ON THE LACK OF LATERAL INHIBITION WITHIN THE RECEPTIVE FIELDS OF GOLDFISH RETINAL GANGLION CELLS’ IS~UEL ABRQIOV’ and MICHAEL W. LEVI& The Rockefeller University, Brooklyn College of the City University of New York. and the University of Illinois at Chicago Circle, Box 4343, Chicago, Illinois 60680. U.S.A. (Receiced 12 Jltly 1974; accepted 21 October 1974) have applied our method of response-summation and sensitivity-summation to the specific question of whether or not one can demonstrate lateral interactions within the centers of the receptive fields of goldfish retinal ganglion cells. We demonstrate that the curvature evident on the response-summation plot cannot be attributed to interactions, for there is no corresponding infexion on the sensitivitysummation plot. We also demonstrate that the response-summation and sensitivity-summation plots are unchanged by changing the spatial separation of the two areas stimulated; since lateral interactions would be expected to have a distance dependence, this indicates a Lack of interactions. Finally, we show that our data are consistent with the previously demonstrated validity of Ricco’s Law, and that the perfect summation of luminous flux within the center of the field does not require the postulation of lateral interactions to be consistent with a non-linear transformation of light intensity. Abstract-We
IXTRODCCTIOX
In our previous paper (Levine and Abramov, 1973, we presented a method for analyzing the processing performed within the receptive fields of individual ganglion cells, and applied it to the summation of spatially distinct areas within the center of the receptive field of goldfish ganglion cells. We found that we could ‘describe the processing by a model in which the influences from each area first undergo a non-linear transformation which may best be described as a power function with an exponent oft; the separate signals then combine (if each exceeds some small threshold level) by linear addition; the summed signal is then further transformed according to a non-linear compressive function. This simple model applies equally well for two areas which are both within the receptive field center, and two areas which are both within the receptive field surround. The model does not include long-term adaptive effects, nor does it include chromatic interactions (our stimuli were of sufficiently long wavelength as to be effective only for the long-wavelength cones, and for no other type of receptor). One of the more striking points about this model was the complete lack of lateral interactions;’ the only
’ Most of this report came from a dissertation submitted by MWL in partial fulfillment of the requirements for a Ph.D. degree from The Rockefeller University, New York, NY. ’ Present address: Department of Psychology. Brooklyn College of the City University of New York. Brooklyn, New York 11210. U.S.A. ’ Address for reprint requests: Department of Psychology, University of Illinois at Chicago Circle, Box 4348, Chicago. Illinois 60680, U.S.A. ’ The terminology, mathematical symbols. and other notation used in this paper are the same as that used in our previous paper (Levine and Abramov. 1975). The reader is urged to peruse that paper before attempting this one. “3
it
‘4
lateral effect was the simple summation of influences from the separate areas into a single final common pathway. While the two signals existed as separate entities, they apparently did not exert influence upon each other. We must be careful to make one additional point when we say there are no lateral interactions. We are speaking in terms of a jimcrional model; that is, we mean there are no features which mathematically distinguish the processing from a model in which there are no interactions. We do not mean to imply that one could not locate lateral processes within the retina, or that the lateral interconnections of the retina are nonfunctional. We simply mean that all the processing which occurs is equit&nt to (may be modelled by) the simple model we have proposed. In this paper, we make use of the methods we described in our preceding paper to explore further the possibility of lateral interactions. We have searched for evidence of lateral interaction through three separate lines of reasoning; in each case, we find no evidence for lateral interaction. METHODS (a) Preparation and apparatus The physiological preparation and apparatus have been described in considerable detail (Abramov and Levine, 1972; Levine and Abramov, 1975). The experiments were performed on the isolated retinae of common goldfish, Carassius auratus. Ganglion cell responses (spikes) were recorded extracellularly from single units using piatinumiridium microelectrodes with glass insulation. Stimuli were 710nm lights of various intensities (expressed in log units relative to 275 x lo’? quanta/cmz/sec) provided by a threebeam optical system. Field stops in each beam were projected directly onto the retina to provide the stimulus patterns; electromagnetic shutters in each beam controlled the duration of stimulus presentation. Stimuli were presented for one second durations, with an intertrial interval of 25 sec. Responses (in adrians) were taken as the total number of action potentials fired by the
791
unit tn 1 see: “on“ responie~ wwz iounred ior the tejic dumtion of the stimulus Hash. “oR’ E+JCXI~S were taken for the l j-x following ~tkt ol the Hash. The retina uas in total dnrkness except for the stimulus dashes. Stimuli werr presented in random order. The data were analyzed b) the mcrhods ue have prepresented (Lzvine and Abramov. 1975). fntensitv-reSPOIM curves were obtained for each of tvvo separate &mums areas stimulated atone. and for sombin~tions of the areas stimulated together. (Combinations in which both areas were stimulated together at the same intensity are called “*equal-intensity”: combinations in which one of the intensities was varied while the other was held at a par&uIar value are called “peg”.] The intensity-response curves Were used to generate both a response-summation plot and a sensitivity-summation plot for each pairing of stimult. The response-summation plot is a graph of the “physiological sum” vs the “algebraic sum”: that is, the responses of theunitwhenbothareasaresimuitaneouslyilluminatedcompared to the sum of the responses when the areas are sepamtely illuminated. SUCKa plot is indicative of non-linearities in the processing of the visual information at or after the first interaction of the two areas; it is unaffected by nonlinearities at earlier stages. It thus gives us information about iaterlll interactions. as well 3s about the processing after the summation of the two areas. (Note that we use “lateral interaction” to mean something other than simple summation or combination of intluences from separate xeas. To be detected by our analyses. lateral interactions must be non-linear.) The sensitivity-summation plot graphs the sensitivity of the unit when both areas are stimulated vs the summed separate sensitivites of the two areas. It is indicative of nonlinearity in the processing 35 or before the final interaction of the tu’o areas: it is unaffected bv non-linearities after summation. It thus gives us informaiion about lateral interactions, as well as about any transformation of the visual information that might occur in the individual receptors or before the spatially separated areas first interact. Lateral interactions. if there are any. must therefore have an influence on both of these graphs. Lateral interactions can be distinguished from other non-linearities by virtue of the fact that only lateral interactions can influence both graphs. The use of both of these plots enables us to identify the processing in the retina in terms of three zones: the “receptor” zone. in which signals from the two areas 3re whoil> independent (this zone affects only the sensitivity-summation plot): the zone of iateral interactions (which affects both the sensitivity-summation and response-summation p~00l; and the tinal common pathway, in which the separate signals have been summed into a single signal (this zone affects onl! the response-summation plot). ViOUSly
Fig. 1. Comparison, for two units. of r~s~ense-summas~on and sensitivity-summation plots with special reference to the point at which the curve on the response-summation plot breaks from linearity (deviates from unity slope). Stimuli (7 10 nm) are sketched at the top of each column. The top graph in each column shovvs the intensity-redone curves for each area stimulated alone (solid symbols and curves) and for both areas stimulated simultaneously with the same intensity (open circles and dashed curves). The middle graph in each is the ~ns~ti~i~~-sum~~on plot (dotted curves); the bottom graph is the respon~summat~on plot. The continuous functions on the sensitivity-summation and response-summation plots were derived from the curves fit to the intensity-response data for each unit. The vertical arrows indicate corresponding points on the three forms of plot (see text).
curves for each area stimulated
alone and for both stimulated together; the middle and bottom graphs present the sensitivity-summation and response-summation curves derived from these responses. The response-summation curves (bottom graphs) start as straight lines of unity slope through the zeroresponse point (the latter is marked by the intersection RESLLTS of the solid lines indicating spontaneous rate). This Our previous paper (Levine and Abramov, 1973;) may be interpreted as implying no non-linearities in the processing at or after the first interaction between presented in some detaif the functional organization of goldfish ganglion ceils as inferred from analysis of re- signais From the two areas stimulated. However. at higher response levels the curves deviate from this sponse-summation and sensitivity-summation plots. Before we consider the possible existence of latera in- “linear” curve; this can be due either to the expression of a non-linear compressive function after summation, teractions, we wiil brieffy review the general properties or the effects of inhibitory lateral interactions maniof the model we have presented; the responses shown in Fig. 1 will serve to summarize for the reader how the fested only at these higher response levels. The sensitivity-summation curves (center graphs) features of the plots led us to postuiate particular are. at the higher intensities. straight lines of unity mechanisms. slope which do not pass through the origin. This imTwo different cells are illustrated in Fig. 1; the unit on the left responded at “OR’ to stimuI~tion in the plies a power function operating on the signal from center of its receptive t’ieid, while the one on the right each area prior to any interaction between the signals from the hvo areas; from the intercept of this portion gave “on” responses to stimulation in the periphery. of the curve the value of the exponent of the power The graphs at the top show the intensity-response
function
bz sho\vn to be approx i. iIt lOWX indeprcssd toward the diagonal. and. dspsnding on the particukir cell. may 2ven Cross th2 diagonal. This may br du2 to “rrceptors”’ becoming more linsar at low light levels (the diagonal is the !ocus for linearity) or to threshold mschanisms on the signal from each &sa b2for2 summation. It could also r:sult from inhibitory lateral intsractions that are manifest2d only at the lowest intensitirs of stimulation. in our prttvious paper u’s concluded that it was not nec2ssa.ry to invoke lateral interactions. The nonLinraritizs apparent in the graphs like these in Fig. 1 can be adequately accounted for by a non-iinrar function at th2 sarlq stages of processing whers th2 rsspons2s from the t\vo arsas stimulatsd x2 still indeprndent and anothzr non-linear ftlnction after th2y have mzrgsd into a common pathway;. We will now examins in morr dstnil th2 possibility rhat therz ars lar2raI intsractions which are not equivalent to the transformations in our earlier simple mod& WC use three ssparats approuchrs to this problem. an
~~njiti~s, the curves cm
Lrt us consider th2 possibility that some of the noniin2arit>- svident on thz respons2-summation and s2nsitivity-summation plots may be du2 to latzrai intsractions that x2 not irrelevant in our model. Ws can show that thz effsct of latzral inhibition would b2 to dzprrss th2 curve on the sensitivity-summation coordinatss ad decrease the slop2 of the curve on th2 response-summation coordinates: if the interaction itself is non-lincnr, it would also introduce curvaturs on both plots (Lzvine. 1972). Since the slope of the low response part of the response-summation plot is always v2ry neari? unity. we can conclude that ther2 is no linrar int2raction (elective at all levels) made nonlinrar bt thrzsholds on the signals entering the final summation. Th2 remaining possibility is that the interactions themselves are non-linear, and must thereforz produce a curvature on both the response-summation and srnsitivity-summation plots. Let us assum that th2 curvature of the respond-su~ation plot. which is generally n fairly sharp “knee” in th2 curve, is due to lateral interactions; in that case, it should be possible to idrnrify a corresponding curvature on the sensitivity-summation plot. Figurs I dcmonstratss two attempts to find an effrct in the sensitivity-summation plot corresponding to ths curvaturr: of the line on the r2sponse-summation plot. Each of the response-summation plots on the bottom of Fig. 1 shows a fairly sharp break away from unity slope at a point indicated by th2 vertical arrow. (The dotted curves were derived from the curves fit to the data points on the intensity-response curves.) This break could be due to one of two causes: (1) either the total response (physiological sum) is compressed when responses exceed some level; or (2) one (or both) pf thz individual intensity-response curves (R , or R2) is acceltrated more rapidly above some response level. rcirhWC (I cotxw?~iic~rrt uccrferarion of tire CWYYfor60th
'WC gsnrraiis refer to the functions before the firs: interils “receptor” functions, but the reader should bear in mind that we have no grounds for dscribmp these non-linearities to any particular anatomical
action Of the t%i.O XCYS
sWuc~ure.
together. Thz latter possibility is far 12~slikslq. but 2\2n if it m.2~2ths casr. th2rs would have to b2 J compression of thz response to both in ordzr to compensate for the acceleration of ths rrceptor functions. It is therefor2 mzaningful to idsntify a response Iev2l R! _ 3 i2.g. Jo adrians. Fig. 1. left) with ths point tarrow on the bottom graph) at which the r2sponsc-summation curve br2aks from linsarity: th2 rssponss 12~21at which this occurs can bt: iotated on th2 intensityr2sponse curve for the rqual-intsnsit! sxp2riment shotvn at rhe top of Fig. 1. This point (arrow on th2 top graph) identifi2s the intznsity \I! _ :) at which the break occurs. Six2 th2 abscissa of thz sensiti~-its-s~tmmation plot is also log IL _ :, it is possible to locat th2 corrssponding point on the ssnsiriiit!-summation plot (middle: of Fig. I : arrow). We have thus looatsd thz point art the sznsitivitysummation plot which corresponds to ths point at which thz responsz-summation plot brsaks from linearity. In both units sholvn in Fig. I. this point is well within the xgmsnt of the curve which is clsarlc a straight line. Thr 2ffect of non-linear lacra’i interactions on rhs s2nsitivit~-summation piot is to shift thz curve: at th2 point correspondins to the break from linearity on th2 response-summation plot. thsre is no change in ths sensitivity-sLimmat~on plot. Thtrzforz, if the curvature on thz rssponsz-summation plot is duz to an int2ruction prior to summation. thr intzracrion must be equivalznt to a comprrssive function after summation. Thr analysis shown in Fig. I could bs p2rformsd on only a f2u units. In most units. the corrssponding point was at a highzr value of i, _ 1 than th2 maximum for which the srnsitivity-summation plot could be drawn (thr: sensiticity-summation plot is limited. since the highest responst- criterion which can bz ussd is the maximum firing of thz least xsponsive area t. However. a numb2r of other units tzndzd toward ths same outcome. and none of 18 units observed had rssponses which contradicted the abov2 conclusions.
Lst us reconsider ths possibility that thsrs are nonlinear latzral intzractions which have sscaped our notice. On the ~nsitiv~t~-summation plot inhibitory intzractions would dspr2ss the curve. On ths responsesummation plots. inhibitory interactions would d2crrase the slope of ths curvs. but the dzcreax might be unnoticeable in thz short region before tht compressive function after summation czpresses its2lf. (Not2 that facilitatory interactions would have just th2 opposit2 tffect.) We shall take advantage of on2 of the expected properties of latsral interactions in ordsr to try to demonstrate their presence: msgmtude of intttraction should change with the separation b2tween the interacting arcas. \V2 shnll compare rssults from stimuli that are close together with those from stimuli that ar2 well wparated. and se2 if there is a difference between the.plots from each casr. An experiment drsigncd to maximize and th2n minimizc iat2rdl interactions is shown in Fig. 2. Th2 stimuli ar2 shown at the upprr right; each is a four-spoksd pinwh22l. or cross, centered on, and confmsd to, the center of the receptive field. (This cross rezmbles the heraldic “cross patr”, but by virtue: of th2 curvatur2 at
Lag
(I?
LOQ iii,?)
Fig. 2. Results of a search for evidence of lateral interactions. Stimuli were four-limbed crosses Ltjshown at the top right. Each war centered on the receptive field, and they aere presented either individuahy or together at equal intensities. One cross (shown lightly stippled) was always in the same position; the other cross could be rotated (before pre~n~tion) to either of two positions, as shown. Open circles refer to responses to the stationary cross stimulated alone; solid circles and dashed curves refer to responses when the rotated cross is in the position adjacent to the stationary cross: triangles and solid curves refer to the rotated cross separated from the stationary cross. On the lower left are the intensity-response data for the crosses presented separately: intensity-response curves for the two simukaneous presentation series are on the upper left. The respond-summation plot is on the upper right (the curves for rhe two situations superimpose): sensitivity-summation on the iower right (solid curve for crosses separated, dashed curve for the crosses nearly touching). All stimuli were 710 nm.
the end of each limb is probably better referred to as a “cross globical”; Franklyn, 1970.) One cross. shown lightly stippled, was held in the same position throughout the experiment. The other cross, shaded with lines in the diagram, could take either of two positions; in one position it was rotated until it was as close as possible to the stationary cross without overlapping it, while in the other position it was as well separated as possible. The crosses take advantage of the approximately radial symmetry of the receptive field: they may be rotated without significantly changing their sensitivities. They also provide a relatively large boundary along which interactions may occur when the two crosses are nearly adjacent. The intensity-response functions
for each cross presented alone are shown in the lower left of Fig. 2. Open circles represent the stationary cross, solid circles represent the moveable cross in the close position, triangles in the separated position. The three curves are nearly identical. Intensity-response functions for both crosses illuminated simultaneously are shown on the upper left. The circles and dashed line are for the case in which the two crosses are adjacent; the triangles and solid lines are for when they are well separated. The response-summation and sensitivity-summation plots for each case are on the right of Fig. 2: There is no substantial difference on the response-summation plots (top), arguing for a lack of lateral interaction. but there is an apparent difference on the sensitivitysummation plots (bottom). In particular, the solid curve. from the case in which the crosses are well separated, is below the dashed curve (crosses nearly touching) at low intensities. Since the effect of inhibi-
tory interaction would be to depress this plot, the implication is that there was greater interaction when the crosses were separated than when they were tangent, and that the interaction was greater at low intensities than high. Neither of these prospects is inviting. for one would expect interaction to increase with intensity, not decrease; moreover, one would expect interactions to be greater with smaller separations, not larger. There are two other possibilities: (1) the differences noted on the sensitivity-summation plots could be the result of a facilitation by which the adjacent crosses aided each other in overcoming the thresholds (which presumably cause the low intensity depression of the curves); or (2) they might be due to random variance. With regard to the latter possibility, it is worth noting that the differences are at the low intensity part of the curve, which is its least reliable part (see Levine and Abramov, 1975); also, the entire difference can be traced to a small number of data points at the lowest intensities. But we may not yet dismiss the possibility that the lateral influences are in fact smaller at close distances. and increase to a maximum at a siiglitly larger separation. This is the kind of distance function that was reported by Barlow (1967) for lateral inhibition in Li&us. We must therefore perform a similar pair of experiments in which the distances considered are somewhat larger. A test for interactions across wider separations is shown in Fig. 3. Three 04rnm spots were focused within the center of this unit’s receptive field. One spot (spot 1) was always used as one of the MO areas stimulated; the second area was either the one quite close to it (spot 2) or the one quite distant (spot 3). Individual
Lack of lateral inhibition on ganglion cells
Fig. 3. Results of a search for evidence of lateral interactions effective over long distances. Stimuli were three spots. 04 mm dia, within the center of the receptive field. Two different experiments were performed: ii) spots 1 and 2 were used (open symbols. dotted curves); and (ii) spots I and 3 were used (solid symbois. dashed curves). Peg experiments are shown by triangles; the solid curve on the intensity-response plot at the upper left fits either set of triangles. Plots arranged as in Fis. 1; all stimuli were 710 nm.
intensity-response curves for each of the three spots are Thus far we have found no consistent evidence to shown m the lower left of the figure: the spots were essentially in equisenritive positions, since the curves compel us to include lateral interactions in our model, but a final point must be considered. Easter i 1965) sugsuperimpose quite closely. The intensity-response curves for the equal-intensity experiments for each gested that lateral interactions account for the differpairing are shown on the upper left of Fig. 3. The ences he noted between the summation of two separate equal-intensity experiment for the close spots is shown areas and the summation of contiguous areas in his esperiment in which spot size was varied to demonstrate by the open circIes (dotted curve), the distant pairing is shown by solid circles (dashed curve); the curves Ricco’s Law. Our data obtained from summing two agree closely. A peg experiment was also performed for areas indicate that no lateral interactions need be poseach of the pairings (triangles, sohd curve); there was tulated, but can the same data be used to verify Ricco’s no difference between the outcomes of the peg exper- Law’? iments for the two separations. The experiment shown in Fig. 4 was designed to The response-summation plots from these data are demonstrate that the relationship of Ricco’s Law is on the upper right in Fig. 3. The curves derived from consonant with the usual results of summing responses the two pairings do not quite superimpose: the curve from two separate areas. Two separate pair of stimuli for the closer pairing fails slightly below that for the were used (upper right). The first pair was a spot more distant pair. This is the result to be expected if @63 mm dia (stimulus 2) and an annulus of equal area there is more Iateral interaction between the closer whose outermost diameter was 1.0mm fstimulus 11; spots. Notice, however, that in this case the interaction this is still just within the center of the receptive field. is evident only at the higher firing levels; furthermore, The second pair consisted of a spot 04 mm dia (stimuit is not evident in the points derived from the peg ex- lus 3) and an annulus whose outer diameter was periments. Moreover, the sensitivity-summation plot. 063 mm (stimulus 3). Intensity-response curves for lower right. shows no substantial difference between stimuli 1 and 1 presented singly and together are the curves; it is probably fair to conclude that there is shown in the upper left of Fig. 4. There was a considerno evidence for lateral interaction in this unit. only able sensitivity difference between region 2 alone {spot) variability in the data. and region 1 alone (annulus); this diflerence is to be If there is lateral interaction its effect must k quite expected, since sensitivity declines with distance from smah, for the differences between the two responsethe midpoint of the receptive field. The response-sumsummation curves in Figs. 2 or 3 are minor compared mation plot derived from these data is shown in the to the difference between any of the curves and a line lower left of Fig. 4. The data from regions 3 and 3 are of unity slope. Other cases in which different stimuli not shown, but they are very similar, and both sets of were used on the same unit show only negligible diffet- curves are typical of those found with orher stimulus ences in the response-summation and sensitivity-sumconfigurations on other units. The sensitivity-summamation plots for each co~guration, indicating that the tion plot for regions 1 and 2. is on the loner right; rhe placement of the stimulus areas is of no real conse- dotted curve is plotted in the usual manner, and is quence. related to the ordinate on the left. This curve also is un-
0
20
(91+
40
RZ!
55
00
100
:2drlonsl
1,’ -30
i-33 -2 0
130
-I 0
!I:+2)
Fig. 4. Demonstration that Ricco’s Law is consistent with a model without lateral interactions. Stimuli (710 nm) were two circles and two annuli: half of each stimulus is shoun on the upper right-all were within the center of the receptive field. Upper left: Intensity-response curves for stimuli 1and 2 presented separately (solid symbols and curves) and together (dashed curve, open circles); these regions are clarly not equisensitive. Lower left: Respond-sum~~on plot for stimuli I and 2, equal-intensity experiment. Lower right: The dashed curve. referred to the ordinate to the left, is the usual sensitivity-summation plot for stimuli I and 1; conventions for this curve and the graphs to the left are the same as for Fig. I. The solid curve is derived from the same data as the dotted, but is plotted according to the method of Easter (1968) and refers to the ordinate to the right; see text for a discussion of these two forms of ptotting the data. Top right: The plot shows iog sensitivity- vs log area for a criterion response of 30 adrians. derived from the intensity-response curves on the left, The stimuli, whose areas correspond to the four values of the abscissa, are indicated above each data point; the line through the points has a slope of one. demonstrating that Ricco’s Law holds up to the largest area used.
exceptional, being a straight line of unity slope with an intercept ofabout 0.27. Now let us examine how these same data would appear if they were plotted exactly as described by Easter (1968). We would assume that regions I and 1 were still within the ptateau at the center of the receptive field where regions are still of equal sensitivity, and make use only of the intensity-response curves for region 2 alone and for both together. (When changing spot sizes in Easter’s experiments, the intensity-response curve for the additional contiguous annuIus would not be measured separately.) When the data are analyzed in this fashion. the ~nsitivity-summation curve is the solid curve whose ordinate is indicated on the right. This ordinate is displaced by 0.3 log units (Easter’s ordinate is a single sensitivity, not a sum of two), so that the implications of the line drawn as the diagonal are the same as for our form of the sensitivity-summation plot; that this curve lies along the diagonal implies complete linear summation and linear receptors. Easter, having shown the receptors were non-linear by su~ing well-separated spots, would have assumed that lateral interactions accounted for the location of the curve. But both the solid curve and the dotted curve were derived from the same set of data, so the difference between them must be due to the method of anatysis. and not the presence or absence of interactions. Finally. it is necessary to show that Ricco‘s Law was
demonstrated by our data. Our stimuii can be considered as four concentric spots of increasing area: stimulus 4 alone is the smallest spot and stir&i 1 and 2 presented together form the Iargest spot. The sensitivity of each of the four circular areas used was derived from the appropriate intensity-response curves using a criterion response of 30 adrians. The log sensitivity ofeach area was plotted as a function of the fog of the area (Fig. 4. upper right) anda Iine of unity slope drawn through the points. If S is sensitivity, A is area. I is intensity, and k and k’ are constants:
6 Easter has also reached this conclusion. based on similar reasoning (personal communication).
analysis
or
Iog s = log A + F:
(1)
=logA+k
(3
= k’
(31
-1ogI rl.i
which is Ricco*s Law, The same data that were used to demonstrate Ricco’s Law gave unexceptional response-summation and sensitivity-summation plots. There is no contradiction requiring us to invoke lateral interactions for its resolution; if there is a square root function at the receptors, it is sufhcient to have the sensitivity across the center of the receptive field fail off as the inverse square of the distance from rhe midpoint6
From these data, we conclude that rhe model we have previously presented (Levine and Abramov, 1975) does not require modification to include lateral interactions. The original model was deriwd from our mation
of response-summation and xnsitivity-sumplots for 17 separate units: in this paper, we
797
Lack of latsrai inhtbition on ganglton ceils examined the evtdence for any hint that lateral interactions might be present. First_ we checked that there was no misinterpretation of the graphs. The sharp bend in the responszsummation plots for two units was correlated with the sensitivity-summation plots from the same units; there was no corresponding break in the sensitivity-summation plots. Since the break was evident in the responsesummation plot only. it must necessarily have been due to a non-linearity after the final combination of the signals from the two areas into a final common pathway. and not to lateral interactions. Secondly. we examined the effect of increasing the spatial separation between the tuo stimulus areas. One would expect lateral interactions to be distance-dependent. but no such effect was found. Finally. we replicated Easter’s (1965) demonstration ofRicco’s Law, and showed that the result is consistent with our model as originally stated. Lateral interactions vvere not required to explain the “perfect summation” implied by Ricco’s Law, despite the demonstrable non-linearity (square root function) at the “receptors”. Our model for the processing within the center of the receptive field (or entirely within the surround) is thus the same as we have previously described (Levine and .Abramov. 1975). Light falling on each sub-area of the receptive field is transformed according to the square root of its intensity (with the possibility of saturation at high intensities, and a somewhat more linear function at very low intensities). Separate contributions which exceed some threshold value are summed linearly into a total signal. This signal is then transformed according to a non-linear compressive function roushiy of the form R=S
(-1)
k+s
where s is the signal and k is a constant; R is the response of the unit. in adrians. VVe must point out again that any model we present is a mathematical formulation in its simplest form; we clearly cannot distinguish between alternatives that are mathematically identrcal. If we found that some part of the processing could be described by a power function vvith an exponent of 1. we obviously could not state that this was not due to the serial application of two square root functions. Similarly. we must point out that certain forms of lateral inhibition, which might be very real on a cellular level, could be wholly equivalent to (and hence indistinguishable from) models in which there is no interaction. Within the range of our investigations. such a lateral interaction would befi~rtctionclll~ nonexistent. The most obvious esampic of a non-functional interaction would be a linear interaction: subtractive inhibition (with no non-linearities such as thresholds or rectification). Such an inhibition could be described as [using the notation of Levine and Abramov (1975) in which -yi is the output of a group of receptors prior to any lateral interactions. and yi is the input. after lateral interactions. to the summing point]: J, = as, - hsl
and
y2 =cx,
-dr,.
(5)
The sum of the two signals would then be 5 = 1’1 + .1’?= IilY, - hx,) + (cxI - ds,)
or 5 = (~1- &Y, i- IC - kJ).y~
(7
which is a linear summation of xL and x1. Each term has merely been multiplied by a constant coefficient: such a form of inhibition would have no effect on either the response-summation or sensitivity--summation plot. Changing the values of the constants would afiect the range of the plots. but not the curves themselves. Another form of inhibition that is less obviously equivalent to a non-inhibitory model is “mulriplicative forward inhtbition” (Furman, 1965 : Griisser. Schaible and Vierkant-&the. 1970). in which each separate signal is divided bi a pooled signal to u hich ~11active elements contribute:
(3). (9) (I, is a constant). But this is in fxt identical to a compressive function in the final common pathauy. for
I IO)
Since it is mathematically equivalent. it must be indistinguishable from compression after sumration; in fact, this could bs the mechanism by which the compression is achieved. Our model is restricted in its application to the domain of our experiments. It does not include adaptation or adaptive (long-term) intluencsj. for our stimuli were relatively brief (1 ssc) Hashes applied simultaneously. It does not include rod-cone interactions or chromatic interactions among dif%rent cone types. for our 710nm stimuli may be presumed effecttve only for the cones containing the long wavelength pigment (625 nm peak; Marks. 1965). i\;or does it include the interactions betvveen the center and iurround of the receptive field, for all our stimuli were chosen so that the two areas under consideration Mere either both within the center or both Kithin the surround. In our future work vve hope to expand the model and remove these restrictions. .-Ick,torvirrly~r,frr~rs-This mvestigarion was supported m part by NIH Training Grant So. GSI 1789 from National Institute of General Medical Sciences. and bv research prants EY 188 from the National Eqe Insiitute and GB 3616s from the National Swncc Foundation. The authors wish to thank Bruce W. Knight of The Rockefeller Cniversit> for his help and encouragement in iormlrlatinp the analytical methods used. We also thank Drs. Floyd RaMand H. K. Hartline of The Rockefeller University. and Dr. Christina Enroth-Cugell. of Northwestsm Lniversity, for their encouragement and advice. REFERESCES
Abramov I. and Levine hl. VV.(1973) The effects of carbon dioxide on ths excised goldfish retina. li~:!~~rRgs. 11. 1X3-1895.
Barlow R. B. (1967) Inhibitory fields in the L:wlus lateral eye. Doctoral dissertation. Thz Rockefeller L-niversitk. (6)
Yew York. NY.
-98
ISRAEL
r\BRAMOv
and
Easter S. S. (1968) Excitation in the goldfish retina: evidence for a non-linear intensity code. J. Physioi.. Lond. 195. 153-271. Franklyn J. (1970) fferaldr). A. S. Barnes. New- York. Furman G. G. (1963) Comparison of models for subtracttve and shunting lateral inhibition in receptor-neuron fields Kybernerik l&251-271. Griisser O.-J., Schaible D. and Vierkant-Glathe J. t 1970) X quantitative analysis of the spatial summation of exci-
~~IcH~EL
w.
LE-,I~.E
tation within the receptive field centers of retinal neurons. Pj?iigers .4rch. y~‘x Ph,vsiol. 319, IOl-I?!. Levine MM. W. (1973) An analysis ofspatiai summation in the receptive fields ofgoldfish retinal ganglion cells. Doctoral dissertation. The Rockefeller University;. New York. Levine M. W. and Abramov I. (1975) An analysis of spatial summation in the receptive fields of goldfish retinal gang_ lion cells. Vision Res. 15. 777-789. Marks W. B. (1969) Visual pigments of single goldfish cones. J. Phvsiol. , 1L.ond 178, 1432.
IXTRODCCTIOX
In our previous paper (Levine and Abramov, 1973, we presented a method for analyzing the processing performed within the receptive fields of individual ganglion cells, and applied it to the summation of spatially distinct areas within the center of the receptive field of goldfish ganglion cells. We found that we could ‘describe the processing by a model in which the influences from each area first undergo a non-linear transformation which may best be described as a power function with an exponent oft; the separate signals then combine (if each exceeds some small threshold level) by linear addition; the summed signal is then further transformed according to a non-linear compressive function. This simple model applies equally well for two areas which are both within the receptive field center, and two areas which are both within the receptive field surround. The model does not include long-term adaptive effects, nor does it include chromatic interactions (our stimuli were of sufficiently long wavelength as to be effective only for the long-wavelength cones, and for no other type of receptor). One of the more striking points about this model was the complete lack of lateral interactions;’ the only
’ Most of this report came from a dissertation submitted by MWL in partial fulfillment of the requirements for a Ph.D. degree from The Rockefeller University, New York, NY. ’ Present address: Department of Psychology. Brooklyn College of the City University of New York. Brooklyn, New York 11210. U.S.A. ’ Address for reprint requests: Department of Psychology, University of Illinois at Chicago Circle, Box 4348, Chicago. Illinois 60680, U.S.A. ’ The terminology, mathematical symbols. and other notation used in this paper are the same as that used in our previous paper (Levine and Abramov. 1975). The reader is urged to peruse that paper before attempting this one. “3
it
‘4
lateral effect was the simple summation of influences from the separate areas into a single final common pathway. While the two signals existed as separate entities, they apparently did not exert influence upon each other. We must be careful to make one additional point when we say there are no lateral interactions. We are speaking in terms of a jimcrional model; that is, we mean there are no features which mathematically distinguish the processing from a model in which there are no interactions. We do not mean to imply that one could not locate lateral processes within the retina, or that the lateral interconnections of the retina are nonfunctional. We simply mean that all the processing which occurs is equit&nt to (may be modelled by) the simple model we have proposed. In this paper, we make use of the methods we described in our preceding paper to explore further the possibility of lateral interactions. We have searched for evidence of lateral interaction through three separate lines of reasoning; in each case, we find no evidence for lateral interaction. METHODS (a) Preparation and apparatus The physiological preparation and apparatus have been described in considerable detail (Abramov and Levine, 1972; Levine and Abramov, 1975). The experiments were performed on the isolated retinae of common goldfish, Carassius auratus. Ganglion cell responses (spikes) were recorded extracellularly from single units using piatinumiridium microelectrodes with glass insulation. Stimuli were 710nm lights of various intensities (expressed in log units relative to 275 x lo’? quanta/cmz/sec) provided by a threebeam optical system. Field stops in each beam were projected directly onto the retina to provide the stimulus patterns; electromagnetic shutters in each beam controlled the duration of stimulus presentation. Stimuli were presented for one second durations, with an intertrial interval of 25 sec. Responses (in adrians) were taken as the total number of action potentials fired by the
791
unit tn 1 see: “on“ responie~ wwz iounred ior the tejic dumtion of the stimulus Hash. “oR’ E+JCXI~S were taken for the l j-x following ~tkt ol the Hash. The retina uas in total dnrkness except for the stimulus dashes. Stimuli werr presented in random order. The data were analyzed b) the mcrhods ue have prepresented (Lzvine and Abramov. 1975). fntensitv-reSPOIM curves were obtained for each of tvvo separate &mums areas stimulated atone. and for sombin~tions of the areas stimulated together. (Combinations in which both areas were stimulated together at the same intensity are called “*equal-intensity”: combinations in which one of the intensities was varied while the other was held at a par&uIar value are called “peg”.] The intensity-response curves Were used to generate both a response-summation plot and a sensitivity-summation plot for each pairing of stimult. The response-summation plot is a graph of the “physiological sum” vs the “algebraic sum”: that is, the responses of theunitwhenbothareasaresimuitaneouslyilluminatedcompared to the sum of the responses when the areas are sepamtely illuminated. SUCKa plot is indicative of non-linearities in the processing of the visual information at or after the first interaction of the two areas; it is unaffected by nonlinearities at earlier stages. It thus gives us information about iaterlll interactions. as well 3s about the processing after the summation of the two areas. (Note that we use “lateral interaction” to mean something other than simple summation or combination of intluences from separate xeas. To be detected by our analyses. lateral interactions must be non-linear.) The sensitivity-summation plot graphs the sensitivity of the unit when both areas are stimulated vs the summed separate sensitivites of the two areas. It is indicative of nonlinearity in the processing 35 or before the final interaction of the tu’o areas: it is unaffected bv non-linearities after summation. It thus gives us informaiion about lateral interactions, as well as about any transformation of the visual information that might occur in the individual receptors or before the spatially separated areas first interact. Lateral interactions. if there are any. must therefore have an influence on both of these graphs. Lateral interactions can be distinguished from other non-linearities by virtue of the fact that only lateral interactions can influence both graphs. The use of both of these plots enables us to identify the processing in the retina in terms of three zones: the “receptor” zone. in which signals from the two areas 3re whoil> independent (this zone affects only the sensitivity-summation plot): the zone of iateral interactions (which affects both the sensitivity-summation and response-summation p~00l; and the tinal common pathway, in which the separate signals have been summed into a single signal (this zone affects onl! the response-summation plot). ViOUSly
Fig. 1. Comparison, for two units. of r~s~ense-summas~on and sensitivity-summation plots with special reference to the point at which the curve on the response-summation plot breaks from linearity (deviates from unity slope). Stimuli (7 10 nm) are sketched at the top of each column. The top graph in each column shovvs the intensity-redone curves for each area stimulated alone (solid symbols and curves) and for both areas stimulated simultaneously with the same intensity (open circles and dashed curves). The middle graph in each is the ~ns~ti~i~~-sum~~on plot (dotted curves); the bottom graph is the respon~summat~on plot. The continuous functions on the sensitivity-summation and response-summation plots were derived from the curves fit to the intensity-response data for each unit. The vertical arrows indicate corresponding points on the three forms of plot (see text).
curves for each area stimulated
alone and for both stimulated together; the middle and bottom graphs present the sensitivity-summation and response-summation curves derived from these responses. The response-summation curves (bottom graphs) start as straight lines of unity slope through the zeroresponse point (the latter is marked by the intersection RESLLTS of the solid lines indicating spontaneous rate). This Our previous paper (Levine and Abramov, 1973;) may be interpreted as implying no non-linearities in the processing at or after the first interaction between presented in some detaif the functional organization of goldfish ganglion ceils as inferred from analysis of re- signais From the two areas stimulated. However. at higher response levels the curves deviate from this sponse-summation and sensitivity-summation plots. Before we consider the possible existence of latera in- “linear” curve; this can be due either to the expression of a non-linear compressive function after summation, teractions, we wiil brieffy review the general properties or the effects of inhibitory lateral interactions maniof the model we have presented; the responses shown in Fig. 1 will serve to summarize for the reader how the fested only at these higher response levels. The sensitivity-summation curves (center graphs) features of the plots led us to postuiate particular are. at the higher intensities. straight lines of unity mechanisms. slope which do not pass through the origin. This imTwo different cells are illustrated in Fig. 1; the unit on the left responded at “OR’ to stimuI~tion in the plies a power function operating on the signal from center of its receptive t’ieid, while the one on the right each area prior to any interaction between the signals from the hvo areas; from the intercept of this portion gave “on” responses to stimulation in the periphery. of the curve the value of the exponent of the power The graphs at the top show the intensity-response
function
bz sho\vn to be approx i. iIt lOWX indeprcssd toward the diagonal. and. dspsnding on the particukir cell. may 2ven Cross th2 diagonal. This may br du2 to “rrceptors”’ becoming more linsar at low light levels (the diagonal is the !ocus for linearity) or to threshold mschanisms on the signal from each &sa b2for2 summation. It could also r:sult from inhibitory lateral intsractions that are manifest2d only at the lowest intensitirs of stimulation. in our prttvious paper u’s concluded that it was not nec2ssa.ry to invoke lateral interactions. The nonLinraritizs apparent in the graphs like these in Fig. 1 can be adequately accounted for by a non-iinrar function at th2 sarlq stages of processing whers th2 rsspons2s from the t\vo arsas stimulatsd x2 still indeprndent and anothzr non-linear ftlnction after th2y have mzrgsd into a common pathway;. We will now examins in morr dstnil th2 possibility rhat therz ars lar2raI intsractions which are not equivalent to the transformations in our earlier simple mod& WC use three ssparats approuchrs to this problem. an
~~njiti~s, the curves cm
Lrt us consider th2 possibility that some of the noniin2arit>- svident on thz respons2-summation and s2nsitivity-summation plots may be du2 to latzrai intsractions that x2 not irrelevant in our model. Ws can show that thz effsct of latzral inhibition would b2 to dzprrss th2 curve on the sensitivity-summation coordinatss ad decrease the slop2 of the curve on th2 response-summation coordinates: if the interaction itself is non-lincnr, it would also introduce curvaturs on both plots (Lzvine. 1972). Since the slope of the low response part of the response-summation plot is always v2ry neari? unity. we can conclude that ther2 is no linrar int2raction (elective at all levels) made nonlinrar bt thrzsholds on the signals entering the final summation. Th2 remaining possibility is that the interactions themselves are non-linear, and must thereforz produce a curvature on both the response-summation and srnsitivity-summation plots. Let us assum that th2 curvature of the respond-su~ation plot. which is generally n fairly sharp “knee” in th2 curve, is due to lateral interactions; in that case, it should be possible to idrnrify a corresponding curvature on the sensitivity-summation plot. Figurs I dcmonstratss two attempts to find an effrct in the sensitivity-summation plot corresponding to ths curvaturr: of the line on the r2sponse-summation plot. Each of the response-summation plots on the bottom of Fig. 1 shows a fairly sharp break away from unity slope at a point indicated by th2 vertical arrow. (The dotted curves were derived from the curves fit to the data points on the intensity-response curves.) This break could be due to one of two causes: (1) either the total response (physiological sum) is compressed when responses exceed some level; or (2) one (or both) pf thz individual intensity-response curves (R , or R2) is acceltrated more rapidly above some response level. rcirhWC (I cotxw?~iic~rrt uccrferarion of tire CWYYfor60th
'WC gsnrraiis refer to the functions before the firs: interils “receptor” functions, but the reader should bear in mind that we have no grounds for dscribmp these non-linearities to any particular anatomical
action Of the t%i.O XCYS
sWuc~ure.
together. Thz latter possibility is far 12~slikslq. but 2\2n if it m.2~2ths casr. th2rs would have to b2 J compression of thz response to both in ordzr to compensate for the acceleration of ths rrceptor functions. It is therefor2 mzaningful to idsntify a response Iev2l R! _ 3 i2.g. Jo adrians. Fig. 1. left) with ths point tarrow on the bottom graph) at which the r2sponsc-summation curve br2aks from linsarity: th2 rssponss 12~21at which this occurs can bt: iotated on th2 intensityr2sponse curve for the rqual-intsnsit! sxp2riment shotvn at rhe top of Fig. 1. This point (arrow on th2 top graph) identifi2s the intznsity \I! _ :) at which the break occurs. Six2 th2 abscissa of thz sensiti~-its-s~tmmation plot is also log IL _ :, it is possible to locat th2 corrssponding point on the ssnsiriiit!-summation plot (middle: of Fig. I : arrow). We have thus looatsd thz point art the sznsitivitysummation plot which corresponds to ths point at which thz responsz-summation plot brsaks from linearity. In both units sholvn in Fig. I. this point is well within the xgmsnt of the curve which is clsarlc a straight line. Thr 2ffect of non-linear lacra’i interactions on rhs s2nsitivit~-summation piot is to shift thz curve: at th2 point correspondins to the break from linearity on th2 response-summation plot. thsre is no change in ths sensitivity-sLimmat~on plot. Thtrzforz, if the curvature on thz rssponsz-summation plot is duz to an int2ruction prior to summation. thr intzracrion must be equivalznt to a comprrssive function after summation. Thr analysis shown in Fig. I could bs p2rformsd on only a f2u units. In most units. the corrssponding point was at a highzr value of i, _ 1 than th2 maximum for which the srnsitivity-summation plot could be drawn (thr: sensiticity-summation plot is limited. since the highest responst- criterion which can bz ussd is the maximum firing of thz least xsponsive area t. However. a numb2r of other units tzndzd toward ths same outcome. and none of 18 units observed had rssponses which contradicted the abov2 conclusions.
Lst us reconsider ths possibility that thsrs are nonlinear latzral intzractions which have sscaped our notice. On the ~nsitiv~t~-summation plot inhibitory intzractions would dspr2ss the curve. On ths responsesummation plots. inhibitory interactions would d2crrase the slope of ths curvs. but the dzcreax might be unnoticeable in thz short region before tht compressive function after summation czpresses its2lf. (Not2 that facilitatory interactions would have just th2 opposit2 tffect.) We shall take advantage of on2 of the expected properties of latsral interactions in ordsr to try to demonstrate their presence: msgmtude of intttraction should change with the separation b2tween the interacting arcas. \V2 shnll compare rssults from stimuli that are close together with those from stimuli that ar2 well wparated. and se2 if there is a difference between the.plots from each casr. An experiment drsigncd to maximize and th2n minimizc iat2rdl interactions is shown in Fig. 2. Th2 stimuli ar2 shown at the upprr right; each is a four-spoksd pinwh22l. or cross, centered on, and confmsd to, the center of the receptive field. (This cross rezmbles the heraldic “cross patr”, but by virtue: of th2 curvatur2 at
Lag
(I?
LOQ iii,?)
Fig. 2. Results of a search for evidence of lateral interactions. Stimuli were four-limbed crosses Ltjshown at the top right. Each war centered on the receptive field, and they aere presented either individuahy or together at equal intensities. One cross (shown lightly stippled) was always in the same position; the other cross could be rotated (before pre~n~tion) to either of two positions, as shown. Open circles refer to responses to the stationary cross stimulated alone; solid circles and dashed curves refer to responses when the rotated cross is in the position adjacent to the stationary cross: triangles and solid curves refer to the rotated cross separated from the stationary cross. On the lower left are the intensity-response data for the crosses presented separately: intensity-response curves for the two simukaneous presentation series are on the upper left. The respond-summation plot is on the upper right (the curves for rhe two situations superimpose): sensitivity-summation on the iower right (solid curve for crosses separated, dashed curve for the crosses nearly touching). All stimuli were 710 nm.
the end of each limb is probably better referred to as a “cross globical”; Franklyn, 1970.) One cross. shown lightly stippled, was held in the same position throughout the experiment. The other cross, shaded with lines in the diagram, could take either of two positions; in one position it was rotated until it was as close as possible to the stationary cross without overlapping it, while in the other position it was as well separated as possible. The crosses take advantage of the approximately radial symmetry of the receptive field: they may be rotated without significantly changing their sensitivities. They also provide a relatively large boundary along which interactions may occur when the two crosses are nearly adjacent. The intensity-response functions
for each cross presented alone are shown in the lower left of Fig. 2. Open circles represent the stationary cross, solid circles represent the moveable cross in the close position, triangles in the separated position. The three curves are nearly identical. Intensity-response functions for both crosses illuminated simultaneously are shown on the upper left. The circles and dashed line are for the case in which the two crosses are adjacent; the triangles and solid lines are for when they are well separated. The response-summation and sensitivity-summation plots for each case are on the right of Fig. 2: There is no substantial difference on the response-summation plots (top), arguing for a lack of lateral interaction. but there is an apparent difference on the sensitivitysummation plots (bottom). In particular, the solid curve. from the case in which the crosses are well separated, is below the dashed curve (crosses nearly touching) at low intensities. Since the effect of inhibi-
tory interaction would be to depress this plot, the implication is that there was greater interaction when the crosses were separated than when they were tangent, and that the interaction was greater at low intensities than high. Neither of these prospects is inviting. for one would expect interaction to increase with intensity, not decrease; moreover, one would expect interactions to be greater with smaller separations, not larger. There are two other possibilities: (1) the differences noted on the sensitivity-summation plots could be the result of a facilitation by which the adjacent crosses aided each other in overcoming the thresholds (which presumably cause the low intensity depression of the curves); or (2) they might be due to random variance. With regard to the latter possibility, it is worth noting that the differences are at the low intensity part of the curve, which is its least reliable part (see Levine and Abramov, 1975); also, the entire difference can be traced to a small number of data points at the lowest intensities. But we may not yet dismiss the possibility that the lateral influences are in fact smaller at close distances. and increase to a maximum at a siiglitly larger separation. This is the kind of distance function that was reported by Barlow (1967) for lateral inhibition in Li&us. We must therefore perform a similar pair of experiments in which the distances considered are somewhat larger. A test for interactions across wider separations is shown in Fig. 3. Three 04rnm spots were focused within the center of this unit’s receptive field. One spot (spot 1) was always used as one of the MO areas stimulated; the second area was either the one quite close to it (spot 2) or the one quite distant (spot 3). Individual
Lack of lateral inhibition on ganglion cells
Fig. 3. Results of a search for evidence of lateral interactions effective over long distances. Stimuli were three spots. 04 mm dia, within the center of the receptive field. Two different experiments were performed: ii) spots 1 and 2 were used (open symbols. dotted curves); and (ii) spots I and 3 were used (solid symbois. dashed curves). Peg experiments are shown by triangles; the solid curve on the intensity-response plot at the upper left fits either set of triangles. Plots arranged as in Fis. 1; all stimuli were 710 nm.
intensity-response curves for each of the three spots are Thus far we have found no consistent evidence to shown m the lower left of the figure: the spots were essentially in equisenritive positions, since the curves compel us to include lateral interactions in our model, but a final point must be considered. Easter i 1965) sugsuperimpose quite closely. The intensity-response curves for the equal-intensity experiments for each gested that lateral interactions account for the differpairing are shown on the upper left of Fig. 3. The ences he noted between the summation of two separate equal-intensity experiment for the close spots is shown areas and the summation of contiguous areas in his esperiment in which spot size was varied to demonstrate by the open circIes (dotted curve), the distant pairing is shown by solid circles (dashed curve); the curves Ricco’s Law. Our data obtained from summing two agree closely. A peg experiment was also performed for areas indicate that no lateral interactions need be poseach of the pairings (triangles, sohd curve); there was tulated, but can the same data be used to verify Ricco’s no difference between the outcomes of the peg exper- Law’? iments for the two separations. The experiment shown in Fig. 4 was designed to The response-summation plots from these data are demonstrate that the relationship of Ricco’s Law is on the upper right in Fig. 3. The curves derived from consonant with the usual results of summing responses the two pairings do not quite superimpose: the curve from two separate areas. Two separate pair of stimuli for the closer pairing fails slightly below that for the were used (upper right). The first pair was a spot more distant pair. This is the result to be expected if @63 mm dia (stimulus 2) and an annulus of equal area there is more Iateral interaction between the closer whose outermost diameter was 1.0mm fstimulus 11; spots. Notice, however, that in this case the interaction this is still just within the center of the receptive field. is evident only at the higher firing levels; furthermore, The second pair consisted of a spot 04 mm dia (stimuit is not evident in the points derived from the peg ex- lus 3) and an annulus whose outer diameter was periments. Moreover, the sensitivity-summation plot. 063 mm (stimulus 3). Intensity-response curves for lower right. shows no substantial difference between stimuli 1 and 1 presented singly and together are the curves; it is probably fair to conclude that there is shown in the upper left of Fig. 4. There was a considerno evidence for lateral interaction in this unit. only able sensitivity difference between region 2 alone {spot) variability in the data. and region 1 alone (annulus); this diflerence is to be If there is lateral interaction its effect must k quite expected, since sensitivity declines with distance from smah, for the differences between the two responsethe midpoint of the receptive field. The response-sumsummation curves in Figs. 2 or 3 are minor compared mation plot derived from these data is shown in the to the difference between any of the curves and a line lower left of Fig. 4. The data from regions 3 and 3 are of unity slope. Other cases in which different stimuli not shown, but they are very similar, and both sets of were used on the same unit show only negligible diffet- curves are typical of those found with orher stimulus ences in the response-summation and sensitivity-sumconfigurations on other units. The sensitivity-summamation plots for each co~guration, indicating that the tion plot for regions 1 and 2. is on the loner right; rhe placement of the stimulus areas is of no real conse- dotted curve is plotted in the usual manner, and is quence. related to the ordinate on the left. This curve also is un-
0
20
(91+
40
RZ!
55
00
100
:2drlonsl
1,’ -30
i-33 -2 0
130
-I 0
!I:+2)
Fig. 4. Demonstration that Ricco’s Law is consistent with a model without lateral interactions. Stimuli (710 nm) were two circles and two annuli: half of each stimulus is shoun on the upper right-all were within the center of the receptive field. Upper left: Intensity-response curves for stimuli 1and 2 presented separately (solid symbols and curves) and together (dashed curve, open circles); these regions are clarly not equisensitive. Lower left: Respond-sum~~on plot for stimuli I and 2, equal-intensity experiment. Lower right: The dashed curve. referred to the ordinate to the left, is the usual sensitivity-summation plot for stimuli I and 1; conventions for this curve and the graphs to the left are the same as for Fig. I. The solid curve is derived from the same data as the dotted, but is plotted according to the method of Easter (1968) and refers to the ordinate to the right; see text for a discussion of these two forms of ptotting the data. Top right: The plot shows iog sensitivity- vs log area for a criterion response of 30 adrians. derived from the intensity-response curves on the left, The stimuli, whose areas correspond to the four values of the abscissa, are indicated above each data point; the line through the points has a slope of one. demonstrating that Ricco’s Law holds up to the largest area used.
exceptional, being a straight line of unity slope with an intercept ofabout 0.27. Now let us examine how these same data would appear if they were plotted exactly as described by Easter (1968). We would assume that regions I and 1 were still within the ptateau at the center of the receptive field where regions are still of equal sensitivity, and make use only of the intensity-response curves for region 2 alone and for both together. (When changing spot sizes in Easter’s experiments, the intensity-response curve for the additional contiguous annuIus would not be measured separately.) When the data are analyzed in this fashion. the ~nsitivity-summation curve is the solid curve whose ordinate is indicated on the right. This ordinate is displaced by 0.3 log units (Easter’s ordinate is a single sensitivity, not a sum of two), so that the implications of the line drawn as the diagonal are the same as for our form of the sensitivity-summation plot; that this curve lies along the diagonal implies complete linear summation and linear receptors. Easter, having shown the receptors were non-linear by su~ing well-separated spots, would have assumed that lateral interactions accounted for the location of the curve. But both the solid curve and the dotted curve were derived from the same set of data, so the difference between them must be due to the method of anatysis. and not the presence or absence of interactions. Finally. it is necessary to show that Ricco‘s Law was
demonstrated by our data. Our stimuii can be considered as four concentric spots of increasing area: stimulus 4 alone is the smallest spot and stir&i 1 and 2 presented together form the Iargest spot. The sensitivity of each of the four circular areas used was derived from the appropriate intensity-response curves using a criterion response of 30 adrians. The log sensitivity ofeach area was plotted as a function of the fog of the area (Fig. 4. upper right) anda Iine of unity slope drawn through the points. If S is sensitivity, A is area. I is intensity, and k and k’ are constants:
6 Easter has also reached this conclusion. based on similar reasoning (personal communication).
analysis
or
Iog s = log A + F:
(1)
=logA+k
(3
= k’
(31
-1ogI rl.i
which is Ricco*s Law, The same data that were used to demonstrate Ricco’s Law gave unexceptional response-summation and sensitivity-summation plots. There is no contradiction requiring us to invoke lateral interactions for its resolution; if there is a square root function at the receptors, it is sufhcient to have the sensitivity across the center of the receptive field fail off as the inverse square of the distance from rhe midpoint6
From these data, we conclude that rhe model we have previously presented (Levine and Abramov, 1975) does not require modification to include lateral interactions. The original model was deriwd from our mation
of response-summation and xnsitivity-sumplots for 17 separate units: in this paper, we
797
Lack of latsrai inhtbition on ganglton ceils examined the evtdence for any hint that lateral interactions might be present. First_ we checked that there was no misinterpretation of the graphs. The sharp bend in the responszsummation plots for two units was correlated with the sensitivity-summation plots from the same units; there was no corresponding break in the sensitivity-summation plots. Since the break was evident in the responsesummation plot only. it must necessarily have been due to a non-linearity after the final combination of the signals from the two areas into a final common pathway. and not to lateral interactions. Secondly. we examined the effect of increasing the spatial separation between the tuo stimulus areas. One would expect lateral interactions to be distance-dependent. but no such effect was found. Finally. we replicated Easter’s (1965) demonstration ofRicco’s Law, and showed that the result is consistent with our model as originally stated. Lateral interactions vvere not required to explain the “perfect summation” implied by Ricco’s Law, despite the demonstrable non-linearity (square root function) at the “receptors”. Our model for the processing within the center of the receptive field (or entirely within the surround) is thus the same as we have previously described (Levine and .Abramov. 1975). Light falling on each sub-area of the receptive field is transformed according to the square root of its intensity (with the possibility of saturation at high intensities, and a somewhat more linear function at very low intensities). Separate contributions which exceed some threshold value are summed linearly into a total signal. This signal is then transformed according to a non-linear compressive function roushiy of the form R=S
(-1)
k+s
where s is the signal and k is a constant; R is the response of the unit. in adrians. VVe must point out again that any model we present is a mathematical formulation in its simplest form; we clearly cannot distinguish between alternatives that are mathematically identrcal. If we found that some part of the processing could be described by a power function vvith an exponent of 1. we obviously could not state that this was not due to the serial application of two square root functions. Similarly. we must point out that certain forms of lateral inhibition, which might be very real on a cellular level, could be wholly equivalent to (and hence indistinguishable from) models in which there is no interaction. Within the range of our investigations. such a lateral interaction would befi~rtctionclll~ nonexistent. The most obvious esampic of a non-functional interaction would be a linear interaction: subtractive inhibition (with no non-linearities such as thresholds or rectification). Such an inhibition could be described as [using the notation of Levine and Abramov (1975) in which -yi is the output of a group of receptors prior to any lateral interactions. and yi is the input. after lateral interactions. to the summing point]: J, = as, - hsl
and
y2 =cx,
-dr,.
(5)
The sum of the two signals would then be 5 = 1’1 + .1’?= IilY, - hx,) + (cxI - ds,)
or 5 = (~1- &Y, i- IC - kJ).y~
(7
which is a linear summation of xL and x1. Each term has merely been multiplied by a constant coefficient: such a form of inhibition would have no effect on either the response-summation or sensitivity--summation plot. Changing the values of the constants would afiect the range of the plots. but not the curves themselves. Another form of inhibition that is less obviously equivalent to a non-inhibitory model is “mulriplicative forward inhtbition” (Furman, 1965 : Griisser. Schaible and Vierkant-&the. 1970). in which each separate signal is divided bi a pooled signal to u hich ~11active elements contribute:
(3). (9) (I, is a constant). But this is in fxt identical to a compressive function in the final common pathauy. for
I IO)
Since it is mathematically equivalent. it must be indistinguishable from compression after sumration; in fact, this could bs the mechanism by which the compression is achieved. Our model is restricted in its application to the domain of our experiments. It does not include adaptation or adaptive (long-term) intluencsj. for our stimuli were relatively brief (1 ssc) Hashes applied simultaneously. It does not include rod-cone interactions or chromatic interactions among dif%rent cone types. for our 710nm stimuli may be presumed effecttve only for the cones containing the long wavelength pigment (625 nm peak; Marks. 1965). i\;or does it include the interactions betvveen the center and iurround of the receptive field, for all our stimuli were chosen so that the two areas under consideration Mere either both within the center or both Kithin the surround. In our future work vve hope to expand the model and remove these restrictions. .-Ick,torvirrly~r,frr~rs-This mvestigarion was supported m part by NIH Training Grant So. GSI 1789 from National Institute of General Medical Sciences. and bv research prants EY 188 from the National Eqe Insiitute and GB 3616s from the National Swncc Foundation. The authors wish to thank Bruce W. Knight of The Rockefeller Cniversit> for his help and encouragement in iormlrlatinp the analytical methods used. We also thank Drs. Floyd RaMand H. K. Hartline of The Rockefeller University. and Dr. Christina Enroth-Cugell. of Northwestsm Lniversity, for their encouragement and advice. REFERESCES
Abramov I. and Levine hl. VV.(1973) The effects of carbon dioxide on ths excised goldfish retina. li~:!~~rRgs. 11. 1X3-1895.
Barlow R. B. (1967) Inhibitory fields in the L:wlus lateral eye. Doctoral dissertation. Thz Rockefeller L-niversitk. (6)
Yew York. NY.
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ISRAEL
r\BRAMOv
and
Easter S. S. (1968) Excitation in the goldfish retina: evidence for a non-linear intensity code. J. Physioi.. Lond. 195. 153-271. Franklyn J. (1970) fferaldr). A. S. Barnes. New- York. Furman G. G. (1963) Comparison of models for subtracttve and shunting lateral inhibition in receptor-neuron fields Kybernerik l&251-271. Griisser O.-J., Schaible D. and Vierkant-Glathe J. t 1970) X quantitative analysis of the spatial summation of exci-
~~IcH~EL
w.
LE-,I~.E
tation within the receptive field centers of retinal neurons. Pj?iigers .4rch. y~‘x Ph,vsiol. 319, IOl-I?!. Levine MM. W. (1973) An analysis ofspatiai summation in the receptive fields ofgoldfish retinal ganglion cells. Doctoral dissertation. The Rockefeller University;. New York. Levine M. W. and Abramov I. (1975) An analysis of spatial summation in the receptive fields of goldfish retinal gang_ lion cells. Vision Res. 15. 777-789. Marks W. B. (1969) Visual pigments of single goldfish cones. J. Phvsiol. , 1L.ond 178, 1432.
