Temporal Properties of Marmoset Lateral Geniculate Cells

Temporal Properties of Marmoset Lateral Geniculate Cells JAN KREMERS,*~ STEFAN WEISS,*t EBERHART ZRENNER* Received 8 August 1996; in revised form 11 December 1996; in final form 11 March 1997

We measured the temporal modulation transfer functions (TMTFs) of cells in the marmoset lateral geniculate nucleus ULGN) at three different luminance levels, and described the responses with a linear model. It was found that qualitatively there are many similarities with the temporal response properties of macaque and marmoset retinal ganglion cells. M-cells displayed stronger attenuation at lower temporal frequencies, and showed more nonlinearities (such as saturation and a contrast gain control) than P-cells. We therefore propose that the temporal properties of the visual system of New and Old World monkeys are similar at least up to the LGN. However, there are 'some quantitative differences, indicating that response alterations take place at the stage of synaptic transmission in the LGN. The most important are an attenuation of the responses to higher temporal frequencies and the smaller differences between parvo- and magnocellular cell responsivities. Cell responses to square-wave modulation were also measured and compared with predictions from a linear systems analysis. The linear systems analysis gave reasonable predicted responses to square-wave modulation, but these predictions were poorer than those for retinal ganglion cells, indicating that additional nonlinearities are introduced at the synaptic transition in the LGN. © 1997 Elsevier Science Ltd

Marmoset Lateral geniculate nucleus Temporal Modulationtransfer function Linear systems analysis

INTRODUCTION et al., 1993). This polymorphism was first discovered by The visual system of New World monkeys (Platyrrhini) Jacobs and colleagues in the squirrel monkey (Saimiri; is in many respects very similar to those of Old World for an overview of the early data on colour vision in New monkeys and apes (Catarrhini). The morphology and World monkeys we refer to the relevant chapter in the dimensions of retinal ganglion cells are very similar book by Jacobs (1981)). In short, di- and trichromatic (Ghosh et al., 1996; Goodchild et al., 1996; Wilder et al., squirrel monkeys were discovered. Moreover, there were 1996). The spectral and spatial retinal processing also three different cone pigments in the middle and long seems to be very similar (Yeh et al., 1995; Kremers & wavelength range (Mollon et al., 1984). A similar Weiss, 1997). The main differences seem to involve the polymorphism was encountered in marmosets (Travis et photoreceptors and their pigments. The extrafoveal cone al., 1988; Tov6e et al., 1992) and other New World density in New World monkeys is higher than in Old monkeys (Jacobs & Neitz, 1987; Jacobs, 1990; Jacobs et World monkeys, whereas the rod density is lower (Troilo al., 1993b, 1987). The polymorphism is apparently sex- et al., 1993; Goodchild et al., 1996). Further, the retinal linked (Jacobs, 1983, 1984; Jacobs & Neitz, 1985), and ganglion cells receive rod input up to relatively high has a genetical origin. On the X-chromosomes, there is retinal illuminances (Yeh et al., 1995; Weiss et al., 1995; only one gene coding for the L-/M-cone pigment. Kremers et al., 1997). This strong rod input is surprising However, there are three different alleles (Jacobs et al., in view of the smaller rod density. Finally, New World 1993b; Hunt et al., 1993; Williams et al., 1992). The L-/ monkeys display an interesting genetically defined M-photopigments of marmosets absorb maximally at polymorphism of the photopigments (Jacobs et al., 543, 556 and 563 nm (Travis et al., 1988; Tov6e et al., 1993b; Mollon et al., 1984; Williams et al., 1992; Hunt 1992). Two platyrrhine species do not exhibit this polymorphism. All howler monkeys (Alouatta) seem to *Department of Experimental Ophthalmology, University of Ttibingen be trichromatic (Jacobs et al., 1996). Further, there are Eye Hospital, R6ntgenweg 11, D-72076 TUbingen, Germany. indications that the owl monkey (Aotus) has only one L-/ tPresent address: Natural and Medical Science Institute NMI, M-cone type (Jacobs et al., 1993a). In addition, they seem Eberhardstral3e 29, D-72762 Reutlingen, Germany. to lack functional S-cones (Wikler & Rakic, 1990; Jacobs STo whom all correspondence should be addressed [Tel +7071 2985031; Fax +7071 295777; Email jan.kremers@uni- et al., 1993a). tuebingen.de]. Although the differences in photoreceptor arrangement 2649 2650 j. KREMERS et al.

in New World monkeys do not result in different spatial animals (three males and three females) had the 563 nm and spectral processing in the retina, not much is known pigment; four animals (three males and one female) had yet about how it might influence the temporal properties the 543 nm pigment; two males and one female possessed : of the peripheral visual system. Yeh et al. (1995) the 556 nm pigment. The animal experiments were described the temporal responsivity of three retinal conducted in accordance with the European Commu- ganglion cell axons of marmosets to luminance modula- nities Council Directive of 24 November 1986 (86/609/ tion, one belonging to the magnocellular (M-) pathway, EEC). The animals were initially anaesthetized by an the other two to the parvocellular (P-) pathway. They intramuscular injection of 15-30 mg/kg ketamine hydro- showed that they were very similar to the temporal chloride (Ketanest ®) and 3.5 mg/kg xylazin hydrochior- responsivity of macaque retinal ganglion cells. Accord- ide with 1.5 mg&g methyl-4-hydroxybenzoate (0.15 ml/ ing to Yeh et al. (1995) the temporal properties of LGN kg Rompun ® 2% solution). Additional doses of ketamine ceils were similar, although no direct data were hydrochloride were administered if necessary. After presented. We measured the temporal modulation tracheotomy, the animals were ventilated through a transfer functions (TMTFs) of marmoset LGN cells to tracheal canula with 70%/30% N20/O2 with 0.2--0.8% provide a more quantitative comparison with data on Enflurane (Ethrane®; 0.4--0.8% during surgery; 0.2- retinal ganglion cells in marmosets and macaques (Lee et 0.4% during measurements). Eye movements were al., 1990; Yeh et al., 1995). A comparison between the suppressed by intravenous administration of gallamin physiological properties of LGN and retinal ganglion triethiodide (Flaxedil®; 5 mg/kg/hr) dissolved in Ringer, cells of Old- and New World monkeys will help to together with glucose and Solu Decortin ® (infusion rate: elucidate the principal functional retinal architecture of 0.6 ml/hr). A rectal probe connected to a thermal blanket both groups. Since many platyrrhines are dichromats, the was used to maintain a rectal temperature of 37.2°C. EEG comparison may also help to disclose the role of and EKG were continuously monitored to check the trichromatic colour vision in the phylogenetic and depth of anaesthesia. ontogenetic development of the retina. Pupils were dilated with atropine sulfate (1%) and We modelled the data with a linear model consisting of neosynephrine (5%). Contact lenses (radius of curvature: a cascade of low-pass filters, two stages of leadlags and a 3.5 mm; diameter: 5 mm), with appropriate correction pure time delay. This model could describe the data (determined with a slit skiascope) to focus the eyes on the reasonably well. As an additional test of the temporal stimulator at 1.14 m distance, protected the eye against linearity of LGN cell responses, we measured responses desiccation. to square-wave modulations, and compared the measured A craniotomy was made and tungsten in glass responses with linear systems predictions on the basis of microelectrodes were lowered into the lateral geniculate the TMTFs. This method has previously been used for the nucleus for extracellular recordings. The stereotaxic lateral eye of horseshoe crab (Brodie et al., 1978) location of the craniotomy was based on an atlas of the macaque retinal ganglion cells (Kremers et al., 1993; Lee marmoset brain (Stephan et al., 1980). et al., 1994), macaque LGN cells (Gielen et al., 1982), cat We measured from cells in the parvo- and magnocel- retinal ganglion cells (Victor, 1987) and cat LGN cells lular layers. Since all the animals were dichromats, it was (Saul & Humphrey, 1990). Therefore, the present study more difficult than for Old World monkeys to determine was not meant as an in-depth investigation of the from which cell we recorded. This is because most usefulness of linear system analysis [for that we refer parvocellular cells do not show any colour opponency the reader, for example, to the review by Watson (1986)]. (Kremers et al., 1997; Weiss et al., 1995; Yeh et al., Rather, the analysis was used as a tool to construct an 1995). We determined the cell type from the sequence of inventory of possible nonlinearities, which may act upon ocular input to the cells (which when entering cranially the responses of LGN cells. is: parvocellular contralateral, parvocellular ipsilateral, magnocellular ipsilateral, and magnocellular contralateral), and from the relative positions of the microelec- METHODS trode to lesions we occasionally made, and which were Animal preparation retraced histologically after the experiments. We recorded from the lateral geniculate nucleus (LGN) After the experiment, normally lasting between 24 and of 13 adult dichromatic marmosets (CaUithrix jacchus; 60 hr, the animals were euthanised with an overdose of 300-350 g; eight males and five females). Seven animals Nembutal ® , and the brains were removed for histological (four males and three females) were chosen randomly. processing after perfusion with 4% paraformaldehyde. The type of photopigments present was established electrophysiologically (Weiss et al., 1995) and by a Visual stimulation posteriori genetic analysis (Kremers et al., 1997). For the The stimuli were generated on a BARCO Calibrator remaining six animals, we were able to do the genetic (vertical frequency: 100 Hz) monitor at a distance of analysis a priori. We selected dichromats because we 114 cm, which was controlled by a VSG 2/2 graphic card wanted to establish more firmly the correlation between (Cambridge Research Systems). The stimuli were the genetic and electrophysiological determination of the spatially uniform, and were circular with a diameter of present photopigment in dichromatic animals. Six 2.5 or 5 deg. We did not find any systematic differences TEMPORAL PROPERTIES OF MARMOSET LATERAL GENICULATE CELLS 2651 in results between both stimulus sizes. The mean purpose of this delay was to avoid recording responses to luminance of the stimulus was 2, 10 or 40 cd/m 2. The the stimulus onset. luminances were recalibrated at regular intervals using the internal calibration of the BARCO monitor and checked with an UDT luminance detector and a RESULTS International Light ILl700 Radiometer. The VSG card Cell responses to sine-wave modulation automatically performed a gamma correction on the The procedure we followed to obtain TMTFs is very calibrations and gave the appropriate output to control the similar to the one used by Lee et al. (1990). The monitor guns. Only the red and green phosphors of the peristimulus time histograms were Fourier analysed. monitor were modulated. The blue phosphor was not Response amplitude and phase were defined as the activated. The mean luminances of the phosphors were amplitude and phase of the first harmonic component of equal and half the total luminance. Therefore, the the stimulus. chromaticity was constant in all conditions. Two mm Figure 1 displays the response amplitudes and phases artificial pupils were positioned in front of the eyes. of an off-centre M-cell as a function of temporal Taking into account the smaller marmoset eye, we frequency and contrast at 615 td retinal illuminance. calculated that the retinal illuminance would be about The cell has a band-pass characteristic which is also seen 4.9-times larger than in humans. Therefore, the mean in macaque retinal ganglion ceils projecting to the retinal illuminance was equivalent to 31, 153 or 615 magnocellular layers of the LGN, when stimulated with human trolands. luminance sine-wave modulation. We displayed sinusoidal modulations of several The optimal frequency is between 4 and 20 Hz [Fig. temporal frequencies and Michelson contrasts (defined I(A)]. In addition, the response phase depends on as contrast. High contrast responses are phase advanced -Lmm) relative to low contrast responses at the same frequency [Fig. I(D)]. The effect of contrast-dependent phase shifts is most obvious at intermediate temporal frequencies where Lm~, and /-~n are the maximal and minimal (between 4 and 20 Hz; Fig. I(B) and (D)]. This was found luminance in the stimulus, respectively). A trigger pulse in most M-cells and indicates the presence of a contrast given by the VSG card was used for the synchronization gain control mechanism in the responses of cells in the of stimulus with the spike recordings. magnocellular layers of the marmoset LGN. The contrast A monitor has inherent temporal restrictions. However, gain control has been described before in cat retinal these had very little influence on the results. Owing to the ganglion cells (Shapley & Victor, 1978, 1981, 1979) and 100 Hz refresh rate and the radiation properties of excited in magnocellular retinal ganglion cells of the macaque phosphors, stimulus frequencies above about 20 Hz are (Benardete et al., 1992; Lee et al., 1994) and the distorted and stimulus frequencies above 50 Hz cannot be marmoset (Yeh et al., 1995). At lower retinal illumi- obtained. However, marmoset LGN cells were not very nances, the phase did not change as a function of contrast, responsive to these frequencies (see Results). An indicating that the contrast gain control mechanism is additional problem was the sequential excitation of only present at high retinal illuminances. monitor pixels. The actual stimulus was displayed on Figure I(C) displays the response amplitude of the M- the centre of the screen and therefore appeared somewhat cell as a function of contrast. Through the data points a Naka-Rushton function was fitted: later than the trigger pulse, which was delivered when the pixels in the upper left corner were excited. This Rm.b introduced a delay of about 5 msec (half the framecycle). R(C) = R(O) + (C + b----~' The phase data were corrected for this delay. To study how useful the monitor is for measuring where R(0) is the spontaneous activity or the response at temporal properties we measured psychophysically the 0% contrast which was measured at each condition, Rm is sensitivity of human observers to sine-wave temporal the maximal response, b is the contrast eliciting a modulation of different frequencies and compared them response which is half the maximal response, and C is with equivalent measurements described in the literature contrast. This function is identical to the one used by Lee using another stimulus source (I_e,e et al., 1990). Since et al. (1990), with the exception that we used a variable the data were very similar, we are confident that the measure for the spontaneous activity instead of a fixed monitor stimulus was suitable for the temporal measure- value. For each temporal frequency, the measurement of ments. spontaneous firing was repeated. We introduced this extra variable because the spontaneous activity of these cells was not as stable as in retinal ganglion cells. The fitted Data acquisition curves show that the responses of the cell saturate at The time of occurrence of each discharge was recorded higher contrasts. This is a typical finding for the majority with a 0.5 msec time resolution on a CED 1401 on-line of M-cells. computer. The CED did not start to acquire the data Figure 2 shows similar data for an off-centre P-cell at before at least one period of the signal was given. The 615 td. The P-cell shows a similar dependency of 2652 J. KREMERS et aL

= 100% C 200 A --4- 75% • 1 Hz ~ ~ ~~__ 50% 100 - - 25% "~ 150 --. .--*.... 10% Q. _E E ~ 100 "0 "0 .=-__ i O.E 10 //-- :!1 X

I I I I 1 10 100 0 25 50 75 100 D Bo 0 loo% 75% -I00 -200 50% ~,~ 1~----~----~---~ = 1 Hz 25% c- • ...... • ...... • ...... • --4-- 2 Hz ~• ,._ .... ---e------e------e --&--- 4 Hz cO -200 c- pe'" / -~- 5 Hz O -400 Q. ~ ..... ~ .... * .... 8 HZ jr=-- --*- 10Hz CC~, 300 .t --o- 15 Hz ,Y -600 I -400 [ I I I 1 10 100 25 50 75 100 Temporal frequency (Hz) Contrast (%) FIGURE 1. Response amplitudes and phases of an off-centre M-cell, 615 td. (A) The response amplitude as a function of temporal frequencyat different stimulus contrasts. (B) Response phase for the same conditions as in (A). All the phases overlap largely, but there is a delay at the lower contrasts and intermediate temporal frequencies. (C) Response amplitude as a function of stimulus contrast at several frequencies. The data are fitted with Naka-Rushton functions, as described in the text. The cell displays some response saturation at higher contrasts. (D) Response phase as a function of contrast and for several temporal frequencies. At low contrasts, the response phase is smaller than at high contrasts, especially at frequencies above 2 Hz, indicative of a contrast gain control mechanism.

response amplitude on temporal frequency as the M-cell. In the model, there is a cascade of N~ low-pass filters with It has a similar optimal frequency. However, a compar- time constant rl and Nz low-pass filters with time ison of the data in Fig. I(A) and Fig. 2(A) reveals that the constant z2. The last term describes a pure time delay of response amplitude of the P-cell is smaller than the D sec. The term between large brackets describes two response amplitude of the M-cell at all stimulus stages of leadlags. The leadlags were introduced to conditions. The response phase does not change system- describe the decrease in response at low temporal atically as a function of contrast at any temporal frequencies. The value ~ of the leadlag stages describes frequency. Thus, in contrast to the M-cell, the responses the amplitude ratio between low and optimal frequency of the P-cell do not show any signs of a contrast gain (Milsum, 1966). A is a constant controlling the overall control mechanism. Further, there is less response gain of the system and can be considered as the response saturation. This is in agreement with other observations at a temporal frequency of 0 Hz. on primate retinal ganglion cells (Benardete et al., 1992; This linear model resembles the model used by Lee et al., 1994; Yeh et al., 1995). Purpura et al. (1990) for describing macaque retinal We modelled the mean responses of the cells with a ganglion cell responses, with the exception that we added cascade of low-pass filters, two leadlag stages and a a pure time delay. The "solver" routine in the Excel 4.0 delay. The amplitude and phase data were fitted with a for Windows program was used for fitting the data. In the linear model with the following mathematical descrip- fit of the model through the average cell response data, tion: we originally used 8 free parameters: A, ct, T, N1, zl, N2,

(1 + io~27rfT)2] x (1 + i27rf'r]) N' x (1 + i27rf'r2) N2 x exp(i27rfD) (1) R(f) ----A I-T1-~ ~-] TEMPORAL PROPERTIES OF MARMOSET LATERAL GENICULATE CELLS 2653

= 100% C 150 A --m-- 75% • 1Hz 100 • 2 Hz • • 4 Hz

• 10% B. 100 • 8 Hz • E ¢P 'o

.."" ",.-' . ~. 50 E 10 < -/'~/.," " i~ ' <

•" i o " I " I I I I I I 10 100 0 25 50 75 100 D a 0- 0

~1~ ---4--= 75%100% g -100 A.~ -J'-~-~ ~--A : 1 Hz -200 - ~_ 50% O9 t~ • ~v ---'~----~----v ---4--2 Hz t- - - 25% t" ~t. e.. --~--4 Hz -10% a~ -200 O9 t- 4 '"* ...... * ...... * ...... * -~--5 Hz O -400 - o .--.-- = .--.--,------e .... e .... 8 Hz o9 -300 ~.. /..~ ...... °-..~...o --*--10 Hz cr rr --•--. 15 Hz

-600 I [ I -400 I I I I 1 10 100 0 25 50 75 100 Tem~x~ralfrequenw(Hz) Contrast (%) FIGURE 2. The same data as in Fig. 1 for an off-centre P-cell, 615 td. In contrast to the M-cell, this cell does not display an obvious saturation and contrast gain control.

z2 and D. We constrained the values of Nl and N2 to That was about 32 msec for M-cells and 37 msec for P- values between 0 and 50. We excepted fits in which the cells. This time difference between a change in the time constants Zl and z2 were between 1 and 1000 msec. stimulus and a change in firing rate may not be solely The rationale for these constraints on the time constants determined by a pure delay. However, it is the best was that filters with other time constants would have cut- estimate we could obtain with the present data. The fits off frequencies outside the measured range. Since ~ of with these fixed delay times had a similar quality as the the leadlag stages describes the low frequency roll-off, fits with variable delay times. we have constrained ~ to values below 500. All free Figure 3 displays the mean response amplitudes and parameters had to be larger than zero. We also varied the phases of 7 P- and 11 M-cells as a function of temporal number of leadlag stages. One stage gave suboptimal fits. frequencies for several contrasts at a mean retinal More than two stages did not improve the fits illuminance of 615 td. The lines are the fits with the significantly. We therefore used two stages for all fits. linear model. Note that the stimulus contrasts all differed Because of the large amount of free parameters, the fits by a factor of 2 or 2.5. The mean response amplitudes of were not constrained well. In particular, delay time D and the P-cells to these contrasts also differed by approx, a numbers of low pass filters N 1 and N2 were variable in a factor of two. This indicates that P-cells show little reciprocal manner: large values of D (>30msec) response saturation. The mean response amplitudes of the occurred in combination with only few (three or less) M-cells are not equidistant for these contrasts, indicating low-pass filters. On the other hand, large values of N 1 substantial response saturation. The response phases are and N2 occurred together with short delay times also given. For clarity, the phases at lower contrasts are (<20 msec). displaced along the (logarithmic) frequency axis by a To obtain better constraints on the fits, we repeated the factor of two. At low temporal frequencies, M-cells fits with a fixed delay time D. This seemed reasonable, respond phase advanced relative to the P-cells. since delay is probably mainly determined by the retinal The values of the free parameters which resulted from wiring and, therefore, should be similar in all conditions. the fits and the number of cells used in the fits are given in To obtain an estimate of the delay, we determined the Table 1. The last column of Table 1 gives a quantification time after the excitatory change in different square-wave for the goodness of fit. It is defined as the sum of squared stimuli at which the firing rate of the cells first changed. distances between measurements and fits in the complex 2654 J. KREMERS et al.

M-cells P-cells A C .E 100 100

E o /" ";"- \ ~- 10 ~/ ]== 10 ~/" • • • " o / .... -'~-- ~.~* o. ,,,/ / \ co ~/// • j/" CE /

I I ~ 1 [ I I 10 100 I 10 100 -- 100% " [B D --- 5o% • 0 ,,_ ~--/" • 0 ---- 25% " -~"x _...... 10% •

", -36o \\ \'

t-- k n -720 ~ -720 - \,

-900 l I I - -900 I I I I 10 100 I 10 100 Frequency (Hz) Frequency (Hz) FIGURE 3. Mean temporal responses of I l M- (A, B~ and seven P-cells (C, D) at several contrasts and at a mean retinal illuminance of 615 td. The data are fitted with a linear model, consisting of a cascade of low-pass filters, two stages of leadlags and a pure time delay of 32 msec for M-cells and 37 msec for P-cells. Further details are provided in the text. The phases at lower contrasts have been shifted along the frequency axis rightwards for clarity. For each decrease in contrast, the temporal frequency was multiplied by a factor of two. The normalized sum of squares were 7.78, 2.25, 2.35 and 0.45 for the 100%, 50%, 25% and 10% contrast M-cell data, respectively (see also Table l). The sum of squares were 3.76, 1.30, 0.79 and 1.37 for the P-cells.

plane, normalized to the maximal response at each characteristics are. The results show that M-cells have a contrast and divided by the number of data points stronger band-pass characteristic than P-cells. included in the fits. Small values indicate good fits: large From the Naka-Rushton functions which were fitted values indicate poor fits. As could be expected from the through the response amplitude vs contrast data for each original fits with variable delay times, just a few low-pass cell [examples of curve fits are shown in Fig. I(C) and filters were needed. Purpura et al. (1990) needed many Fig. 2(C)], we obtained the cells' responsivity, which was more low-pass filters to describe their data, probably defined as the contrast gain (expressed as imp/sec/% because they did not introduce a delay in their model. contrast), being the initial slope of the Naka-Rushton Often, a single type of low-pass filter was sufficient to function. The responsivity was used for the amplitude describe the data. plot of the TMTF. The mean response phase of the first The fits to the M-cell data at high contrasts were three responses above a level of 10 imp/sec was used for relatively poor at low frequencies (Fig. 3), probably the phase plot. Figure 4 shows the mean TMTFs of P- and owing to nonlinearities in the M-cell responses, which M-cells at three different retinal illuminance levels. cannot be described by our linear model. The model fits were better for the P-cell responses at most conditions. Linear systems analysis M-cells had larger values for parameter ~ than P-cells We used a linear systems analysis to predict responses at all conditions. As described before, ~ quantifies the low to square-wave stimuli using the TMTFs. The procedure frequency roll-off of the response amplitudes. The larger for calculating the predicted responses are the same as the value of ~, the more band-pass the response described by Kremers et al. (1993) and Lee et al. (1994). TEMPORAL PROPERTIES OF MARMOSET LATERAL GENICULATE CELLS 2655

TABLE 1. Fitting parameters of the linear model to response data for P- and M-cells, at different mean retinal illuminances and contrasts

Retinal Normalized illuminance Number Contrast zx t2 T sum of Cell type (td) of cells (%) N1 (msec) N2 (msec) ~t (msec) squares M-cells 615 11 100 0 -- 3 5.97 7.59 3.35 7.78 75 0 -- 2 31.06 19.56 11.67 4.41 50 1 14.88 1 92.48 50.46 10.70 2.25 25 0 -- 2 23.89 37.84 6.66 2.35 10 2 19.77 0 -- 6.95 4.64 0.45 153 9 100 2 8.12 1 1.00 6.04 2.00 5.98 75 2 11.14 0 -- 6.21 2.49 4.79 50 2 14.32 0 -- 6.86 3.22 3.72 25 2 23.36 0 -- 6.66 4.73 3.38 10 2 30.74 0 -- 7.60 5.19 0.52 31 3 100 3 24.19 0 -- 5.85 6.59 6.88 75 0 -- 2 35.89 6.10 4.43 7.20 50 2 37.75 1 2.10 6.10 4.22 5.71 25 2 10.54 1 49.71 6.87 3.20 5.21 P-ce~s 615 7 100 1 40.12 1 1.00 3.05 4.66 3.76 75 1 1.16 1 36.15 3.10 4.40 2.63 50 1 19.45 3 1.00 2.70 2.76 1.30 25 4 1.00 1 39.63 3.06 4.02 0.79 10 1 26.92 0 -- 2.34 5.91 1.37 153 14 100 2 4.80 1 23.34 2.00 2.30 1.61 75 1 20.99 2 5.30 2.23 1.86 1.28 50 1 36.86 2 4.60 2.79 2.34 0.74 25 1 12.09 1 50.35 2.92 3.16 0.90 10 1 54.37 1 2.94 2.82 2.93 0.38 31 4 100 2 20.64 0 -- 2.28 4.58 4.01 75 2 17.86 0 -- 2.57 2.63 3.86 50 1 20.99 3 8.05 3.30 4.06 3.04 25 3 15.85 1 13.93 4.37 9.90 3.45

The TMTFs were completed by fitting a third-order Figure 5 displays the response of a P-cell to square- polynomial through the amplitude data and a second- wave luminance modulations at three different contrasts order polynomial through the phase data. The predicted and at four different frequencies. In the same plot, linear responses to the square-wave stimuli were obtained by systems predictions of cell responses are given. The multiplying the Fourier expansion of the square-wave goodness of prediction was quantified by the standard stimulus with the sensitivities of the cells at the error between the predicted and actually measured appropriate temporal frequency and by shifting according response. Generally, there is a reasonable correspondence to the response phases. The mathematical description of between predictions and actual responses, although linear the calculated response to the square-wave stimuli systems predictions of responses of macaque retinal (Rsq,C(t)) is: ganglion cells (Kremers et al., 1993; Lee et al., 1994) are

4 o~ S2n+ 1 Rsq,c(t) = R0 + C~y~ 2(2n + 1)" sin ((2n + 1)27rfl + ~2n+1) (2) where Ro is the maintained firing rate at the mean generally more accurate. This indicates that more luminance, which was measured by including stimuli nonlinearities are involved in the responses of marmoset with 0% contrast; C is the contrast of the square-wave LGN cells. stimulus; S2n+~ and qo2n+l are the sensitivities and the Figure 6 displays the responses and predictions for an phases of the cells at the (2n + 1)th harmonic, respec- off-centre M-cell at a mean retinal illuminance of 615 td. tively. The expansion was truncated at harmonics for The predictions are poorer than those for the P-cell shown which the sensitivity of the cell was smaller than 0.1 imp/ in Fig. 5. This confirms the previous conclusion that M- sec/%. Negative predicted responses were set to zero. cells are more nonlinear than P-cells. Especially at the We did not average the responses of all the cells of the higher contrasts and at 4 and 8 Hz the predictions are same type, as has been done with the data on ganglion relatively poor. At these conditions, magnocellular cells (Kremers et al., 1993), since the responses of the retinal ganglion cell responses of macaques were also cells were more variable than the ganglion cell responses, less well described by the linear systems analysis making averaging less useful. Instead, we calculated the (Kremers et al., 1993; Lee et al., 1994). However, predictions for each individual cell. similar to the parvocellular cell responses, the predictions 2656 J. KREMERS et al.

31 td 153 td 615 td

A E o o 10 P-cells -- M-cells

r~ E 1

0.1 I I I I I I I I 0 o B • on-centre P-cells A _ ,~. --,-- off-centre P-cells "* ------on-centre M-cells -100 - .-,,-- off-centre M-cells

e- r~ -200 - ~ c- -300 - O O. -400 - r~

-500 I I I I I I 1 10 100 1 10 100 1 10 100 Frequency (Hz) FIGURE 4. Mean contrast gain (A) and phase (B) of M- and P-cells at different mean luminance levels. The TMTFs were obtained by averaging the TMTFs of individual cells (number of cells at 615, 315 and 31 td, respectively. M-cells: I l, 10, 3; P- cells: 7, 13, 4). The range of frequencies to which the cells respond becomes narrower when retinal illuminance decreases. They become less responsive, especially at higher frequencies, and their phases show that they respond more sluggishly.

of the magnocellular LGN cell responses in the marmoset DISCUSSION were poorer than those of the macaque retinal ganglion cells. Comparison of P- and M-cell data We calculated the mean goodness of fit for five M-cells The responses and the TMTFs of marmoset P- and M- and five P-cells at three different luminance contrasts and cells have band-pass characteristics for luminance at four different temporal frequencies. The mean retinal modulation. The factor c~ of the leadlags in the fits with illuminance was either 153 or 615 td. We measured one M-cell and one P-cell at both illuminances and found the linear model were larger for the M-cells than for P- only negligibly small differences in the goodness of fit. cells, indicating a stronger attenuation at low frequencies. The mean standard errors of the predictions for P- and M- The fits with the linear model to M-cell responses were cells at different stimulus contrasts are displayed in Fig. not very satisfactory at low temporal frequencies and 7, as a function of temporal frequency. For all conditions, high mean retinal illuminances, indicating that M-cells the predictions are poorer for M-cells than for P-cells, display some nonlinearities. The linear model gives confirming the conclusion from the fits with the linear better descriptions of the P-cell responses, indicating that model that M-cells are temporally more nonlinear than P- P-cells are more linear. In agreement with this result, the cells. Especially in M-cells, the mean standard error linear systems analysis gives better predictions of increases with increasing contrast. Possible nonlinearities responses to square-wave modulations in P-cells than in at these high contrasts are response saturation and M-cells. rectifying nonlinearities. Further, M-cells are more The contrast gain control is a nonlinearity present in nonlinear at 4 and 8 Hz temporal frequency than at I M-cell responses, but not in P-cells. The cell responses of and 2 Hz. The frequency dependency of the mean the marmoset retinal ganglion cell belonging to the M- standard error is possibly caused by the contrast gain pathway described by Yeh et al. (1995) indicate the control mechanism, which is also stronger at intermediate presence of a contrast gain control mechanism. The cells frequencies (Shapley & Victor, 1978). The frequency belonging to the P-pathway do not display a contrast gain dependency is not so strong in P-cells, which confirms control. The response data in Fig. 3 also indicate that P- our previous conclusion that P-cells lack a contrast gain cell responses do not saturate, whereas M-cell responses control mechanism. show significant saturation. We conclude that marmoset TEMPORAL PROPERTIES OF MARMOSET LATERAL GENICULATE CELLS 2657

200 1 Hz 2 Hz 4 Hz 8 Hz

150 I%1 100 T I T

0 fl[~ nrl

"~ 2oo

ol I Io

0 n n~ J

200

150

100 I 50 lon J" 250 500 750 1000 0 125 250 375 500 0 125 250 0 25 50 75 100 125 Time (msec) FIGURE 5. Some examples of responses of an on-centre P-cell to square-wave stimuli (histograms) at three different contrasts (25, 50 and 75%) and four different temporal frequencies (1, 2, 4 and 8 Hz) together with a linear systems prediction (drawn lines), using the TMTFs of the cell. The linear systems analysis gives reasonable predictions of the time of response and of the response form. The bars indicate the standard error of the prediction. Mean retinal illuminance was 153 td.

M-cells are temporally more nonlinear than P-cells, described for macaque retinal ganglion cells (Kremers partly due to a contrast gain control mechanism. et al., 1993; Lee et al., 1994). We found that the responsivity of M-cells was larger Comparison of macaque and marmoset data than that of P-cells (Fig. 4). This has also been observed Qualitatively there is a good resemblance between in macaque retinal ganglion cells (Lee et al., 1990; primate LGN and retinal ganglion cells in many aspects Kaplan & Shapley, 1986). We further found that the of their temporal responsivity to sine-wave modulation. temporal frequency range to which the marmoset LGN The temporal responses and the TMTFs of P- and M-cells cells respond decreases with decreasing mean retinal to luminance modulation are band-pass in marmosets and illuminances, mainly owing to a loss of responsivity, macaque retinal ganglion cells, but also in marmoset especially at high temporal frequencies, which has also LGN cells. However, as the fits with the linear model been described for macaque retinal ganglion cells (Lee et indicate, M-cells display a larger low frequency roll-off, al., 1990; Purpura et al., 1990). which was also found for macaque retinal ganglion cells From these similarities between marmoset and maca- (Purpura et al., 1990). The colour-opponent P-cells of the que data, we conclude that the visual system of macaque are band-pass because of a latency difference marmosets and Old World monkeys are very similar in between centre and surround response (Gouras & the temporal domain. All the marmosets included in this Zrenner, 1979; Lee et al., 1989a; Smith et al., 1992). study were dichromats. Dichromacy, therefore, seems to From our data it is not possible to conclude whether the have no or only a minor influence on the temporal responses of P-cells to luminance modulation in the response of the cells. In another study, we have found that dichromatic marmoset are also influenced by these phase the spatial receptive field dimensions of marmoset LGN differences between centres and surrounds. As in cells are, after correction for the eye size, similar to those macaques, the M-cells in the marmoset LGN are of Old World monkeys (Kremers & Weiss, 1997). temporally more nonlinear than P-cells. M-cells have a Anatomical studies on the marmoset retina have not contrast gain control mechanism and their responses revealed major differences with Old World monkey saturate more strongly than P-cells, which is also retinae either (Ghosh et al., 1996; Goodchild et al., 1996; 2658 J. KREMERS et al.

300 1 Hz 2 Hz 4 Hz 8 Hz

225 PO 01 150 1 T 75 j"

0 \[1 "~ fl n , ,

300 (I,1

225

o~E O1 q,t 150 O

75 tR 0 nnnnn

300

225

150

0 L I nr]n ~n n , , 0 250 500 750 1000 0 125 25O 375 50O 0 125 250 0 25 50 75 100 125 Time (msec) FIGURE 6. Similar data as in Fig. 5 for an off-centre M-ce]] at a mean retinal i]luminance of 6]5 td.

P-cells M-cells Wilder et al., 1996). It therefore, very probable that the 70 is, 25% contrast 0 retinal organization is very similar in Old and New World 50% contrast --Et-- monkeys, despite the difference in photoreceptor ar- ~L-- 75% contrast --~-- / ~ ..... A 60 / rangements between the two groups (see Introduction). / Thus, it seems likely that the phylogenetically recent / / development of trichromatic colour vision in primates o 50 / (Tovre, 1994) made use of an existing retinal organiza- / / tion. The P-cell system does not seem to be a "1o specialization in primates developed to subserve colour / / / vision, as has been proposed by Shapley & Perry (1986). Instead, it seems originally to have had another function, / • 30 possibly in spatial vision or as a channel for a brightness (though not luminance!) signal. However, it cannot be excluded that secondary specializations for colour vision 20 have occurred in the P-pathway (for instance the "private line" of one cone connected to one bipolar cell to one midget ganglion cell). 10 ' ' ' ' I ' ' ' ' ' ' [ 1 10 The comparison between the response characteristics of marmoset LGN cells and those of macaque and Frequency (Hz) marmoset (Yeh et al., 1995) retinal ganglion cells reveals FIGURE 7. The mean standard error between response prediction and actually measured response to square-wave stimuli as a function of some quantitative discrepancies. temporal frequency. The larger the value, the less good the prediction. Although M-cells are more responsive than P-cells to These are mean data of five P- and five M-cells. The measurements luminance modulation, the difference in marmoset LGN were performed either at 153 or at 615 td mean retinal illurninance. P- cells was not more than a factor of two. In macaque cells have lower mean standard errors than M-cells at all conditions. Thus, P-cells are temporally more linear than M-cells. The deviations retinal ganglion cells, the difference in responsivity can between predictions and measurements increase with increasing be up to a factor of ten (Lee et al., 1990). contrast. Moreover, M-cells are more nonlinear at 4 and 8 Hz than at We found that the optimal frequency (between 4 and 1 and 2 Hz. 10 Hz) and the maximal frequency to which the LGN TEMPORAL PROPERTIES OF MARMOSET LATERAL GENICULATE CELLS 2659 cells of marmosets respond (between 15 and 50 Hz) are support for the idea that many observed nonlinearities in lower than in retinal ganglion cells. We propose that M-cells have very similar origins. these response characteristics are probably altered at the synaptic transmission in the LGN. Similar alterations REFERENCES have been observed in cat and monkey LGNs, when comparing responses of LGN cells with their S-potentials Benardete, E. A., Kaplan, E. & Knight, B. W. (1992). Contrast gain control in the primate retina: P cells are not X-like, some M cells are. et al., et al., et (Hamamoto 1994; Kaplan 1987). Kaplan Visual Neuroscience, 8, 483--486. al. (1987) do not find any relation between cell type and Brodie, S. E., Knight, B. W. & Ratliff, F. (1978). The response of the signal transmission, but our finding that the contrast gain Limulus retina to moving stimuli: a prediction by Fourier synthesis. difference between P- and M-cells in the LGN (Fig. 4) is Journal of General Physiology, 72, 129-166. Ghosh, K. K., Goodchild, A. K., Sefton, A. E, & Martin, P. R. (1996). not as large as in retinal ganglion cells suggests that the The morphology of retinal ganglion cells in the new world marmoset signal transmission might be different for the two monkey Callithrix jacchus. Journal of Comparative Neurology, 366, systems. The amount of signal loss might depend on 76-92. the depth of anaesthesia (Kaplan et al., 1993). However, Gielen, C. C. A. M., Gisbergen, J. A. M. & Vendrik, A. J.H. (1982). the LGN is possibly a site where temporal filtering might Reconstruction of cone-system contributions to responses of colour- opponent neurones in monkey lateral geniculate. Biological occur also in an aroused state (Hamamoto et al., 1994; Cybernetics, 44, 211-221. Sherman & Koch, 1986). It has been proposed that a Goodchfld, A. K., Ghosh, K. K. & Martin, P. R. 0996). A comparison temporal filter must act on the responses of retinal of photoreceptor spatial density and ganglion cell morphology in the ganglion cells, since the retinal ganglion cells respond up retina of human, macaque monkey, cat, and the marmoset Callithrix jacchus. Journal of Comparative Neurology, 366, 55-75. to higher temporal frequencies than the flicker fusion Gouras, P. & Zrenner, E. (1979). Enhancement of luminance flicker by frequency of human subjects (Lee et al., 1990; Kremers color-opponent mechanisms. Science, 205, 587-589. et al., 1992, 1993). The synaptic stage in the LGN might Hamamoto, J., Cheng, H., Yoshida, K., Smith I]I, E. L. & Chino, Y. M. be this filter or part of it. (1994). Transfer characteristics of lateral geniculate nucleus X- The linear systems analysis provided reasonable neurons in the cat: effects of temporal frequency. Experimental Brain Research, 98, 191-199. predictions of the marmoset LGN cell responses to Hunt, D. M., Williams, A. J., Bowmaker, J. K. & Mollon, J. D. (1993). square-wave modulations, but they were poorer than Structure and evolution of polymorphic photopigment gene of the those for retinal ganglion cells of macaques (Kremers et marmoset. Vision Research, 33, 147-154. al., 1993; Lee et al., 1994). A similar difference in the Jacobs, G. H. (1981). Comparative color vision. New York, London: Academic Press. quality of the predictions can be inferred from cat data Jacobs, G. H. (1983). Differences in spectral response properties of [compare the quality of the response predictions for LGN LGN cells in male and female squirrel monkeys. Vision Research, cells (Saul & Humphrey, 1990) with those for retinal 23, 461-468. ganglion cells (Victor, 1987)]. Although it has previously Jacobs, G. H. (1984). Within-species variations in visual capacity among squirrel monkeys (Saimiri sciureus): color vision. Vision been found that using high contrasts, as have been used in Research, 24, 1267-1277. the present study, leads to gross failures of the linear Jacobs, G. H. (1990). Discrimination of luminance and chromaticity systems analysis also in retinal ganglion cells (Kremers et differences by dichromatic and trichromatic monkeys. Vision al., 1993; Lee et al., 1994), extra nonlinearities are Research, 30, 387-397. probably introduced at the stage of signal transmission in Jacobs, G. H., Deegan II, J. F., Neitz, J., Crognale, M. A. & Neitz, M. (1993a) Photopigments and color vision in the nocturnal monkey the LGN, resulting in poorer predictions. Aotus. Vision Research, 33, 1773-1783. Jacobs, G. H., Deegan II, J. F., Neitz, M. & Neitz, J. 0996). Presence of routine trichromatic color vision in new word monkeys. Contrast gain control in M-cells Investigative Ophthalmology and Visual Science, 37 Suppl. As stated earlier, M-cells display a distinct contrast Abstract, 346. gain control mechanism at high retinal illuminances. Jacobs, G. H. & Neitz, J. (1985). Color vision in squirrel monkeys: sex- related differences suggest the mode of inheritance. Vision However, we observed that the contrast gain control is Research, 25, 141-143. less obvious at lower retinal illuminances. In agreement Jacobs, G. H. & Neitz, J. (1987). Polymorphism of the middle with this, the fits with the linear model were better at low wavelength cone in two species of South American monkey: Cebus retinal illuminances (Table 1). Thus, contrast gain control apella and Callicebus moloch. Vision Research, 27, 1263-1268. Jacobs, G. H., Neitz, J. & Crognale, M. (1987). Color vision and probably other nonlinearities become larger when the polymorphism and its photopigment basis in a callitrichid monkey mean retinal illuminance increases. Shapley & Victor (Saguinus fuscicollis). Vision Research, 27, 2089-2100. (1978) have shown that in cat Y-cells the contrast gain Jacobs, G. H., Neitz, J. & Neitz, M. (1993b) Genetic basis of control is probably linked to odd-order spatial nonlinea- polymorphism in the color vision of platyrrhine monkeys. Vision rities. Other nonlinearities such as the frequency doubled Research, 33, 269-274. Kaplan, E., Purpura, K. & Shapley, R. M. (1987). Contrast affects the component in M-cells to isoluminant chromatic modula- transmission of visual information through the mammalian lateral tion might, however, also have the same origin, since geniculate nucleus. Journal of Physiology, 391,267-288. they depend on the spatial extent of the stimulus (Lee et Kaplan, E., Mukherjee, P. & Shapley, R. (1993). Information filtering al., 1989b). The fact that both the strength of the in the lateral geniculate nucleus. In Shapley, R. & Lam, D. M.-K. (Eds), Contrast sensitivity (pp. 183-200). Cambridge, Massachu- frequency doubled response at isoluminance (Lee et al., sets: MIT Press. 1989b) and the contrast gain control nonlinearity (in this Kaplan, E. & Shapley, R. M. (1986). The primate retina contains two study) decrease with decreasing retinal illuminance is types of ganglion cells with high and low contrast sensitivity. 2660 J. KREMERS et al.

Proceedings of the National Academy of Sciences USA, 83, 2755- Sherman, S. M. & Koch, C. (1986). The control of retinogeniculate 2757. transmission in the mammalian lateral geniculate nucleus. Experi- Kremers, J., Lee, B. B. & Kaiser, P. K. (1992). Sensitivity of macaque mental Brain Research, 63, 1-20. retinal ganglion cells and human observers to combined luminance Smith, V. C., Lee, B. B., Pokorny, J., Martin, P. R. & Valberg, A. and chromatic modulation. Journal of the Optical Society ~ (1992). Responses of macaque ganglion cells to the relative phase of America A, 9, 1477-1485. heterochromatically modulated lights. Journal of Physiology, 458, Kremers, J., Lee, B. B., Pokorny, J. & Smith, V. C, (1993). Responses 191-221. of macaque ganglion cells and human observers to compound Stephan, H., Baron, G. & Schwerdtfeger, W. K. (1980). The brain of periodic waveforms. Vision Research, 33, 1997-2011. the common marmoset ( Callithrix jacchus) Berlin: Springer. Kremers, J., Weiss, S., Zrenner, E. & Maurer, J. (1997). Spectral Tovre, M. J. (1994). The molecular genetics and evolution of primate responsivity of lateral genieulate cells in the dichromatic common colour vision. Trends in Neuroscience, 17, 30-37. marmoset (Callithrix jacchus). In Drum, B. (Ed.), Colour vision Tovre, M. J., Bowmaker, J. K. & Mollon, J. D. (1992). The deficiencies XIII (pp. 87-97). Dordrecht, Boston, London: Kluwer. relationship between cone pigments and behavioural sensitivity in Kremers, J. & Weiss, S. (1997). Receptive field dimensions of lateral a New World monkey (Callithrixjacchusjacchus). Vision Research, geniculate cells in the common marmoset (Callithrix jacchus). .72, 867-878. Vision Research, 37, 2171-2181. Travis, D. S., Bowmaker, J. K. & Mollon, J. D. (1988). Polymorphism Lee, B. B., Martin, P. R. & Valberg, A. (1989a) Amplitude and phase of visual pigments in a callitrichid monkey. Vision Research, 28, of responses of macaque retinal ganglion cells to flickering stimuli. 481-490. Journal of Physiology, 414, 245-263. Troilo, D., Howland, H. C. & Judge, S. J. (1993). Visual optics and Lee, B. B., Martin, P. R. & Valberg, A. (1989b) Nonlinear summation retinal cone topography in the common marmoset (Callithrix of M- and L- cone inputs to phasic retinal ganglion cells of the jacchus). Vision Research, 33, 1301-1310. macaque. Journal of Neuroscience, 9, 1433-1442. Victor. J. D. (1987). The dynamics of the cat retinal X cell centre. Lee, B. B., Pokorny, J., Smith, V. C. & Kremers, J. (1994). Responses Journal of Physiology, 386, 219-246. to pulses and sinusoids in macaque ganglion cells. Vision Research, Watson, A. B. (1986). Temporal sensitivity. In Boff, K. R., Kaufman, 34, 3081-3095. L. and Thomas, J. P. rEds), Senso~ processes and perception Vol 1 Lee, B. B., Pokorny, J., Smith, V. C., Martin, P. R. & Valberg, A. q]handbook of perception and human performance (pp. 6-1-6-43). (1990). Luminance and chromatic modulation sensitivity of New York: Wiley. macaque ganglion cells and human observers. Journal qf the Weiss, S., Kremers, J. & Zrenner, E. (1995). Cone and rod input of Optical Society of America A, 7, 2223-2236. parvocellular lateral geniculate cells in the common marmoset Milsum, J. H. (1966). Biological control systems analyis. New York: (Callithrix jacchus). Investigative Ophthalmology and Visual McGraw-Hill. Science, 36 Suppl. Abstract, 691. Mollon, J. D., Bowmaker, J. K. & Jacobs, G. H. (1984). Variations of Wikler, K. C. & Rakic, P. (1990). Distribution of photoreceptor colour vision in a new world primate can be explained by subtypes in the retina of diurnal and nocturnal primates. Journal of polymorphism of retinal photopigments. Proceedings ~( the Royal Neuroscience, 10, 3390-3401. Society of London B, 222, 373-399. Wilder, H. D., Grtinert, U., Lee, B. B. & Martin, P. R. (1996). Purpura, K., Tranchina, D., Kaplan, E. & Shapley, R. M. (1990). Light Topography of ganglion cells and photoreceptors in the retina of a adaptation in the primate retina: analysis of changes in gain and new world monkey: the marmoset (Callithrix jacchus). Visual dynamics of monkey retinal ganglion cells. Visual Neuroscience, 4~ Neuroscience, 13, 335-352. 75-93. Williams, A. J., Hunt, D. M., Bowmaker, J. K. & Mollon, J. D. (1992). Saul, A. B. & Humphrey, A. L. (1990). Spatial and temporal response The polymoiphic photopigments of the marmoset: spectral tuning properties of lagged and nonlagged cells in cat lateral geniculate and genetic basis. EMBO Journal, 6, 2039-2045. nucleus. Journal of Neurophysiology, 64, 206-224. Yeh, T., Lee, B. B., Kremers, J., Cowing, J. A., Hunt, D. M., Martin, P. Shapley, R. M. & Perry, V. H. (1986). Cat and monkey retinal ganglion R. & Troy, J. B. (1995). Visual responses in the lateral geniculate cells and their visual functional roles. Trends in Neuroscience, 9, nucleus of dichromatic and trichromatic marmosets (Callithrix 229-235. jacchus). Journal of Neuroscience, 15, 7892-7904. Shapley, R. M. & Victor, J. D. (1978). The effect of contrast on the transfer properties of cat retinal ganglion cells. Journal ~/ Physiology, 285, 299-310. Shapley, R. M. & Victor, J. D. (1979). Nonlinear spatial summation and the contrast gain control of cat retinal ganglion cells. Journal ~?" Ackmm'ledgements--The authors wish to thank Eva Burkhardt for Physiology, 290, 141-161. technical assistance and Dr Lindsay T. Sharpe for comments on the Shapley, R.M. & Victor, J.D. (1981). How the contrast gain control manuscript. The work was supported by DFG grant Zr 1/9-2. We thank modifies the frequency responses of cat retinal ganglion cells. Prof. Vivianne Smith for advice on the fits with the complex function Journal of Physiology, 318, 162-179. and for providing the Excel worksheet.