The Size of the Horizontal Cell Receptive Fields Adapts to ... · Ophthalmic Research Institute, Department of Visual System Analysis, Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands

Pergamon PII: S0042-6989(96)00143-5

Vision Res., Vol. 36, No. 24, pp. 4105--4119, 1996 Copyright © 1996 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0042-6989/96 $15.00 + 0.00

The Size of the Horizontal Cell Receptive Fields Adapts to the Stimulus in the Light Adapted Goldfish Retina M. KAMERMANS,*t J. HAAK,* J. B. A. HABRAKEN,* H. SPEKREIJSE*

In this study the dynamic properties of goldfish horizontal cell (HC) receptive fields were evaluated. The size of HC receptive fields increases up to about 60 msec after stimulus onset, and then reduces to a smaller end value. They can therefore not adequately be described by the cable equation. Estimates of the length constant of the HC network based on the sustained responses are about 43% smaller than those based on the initial part of the response. This difference can be accounted for by feedback connections from HCs to cones because negative feedback reduces the receptive field size. The implication is that HCs are strongly coupled when the retina is stimulated more or less homogeneously but that they partly uncouple from the rest of the HC network when they are stimulated differently than the rest of the retina. The HCs thus generate a feedback signal based on the "local" stimulus properties. The size of the HC receptive fields depends on the spatial detail of the stimulus. Copyright © 1996 Elsevier Science Ltd.

Horizontal cell Goldfish Retina Receptive fields

INTRODUCTION

Retinal horizontal cells (HCs) have large receptive fields because they are strongly electrically coupled by gap- junctions (Naka & Rushton, 1967; Kaneko, 1971). The size of the HC receptive field changes with the adaptational state of the retina (Mangel & Dowling, 1985; Yang et al., 1988a,b; Tornqvist et al., 1988), and with background illumination (Baldridge & Ball, 1991; Umino et al., 1991). Furthermore, HC receptive fields strongly depend on the stimulus intensity (Lamb, 1976; Kamermans et al., 1989a,b; Lankheet et al., 1990) and are wavelength-dependent (Lipetz & Kaneko, 1984; Kamermans et al., 1989a,b). For intense stimuli they are larger than for dim stimuli and for green stimuli they are smaller than for red stimuli. Finally, HC receptive fields do not have their final shape immediately after stimulus onset (Byzov &: Shura-Bura, 1983) but develop with time as can be seen in Fig. 1 that was recorded in our own laboratory. In this experiment the receptive field profile of a monophasic HC in goldfish retina was determined with a slit of 500 #m width of 565 or 660 nm wavelength, flashed on for 500 msec at various locations in the receptive field. Figure 1 gives the half-width of the

*Graduate School Neurosciences Amsterdam, The Laboratory of Medical Physics, University of Amsterdam and The Netherlands Ophthalmic Research Institute, Department of Visual System Analysis, Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands.

tTo whom all correspondence should be addressed [Email M.KAMER [email protected]].

receptive field profile at 1/e of the maximal response amplitude (receptive field size) for the two stimulus wavelengths as a function of time. The receptive field size for 660 nm stimulation increases monotonically, whereas the receptive field for 565 nm firstly shows an increase and then a reduction, being the largest around 60 msec after response onset. Although there was a large variability between the cells, most cells showed a transient receptive field increase for 565 nm stimuli and a sustained receptive field increase for 660 nm stimuli. This experiment illustrates unambiguously that HC receptive fields are strongly dynamic structures.

It has been argued that the receptive field size of HCs depends on the square root of the ratio of the membrane and the gap-junction resistance and that for slit stimuli the spread of potentials in the unilluminated part of the HC layer will be an exponential function with a length constant (2) as given by Eq. (1) (Lamb, 1976). Although many of the features of the HCs can be accounted for with this approach, a large number of experimental results cannot be explained.

V ( x ) = E . s i n h ( ~ ) . ( a - e x p ( - I ~ l ) ) I x l > a

A__ R~ (1)

with 2, the length constant; Re, the coupling resistance in the HC layer in (MQ); Rm, the leak resistance of the HC layer in (Mr2 cm2); E, the driving potential (mV); a, the half-width of the stimulus slit (cm).

4105

4106 M. KAMERMANS et al.

2500

2000

1 5 0 0 '

1000

500

i " 660 n m

565 n m

0 200 400 600 800

Time (ms)

FIGURE 1. Kinetics of the HC receptive field size for two stimulus wavelengths (565 and 660 nm) in an isolated light adapted goldfish retina• The receptive field size is defined as the half-width of the receptive field profile at 1/e of the maximal response amplitude. The receptive field size was determined every 4 msec and plotted as function of time. Stimulus intensity: - 1.9 log for the 565 nm stimulus and - 1.8 log for the 660 nm stimulus (0 log = 4.0 x 1018 quanta m - 2 sec-1 ; the slit (width 500 #m, length 8 mm) was displaced in steps of 300 pm. For a description of the experimental setup and of the Ringer 's composition see

Verweij et al. (1996b).

The aim of this study

In the present model study, the effect of network connections in the outer plexiform layer on the spatial properties of the HCs will be investigated. Firstly, using an analytical model, we will estimate the linear electrical properties of the HC layer without influence of the synaptic pathways. Secondly, we will present a computational model of the HC layers that generates the dynamic changes of the HC receptive fields. Finally, the computational model will be expanded such that it also shows the wavelength-dependence of the HC receptive fields. With these simulations we illustrate how the network connections of the outer plexiform layer strongly influence the spatial properties of the HCs.

MATERIAL AND METHODS

Analytical model

If only the linear electrical properties of the HC layer and slit stimuli are considered and the connections with other neurons are ignored, a HC layer can be treated as a cable, for which Eq. (2) gives the potential distribution in space and time (Jack et al., 1975).

A2 O2V(t,x) OV(t,x) • Ox 2 -- ~-nc" 0----~ + V(t ,x) (2)

with 2 = ~f(Rm/Rc); ZHC = Cm'Rm; V(t,x), the membrane potential at time t and position x (in mV); Cm, the leak

capacitance of the cable (in nF cm-1); Rm, the leak resistance of the cable (in Mf~ cm- 1); Re, the coupling resistance of the cable (in MQ cm-1).

With the impulse response of such a system, the response of this system to any stimulus can be calculated. The impulse response is the response to an infinitely short current injection into the network which can be represented as a charge, Qo, delivered at t = 0 and x = 0. The intensity of the stimulus is proportional to Qo. Because the system under consideration is linear, Qo will only scale the amplitude of the resulting receptive field profile• The response to this impulse stimulus is given by Eq. (3) (Jack et al., 1975):

V(T,X)= Qo (-x2- 4T 2 ) 2Cm AX/~ "exp 4T

(3)

with X = x/2; T = t/Thc; X, the position (in cm); t, the time (in msec); Qo, the charge delivered at the cell at x = 0 and t = 0 (in pC). Qo = 1 pC.

To obtain the response of the HC layer in space and time, the input to the HCs (the output of the photoreceptors) must be convoluted with the transfer function of this system [Eq. (3) divided by Qo]- We have assumed that the photoreceptors filter the light stimulus with a first order RC-filter (time-constant: Tph ) and that the photoreceptors are not electrically coupled. The input we used is a step of light of a 500 #m wide slit [Eq. (4)]. Because the convolution of Eq. (4) with the transfer function of

RECEPTIVE FIELD SIZE RESPONSE TO STIMULUS 4107

this system cannot be solved analytically, we solved it numerically.

fort___ 0

forx> 0andx <_0.25cmthen:l( t ,x)= Io. ( 1 - exp(~--~))

forx > 0.25cmthen : l(t,x) = 0 (4)

with Zph, the time-constant of the photoreceptors (msec); Io, constant (nA) (•0=2 p A).

Computational model

The computational model, formulated in this study, describes the goldfish retina. In goldfish retina four cone types (L*-, M-, S- and UV-cones) and three cone driven HC types (mono-, bi- and triphasic HCs) can be found (MacNichol & Svaetichin, 1958; Norton et al., 1968). For simplicity reasons, the model only consists of a L- and M- cone system and a monophasic HC (MHC) and a biphasic HC (BHC) layer. Although the general conclusions will not be influenced by the omission of the UV- and S-cones and the triphasic HC (THC), the model is easily expandable to a system that includes these cone and HC-systems. The distribution of the HCs in each layer is taken as a homogenous hexagonal structure (Kaneko, 1971; Wagner, 1976; Kaneko & Stuart, 1980). Because only experiments with full-field or slit stimuli are considered, the HC network will be reduced to a one- dimensional structure (Kamermans et al., 1989a). In this reduced model, each HC is coupled to its two neighbours via a coupling conductance (go), except for the first cell (the centre of the slit), which is coupled to only one neighbour with a coupling conductance of 2gc.

As the emphasis of the model is on the pathways and the connection strength instead of on the non-linear properties of the cells involved, we did not include the voltage-gated currents in both the cones and the HCs. Because only responses to slit stimulation are used in the receptive field measurements, this latter simplification seems appropriate since the tonic glutamate input to the HCs and the large coupling conductances between the HCs shunt the voltage-gated currents in the unilluminated areas (Winslow & Ma, 1990).

A diagram of the pathways incorporated in the model is given in Fig. 2. The left, middle and right columns represent the physiologiLcal process, the variable in the model and the relevant equations, respectively. Next we will discuss the various components of the model in detail.

The cones. In the outer segment of the cones, light is converted into a change of cGMP concentration ([cGMP]) via the pho~otransduction pathway, cGMP increases a conductance permeable to sodium and calcium [for a review see: Yau (1994)]. The relation between [cGMP] and the conductance of this channel

*Throughout this paper L-cone denotes long wavelength sensitive cone, M-cone middle wavelength sensitive cone, S-cone short wavelength sensitive cone and UV-cone ultra short wavelength sensitive cone.

(gcat,q,n) is given by a Hill function with a cooperativity of 2 (Lyubarskii & Fesenko, 1987). The wavelength- dependency of the transformation of light into conductance change is given by the spectral sensitivity curves Sq(2) of the cones and were taken from Van Dijk and Spekreijse (1984). All these transformations are combined in Eq. (5).

( (Iq'n~g')2 2 -~ gcat ,q ,n=gcat , max" 1 -- (iq, n 2

-- KI) ~- KcGMPJ (5)

d -- Tph " ~-~ gcat,q,n

with Iq,, = I* • Sq(A); Iq,n, the effective stimulus intensity for the nth cone of type q (in log quanta m -2 sec-1); I*, the absolute stimulus intensity for the nth cones (in log quanta m -2 sec-1); K~, intensity offset (in log quanta m -2 sec-1); KeGMP, the effective intensity at which the cGMP sensitive conductance is half maximum (in log quanta m -2 sec-1); Sq(,~), the spectral sensitivity function of the photopigment of cone type q; 2, the stimulus wavelength (in nm); "Cph, the time-constant of the phototransduction process (in msec); gcat,q,n, the conductance of the cGMP modulated channels in the nth cone of type q (in pS); gcat,cone,max the maximum cGMP sensitive conductance in the cones (in pS).

The membrane potential of the nth cone of type q (Vq,n) depends on two conductances and their equilibrium potentials and is given in Eq. (6)• q can denote the L- or M-cones.

Vq,n = Ecat, cone • gcat,q,n -+- eK,cone " gK,cone (6) gcat,q,n "~- gK,cone

with Vq,,,, the membrane potential of nth cone of type q (in mV); Ecat,cone, the equilibrium potential of the cGMP gated conductance in the cones (in mV); EK,cone, the potassium equilibrium potential in the cones (in mV); gK,cat, the conductance of the potassium channels (in #S).

The glutamate concentration in the synapse of the nth cone of type q, [Glu]q,n, is a function of Vq,n and is given by Eq. (7). It has been assumed that: (1) the Ca-current can be described by a linear current multiplied by an activation function [see: Hille (1992)]; and (2) that the glutamate release depends on the third power of the Ca- current (Augustine & Charlton, 1986; Dodge & Raha- mimoff, 1967; Stanley, 1986; Dudel, 1981). HCs feed back to cones by shifting the activation range of the Ca- currents of the cones (Verweij et al., 1996). This is incorporated in the model by the function f([F]q,n) and will be discussed in the next section.

[Glu]q,n Cma X [

• ( - - ( V , n -- Eca,cone) \

d 1 -- TG1 u "~-~ [G U]q,n

1 ) 1 + e x p [F]q.n)-Vq,n'~

\ ac~ / /

(7)

with [Glu]q,n, the glutamate concentration in the synaptic terminal of the nth cone of type q (in pM); Cm~x, a

4108 M. KAMERMANS et aL

Physiological process Variable Equation

Stimulus

Photo- transduction

Cone membrane potential

I* n

J [cGMP]

l gcat,q,n

l V q,n

Equ (5)

Equ (6)

Cone" calcium current

Glutamate concentration

Feedforward connectivity

Glutamate receptors HCs

HC membrane potential Vp,n_ 1 ~- ~ Vp,n+ 1

Feedback neurotransmitter

concentration

Feedback connectivity

Shift in cone calcium current

f

Ica,q,n

1 [GlU]q, n

l geat,qp,n

1 ~Cff,qp

gcat,p,n

l ~ V ~ p,n

[F]p,n

~Cfb.pq

1 [F]q,n

J f([Flq,n)

I

t

t

Equ (7)

Equ (8)

Equ (9)

Equ (10+11)

Equ (12)

Equ (13)

Equ (14)

FIGURE 2. A schematic representation of the computational model. Left column, physiological parameter; middle column, model variable; right column, corresponding equation. The stimulus light is transformed into a change in cone membrane potential. This change in membrane potential modulates the glutamate concentration. Glutamate modulates cation conductances in the HC membrane that will lead to changes in the HC membrane potential. The changes in HC membrane potential will yield changes in the release of feedback neurotransmitter. The feedback neurotransmitter will affect the cone calcium current in a

negative feedback loop.

constant indicating the maximal glutamate release by the cones (in pM m V - 3 ) , Eca,cone, the equilibrium potential for Ca (in mV); f(Fq,,), the feedback activity function (in mV); rGlu, the time-constant o f the feedback pathway (in

msec); O'ca , the slope o f the activation curve o f the calcium current (in mV).

The horizontal cells. The glutamate released by the cones will affect the glutamate receptors in the HCs. The


relation between [Glu]q and gcat,v, n is taken as a Hill function with a cooperativity of 2 (Shiells & Falk, 1995; Zorumski & Thio, 1992) and is given by Eq. (8):

[GluJ2q'n (8) gcat,qp,n z gcat,HC,max . [GlU]2q, n ÷ g21u

with gcat,qp,n, the cat-ion conductance in the nth HC of type p driven by cone type q (in #S); gcat,HC,max, the maximum cation conductance in the HCs (in #S); Kclu, the glutamate concentration at which the cation; conductance is half maximum (in #M).

The spectral sensitivity of the HCs is completely determined by the feedforward strength from cone type q to HC type p (Ctt, qp) and the feedback strength from HC type p to cone type q (Cfb,pq). Because both MHCs and BHCs can have dendrites in both cone types [see for a review: Kamermans & Spekreijse (1995)], two groups of cation channels in the HC will be modulated indepen- dently. This means, for instance, that it is not possible to close the glutamate-sensitive channels driven by the M- cones by stimulating only the L-cones strongly. The total conductances of the cation channels in a HC is given in Eq. (9).

gcat,p,n ---~ I E Cff,qp. gcat,qp,n (9) q

with gcat,q,n, the total cation conductance of the nth HC of type q; gcat,qp,n, the cation conductance of the nth HC of type p driven by cone type q; Cff, qp, the feedforward connection strength between cone type q and HC type p.

We have assumed that, apart from the differences in morphology and connectivity, all HCs have equal properties. The size of the MHCs and BHCs have been taken to be 50 pm anti 75/am, respectively (Stell & Lightfoot, 1975). HCs of the same type are strongly electrically coupled by gap junctions (Kaneko, 1971). The gap-junction conductance (gc) is voltage indepen- dent, and linear in the physiological voltage range (DeVries & Schwartz, 1992) and is kept constant during the simulations. The membrane potential (Vp,n) of the nth HC of type p is determined by two membrane conductances and their equilibrium potentials, by gc and the potentials of both neighbouring HCs (Vp,n- 1 and Vp,n+l). Because we need to know the potential of all HCs before we can calculate the effect of the gap-junction coupling on the membrane potential of each HC, the membrane potential of the nth HCs of type p without the influence of the neighbours (Eun,p,,,) will be calculated first using Eq. (10).

Ecat,HC " ~cat,p,n ÷ EK,HC " gK,HC Eun,p,n =

gun,p,n (10)

gun,p,n = gcat,p,n ÷ gK,HC

with Eun,p,,,, the membrane potential of the uncoupled nth HC of type p (in mV). p denotes MHC or BHC; gu~,p,,,, the membrane conductance of the uncoupled nth HC of type p (in #S); Ecat,HC , the equilibrium potential of the glutamate gated cation conductance in the HCs (in mV);

EK, HC, the potassium equilibrium potential in HCs (in mV), gcat,p,n, the conductance of the glutamate gated cation channels of the nth HC of type p; gK, HC, the conductance of the potassium channels of the HC.

Knowing the membrane potentials of the uncoupled HCs (Eun,p,n), we can calculate the behaviour of the HCs in the network, using the Laws of Kirchhoff leading to Eq. (11).

Eun,p,n - gun,p,n + Vp,n+l "gc + Vp,n-1 "gc gp~n

gun,p,n ÷ 2gc Cm d (11)

gun,p,n -~- 2gc " dt gp,n

with Vp,_ 1 = Vp,1; Vp,n, the potential of the nth HC of type p. For the central cell n=0; Cm, the HC membrane capacitance (in nF).

HCs feed back to cones by shifting the activation curve of the cone calcium currents via an unidentified neurotransmitter (Verweij et al., 1996a). Because in dendrites of HCs no synaptic vesicles are found we have assumed that the unidentified neurotransmitter is released in a similar way as GABA is released by the HCs (Schwartz, 1987; Kamermans & Werblin, 1992). Therefore, the relation between Vp,n and the release of the unidentified neurotransmitter has been taken as an exponential function [Eq. (12)].

(~p~n) dFp,n (12) [F]p,n = C 1 .exp - fro d--~-

with [F]p,n, the concentration of the feedback neurotransmitter determined by the nth HC of type p (in/aM); C1, a constant determining the size of the feedback signal (in gM); C2, a constant; determining the slope of the feedback signal (in mV); zfbthe time-constant of the feedback pathway (in msec).

Because all HCs can feed back to all cone systems, the concentration of the feedback neurotransmitter near cone type q ([F]q,n) is determined by all HCs and is given by Eq. (13).

[F]q, n ~- E Cfb,qp. [F]p,n (13)

with [F]q,n , the concentration of the feedback neurotransmitter near the nth cone of type q (in/aM); Cfb,qp the feedback connection strength between HC type p and cone type q.

[F]q,n shifts the activation curve of the cone calcium current according to a sigmoidal curve. Eq. (14) gives the relation between [F]q,n and the shift in activation range of the cone calcium current. We have introduced this sigmoidal curve to mimic the interaction of the feedback neurotransmitter with its receptor on the cone. In this way the shift of the activation curve of the cone calcium current remains between strict boundaries.

f([F]q'n)=C3+(1 + exp i ~ ) ' ) "Dact (14)

with C3, the offset of the feedback function (in mV); Dact, the maximal shift of the activation curve (in mV); KF, the

4110 M. K A M E R M A N S et al.

TABLE 1. Parameters of the computational model

Computational model

Cone--parameters HC--parameters

Ecat, q 50 mV Ecat,HC 0 mV EK . . . . . -- 80 mV EK,HC -- 80 mV

Eca,con e 30 mV gcat,HC,max 2 ,uS gcat,cone,max 0.7 ,uS gK 1 ,uS gK 1 ,uS g~ 2210 ,uS Cmax 0.0003 ,uM m V - 3 C1 80 KI - 7 log quanta m - 2 s e c - 1 C2 26.6 ,uM K~GMP 4 log quanta m 2 s e c - a C3 - 65 pM KGI u 40 'UM Oac t 50 mV "Cph 60 msec ZGIu 3 msec KF 16 'UM O'Ca 6.5 mV aF - -3 #M

Zfb 100 msec Cm 60 nF

concentration of the feedback neurotransmitter for which the shift of the activation curve is half maximal (in/~M); aF, the slope of the sigmoidal curve (in #M).

The parameters. The parameters used for the analytical model are given in the Results section. Table 1 give the values of the parameters used in simulations with the computational model. The parameters describing the connectivity of the model are given in the Results section.

Cones. The conductance of the potassium channel (gK) is not modulated by any transmitter and its value is 1/~S. All the other conductances are relative to this value. The parameters in Eq. (5) which describe the relation between the light intensity and gcat,q,n are not crucially important for the model behaviour. They set the sensitivity of the cone systems. Equation (5) scales the light intensities.

The relation between Vq,n and [Glu]q,n is based on the notion that the calcium current determines the release of glutamate. The parameters describing the cone calcium current were based on our own estimates of this current in goldfish retinal slices (Verweij et al., 1996a). Although there may be a large influence of the neurotransmitter re- uptake systems on the cone/HC synaptic transmission (Vandenbranden et al., 1996), we have not included this in the model and assumed that the release of glutamate is the only factor determining the cone/HC synaptic dynamics. This is not a critical assumption. The same model behaviour would have been obtained when the neurotransmitter concentration in the synaptic cleft was modulated by the activity of neurotransmitter transpor- ters. The parameters that determine the amount of shift of the activation curve by the feedback neurotransmitter from the HCs (Oac t and aro) are the crucial parameters in the model. They set the strength of the feedback from the HCs to the cones.

As in cones, the conductance of the gK,HC is 1 #S and all other conductances are relative to that value. HCs receive only input from and feed back to L- and M-cones directly above them. This means that the dendritic fields of the HCs do not overlap. The size of the HC receptive fields is primarily determined by gc, which is chosen such

(a) 6.0

E 5.0

4.0

3.0

~ 2.0 0

~ 1.o

0

(b)

~ 1.0

~ 0.8

o 0.6 g

0.4

0.2

Z 0

0

(c) 2400

2200

E 2000 1800

1600 1400

o 1200

lOOO '~ 800

600

400

200

0

1000 2000 3000 4000 5000 6000 7000 8000

Space (~m)

1000 2000 3000 4000 5000 6000 7000 8000

Space (~m)

/ 0 50 100 150 200 250 300 350 400 450 500

Time (ms)

FIGURE 3. Development of the receptive field profile. (a) The various curves represent 10, 20, 30, 40, 50, 70, 100 and 500 msec after s t imulus onset. (b) The curves from Fig. 3(a) normalized. (c) Receptive field size vs time based on the receptive field profiles of Fig. 3(b). Qo = 1 pC, lo = 2 pA, 2 = 2100 'um, Zph = 60 msec and THC = 40 msec.

that 2 in the computational model yields values about equal to the estimates in the analytical model. The number of cells was always taken such that the length constant of the HC network was one-third of the total simulated length of the model.

Two of the equations for the model cone represent time-dependent processes: (1) the phototransduction pathway ('Cph); and (2) the release of glutamate ('/TGlu). The time-dependent steps for the HCs are the neurotransmitter release (Zfb) and the membrane capacitance of the HCs; The time-constant of the HCs (ZHC,n = Cm/ gun,p,n) will be estimated using the analytical model.

Computations. Both the analytical and the computational model were programmed in C in a XWindows


2500

E ---- 2000

N 1500

. i

1000

o . . 500

0

, ' / ' a

X = 1900 rtm X = 2100 lam ~. 2300 rtm ---

2500

E --- 2000 ® N . I

1500 I

1000

500 O

0

"~p. = 4 0 m s - - :o. = 6 0 m s - - - ~p. = 8 0 m s

2500

E ,-- 2000

1500

1000 ®

~- 500 O

0 0

~ ~ q - 7 7 - - : . . . . . . . .

/ , / •

i , / " %c = 20 ms - -

, "I~Hc = 40 ms - - - ', "I~Hc = 6 0 m s - - -

100 200 300 400 500

t ime (ms)

FIGURE 4. Dependence of the analytical model on the parameters 2, Zph and ZHC. (a) The effect, of changes in 2. Tph = 60 msec and Znc = 40 msec. (b) The effect of changes in Zph. 2 = 2100 #m and ZHC = 40 msec. (c) The effect of changes in zHc. 2 = 2100 #m and

zpa = 60 msec.

environment on a Silicon Graphics INDY work station. Integration is done by the Runga Kutta method of order 4 (Hamming, 1962). In the analytical model At and Ax were 1 msec and 1/~m, respectively. In the computational model At was 1 msec. In the computational model, firstly, the concentrations of the transmitters were calculated according to the membrane potentials of the various cells in the preceding time step [Eqs (7), (12) and (14)]. Secondly, the various neurotransmitter concentrations near the postsynaptic receptors and the resulting conductance changes were calculated according to the connections determined by Cff,qp and Cfb,pq using Eqs (5), (8), (9) and (13). This enables the calculation of the cone membrane potential [Eq. (6)]~ the HC equivalent equilibrium potential and the HC equivalent membrane conductance using Eq. (10). Finally, using an implicit

method, the HC membrane potentials in the network were calculated [Eq. (11)].

RESULTS

The analytical model

Figure 3(a) gives the receptive field profile of the analytical model to a 500/~m width slit stimulus for various moments in time (10, 20, 30, 40, 50, 70, 100 and 500 msec). In Fig. 3(b), these profiles are normalized illustrating that the length constant (2) increases with time. In Fig. 3(c) the half-width of the receptive field profile at 1/e of the maximal response amplitude is given as a function of time. Three parameters determine the speed of development and the size of these receptive field profiles: 2, Zph and ZHC. Figure 4 shows the dependence of the HC receptive field profile on these parameters. As is obvious from Fig. 4, 2 and ZHC are the main factors determining the initial slope of the receptive field size vs time plot, whereas Zph has its main influence in the trajectory between 50 and 100 msec after stimulus onset.

The feedback signal from the HCs to the cones is slower than the feedforward from the cones to HCs. This means that the first part of the response is mainly determined by the feedforward signal and that the second part of the response is determined by both the feedforward and the feedback signal. Because the equations used in the analytical model only hold for a situation without feedback from HCs to cones, ZHC and 2 can only be estimated based on the kinetics of the receptive field shortly after the response onset. Although a whole family of curves can be fitted through the first part of the curve, it is still possible to estimate a minimal value for 2 of the HC network and its corresponding ZHC.

Figure 5 shows the development of the measured receptive field profile (taken from Fig. 1) and of the theoretical receptive field profiles for various values of ZHC and 2. Zph was taken as 60 msec, which is close to the value Baylor et al. (1974) estimated in turtle, and ZHC ranged from 20 to 80 msec, yielding space constants ranging from 1900 to 2600 #m.

The fits using values for ~HC and 2 of 20 msec and 1900 ~m, respectively, are very poor, indicating that ZHC and 2 must be larger than these values. For this cell ZHC of about 40 msec and 2 of about 2100 #m yielded a reasonable fit. Note that these values for 2 and ZHC ate minimal estimates of the length constant and the time- contant of the isolated linear electrical HC network. Because we have defined the receptive field size as the half-width of the receptive field profile at 1/e of the maximal response amplitude, we cannot compare 2 and the receptive field size directly but have to correct for the stimulus width. The relation between 2 and the receptive field size for a 500 #m wide slit was derived using the analytical model and is given in Fig. 6. The sustained part of the HC responses from the experimental data of Fig. 1 was used to estimate the receptive field size yielding 900 and 1750/~m for 565 and 660 nm stimuli, respectively. If this receptive field size was determined purely by the

4112 M. KAMERMANS et al.

.-1

"o

-=

u

2500

2000

1500

I000

500

_-.-:_- --p---

565 nm

0 [ 0 200 400 600 800

T i m e ( m s )

FIGURE 5. Kinetics of the HC receptive field size for two stimulus wavelengths (565 and 660 nm). The receptive field size is defined as the width of the receptive field profile at 1/e of the maximal response amplitude. Stimulus intensity: - 1.9 log for the 565 nm stimulus and - 1.8 log for the 660 nm stimulus; the slit with a width of 500 #m was displaced in steps of 300 #m. The continuous curves represent the theoretical receptive field profiles. For all curves Zph. was 60 msec. (a) ZHC = 20 msec, 2 = 1900/tm; (b) ZHC = 40 msec, 2 = 2100 #m; (c) ZHC = 60 msec, 2 = 2300 #m; (d) ZHC = 80 msec, 2 = 2600 #m. For details

see text.

linear electrical properties of the HC network, 2 should have been around 760 #m for the 565 nm and around 1625 #m for 660 nm stimuli (Fig. 6). In this cell, the network properties reduce 2 to about 36% for the 565 nm stimulus and to about 77% for the 660 nm stimulus relative to the estimated minimal 2 o f the linear electrical network. This analysis was performed on 12 MHCs. On average, feedback reduced the receptive field to about 57.0 ___ 13.1% (12) for the 565 nm stimuli and to about 69.0 ___ 10.2% (12) for the 660 nm stimuli.

1800

.1 oo . . . . . . . . . . . . . . .

~i 1400

/ J ' ~ 1200 I

" 0 1000 I

800 I I (1) 2 / ' , ~ . 600 / / I I

400 ~. I I

J I 200 i o

I I

0 i • • 0 200 400 600 800 1000 1200 1400 1600 1800

s p a c e c o n s t a n t ( lam)

FIGURE 6. Relation between the width of the horizontal cell profile at 1/e of the maximal response (the receptive field size) and the space constant (2) of the HC layer derived using the analytical model. The

slit width is 500 #m.

If these changes were to be attributed to changes in the membrane or gap-junction resistance what changes in these resistances would be needed? Based on the estimates of 2 we can calculate the ratio rm/r c. For a hexagonal network with a mean cell spacing of d the relation between 2 and rm/rc is given by Eq. (15) (Lamb & Simon, 1976).

/ h,V d, ) rmrC ___ 4. cos 1 (15)

with: d, the HC cell spacing in cm [50 #m, Stell & Lightfoot (1975)]; rm, the membrane resistance o f a HC (in Mff~); re, the coupling resistance between to neighbouring HCs (in MQ).

For the cell used in Fig. 5, the minimal estimated value for 2 is 2100 #m. This means that rm/rc must be at least 1176. The estimate o f the apparent length constant using the steady state values o f the receptive field measured with 565 nm light suggests that rm/rc is only 171. The effect o f network connections in this cell changes the apparent coupling conductance by a factor o f 6.9. For the 12 cells tested, the mean change in apparent coupling was a factor o f 6.0 ranging from 1.6 to 13.6 for 565 nm stimuli.

Computat ional model

Although the analytical model yields minimal estimates of 2 and ZHO it does not account for the HC behaviour after the initial 60msec . Therefore, we formulated a computational model. First, the receptive


gcat,L-con¢,O

[GlU]L.cone,O

[F]MHC,O

565 nm 660 nm

\ ' \ gcat,L-cone,0

[GlU]L_cone

/ IF]MltC,O

f([F])L.con¢,O

VMHC,O

/ 0mvi- / , : , . . . .

/ ~ 200 ms I _ _

FIGURE 7. Changes in gcat,q,o, [Glu]q,o, [F]p,o, f([F]q,o) and Vp,o of the model to full-field stimuli in the computational model cortsistingof only one cone type and one HC type. Note that the dynamic behaviour of the HC responses to 565 and 660 nm light

stimulation are equal.

field properties in a cone/HC network with only one cone and HC type will be simulated in order to study the effect of the feedback pathways without the influence of spectrally coded connections. The analytical model has yielded estimates for the ratio membrane conductance and gap-junction conductance and for ZHC. These values will nowbe used in the computational model consisting of only one functional HC layer driven exclusively by the L-cones and feeding back exclusively to the L-cones.

1600

1400 E ::L

1200

N • ~ 1ooo

.~_ 800

600 ._ Q_

400 0

200

50 100 150 200 250 300 350 400 450 500

time (ms)

FIGURE 8. Receptive field size vs time plot for the model HC of Fig. 7. The receptive field first increases and than decreases to stabilize on a

smaller value.

Figure 7 shows the change in gcat,q,O, [Glu]q,O, [F]p,O, f([F]q,O ) and Vp,o of the model to full-field 500 msec light flashes of 565 or 660 nm. The wavelengths were chosen such that for both stimuli the cones absorb an equal number of quanta. The intensity was - 3 log units. For both stimulus wavelengths, the HC response shows a secondary depolarizing phase during light on. Evaluation of the various pathways in the model shows that the feedback pathways completely account for this secondary depolarizing phase in the HC response.

Figure 8 shows the receptive field size vs time plot of this network for a 500 msec flash of a 500/~m wide slit of 660 nm. This model gives an increase of the receptive field size up to around 60 msec and then a reduction. The receptive field size stabilizes on a value of about 900 #m. Note that for this model network 2 was set to 2100 #m. As will be clear from Fig. 8, the deviation of the receptive field size from the analytically derived curve, as found in the experimental data, is also generated by this very simple cone/HC network and can be fully accounted for by feedback from HCs to cones.

TABLE 2. Connectivity of the cone/HC network with spectrally coded HCs

Cff MHC BHC Cro MHC BHC

L-cone 0.7 0.1 L-cone 0.3 0.7 M-cone 0.3 0.9 M-cone 0.5 0.5

4114

[GlU]L'c°ne,0 ~ 20 p~M

[F]MHC,0 4 p~M

M. KAMERMANS et al.

550 nm

I

I I 200 ms

S gcat,M-cone,0

[GlU]M_cone,0

[F]BHC,0

f([F])M-cone,0

VBHC,O

700 nm

gcat,L-cone,0 ~ . - - f 0.4 I~S I

[GlU]L-c°ne'O ~ ~ ~ 2 0 mM t

L,l..c.o I

gcat,M-cone,O

[Glu]M.cone,0

[F]BHC,0

f([F])L.cone,0 10 mM

ff I VMHC'0 4 mV I I 200 ms

f([F])M-con¢,0

VBHC,O

FIGURE 9. Changes in gcat,q,0, [Glu]q,0, [F]p,o, f([F]q,o) and Vp,o of the model to full-field stimuli in the computational model consisting of L- and M-cones and MHCs and BHCs. Note that the dynamic behaviour of the HC responses to red and green light

stimulation differs strongly.

Next, we studied the effect of the spectral coding of the HCs on the receptive field properties. The connectivity of the cone/HC network was changed according to Table 2. This connectivity is qualitatively equal to the connectiv-

ity of the cone/HC network derived by us (Kamermans et al., 1991).

Figure 9 shows the changes in gcat,q,o, [Glu]q,o, [F]p,O, f([F]q,O) and Vp,o of the model to full-field light stimuli of


1600

1400

1200

1ooo

"~ 800

600 .=

o 400

200

/

660 n m . . . . .

565 n m _ _

50 100 150 2 0 0 250 300 350 4 0 0 4 5 0 5 0 0

Time (ms)

FIGURE 10. Receptive field size vs time plot for the model HC of Fig. 9. For green light stimulation the receptive field first increases and then relaxes to stabilize on a smaller value. For red light stimulation, the

receptive field does not show this secondary reduction.

550 and 700 nm. The intensity was - 3 log units. For 500 nm stimuli, both tiCs hyperpolarize whereas for 700 nm stimuli the MHC hyperpolarizes and the BHC depolarizes. This is in accordance with the physiological behaviour of these cell types [see Kamermans & Spekreijse (1995)]. The MHC response has a secondary depolarizing phase during light on for the 500 nm stimulus whereas this secondary depolarization is absent for 700 nm stimuli. Also this behaviour is physiological showing that the HCs in the simulated retina and the physiological retina behave similarly (Fig. 12).

Figure 10 clearly shows that adding the spectral coding of HCs to the model, results in a spectral dependence of the HC receptive field size. Varying the connectivity strongly influences the difference between the receptive field size for 565 and 660 nm stimuli. The most critical parameter is the feedback from the BHCs to the L-cones. This feedback pathway is essential for the wavelength- dependence of the MHCs receptive fields.

To investigate the importance of the feedback connections from the BHCs to the L-cones we determined the difference between the receptive field size simulated with a 565 and a 660 nm slit stimulus for various values for the feedback strength from the BHCs to the L-cones (Fig. 11). It is clear that the feedback strength from the BHCs to the L-cones has a major effect on the difference between the receptive field size for the two stimulus wavelengths. Note thai: the condition with Cfb,BHC,L- cone = 0 equals the connectivity of the Stell model (Stell et al., 1975; Stell & Lightfoot, 1975). In that condition, the spectral dependence of HC receptive fields is almost completely absent. Furthermore, the difference in dynamics of the MHC responses to 565 and 660 nm is absent with these parameter settings i.e.: both responses show a pronounced secondary depolarizing phase which is not found in the physiological retina (Fig. 12).

(a) 1800

"~" 1600

N ' 3 1400 --o

t~ 1200 >

1ooo

800

(b)

.__.

2 ID g

®

r"

700

t

600 i

5o0

40O

3OO

200

100

0

, i ' i i

i I

I

; I i I I I I

0.0 0.2 0.4 0.6 0.8 1.0

I : I

q i

I

I I I I I I t

O0 0 2 0.4 0.6 0.8 1.0

feedbackstrength BHCto R-cones(C.B~C ..... )

FIGURE 11. The relation between the feedback strength from the BHCs to the L-cones and the difference in receptive field size between red and green light stimulation. (a) Receptive field sizes for 565 (dashed line) and for 660rim light (solid line) stimulation. (b) Difference between the receptive field sizes measured with

660 nm and 565 nm stimuli as function of Cff, BHC,L.cone.

D I S C U S S I O N

In this study we have shown that the behaviour of the

HC receptive fields are not described by a simple cable

model. The results of the analytical model indicate that

the HCs are coupled much stronger than was expected on

the basis of the sustained responses of the HCs. With the

computational model consisting of one cone type and one

HC type, we showed that negative feedback from the HCs

to the cones always reduces the receptive field size.

Expanding the computational model such that it consists

of L- and M-cones and MHCs and BHCs resulted in wavelength-dependent receptive fields. This behaviour

strongly depends on the feedback pathways from the

BHCs to the L-cone. Thus the spectral coded feedback

pathways not only set the spectral sensitivity of the HCs

but strongly influence the spatial and dynamic properties

of the HC layer as well. In the next section, we will

discuss the mechanism for these receptive field changes

in more detail.


520 nm 694 nm

10 mV

J I J I 500 ms 500 ms

FIGURE 12. Responses of a MHC in carp retina to full-field stimuli of 520 and 694 nm and of 500 msec duration. Timing and scaling are indicated in the figure. [Taken from Kamermans & Spekreijse (1995)].

The mechanisms of the delayed reduction of the HC receptive field size

We start the discussion with the simple model, consisting of only one HC type and one cone type. The slit stimulus hyperpolarizes the cones at the location of the slit. These cones will reduce their neurotransmitter release and thus hyperpolarize the HCs directly below them. Because HCs are strongly electrically coupled, the hyperpolarization will spread to neighbouring HCs. Due to the electrical coupling, the feedback signal will be much wider than the direct light stimulus. The cones receiving the feedback signal will increase their neurotransmitter release and thus will tend to depolarize the HCs. The membrane potential of the HCs will now be determined by three factors:

1. The direct cone input; 2. The current flowing through the gap-junctions; and 3. The change in the glutamatergic input from the

cone due to the negative feedback.

Because feedback is slower and spatially more extensive than the feedforward signal, this organization will result in a delayed reduction of the HC receptive field size. This is a general feature of the cone/HC network. In the model, HC receptive fields showed this delayed receptive field size reduction with almost any parameter setting.

Spectral dependence of the HC receptive field size The next question is: what is the mechanism for the

spectral dependence of the HC receptive field? First of all, can the spectral dependence of the HC receptive field size be due to straylight? As is obvious from Fig. 1, the dynamic behaviour of the HC receptive fields differs for 565 and 660 nm slit stimuli. The first part of the response is equal for both wavelengths. The responses start to deviate at around 60 msec after stimulus onset. There- fore, straylight as the source of the wavelength dependence can be excluded.

The first question that comes to mind is whether the feedback strength could be different for the different stimulus wavelengths. An indication that this is indeed the case can be found in the dynamic behaviour of the MHCs for different wavelengths. Figure 12 shows the responses of a MHC to full-field light responses of 520 and 694 nm. The intensities were chosen such that the responses had almost equal sustained response ampli- tudes. For 520 nm the light response of the MHC shows a clear secondary depolarizing phase during light on whereas for the 694 nm stimulus the response is rather sustained. Similar behaviour is also found in the model MHCs (Fig. 9) and can completely be accounted for by the feedback connections. This suggests that the effective feedback is reduced for red stimuli. In the model the mechanism responsible for this reduction of the effective feedback strength is as follows. For green light both HCs hyperpolarize and thus both HCs feed back to the cones with the same sign. The total feedback strength will be relatively large. For red light, however, the MHCs hyperpolarize and the BHCs depolarize. Because the feedback signal from the BHCs to the cones has an opposite sign as the feedback signal from the MHCs to the cones, it reduces the effective feedback strength. Therefore, for red light stimulation the total feedback from the HCs to the cones is less than for green light stimulation, and the reduction of the receptive field size by feedback will be less for red light stimulation compared with green light stimulation. This is completely in accordance with the results in Fig. 1.

Dynamics of the HC receptive field profile in systems without negative feedback

Rod driven HCs in goldfish exclusively contact rods (Stell, 1975). No report exists that shows conclusively that rods receive a feedback signal from HCs [for a review see Piccolino (1995)]. Because feedback may be very limited or even absent in the rod/HC network, the kinetics of the rod HCs receptive field would approach the theoretical predictions. We measured the kinetics of rod driven HCs with the same protocol as was used for the cell in Fig. 1. As is obvious from Fig. 13, the receptive field profile follows the theoretical prediction closely. For the parameters see the figure legend. We studied this effect in two rod driven HCs, both of which behaved similarly.

The size of the effect of feedback on the HC receptive field size

As was calculated in the Results section, the change in apparent coupling conductance due to the network properties is on average a factor of 6.0. In these calculations we have assumed that the network properties do not influence the initial part of the response at all. However, this is unlikely since there does not seem to be a "pure" delay in the feedback pathway (Kamermans & Spekreijse, 1995). As the network connections reduce the receptive field size, even the estimates of 2 based on the initial part of the response will be underestimated.


:=L

~v

800

600

400

200

/I

, "''

/t

I I 0 0 200 400 600 800

Time (ms)

FIGURE 13. Kinetics of the receptive field profile of a rod driven HC combined with the theoretical receptive field profile. Stimulus: 500 nm wavelength, - 3 . 6 log intensity, 500/~m slit. Zph = 60 msec; ZHC = 140 msec; 2 = 690/tm. For details see

text.

How significant is the effect of feedback on the HC receptive fields? The reported reduction in coupling conductance due to dopamine application in pairs of dissociated coupled HCs is around a factor of 10 (Lasater & Dowling, 1985), whereas dopamine increases the sensitivity of the glutamate modulated conductance in dissociated HCs by a factor of about 2-3 (Knapp & Dowling, 1987; Liman et al., 1989). We have found that feedback reduces the apparent coupling conductance by a factor of 6. Thus, the effects of feedback and of dopamine on the receptive field size are of the same order of magnitude.

HC receptive field sizes change during light/dark adaptation (Mangel & Dowling, 1985; Yang et al., 1988a,b; Tornqvist et al., 1988). These changes are generally attributed to the effect of dopamine on the gap- junction conductances. However, light/dark adaptation is also known to modulate the strength of feedback from the HCs to the cones (Weiler & Wagner, 1984; Kirsch et al., 1990; Djamgoz et al., 11988). Therefore, it cannot be excluded that part of the effect of dopamine on the

receptive fields is due to modulation of the feedback pathways.

Functional implications

This study suggests that HCs are strongly electrically coupled and that feedback from HCs to cones reduces the effect of this coupling. Why are the HCs coupled so strongly when negative feedback will reduce the effect of coupling again? When all HCs in one class have more or less the same membrane potentials, for instance during homogeneous illumination or in the dark, feedback will not have any effect on the spatial properties of the HC layers and thus the apparent coupling will be large. Small differences in local input will be spread out in the HC network and thus will hardly influence the membrane potential of individual cells, making sure that all HCs feed back to the cones more or less equally. When the retina is stimulated with a spot of light, the HCs stimulated by the spot will hyperpolarize. The potential will spread through the HC layer and the HC layer will respond over a large area. As discussed in this paper, in


this condition feedback will reduce this spread of potential and keep the HC activity relatively local.

This mechanism ensures that the HCs are extremely coupled when the HCs are stimulated more or less equally but that the HCs partly uncouple when part of the HC layer is stimulated differently from the rest of the retina. This has two advantages: (1) for homogeneous stimuli or stimuli with very fine detail the strong coupling of the HCs reduces the spatial noise; and (2) for intermediate sized stimuli, the stimulated HCs partly uncouple from the rest of the HC network and generate a feedback signal based on the "local" stimulus properties instead of the "global" properties, thus generating the bipolar cell surround based on the stimulus properties near the stimulated bipolar cell.

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Acknowledgements We would like to thank J. Verweij for his comments on the manuscript. M. K. is a fellow of the Royal Netherlands Academy of Arts and Sciences (K. N. A. W.). This work was supported by the Human Frontier Science Program (HFSP; PI: H. Spekreijse) and by the Netherlands Organization for Scientific Research (N.W.O.)

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