of the SACs, stimulationby light resulted in a graded depolarization on which spike-like events were super-

imposed. These were unaffected by TTX, but were abolished by adding quinoxa-line (NBQX, CNQX) (an AMPA antagonist), i.e., they depended on postsynapticglutamate receptors (AMPA). The characteristic response consisted of a transientdepolarization at the onset of light with a burst of spikes riding at the top of alarge-amplitude, sustained depolarization [7]. The prolonged inward Na + current iseither a persistent current or an exchanger current. Such a sustained depolariza-tion may also arise from TTX-resistant Na + channels (NaV 1.8) expressed on thesomata of SACs in mouse [79] and rabbit retina [80]. Therefore, Na + currents inthese neurons do not propagate spikes, but enhance the voltage responses evoked

by visual stimulation. Although persistent Na+

and Ca2+

currents are known to beacting synergistically in cultured amacrines (see [63]) in order to prolong subthresh-old depolarization, the evoked depolarization in SACs is suprathreshold. But noevidence of Na+ spikes has been found by recording from somata of SACs, [85 , 105]



310 Poznanski

suggesting that another type of channel could be involved. One such possibility isa Na+ -dependent current restricted to the soma that activates the sustained depo-larization, while an exchanger current maintains the prolonged threshold wave of excitation in the somatofugal direction.

A summary of possible activities explaining how SACs come to favor somatofu-gal motion is illustrated in Fig. 2. This provides a plausible explanation why thecytosolic Ca 2+ signal is somatofugally motion biased [32].

Fig. 2. Conceptual model of a SAC dendrite and the role of various ionic currents underlyingsomatofugal motion bias of Ca 2+ . The biophysical substrate of the nonlinearity favoring somatofu-gal motion of Ca 2+ remains the input–output segregation, with glutamatergic excitatory synapticinputs from cone bipolar cells mediated by AMPA receptors [41], and cholinergic/GABAergic out-puts localized at the distal end of the starburst dendrite [88 , 91]. The cytosolic Ca 2+ is locallydistributed near the terminal end (as indicated by the arrows). The distal varicosity is capableof mediating conjoint excitation and inhibition, also known as co-transmission. However, in thesomatofugal direction the cholinergic input is preferentially suppressed by self-inhibitory autore-ceptors (i.e., extrasynaptic GABA A receptor [122]) mediating a prolonged negative feedback thatprevents an increase in the Ca 2+ -dependent responses in the somatopetal direction [32]. Glossaryof various ionic currents found in SACs of the rabbit’s retina: I Na — transient inward sodiumcurrent [13 , 23]; I GABATransporter — GABA transporter current [82]; I Ca — calcium inward cur-rent (P/Q-type) [23]; I K(Ca) — calcium-activated potassium outward current [8, 116, Zhou (per-sonal communication)]; I Cl — chloride inward current mediated by synaptic GABA A receptor[120] and extrasynaptic GABA A receptors [122]; I GABA — tonic inhibitory inward current medi-ated by extrasynaptic GABA B receptors [126]; I Na − K − Cl — the sodium, potassium, and chlorideexchanger current (passive transport) [44]; I K − Cl — potassium and chloride exchanger current

(passive transport) [44]; I Na − Ca — sodium and calcium exchanger current (passive transport) [44];I Ca(ATP) — calcium co-transporter current (ATP) [44]; I K — inward-rectifying potassium current[84] (in the mouse retina); I NaP — prolonged inward sodium current (TTX-resistant) [79] (in themouse retina) [80].




Step 1: Somatic transient Na + spikes are initiated by light but rapidly shuntedby the very thin proximal dendrite diameter under 1 µm; subsequent activation of glutamatergic excitatory synaptic inputs from cone bipolar cells mediated by AMPAreceptors onto SACs [41] results in a sustained (prolonged) depolarization. ThoseSACs without direct connectivity with overlaying cone bipolars, the somatic Na +

spikes, although shunted by the thin dendrites, may still contribute to the sustainedprolonged depolarization within the dendrites.

Step 2: Na+ –K+ –Cl− co-transporters depolarize the membrane by inux of K +

and Na + , and efflux of Cl− , driving the sustained depolarization in the somatofugaldirection to the distal dendrites of SACs [44]. Bipolar glutamatergic input activityvia AMPA receptors while urther maintains the depolarization in the somatofugaldirection while Na + P TTX-resistant channels depolarize the membrane and drivethe sustained depolarization in the somatofugal direction [80].

Step 3: If the sustained depolarization arriving at the distal varicosity of a SACdendrite is of sufficient magnitude, it will increase the [Ca 2+ ]i level as a result of Ca2+ inux through P/Q-type Ca 2+ channels. There is also an efflux of K + ionsthrough I K(Ca) channels (Ca 2+ -activated K + channels), resulting in a build-up of extracellular K + . If there is presynaptic GABAergic synaptic input onto receptorsof postsynaptic SACs activated through GABA A receptors, then a hyperpolarizationof the distal tip can prevent GABA from being released by the cytosolic Ca 2+ signal.

A reduction in the cytosolic [Ca 2+ ]i could also result in little or no release of AChfrom the same SAC, due to a low concentration of Ca 2+ available for vesicular releaseof ACh. Thus, GABAergic synaptic input suppresses the effect of released AChonto the membrane of a postsynaptic SAC through nicotinic cholinergic receptors.Although low levels of nicotonic cholinergic autoreceptors on SAC were observedpossibly as a result of ACh transmission at distal varicosities being difficult to detectat the soma due to poor space-clamping of dendrites [120]. In addition, ACh isreleased through ligand-gated cationic channels, which occur only if an overlappingpresynaptic SAC is rst activated in the somatopetal direction. The light evoked

release of ACh is Ca 2+ -dependent for vesicular release [120].Step 4: Depolarization activates signal transduction pathways, causing non-vesicular, carrier-mediated GABA release by activating GABA transporters [82].The quantity of GABA transmitter released from a varicosity is greater for largerdepolarizations, since the enzyme, glutamic acid decarboxylase (GAD) synthesizesGABA inside the SACs [18]. Upon release, exogenous GABA binds to either extrasy-naptic GABA A autoreceptor or postsynaptic GABA A receptors (i.e., ligand-gatedCl− channels), causing the membrane to hyperpolarize either in the presynaptic or

postsynaptic SAC, respectively.Step 5: The GABA A receptor-mediated postsynaptic inhibition current ( I Cl ) medi-ated by GABA A synaptic receptors causes a suppression of tonic input from bipolar



312 Poznanski

cells, and therefore decreases cytosolic Ca 2+ , while the presynaptic inhibition cur-rent ( I Cl ) mediated by GABA A autoreceptors suppresses the ACh response. Bothare mediated by Ca 2+ -dependent vesicular release of endogenous GABA. This exem-plies the functional role of Ca 2+ in transduction, because direct depolarizationwithout Ca 2+ inux does not release GABA.

Step 6: The negative feedback that produces a prolonged release of GABA (light-independent) [120] is deactivated by K + –Cl− co-transporters that extrude Cl − ionsfrom the inside of SACs and deliver K + back into the cell preventing further GABAfrom being released. In the mouse retina, an I K current tends to stabilize the restingmembrane potential and contribute to the functional independence of the distaldendrites of SACs [62 , 84].

Step 7: The dendritic Ca 2+ signal is further reduced in the somatopetal direction byNa+ − Ca2+ exchanger current which uses the Na + concentration gradient as energyto move Ca 2+ back into the extracellular space (primary efflux mechanism) andNa+ into the cytoplasm; Ca 2+ efflux is also aided by a Ca2+ transporter along thedendrite using adenosine triphosphate (ATP). In the mouse retina, it is proposedthat a I K current provides voltage-dependent shunt that limits depolarization tothe soma and provides a mechanism likely to contribute to the electrical isolationof individual SAC dendrites. This is achieved in the somatopetal direction, becausethe closer the hyperpolarization activated I K current is to the soma, the higher itsdensity.

4. Directional Selectivity Emanating from the Starburst Network

The laser-irradiated partial disruption of a limited number of rabbit SACs (e.g.,20–60 displaced SACs in the ganglion cell layer, not in the inner nuclear layer) stillmaintained the directionally selective responses of ganglion cells [49]. However, acomplete SAC ablation abolished directional selectivity in the mouse [119] and rab-bit retina [3] suggest that the substrate of retinal direction selectivity resides in anextremely dense plexus of SACs. Starburst amacrine cells have a high degree (20- to30-fold) of dendritic overlap [34–36]. It is therefore unlikely that SACs convey direc-tion selective information to DSGCs by virtue of their radial geometry, since theminimum distance over which directional selectivity can be detected is 43–86 µm,which corresponds to the length of dendritic segments in the distal dendritic zone of SACs [40]. The functional independence of such SAC “subunits” as originally postu-lated by Miller and Bloomeld [76] is partly a question of their network organization.

Earlier models postulated that the direction selective mechanism is associatedwith the SAC morphology in which a single SAC during somatofugal motion requiredthe null direction for releasing of GABA, and during somatopetal motion the pre-ferred direction is required for releasing ACh [109]. The problem with such a schemeis that each SAC must use some developmental specialization to preferentially con-nect synaptically to DSGCs. If SACs release GABA inhibition somatofugally, then




they can only connect with different DSGCs, so that each side of the SAC releasestransmitters in the null direction of a particular DSGC. This was necessary to avoidthe SAC from responding diphasically in time to a conjoint excitatory and inhibitoryinput. However, a study by Dong et al. [28] showed that directional selectivity doesnot result from geometrically asymmetrical contacts between the DSGCs and SACsprocesses. This nding, which shows DSGCs having tight cofasciculation, with noapparent specic SAC connectivity for induction of directional selectivity, suggeststhat the substrate of retinal direction selectivity is within a network. Therefore,such unnecessary complexity is not a requirement in the model where the requiredasymmetry is the spatial arrangement of glutamatergic cone bipolars on dendritesof SACs within the network of SACs, as opposed to a single SAC. Hence, the mech-anism of directional selectivity does not correlate with the morphology of singleSACs, but is a manifestation of the topology of the network, yet dependent on theintrinsic electrical characteristics of individual SACs.

When the network of SACs generates a directional response, this response mustbe maintained in all the four known directions (for ON–OFF DSGCs), requiringSACs in this network to acquire the same type of specic synaptic connectivity toDSGCs unless the SACs plexus is randomly arranged [100]. Inhibition is transmittedwithin the sphere of either the ON or OFF pathways. A single SAC makes contactwith ON and ON–OFF DSGCs that differ in their preferred directions [40]. One pos-sible hypothesis to explain this cross-pathway inhibition is that networks of multipleSACs are involved (cf. [68]). Interactions between SACs have not been taken intoconsideration nor does it explain why, in the preferred direction, excitation throughACh release has only been observed in the OFF sublamina [43].

The synaptology of the rabbit DSGC was examined with electron microscopy byDacheux et al. [27]. The mosaic of overlapping SAC dendrites was shown to displaya regular lattice containing up to ve dendrites bundled together to form a fasciclein the inner plexiform layer strata [101 , 110, 112]. Clusters of SAC processes providesynaptic input to DSGCs. These processes established a chain of three neuronsserially synapsing with each other before synapsing directly with the DSGC dendrite.

Another presynaptic conguration providing synaptic input to the DSGC dendritesconsisted of a “ synaptic couplet ” in which two SACs synapsed with each otherbefore one of them synapsed with a DSGC (see also [73]). Other studies advocatedthat the synapses between two cholinergic processes may not be cholinergic andare most likely GABAergic [19 , 81], because SACs express GABA receptors [45 , 120]and GABA-activated currents [122]. GABAergic amacrines play an essential rolein directional selectivity since blockade of GABA A receptors on DSGCs eliminatesdirection selectivity [6 , 20, 60], but see the contradictory result of Euler et al. [32].Brief lasting postsynaptic potentials are mediated by GABA A receptors [99].

Cholinergic input onto DSGCs is not required for directional selectivity sinceACh antagonists (e.g., curare) do not abolish it [60]. In other words, blockingcholinergic input does not abolish directional selectivity [6 , 24, 59]. Most of thenon-cholinergic excitatory input to DSGCs appears to be mediated by NMDA



314 Poznanski

(N-methyl-D-aspartic acid) receptors. Cohen and Miller [24] report that most of the glutamate input to DSGCs is mediated by NMDA receptors (slow). While nico-tinic antagonists did not abolish directional selectivity in DSGCs, the application of quinoxaline AMPA (non-NMDA) antagonists (CNQX, NBQX) did so [24 , 60]. Non-NMDA receptors (e.g., AMPA) are involved in the presynaptic release of endogenousGABA on SACs [72]. Although both NMDA and ACh antagonists did not abolishdirectional selectivity, non-NMDA antagonists, however, abolished directional selec-tivity by blocking ACh release onto SACs, causing no GABA release; while AMPAantagonists block the transmission of the directional signal from the SAC to theDSGC it does not block directional responses in the SAC [80]. Postsynaptic poten-tials up to several seconds of duration may be mediated by GABA B or GABA Creceptors on DSGCs [99]. The long duration of null inhibition remains 1–2s afterthe passage of the wave through a given location (see [2 , 99, 117]). The presynap-tic co-transmission model does not preclude a postsynaptic synaptic circuitry, suchas direct excitatory input from overlying cone bipolars through NMDA glutamatereceptors [24 , 60] and the contributions of both GABA A and GABA c receptors toACh release [70]. The precise wiring leading to a postsynaptic scheme of directionalbehavior involves non-robust directional selectivity, which has been suggested tooperate in the rabbit retina [43 , 103].

To decipher how the directional signals emerge from network interactions of SACs, a network model must incorporate the intrinsic electrical properties of indi-

vidual SACs and their cellular behavioral responses. In order to produce such anetwork model, in which the SAC also uses GABA to provide the inhibitory vetoonto DSGCs by releasing ACh to somatopetal motion and GABA to somatofugalmotion (see Section 3), extremely complex rules for controlling the spatial distribu-tion of synaptic and extrasynaptic receptors are required. Earlier network models,such as those advanced by Dowling [29], were dependent on the idea of a “ push– pull ” synaptic topology favoring an excitatory (cholinergic) model onto a DSGC(see also Fig. 1 in Ref. 88). A “ push–pull ” conguration refers to the inhibitory orexcitatory input activation onto a DSGC when it is in the null direction or pre-

ferred direction, respectively. Since GABA antagonists unmasked excitatory inputs(ACh) to symmetrically facilitate the responses of DSGCs [22], there is clear evi-dence against the “ push–pull ” synaptic topology on a SAC. The Dowling model andother models such as the co-transmission model [110 , 111] and “push–pull ” modelsof directional selectivity which depend on SACs for the excitatory drive in the pre-ferred direction, all require a very specic connectivity pattern to be imprinted ontoa single DSGC in order for the mechanism to be functional; commensurate withstarburst–starburst interactions in producing the lateral offset inhibition requiredfor negating glutamatergic bipolar inputs onto the DSGCs in a “ push–pull ” con-

guration [43 , 91, 104] seems also to be in doubt as it requires the GABAergicinputs to fully negate the glutamatergic inputs distributed on the dendritic treesof DSGCs. Hence, it is unlikely that SACs convey directionality to DSGCs as they




appear to be indiscriminate in their connections with DSGCs [40]. However, suchwiring complexity can be omitted if a preferential release of ACh is suppressed byautoreceptors through a negative feedback mechanism and GABA A receptors. Inthe primate retina, SACs make inhibitory synapses onto bipolar cell axon terminals(but not yet evident in the rabbit retina) and processes of other SACs [118]. If thesebipolars are inputs to DSGCs then instead of a “ push–pull ” conguration, therewould be SACs directionally inhibiting bipolars. Further experimental study in therabitt retina is needed.

The topography of directional selectivity resides in a plexus of criss-crossingdendrites of juxtaposed SACs consisting of: (i) serial synapses of three or moreSACs and a ON–OFF DSGC; (ii) a synaptic couplet between two SACs; (iii) areciprocal synapse between SACs and cone bipolars; and (iv) a conventional synapsebetween a SAC and a ON–OFF DSGC. The generation of retinal direction selectivityemanating from the dendritic plexus of SACs is schematically illustrated in Fig. 3.Cholinergic and GABAergic monosynaptic interactions between SACs, includingglutamatergic interactions with cone bipolar cells, are involved in robust directionalresponses. How the behavior of a network of individual SACs allows for asymmetrieson the spatiotemporal integration acquired in retinal direction selectivity? There isno asymmetry at the single SAC level, but at a network level the SACs producethe required asymmetrical synaptic input to DSGCs as a result of the geometricallyasymmetrical contacts between the SACs and cone bipolars. Indeed, the spatial

distribution of cone bipolar inputs is the missing link underlying the asymmetryneeded for directional behavior in the SAC network. The cofasciculation of SACdendrites within the plexus appears to be random [100], and therefore the precisionof which SAC drives a directionally response in a local region cannot be teased outexperimentally. The model of directional selectivity is consistent with the extantliterature on ON–OFF DSGCs of rabbit. The retinal network as shown in Fig. 3 is asummary of known physiological and pharmacological evidence on the retinal circuitin rabbit, yet it excludes contributions from cells in the outer retina (e.g., horizontalcells, Mueller cells) because the dendritic plexus of the starburst network model can

non-arbitrarily represent the distributed properties of the input from either bipolarcells or other SACs whose outputs are localized to the distal varicosities [40]. Retinaswhose ganglion units show no motion sensitivity are assumed to employ the samecircuitry with the exception that they are rarely encountered in such species.

The connectivity of the neural network is through electrogenic synapses, includ-ing mostly ligand-gated channels, but there are also other types of connectivitiesexisting in the SAC network, such as a non-cholinergic GABAergic amacrine cell(disinhibition) as proposed by Massey and Redburn [72], cholinergic excitation of a presynaptic GABAergic or glycinergic amacrine cell (indirect inhibition) as pro-

posed by Neal and Cunningham [78] involving non-electrogenic synapses includingG-protein-coupled receptors (i.e., metabotropic receptors) such as muscarinic recep-tors, metabotropic glutamate receptors (mGluRs), and GABA B receptors which



316 Poznanski

Fig. 3. A schematic illustration of the presynaptic co-transmission model’s neural circuitryunderlying the formation of directional selective responses in the rabbit retina. The synapticorganizational arrangement of a series of multineuronal synapses between SACs is random and doesnecessitate the somatofugal release of endogenous GABA. The network remains robust to direc-tionally selective responses even if GABAergic and nicotinic interactions between SACs were to bepharmacologically blocked. ACh release from SACs is non-directional [22] due to its suppression by

GABA in both null direction and the preferred direction. In the somatopetal direction, endogenousACh release from presynaptic SACs onto nicotinic cholinergic receptors (NicACh) of postsynapticSACs or the DSGC leads to direct excitation. However, in the somatofugal direction, autoreceptors(on the presynaptic SACs) through a negative feedback mechanism and concurrent GABA releasefrom the presynaptic SACs onto GABA A receptors on the postsynaptic SACs suppress ACh releasefrom the presynaptic SACs. Endogenous GABA release is mediated via carrier-dependent mecha-nisms, which can contribute to a temporal delay, required of directional selectivity to suppress theACh input (ACh release from the presynaptic SACs) onto the DSGCs when activated rst in thesomatofugal direction. The release of exogenous GABA from the SACs is through GABA trans-porter ux from the distal varicosities of SACs. Robust DSGCs are generated by a combination of increased inhibition and suppressed excitation in the null direction. Bipolar cell excitatory input

onto DSGCs is vetoed by SACs releasing GABA at its axon terminals in the null direction (seeFig. 4). Note that although only a single stratum is shown for schematic purposes, a bistratieddendritic tree is required with one stratum in each of the ON and OFF layers.

are all expressed in a variety of different amacrine cells. Using immunohistochem-ical labeling Zucker et al. [126] have shown that GABA B receptors are expressedon SACs, and a model of a SAC network underlying retinal direction selectivitywas proposed by Grzywacz and Zucker [46]. In their model, GABA B autorecep-tors on SAC would suppress voltage-gated Ca 2+ channels by activation of second-messenger cascades, thereby reducing the release of ACh, having a similar effect toGABA A autoreceptors [91]. However, the time course of action (decrease of releaseof ACh from SAC) after activation of GABA B is much slower than release of ACh by




Fig. 4. A schematic illustration of the retinal circuitry possibly underlying direction selectivity inthe primate retina. The synaptic organizational arrangement consists of an excitatory dyad synapseinvolving a cone bipolar cell which synapse onto a DSGC and a SAC. The SAC has a juxtaposedreciprocal synapse with the cone bipolar cell [118]. In the null direction (from right to left for theleft-hand side inset), the cone bipolars excite the SACs mediated by AMPA receptors on SACS,but are in turn inhibited by GABA A receptors on their axonal terminals, which in turn suppressdirect release of ACh mediated by nictonic cholinergic receptors on DSGCs.

hyperpolarization of SAC after GABA A receptors. In view of this, GABA B receptorswould be insensitive to the directionally selective stimuli for it to play the role in

the operation of directional selectivity. It is likely that GABA B receptors play therole in tonically inhibiting SACs as proposed by Massey and Redburn [72] to limit[Ca2+ ]i levels by suppressing voltage-gated Ca 2+ channels.

5. Somatofugally Propagating Potentials Inuencing Calcium-dependent Transmitter Release

Poznanski [88] hypothesized that the terminal processes of SACs could be activatedmore strongly by somatofugal motion than somatopetal motion which in turn wouldallow individual dendrites of SACs to be directionally selective. However, the endresult was produced with a uniform dendrite. The dendrites of SACs have uniquemorphology showing aring from 0.28 to 3 µm in diameter as one proceeds fromthe soma to the distal end near where the varicosities are located [88]. On thisbasis, a non-uniform dendrite may be more appropriate. Tapering models investi-gated earlier by Poznanski [86] were appropriate for some typical tapering geome-try representing dendrites. These models show that the voltage attenuates with abias in the somatofugal direction. Therefore, the SAC’s unique morphology is wellsuited for sensitivity modications with respect to somatofugal current ow, but notsomatopetal current ow.

Synaptic connectivity between the dendrites of SACs and bipolar inputare dependent on tonic transmitter release [32 , 70]. The same can be said for



318 Poznanski

GABAergic–GABAergic connectivity between SACs (Zhou, private communica-tion). A synaptic mechanism of transmitter release, based on tonic transmitterreleasing synapses, needs to be included in models of SACs. Synaptic transfer inretinal networks usually entails the inclusion of dendro-dendritic synapses, whichare deactivated by depolarization. This results in a suppression of the continuousrelease of transmitters, producing a conductance change. The process entails a sig-moidal dependence of the synaptic transfer function between the pre- and postsy-naptic membrane potential and is based on two assumptions: rstly, that the rate of transmitter release is proportional to the exponential of the presynaptic membranepotential, thus occurs only in the event of synaptic transmission generated in theabsence of spikes [1]. The second is that the rate of neural transmitter release is pro-portional to the transmitter-induced conductance change. Figure 5 illustrates theprocess of synaptic transfer of signals between neurons in retinal networks that donot rely on “ synaptic weights ” but are chemically tuned. This means that the trans-mitter release does not rely on nerve impulses, but depends on a prolonged depolar-ization leading to light-independent vesicular release that is Ca 2+ -dependent, whichcauses a subsequent drop in the conductance of the postsynaptic SAC.

A comprehensive treatment of dendritic cables including the distribution of Na+ P ionic channels in neural networks of biologically constrained “ sequential conguration ” models was successfully undertaken in 2001 [89], but networks thatinclude multicylinder cables for the asymmetry needed to model the SACs are yet to

time0

g

g tonic g tonic

0

Complement of Sigmoidal function

Exponentialfunction

Vpre

g

Fig. 5. A schematic illustration depicts the Ca 2+ -dependent synaptic release of neural transmitter(e.g., ACh, GABA, glutamate) that would occur at a cone bipolar and SAC synapse when lighttriggers a cascade which produces a conductance change to drop from a constant tonic releasemaintained in darkness. The presynaptic release of transmitter is represented on the RHS eitherin terms of a sigmoidal dependence on the presynaptic voltage ( V pre ) in the cone bipolar cell,resulting in a continuous saturating process of transmitter release. The concentration of releasedneural transmitter causes a change in the conductance of the postsynaptic SAC (g) resembling an

exponential function as a result of the bound transmitter on the postsynaptic membrane [1]. Theconductance change in turn invokes a synaptic current ( I syn ) in the postsynaptic SAC. The SACsrelease ACh and GABA in a prolonged and Ca 2+ -dependent manner (vesicular release) at nicotiniccholinergic receptors and GABA A autoreceptors, respectively.




be constructed. Consider a synaptic couplet of SACs as shown in Fig. 3 where SAC2 and 3 are aligned and SAC 1 is misaligned to the left by half of the total physicallength (2 ) of the double dendritic cable where the membrane potential V i (x, t ) canbe represented by equations of the form:

C m V it = ( d/ 4ρi )V ixx + gl(V l − V i ) +ψ

j =1I iion (x, t ;[Ca2+ ]i , V i )[δ(x − x j ) + δ(x + x j )]

+2N

b=1

I ibipolar (x, t )[δ(x − xb) + δ(x + xb)] i = 1 , t > 0


j =1

I iion (x, t ;[Ca2+ ]i , V i )[δ(x − x j ) + δ(x + x j )]

+M

k=1

I isyn (x, t ;[Ca2+ ]i , V i)δ(x + xk) +N

b=1

I ibipolar (x, t )δ(x − xb)

i = 2 , t > 0


j =1I iion (x, t ;[Ca2+ ]i , V i )[δ(x − x j ) + δ(x + x j )]

+ 2M

k=1

I isyn (x, t ;[Ca2+ ]i , V i )[δ(x − xk ) + δ(x + xk )] i = 3 , t > 0

where C m is the membrane capacitance per unit area (A 2s2/Jcm 2); d is the diameterof the cable (cm); ρi is the internal cytoplasmic resistivity (Ω cm); gl is the leakconductance (S/cm 2); V l is the reversal potential (mV); the superscripts correspondto the ith SAC where subscripts x and t represent partial derivatives with respectto space x (cm) and time t (sec), respectively; I iion is the current density of variousionic channels superimposed on a passive linear cable from − < x < 0 and from0 < x < of the ith SAC at the locations x = x j and x = − x j , respectively with δ(·)representing the Dirac-delta function; ψ is the number of clusters of ionic channels(assumed to be identical for each cell); I ibipolar is the bipolar current input representedas α-functions [86] and distributed along the dendritic cables from − < x < 0 and0 < x < of the ith SAC at the locations x = xb and x = − xb, respectively; N is the number of bipolar inputs; M synapses control the ionic channels at speciedpositions along a dendritic cable emanating from the soma ( x = 0) and terminatingat ( x = ) positioned at x = xk and the other dendritic cable emanating from thesoma (x = 0) and terminating at ( x = − ) positioned at x = − xk and I i

synis the

synaptic current from the presynaptic SACs treated as a conductance change:

I isyn (x, t ;[Ca2+ ]i, V i ) = gi (x, t ;[Ca2+ ]i , V i− 1)[V irev − V i (x, t )], i = 2 , 3



320 Poznanski

where V irev is the reversal potential of the ith SAC (mV), gi (xk , t ) representing thesynaptic conductance change (nS/cm) of the ith SAC at the kth synapse activatedby voltage in the presynaptic ( i − 1)th SAC expressed by

gi (xk , t ;[Ca2+ ]i , V i− 1) = gtonic e[− σ R t

0 |V i − 1

(x l ,s )|ds ]where x l (cm) represents the location of synapses in interconnected presynapticSACs (note gi(− xk , t ;[Ca2+ ]i , V i− 1) denotes synaptic conductance change occur-ring on the LHS of SAC with soma positioned at x = 0), gtonic represents a constantconductance per unit length associated with a tonic or continuous release of neuro-transmitter per unit membrane surface of cable (nS/cm), σ is a positive constantgoverning the net gain during synaptic transmission. It is assumed that all synapsesimpinging on the postsynaptic SAC generate a conductance change of different valuedue to the non-constant presynaptic voltage. We have assumed the less realisticsymmetry of synaptic connections, namely the homogeneity condition of synapticconnections are symmetric. This implies that the distance from the cell bodies of all synaptically connected SACs is the same. The inclusion of the soma is modeledthrough a lumped soma-boundary condition modied for the SAC with inclusion of various ion channels. A full outline of a model with two cables emanating from acommon soma will be presented elsewhere (cf. [21]).

Many challenges remain to be learned how each DSGC integrates the activity of a population of SACs to yield the underlying directional responses. One such chal-

lenge is to simulate the model of a SAC network described herein including both ONand OFF sublamina. The exploration of topological arrangements of excitatory andinhibitory synapses onto SACs and the transfer of synaptic action onto bistratiedDSGCs has not been previously simulated with multi-hierarchical biophysical neuralnetworks. A minimum parametrization of the required electrical characteristics of the SACs will need to be performed to facilitate iterative adjustments in matchingmodeled output to the physiological performance of the neural system at the DSGClevel. The model will need to show how cell-level biophysical properties of singleSACs as biophysically distinct entities are explicitly incorporated into a mathemat-

ical formalism that predicts network-level functionality. The simulation of such aSAC network model will be the focus of a forthcoming paper.

6. Discussion

The paper presented a model of a generic neural network, specically the SACnetwork in order to elucidate how retinal direction selectivity occurs in the dendriticplexus of SACs. Consequently, a network model of interactions between triplets orpairs of SACs and between SACs, cone bipolar cells and DSGCs was proposed to

explicitly describe how direction selectivity can be produced from the integrationof subtle, isolated preferences for somatofugal motion in autonomous regions of thedendritic plexus of these SACs. This dwarfs the simplied discrete network modelof Encisco et al. [31]. They investigated a model of directional selectivity in an SAC




network using sustained GABA inhibition and sequential glutamate and GABAinputs that relied on the location of ion co-transporters for the asymmetry. Althoughthe ionic uxes generated by co-transporters are necessary, they are too sluggish fordetection of moving stimuli in the operation of directional selectivity.

Direction selectivity in SACS is experimentally proven to be due to the somatofu-gal bias of calcium in the dendritic plexus of SACs which is independent of co-fasciculation of the dendrites of the networks of SACs. While the model of retinaldirectional selectivity is consistent with the extant literature on ON–OFF DSGCsof rabbits (see [104] for an overview), the proposed directionality at the DSGClevel arises as a consequence of the cofasciculation of the dendrites of SACs (see[101]), resulting in geometrically asymmetrical contacts between SACs and conebipolars, the preferentially suppressed ACh by negative feedback mechanism involv-ing extrasynaptic GABA A autoreceptors, and the requirement of laterally offset inhi-bition onto DSGCs from SACs releasing ACh to somatopetal motion and GABA tosomatofugal motion as proposed by Fried et al. [43]. Furthermore, if the model is toremain robust then glutamatergic bipolar inputs impinging directly on DSGCs willneed to be inhibited not by the effects of GABAergic synaptic contacts in a push–pullconguration with SACs, but as a result of reciprocal synapses from SACs directlyinhibiting glutamatergic bipolars impinging onto DSGCs. Direct physiological obser-vation of such connectivity observed in the primate retina is yet to be found in therabbit retina. Paired-cell recordings between SACs and other synaptically connectedneurons will be required. No experimental technique has yet revealed the answer tothe question of SAC displaying directional bias in the membrane potential becauserecordings from both the proximal dendrites and the distal tips of single SACs havebeen difficult to perform using a dual-patch clamp technique due to the gossamernature of the the SAC dendrites. Euler et al. [32] optically measured [Ca 2+ ]i lev-els throughout the dendrite of a SAC, but not the membrane potential. Therefore,simulation of a single SAC within a network of SACs based on the retinal circuitrydescribed herein could help eludicate the functional properties of SACs underlyingthe formation of direction selectivity in the inner plexiform layer of the mammalian

retina.

Appendix

(i) Equations for calcium-dependent ionic membrane currents in SACs

There is a plethora of ionic membrane currents in the dendrites of SACs that differ inprincipal carrier, voltage- and time-dependence, and dependence on cytosolic Ca 2+ .In order to understand the functional role of these currents in determining membraneresponses, a description of the various inward and outward currents using equations

with exact tting of data from SACs is required. Equations often approximate thebehavior of the ionic currents under physiological conditions when simulation of theequations are compared with the physiological responses of SACs. By assuming thatall ionic current ow occurs through channels or pores and that the ionic current is



322 Poznanski

related to the voltage across the membrane by Ohm’s law:

I (x, t ;[Ca2+ ]i , V ) = g(x, t ;[Ca2+ ]i , V )[V − E ],

where E is the Nernst potential for the ionic current under investigation (mV)and g(x, t ;[Ca2+ ]i, V ) is the conductance per unit length associated with the chan-nel (S/cm). This conductance depends also on various chemical mediators such asintracellular or cytosolic Ca 2+ . It should be mentioned that voltage-dependent ionchannels constitute active point sources of current, imposed on a passive cable.Membrane and synaptic currents are dened as positive when they are outward.

Inward Currents: The Ca 2+ current is smaller and slower than the sodium current,contributing little to the electric charge entering the cell during an action potential.In addition, the Ca 2+ current inactivates slowly, reecting a channel block as a resultof internal Ca 2+ accumulation in the space just below the cell membrane. Removalof inactivation (via internal Ca 2+ ) takes several seconds following closure of theCa 2+ channels. The voltage-dependent inactivation present in other Ca 2+ currentsis present in SACs and is described as

I Ca (x, t ;[Ca2+ ]i , V ) = gCa mCa hCa P Ca [E Ca − V ], (1)

where E Ca is the equilibrium potential for calcium (mV), gCa is the conductanceper unit length describing the ow of Ca 2+ ions through the membrane (S/cm).It is expressed in terms of a voltage-dependent conductance with one activation(mCa ) and one inactivation ( hCa ) gate, and P Ca is the active transport through themembrane, viz,

P Ca = χ [1 − exp− V Ca ([Ca2+ ]i − [Ca2+ ]rest )] + σCa (2)

with χ being a dimensionless constant, V Ca being the reversal potential for Ca 2+

channels (mV), [Ca 2+ ]rest the resting concentration of Ca 2+ (µM), and σCa a pump-ing rate needed to maintain the resting levels of Ca 2+ ions (dimensionless).

The inactivation ( hCa ) is dependent on the concentration of intracellular Ca2+

below the membrane by using the simplest Michaelis-Menton equation: hCa = p/ ( p + [Ca 2+ ]i ) where p is a constant which is the half-way inactivation concen-tration ( µM). Activation ( mCa ) is modeled through rst-order kinetics of the formdm Ca (t, V )/dt = [m∞ (V ) − mCa (t; V )]/τ Ca (V ) where the steady-state value of mCaand time-constant are expressed in terms of rate constants which depend on the[Ca2+ ]i . These rate constants must be measured experimentally on the basis of extensive voltage-clamp experiments in conjunction with the application of phar-macological agents to block other currents. Such data was used for the Q-type Ca 2+

channels by Tukker et al. [107].Outward Currents: The SAC has two ionic currents which are potassium currentsactivated as a result of Ca 2+ inux. The non-inactivating Ca 2+ -dependent potassium




current density per unit length I KCa and the voltage-independent Ca 2+ -dependentpotassium current density per unit length I Ca(ATP) are described by the followingequations in A/cm:

I KCa (x, t ;[Ca2+ ]i , V ) = gKCa mKCa hKCa [V − E KCa ] andI Ca(ATP) (x, t ;[Ca2+ ]i , V ) = gCa(ATP) mCa(ATP) hCa(ATP) [V − E Ca(ATP) ],

(3)

where the activation variable for both currents depends on both voltage and [Ca 2+ ]ilevels.

For Ca 2+ homeostasis to be maintained, Ca 2+ -buffering machinery is required.Given SACs have pre- and postsynaptic sites that are spatially intermingled andsynaptic mitochondria can be found within 1 µm of presynaptic active zones, itappears that mitochondrial Ca 2+ transport (efflux of Ca 2+ from mitochondria) inter-acts with Ca 2+ transport mechanisms on the endoplasmic reticulum by stimulatingthe opening of adjacent ryanodine receptors via Ca 2+ -induced Ca 2+ release in cul-ture [96]. However, there is limited information whether buffers from SACs in theintact retina are released from endoplasmic reticulum in the distal varicosities orwhether they originate in the soma through mitchondrial transport. Calcium ionsin SACs are known to be removed by two transport systems: (i) a Na + -dependentCa2+ efflux (i.e., Na+ –Ca2+ exchanger system) in which the energy required forthe extrusion of Ca 2+ ions is derived from the inward movement of Na + ions downtheir concentration gradient; (ii) Ca 2+ -extrusion system (ATP-driven pump) thatrequires ATP, an energy source activated at low values of [Ca 2+ ]i .

The extrusion or re-uptake into stores of Ca 2+ is through pumps which isassumed to be homogeneously distributed over the plasma membrane, active atlow values of [Ca2+ ]i . The Ca 2+ pump (powered by ATP-operated) is a high affin-ity, low capacity pump that can cause an outward current density and is governedby Hill-type equation:

P ([Ca2+ ]i) =4P m

dK p

[Ca2+ ]i[Ca2+ ]i + K p

, (4)

where P m is the membrane pump parameter (m/sec), K p(µM) is the pump dissoci-ation constant and the factor 4/d is the surface area-to-volume ratio for a dendriticcable of diameter d (cm).

The Na + –Ca2+ exchanger current density per unit length (A/cm) is given by

I NaCa = kNaCa πd([Ca2+ ]i[Na+ ]3oe(β − 1)FV/RT − [Ca2+ ]o[Na+ ]3i eβ FV/RT ), (5)

where kNaCa is a concentration-to-current density per unit area conversion factor(µAmM − 4 cm− 2) and [Ca 2+ ]i , [Ca2+ ]o, [Na+ ]i , [Na+ ]o are the internal and exter-nal concentrations ( µM), for calcium and sodium, respectively; β is a partitionparameter (dimensionless), R is the gas constant (J/Kmol), F is Faraday’s constant(C/mol), and T is temperature (K).




The equation governing the dynamics of Ca 2+ diffusion just below the membraneis given as

∂ [Ca2+

]i∂t = DCa ∂ 2

[Ca2+

]i∂x2 + J leak − P ([Ca2+ ]i)

+ d2N NaCa

j = i

4I NaCa (x, t ;[Ca2+ ]i , V )F

δ(x − xNaCa j )

+ d2N Ca

j = i

2I Ca (x, t ;[Ca2+ ]i , V )F

δ(x − xCa j ), (6)

where D ca is the diffusion coefficient (µm2/sec), J leak is the leakage rate ( µM/sec),and δ(·) represents the Dirac-delta function. In the absence of leakage, the resting

value of [Ca2+ ]i is taken to be zero. Note that a constant leakage rate can alwaysbe absorbed by dening the concentration relative to the resting level.

The Ca 2+ pump under the assumption that the value of [Ca 2+ ]i is much lowerthan the dissociation constant K D (i.e., [Ca2+ ]i K D ) and that the pump extrudesendogenous Ca 2+ in a linear fashion, viz.

lim[Ca

2+

]i K D

P ([Ca2+ ]i ) → γ [Ca2+ ]i , (7)

where γ = 4P m /d is a constant (sec − 1). By including the effect of pumps, we canstudy how the wavefront propagates (via ∂ [Ca2+ ]i /∂x ), as well as describing how[Ca2+ ]i goes back to its basal level. Note that in the absence of leakage, the basallevel is zero, so the basal level can be dened by letting [Ca 2+ ]o = 0, i.e., Ca 2+

isopotentiality in the extracellular medium. Finally, this leads to a reaction-diffusionequation for the dynamics of Ca 2+ diffusion just below the membrane:

∂ [Ca2+

]i∂t

= DCa ∂ 2[Ca

2+]i

∂x2 − γ [Ca2+ ]i

+ d2N NaCa

j = i

4I NaCa (x, t ;[Ca2+ ]i , V )F

δ(x − xNaCa j )

+ d2N NaCa

j = i

2I Ca (x, t ;[Ca2+ ]i , V )F

δ(x − xCa j ). (8)

Consider a single SAC as a double dendritic cable of physical length (2 ) in theregion (− < x < ) with x = 0 representing an isopotential soma. The reaction-diffusion equation follows with the inclusion of all the ionic currents present in its



326 Poznanski

dendritic processes:

C m∂V ∂t

=d

4ρi

∂ 2V ∂x2 + gl(V l − V )

+N NaKCl

j =1

I NaKCl (x, t ; V )[δ(x − xNaKCl j ) + δ(x + xNaKCl

j )]

+N GABA

j =1

I GABA (x, t ; V )[δ(x − xGABA j ) + δ(x + xGABA

j )]

+N KCa

j =1

I KCa (x, t ;[Ca2+ ]i , V )[δ(x − xKCa j ) + δ(x + xKCa

j )]

+N K

j =1

I K (x, t ; V )[δ(x − xK j ) + δ(x + xK

j )]

+N NaCa

j =1

I NaCa (x, t ;[Ca2+ ]i , V )[δ(x − xNaCa j ) + δ(x + xNaCa

j )]

+

N Ca

j =1

I Ca (x, t ;[Ca

2+

]i , V )[δ(x − xCa

j ) + δ(x + xCa

j )]

+N Cl

j =1

I Cl (x, t ; V )[δ(x − xCl j ) + δ(x + xCl

j )]

+N KCl

j =1

I KCl (x, t ; V )[δ(x − xKCl j ) + δ(x + xKCl

j )]

+N Ca(ATP)

j =1

I Ca(ATP) (x, t ;[Ca2+ ]i , V )[δ(x − xCa(ATP) j ) + δ(x + xCa(ATP)

j )]

(9)

where I µion is the ion current density per unit length (A/cm) for channel µ. The ioniccurrent density per unit length generated at x i by channel µ can be expressed as

I µion (x, t ;[Ca2+ ]i , V ) = gµN µ(x i )(mµ) p(hµ)q(E µ − V ), (10)

where gµ is the single channel conductance (nS), N µ(x i ) = θµ(x i )/πd is the numberof channels per unit length (cm − 1) and θµ(x i ) represents the number of channelsin the ith hot-spot, E µ is the equilibrium potential of ions (mV), and the dynam-ics of each of the gating variables, namely the voltage-dependent activation and




inactivation of ionic channels is governed by rst-order reaction kinetic:

∂m µ /∂t = [mµ∞ (V ) − mµ]/τ µ

m (V ) (11)

∂hµ

/∂t = [hµ∞ (V ) − h

µ

]/τ µ

h (V ) (12)with steady-state activation variable and its time constant respectively, for each of the channel gates given by

mµ∞ =

α µm (V )

α µm (V ) + β µm (V )

(13)

τ µ∞ =1

α µm (V ) + β µm (V )

, (14)

where αµm (V ) and β

µm (V ) are the voltage-dependent forward and backward reactionrates. Similar expressions apply to the steady-state inactivation variable and its

corresponding time constant.Equation (9) can be transformed into the following system of nonlinear Volterra

integral equations:

V (x, t ;[Ca2+ ]i ) = ε t

0

N NaKCl

i=1(f NaKCl [V (xNaKCl

i , s )]G(x, x NaKCli ; t − s)

+ f NaKCl [V (− xNaKCl

i, s )]G(x, − xNaKCl

i; t − s))

+N GABA

i=1(f GABA [V (xGABA

i , s )]G(x, x GABAi ; t − s)

+ f GABA [V (− xGABAi , s )]G(x, − xGABA

i ; t − s))

+N KCa

i=1(f KCa [V (xKCa

i , s ), [Ca2+ ]i (xKCai , s )]G(x, x KCa

i ; t − s)

+ f KCa [V (− xKCai , s ), [Ca2+ ]i (− xKCai , s )]G(x, − xKCai ; t − s))

+N K

i=1(f K [V (xK

i , s )]G(x, x Ki ; t − s)

+ f K [V (− xK i , s )]G(x, − xK

i ; t − s))

+N NaCa

i=1(f NaCa [V (xNaCa

i , s ), [Ca2+ ]i (xNaCai , s )]G(x, x NaCa

i ; t − s)

+ f NaCa [V (− xNaCai ), s ), [Ca2+ ]i (− xNaCa

i , s )]G(x, − xNaCai ; t − s))

+N Cl

i=1(f Cl [V (xCl

i , s )]G(x, x Cli ; t − s)



328 Poznanski

+ f Cl [V (− xCli , s )]G(x, − xCl

i ; t − s))

+N Ca

i=1

(f Ca [V (xCai , s ), [Ca2+ ]i (xCa

i , s )]G(x, x Cai ; t − s)

+ f Ca [V (− xCai , s ), [Ca2+ ]i (− xCa

i , s )]G(x, − xCai ; t − s))

+N KCl

i=1(f KCl [V (xKCl

i , s )]G(x, x KCli ; t − s)

+ f KCl [V (− xKCli , s )]G(x − xKCl

i ; t − s))

+

N Ca(ATP)

i=1(f Ca(ATP) [V (xCa(ATP)

i, s ), [Ca2+ ]i (xCa(ATP)

i, s )]

× G(x, x Ca(ATP)i ; t − s) + f Ca(ATP) [V (− xCa(ATP)

i , s ),

[Ca2+ ]i (− xCa(ATP)i , s )G(x, − xCa(ATP)

i ; t − s)) ds, (15)

where ε is a scaling parameter (cm/C), and

f µ[[Ca2+ ]i (xµi , s ), V (xµ

i , s )] = gµN µ(xµi )mµ[[Ca2+ ]i (xµ

i , s ), V (xµi , s )] p

× hµ[[Ca2+ ]i(xµi , s ), V (xµ

i , s )]q(E µ − V (xµi , s )) . (16)

The Green’s function for the voltage system G(x, ξ ; t) satises the following equationwith appropriate boundary and initial conditions:

C m ∂G (x, ξ ; t)/∂t = d/ 4ρi ∂ 2G(x, ξ ; t)/∂x 2 + gl(V l − G(x, ξ ; t)) .

Furthermore, we assume resting value of [Ca 2+ ]i is zero and Eq. (8) is recast intothe following system of nonlinear Volterra integral equation:

[Ca2+

]i (x, t ) = −4kNaCa πd 2

F t

0

N NaCa

j =1 (e

βF V ( x NaCaj ,s )

RT

[[Ca2+

]o[Na+

]3i

− [Na+ ]30e− F V ( x NaCa

j ,s )RT [Ca2+ ]i (xNaCa

j , s )]H (x, ξ NaCa j ; t − s)

+ eβF V ( − x NaCa

j ,s )RT [[Ca2+ ]o[Na+ ]3i − [Na+ ]3o e

− F V ( − x NaCaj ,s )

RT

× [Ca2+ ]i (− xNaCa j , s )]H (x, − ξNaCa

j ; t − s))ds

+2dF

t

0

N Ca

j =1(gCa P Ca mCa hCa [E Ca − V (x

Ca j , s )]H (x, ξ

Ca j ; t − s)

+ gCa P Ca mCa hCa [E Ca − V (− xCa j , s )]H (x, − ξCa

j ; t − s))ds, (17)



330 Poznanski

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