The receptive field organization of X-cells in the cat: Spatiotemporal coupling and asymmetry

0042-698984 53.00to.00 Copyright i: 1981 Pergamon Press Ltd

b'mm Rrs Vol.21. So. 6. pp 549-561.1981 Pnnted I" Gr:at Bntam All nghts resmed

THE RECEPTIVE FIELD ORGANIZATION OF X-CELLS IN THE CAT: SPATIOTEMPORAL COUPLING AND ASYMMETRY S. DAWIS*,

R.

SHAPLEY~.

E.

RockefellerUniversity.1230 (Rewired

23 .\fo~

KAPLAS

and D.

TRANCHIXA

York Ave. NY 10021. U.S.A.

1983; in recised form

I I Oc~oher

1983)

.ibstract-The visual responses of retinal X-cells. recorded as s-potentials in the lateral geniculate nucleus (LGN). and geniculate X-cells. recorded as action potentials in the LGN, were studied in urethane anesthetized cats. We tested two widely held hypotheses: that the receptive field is (I) separable into spatial and temporal factors. and (2) even symmetric in space. The tests were applied to the amplitudes and phases of responses to sinusoidal gratings which drifted across the receptive field. The X-cell responses failed the test for spatiotemporal separability. and cases of spatial asymmetry were observed. A modified “Difference of gaussians” (mod DOG) model proved useful in the interpretation of the amplitude and phase data, Application of the mod DOG model to the amplitude and phase data revealed the existence of three forms of spatiotemporal coupling. Changes in the temporal frequency of the stimulus can change (I) the ratio between center and surround strengths, (2) the difference between center and surround phases, and (3) the spatial extent of the surround mechanism. Our results led us to a new view of receptive field oreanization in the X-cell: the center mechanism is spatially homogeneous in its dynamics whereas the surround mechanism is spatially inhomogeneous. X-cell

Receptive tields

measures has previously relied on convenient assumptions about receptive field organization. This paper examines two of these assumptions: spatiotemporal separability and spatial symmetry. The first assumption, spatiotemporal separability, is that the neural response can be separated into spatial and temporal factors. For example, suppose one probed the receptive field with a small spot of light that was flashed as an impulse of constant strength. Let I (s, y, t) denote the response when the light impulse was presented at position (x, y) in space and at time f = 0. If the response were spatiotemporally separable, one could write a mathematical expression for I (x, y, 1) in which the spatial dependence and temporal dependence were factored. In other words, I(x,g, 1) could be expressed in the following form

INTRODUCTION

One method of characterizing a visual neuron is to map its receptive field with a small flashing spot. Kuffler (1953) and Rodieck and Stone (1965) mapped concentric center and surround regions in the receptive field of cat retinal ganglion cells. The responses in the surround region were opposite in sign and antagonistic to those in the center region. Rodieck (1965) described the receptive field organization quantitatively with a “Difference of Gaussians” (DOG) model: the response at a point in the receptive field is given by the diRerence of the responses of center and surround mechanisms at that point, with the size of response of each mechanism being determined by a Gaussian spatial sensitivity profile. Another useful approach to studying the spatial organization of the receptive field of a visual neuron is to measure its contrast sensitivity as a function of spatial frequency (Enroth-Cugell and Robson, 1966). In this approach the visual neuron is stimulated with drifting sinusoidal gratings. At each spatial frequency. the contrast that gives a criterion response is determined. The contrast sensitivity function is the dependence of the inverse of the determined contrast, the contrast sensitivity, on spatial frequency. Both the receptive field sensitivity profile and the contrast sensitivity function are measures of spatial summation by the visual neuron. The interpretation of these

*Present address: Neurophysiologie Laboratorium, Universitaets Augenklinik. 8091 Zurich, Switzerland. +To whom correspondence should be addressed.

((X,Y, 1) =g(.r,y)h(O

(1)

where g&y) is a spatial function and /l(t) is a temporal function. Although convenient, the assumption of spatiotemporal separability is probably invalid. Maffei et al. (1971) and Schweitzer-Tong ef al. (1981) have found that the receptive field map and contrast sensitivity function of the retinal ganglion cell depend on the temporal nature of the stimulus. In this case the neural response is not spatiotemporally separable and is said to be “spatiotemporally coupled”. The second assumption to be examined is that the receptive field has ecen symmetry. Rodieck (1965) and Enroth-Cugell and Robson (1966) made this assumption explicit in the DOG model by constraining center and surround Gaussians to be concentric. An

549

ajsumption of symmetry is invoked implicitly whenever a contrast sensitivity function is measured with contrast-reversing gratings or with gratings drifted in one direction. There is, however. evidence of receptive field asymmetry (Rodteck and Stone. 1965: Maffei er a(.. 1971). S~gni~~ant receptive field asymmetry would preclude the measurement of contrast sensitivity functions with contrast-reversing gratings drifted in one direction. The major result of this paper is that by drifting gratings in two opposite directions. spatiotemporal coupling and receptive tield asymmetry can be detected with phase measurements. This result has led us to the formulation of a modified “DifTerence of Gaussians” (mod DOG) model. This model is a useful tool for interpreting amplitude and phase data and leads to new ideas about retinal and geniculate receptive fields. SIETHODS

Cats were anesthetized with urethane, paralyzed with gallamine. artificially respirated. and maintained at 38 C. Physiological indicators (EEG, arterial blood pressure. heart rate. and expired CO,) were monitored. Details of physiological methods are given in So & Shapley (1979).

The computer-controlled visual stimulator described by ivlilkman CI al. (1980) was used to generate drifting sinusoidal gratings. The visual display subtended a visual angle of IO’ x IO’. The mean luminance was typically 105 cd:m’. Contrast is defined as CL,,,dx- &,,,A’(&,,,, f .&,,,A where L,,, and &,,, are the maximum and minimum fuminance levels in the grating. Spatial frequency (in cycIes/visual degree) is defined to be the inverse of the distance between two neighbouring luminance peaks in &he grating. The grating (in visual velocity of a drifting degrees/second) is given by the distance traveled per unit time by a luminance peak. We have considered both spatial frequency and velocity to be sigtred quantities; the convention we used was positive for gratings drifting from right to left and negative for ieft to right. Temporal frequency (in cycles/second or Hertz) is defined as the product of spatial frequency and velocity. The temporal period of the sinusoidal grating is defined as the inverse of its temporal frequency. For our work, it was crucial to use positive and negative spatial frequencies and to include very low (0.05 to 0.1 cjdeg) spatial frequencies. The frame rate of the electronic visual stimulator was 270 Hz. Recoding

Extracellular recordings with glass micropipettes were made of s-potential and spike activity from cat

LGN units. Methods of recording from the cat LGN are detailed in So and Shapley (1979). As did So and Shapley (1951). we regarded the s-potentials to be synaptic potentials produced by an incoming retinal ganglion fiber and the spikes to be the outgoing activity of the LGN neuron (Bishop rr (II’.. 1962). Since the spike can sometimes be seen rising from the smaller s-potential. the occurrence of a spike was also counted as the occurrence of an s-potential. Receptive fields were mapped onto a tangent screen and then were repositioned with a mirror onto the ttsual display. Units were typed as X and Y using a modified null test (Enroth-Cugell and Robson. 1966; Hochstein and Shapley. 1976: So and Shapley. 1979). Sputiotemporal

analysis

Response histograms were formed by averaging the neural responses to various sinusoidal drifting gratings for 15 sec. The histograms consisted of 256 bins, each bin corresponding to i/256th of the temporal period of the sinusoidal grating. Fourier analysis gave the amplitude and phase of the fundamental (first harmonic) component of each histogram. The response phase is defined as follows: as a sinusoidal grating drifts across the screen, both the fundamental component of the neural response and the luminance of the vertical line at the middle of the screen are sinusoidally modulated in time with the same temporal frequency as the drifting sinusoidal grating; the response (temporai) phase is defined to be the phase difference between the two temporal sinusoids. The reference line defined the “spatial origin” of the axis along the direction of movement of the gratings. A set of amplitude and phase measurements at a given temporal frequency were made with sinusoidal gratings of various spatial frequencies and constant contrast drifted at a constant temporal frequency (velocities inversely proportional to their spatial frequencies). The sinusoidal gratings were also drifted in the opposite direction, with corresponding negative velocities (and, according to our convention, negative spatial frequencies), and corresponding measurements of response amplitudes and phases were made. The “spatial amplitude curve” at a given temporal frequency and contrast was constructed from the above set of measurements by piotting the logarithm of the response amplitude as a function of the logarithm of the spatial frequency. Response amplitudes obtained with negative velocities were plotted vs “negative spatial frequencies”. The “spatial phase curve” at a given temporal frequency was constructed from the above set of measurements by plotting the response phase as a function of the logarithm of the spatial frequency. Together, the spatial amplitude curve and the corresponding spatial phase curve at if given temporal frequency and contrast constitute the “spatial transfer function” at the given temporal frequency and contrast.

551

X-cell receptive fields MODELS

The *‘D#erence

of Gaussians” (DOG) model

The DOG model presented by Rodieck (1965) and Enroth-Cugell and Robson (1966) is spatiotemporaffy separable and spatially symmetric. Since the model is spatiotemporafly separable, the spatial and temporal factors of the response can be considered separately. The ~~ar~af~~ncf~o~.In the DOG model, illustrated in Fig. f(a). the spatiaf profile of the center mechanism is a narrow Gaussian and that of the surround mechanism a broad Gaussian. The two Gaussians are positioned concentrically, and the spatial profile of the receptive field is given by subtracting the surround Gaussian from the center Gaussian. The DOG spatial profile is characterized by four parameters: the strengths and extents of the center and surround mechanisms. The strength of a mechanism is defined to be the area under its Gaussian profile; the extent of a mechanism is defined to be the distance at which its Gaussian profile falls to a value of I.‘e of the peak value. A fifth parameter is the position in space of the receptive field; the position of the peak of the center Gaussian is a convenient measure of receptive field position. Tfw temporal jirnction. The basic assumptions of the DOG model are that responses of the center and surround mechanisms are linear, have identical timecourses, are opposite in sign, are weighted in space by Gaussian functions, and are summed to give the observed response. Given these assumptions, suppose that the stimulus is a line with a sinusoidally modulated luminance. The responses of the center and surround mechanisms will be sinusoids that are 180’ out of phase. The temporaf phase of the center response serves as a sixth parameter of the DOG model. Although the temporal phase of the center response depends on the temporal frequency of the sinusoidal stimulus, the phase difference between center and surround responses will always be 180’. The phase information is displayed in a phase plot in the upper right corner of Fig. I(a). The mo&fied “D@jCerenceoif Gaussims” (mod DOG)

modef We propose the mod DOG model shown in Fig. I(b). As in the DOG model, the spatial profiles of the center and surround mechanisms are given by narrow and broad Gaussians, respectivety. However, unlike the DOG model, the Gaussians of the modified model may be nonconcentric. That is, we do not assume even spatial symmetry at the outset. Furthermore, the temporal phase difference between center and surround mechanisms may be other than 180’ (see also Kaplan et a/., 1979 and Schweitzer- Tong et af., 1981). That is, we do not assume spatiotemporal separability at the outset. When the center and surround Gaussians of the modified model are vettorially added, the resutting spatial profile is a com-

pfex valued, rather than a real valued. function of position. Experimentally, the complex valued spatial function, known as the fine spread function, is obtained by sinusoidally modulating the luminance of a thin bar of light, placing it at various points in the receptive field, and at each point, measuring the amplitude and phase of the response. A full characterization of the mod DOG model requires eight parameters: the strengths, the extents, the temporal phases, and the spatial positions of the center and surround mechanisms. “Difference

of goussions”

model

(0)

Center

Phase

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Fig. I. (a) The original DOG model. ft is assumed that the spatial profiles of the center and surround mechanisms are Gaussians that are concentrically positioned. This means that the receptive field is spatially symmetric. Typically, the model is characterized by four parameters: KC, the strength of the center mechanism; KS, the strength of the surround mechanism; rr, the extent of the center mechanism; and r,, the extent of the surround mechanism. When the model is stimmated with a sinusoidally modulated luminance, one can also specify 6, and B,, the phases of the center and surround responses, respectively. The phases will be a function of the temporal frequency of the stimulus. The DOG model is assumed to be spatiotemporafly separable. Therefore, as the temporal frequency of the sinusoidal stimulus is changed, the strength ratio KJK, remains constant and the phase difference 0, - 0, remains equal to 180”. (b) The mod DOG model. The modified model is a DOG model that allows for: (I) spatial asymmetry by allowing the position of the center, ,u,, to be different from the position of the surround, x,; and (2) spatiotemporal coupling by allowing the strength ratio KJKc and the phase difference 8, - Q, to change as the temporal frequency of a sinusoidal stimulus is changed.

s. D4WIS

ii7 _--

In the mod DOG model the tzmporul phase ditference between center and surround responses depends 0x1 the temporal frequency of the sinusoidal stimulus. If there is a range of temporal frequencies over which the phase difference is not 180 . then the responses of the center and surround mechanisms must have different time-courses. Consequently. the time-course of the observed response to a line stimulus which is briefly Hashed will vary from point to point in the receptive field. In other words, the response is spatiotemporslly coupled.

In the following. tests for spat~otempor~ll separability and spatial symmetry are presented. Each test is illustrated by applying it to hypothetical cases generated by the mod DOG model. The tests then are applied to data from cat retinal and LGN units to determine whether the receptive fields are spatiotemporally separable and spatially symmetric. The conclusion-that the receptive fields are spatiotemporally coupled and that spatial asymmetries occasionally arise-point to the need for a simultaneous analysis of amplitude and phase data. In this paper, two units are analyzed in detail. One unit is a retinal X-ceil, recorded as an s-potential in the LGN, and the other is a geniculate X-cell. recorded as an action potential in the LGN. These units are representative of a sample of 38 s-potential and 33 LGN X-cell recordings made in 16 cats.

Spatiotemporal

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SPATIAL AMPLITUOE CURVES AT HIGH, MEDIUM, AN0 LOW TEMPORAL FREOUENCIES

separability

Amplitude. In the past, investigators have tested for spatiotemporal separability by measuring response amplitudes alone. The test consists of measuring the spatial amplitude curve at various temporal frequencies. If the shape of the amplitude curve on a logarithmic amplitude scale is retained when the temporal frequency of the stimulus is changed, then the results support, but do not prove, spatiotemporal separability (see the Appendix). On the other hand, if the shape of the amplitude curve (on a logarithmic amplitude scale) changes when the temporal frequency of the stimulus is changed, then the spatial organization of the response must be coupled to temporal influences. The eRect of spatiotemporal coupling on the shape of the spatial amplitude curve as the temporal frequency of the stimulus is varied can be illustrated with the mod DOG model. The responses of this model are spatiotemporally coupled when the phase difference between center and surround responses deviates from 180’. Figure 2(a) shows the effect on the spatial amplitude curve of a phase difference that begins at 180’ at low temporal frequencies and increases to 230’ at high temporal frequencies, In Fig. 2(a), the spatial amplitude curve is more spatially tuned at low than at high temporal frequencies-that is, the attenuation of the amplitudes at low spatial

Fig. 2. The effects on spatial tuning of two iorms of sp~t~otemporalcoupling. (a) The effect on spatial tuning of a phase lag between center and surround responses that increased as temporal frequency increased. On the left side of panel (a) are the model choices used to generate spatial amplitude curves at low. medium and high temporal frequencies. As the temporal frequency was changed. the center and surround phases were varied while the center and surround strengths and extents were held constant. Note that the phases were varied in such a way that the phase lag (the angle subtending the continuous arc in the unit circle) increased with increasing temporal frequency. The result was that the spatial amplitude curves became less tuned as the temporal frequency increased. (b) The effect on spatial tuning of a surround strength that decreased, relative to the center strength, as temporal frequency increased. On the left side of panel (b) are the model choices used to generate spatial amplitude curves at low, medium and high temporal frequencies. As the temporal frequency was increased, the strength of the surround mechanism was reduced while the strength of the center mechanism was held constant. The extents of the center and surround m~hanisms were also held fixed, and the phase difference between center and surround responses was fixed at 190’. The effect on spatial tuning of reducing the ratio of surround to center strength was very similar to the effect of increasing the lag between center and surround phases.

N-cell receptive fields

frequencies relative to the peak amplitude diminishes as the temporal frequency is increased. The modified model can be spatiotemporally coupled in another manner. In the DOG model the ratio of the surround to center strength must remain constant as temporal frequency is varied, a restriction not imposed on the modified model. Figure 2(b) illustrates the effect on the spatial amplitude curve of a surround/center strength ratio that decreases as temporal frequency increases. This would be the case if the surround response underwent more temporal integration than the center response. The effect on spatial tuning is similar to that shown in Fig. 2(a). Previous application of the amplitude test has suggested spatiotemporal separability for certain visual responses: in simple and complex cells in cat striate cortex (Tothurst and Movshon, 1975); in cat LGN cells (Lehmkuhle er al., 1980); in a transient mechanism of the human visual system (Wilson, 1980). The opposite conclusion, that visual responses are spatiotemporally coupled, was reached by other investigators: Maffei et al. (1971) and SchweitzerTong et af. (I981). for retinal ganglion cells, and Lee el al. (1981), for cat retinal ganglion and LGN units. The measurements shown in Fig. 3 agree with the earlier reports of spa~iotempora[ coupling. The data in Fig. 3(a) were taken from the spike train of an LGN X-cell, and the data in Fig. 3(b) were obtained From the s-potentials of another LGN X-cell. It is

553

clear that at low temporal frequencies there is greater spatial tuning than at high temporal frequencies. The data in Fig. 3(c) were taken from the s-potentials of a third LGN X-cell. For this response. the amplitude curves at different temporal frequencies changed only slightly. However, a test based on phase measurements. given below. demonstrates that the unit of Fig. 3(c) was spatiotemporaily coupled. Amplirude undphasr. The amplitude test for spatiotemporal separability is incomplete; the complete test for spatiotemporal separability is as follows: a response is spatiotemporally separable only if (I) at each temporal frequency (a) the spatial amplitude curve has eren sentmefr_r with respect to spatial frequency and (b) the spatial phase curve has point symmetry around the phase point at zero spatial frequency and (2) across temporal frequencies (a) the spatial amplitude curves are shapeinvariant on a logarithmic amplitude scale and (b) the spatial phase curves are shape-invariant on a linear phase scale. By definition, the spatial amplitude curve, A (v), has eren symmetry with respect to spatial frequency, V, if A (v) = A c-v); in other words, the amplitude

SPATIAL FREQUENCY (cyclesldegi Fig. 3. Spatial amplitude curves measured at various temporal frequencies for three units. (a) Amplitude data from an LGN on-center X-cell, recorded as spikes, at a mean intensity of 102 cd/m? and with stimulus contrasts of 0.14. (b) Ampfitude data from a retinal off-center X-cell, recorded as an s-potential in the LGN, at a mean intensity of 107.4 cd/m? and with stimulus contrasts of 0.076. (c) Amplitude data from a retinal on-center X-cell, recorded as an s-potential in the LGN, at a mean intensity of 104.1 cd/m? and with stimulus contrasts of 0.14.

value at any given spatial frequency amplitude value at the corresponding frequency.

is equal to the negative spatial

By definition. point

the spatial phase curve, U(Y). has around the phase point at zero spatial [O. If (O)] if

s~~~~r~tr~

frequency

II(V)--1!(O)=

-[O(-r)-i)(O)].

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in other bvords. if a horizontal line rvere drawn through the phase point at zero spatial frequency, the phase value at an! given spatial frequency would lie above (below) the drawn line by the same amount that the phase va1~1e at the corresponding negative spatial frequency lay below (above) the line. Later we will make use of vtld s)mmcrr~: a function*&,{ 1’). has utin’s>~trmlerr_rwith respect to spatial frequency. 1‘. ifj,,,(r) = -&,(-v); in other words. an odd function has point symmetry around the origin. (0,O). The basis of the first part of our proposed test is that if the neural rssponse is spatiotemporali~ separable, then its spatial transfer function is the Fourier transform of a real function. The basis for the second part of the test is that if the response is spatiotemporally separable. then its spatiotemporai transfer function can be factored into spatial and temporal functions. A proof of the test is supplied in the Appendix. Sp~~~~le~tp~r~t~ sepiircthilify and cotcpiirtg in ntociri .~~.rtenr.s. We use the mod DOG model to illustrate that the spatial transfer function of a spatiotemporally separable receptive fisld meets the above symmetry requirements even if the receptive field is spatially asymmetric. Figure 4(a) shows the spatial transfer function for the modified model with parameters chosen to give a spatiotemporally separable but spatially asymmetric receptive field. Note that even though the receptive: field is spatially asymmetric. the spatial amplitude curve has even symmetry and the spatial phase curve is point-symmetric around the phase point at zero spatial frequency (the phase point measurrd when the luminance of the full field is sinusoidally modulated). As shown in Fig. 4(b) the symmetries disappear when the receptive field is made spatiotemporally coupled. The receptive field in Fig. 3(b) has the same offset between the center and surround positions as that in Fig. 3(a), but the phase difference between center and surround responses is no longer 180’. In this case, neither the spatial amplitude curve nor the spatial phase curve possess the appropriate symmetries for spatiotemporai separability. Of course, the spatial transfer function is most symmetric when the receptive field is both spatiotemporaily separable and spatially symmetric as shown in Fig. J(c). One might worry that symmetries in the spatial transfer function resulting from spatial symmetry of the receptive field could be misconstrued as evidence for spatiotemporal separability. Figure 4fd) shows the spatial transfer function for a parameter choice

that is spatiotemporall~ coupled and spatlai]y ,!nlmetric. In this case. the spatid ~~$irt& curve has even symmetry: however. the spatid phrtse cur\? is not point-symmetric. Thus. the combination of amplitude and phase measurements allows on2 to distinguish spatiotemporai separability from spatial s>mmetry. Sputiotemporul

coupling

in cut s-potentid

durti. Let

US apply

the test for spatiot2mpora~ separability to real data: the 1 and 12 HZ spatiai transfer functions of the retinal X-cell that gave the amplitud2 curves in Fig. 3(c). With Fig. 3(c) we have already examined item 31 of the test: since the spatial amplitude curves changed shape slightly as the stimulus frequency kvas changed from 1 to 12. Hz. spatiotemporal coupling seemed to b2 indicated. To test symmetry of the spatial amplitude curve (item la), the negative spatial frequency portion of the amplitudr curve was reflected onto thr positike spatial frequency portion in Fig. 5(a) for Z Hz and in Fig. 3(b) for 12 Hz. Figure jib) indicates that the 12 Hz spatial amplitude curve is slightly asymmetric, but stronger support for spatiotemporai coupling can be found in the phase data. The test of point symmetry of the spatial phasr curve (item lb) is shown in Fig. 6. Figure 6(aj illustrates that the 2 Hz phase curvt‘ is pointsymmetric. but Fig. 6(b) clearly shows that the 12 Hz phase curve is not point-symmetric. Since there is a temporrtl frequency ( It Hz) at which point symmetry breaks down. the response is sp~~t~or~rnpo~~~~~COW pled. The lack of point symmetry of the I:! Hz phase curve cannot bz attributed to changes in eye position because the contribution of eye position to the spatial phase curve is odd symmetric (see next section). Thus. the amplitude measurements in Fig. 3(c) gave weak evidence for spatiotemporal coupling, but strong confirmation of the coupling was found in tht: corresponding phase measurements.

Since in the experimental situation the placement of the receptive field relative to the spatial origin may be arbitrary. ~‘2 should explain the effect th;tt trimslation in space has on the spatial transfer function. Suppose that a spatial transfer function was .measured when the receptive field was centered at the spatial origin and then again when the receptive field was centered +-so visual degrees from the spatial origin. The spatial amplitude curves of the two measurements will not differ: however, the spatial phase curve of the second measurement will differ from the first measurement by -360.~~~ degrees of spatial phase angle at each spatial frequency 1 (cjdeg). The difference is proportionai to spatial frequency because the number of cycles that can be fitted into the distance x0 is proportional to spatial frequency. One can still tell that a receptive field passes the test for spariotemporal separability even if it is offset from position .x = 0. Item lb of the test for spatiotemporal

X-cell receptive fields

555

(b) SWARASLE

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Fig. 4. Spatial transfer functions produced by the mod DOG model For cases of (a) a spatiotemporally separable, spatially asymmetric receptive field, (b) a spatiotemporally coupled, spatially asymmetric receptive field, (c) a spatiotemporally separable, spatially symmetric receptive field, and (d) a spatiotemporally coupled, spatially symmetric receptive field. (Note that the spatial frequency scales are linear). The receptive fields used to generate the spatial transfer functions are shown in the upper right hand comer of each panel. All four receptive fields were positioned with the peak of the center Gaussian at the origin. Since the receptive fields in (a) and (c) are spatiotemporally separable, and spatial amplitude curves are even functions of spatial frequency, and the spatial phase curves are point symmetric around the phase point at zero spatial phase (indicated by the arrow). Since the receptive fields in (c) and (d) are spatially symmetric, both the amplitude and phase curves are even functions of spatial frequency.

separability states that the spatial phase curve must be point-symmetric around the phase point at zero spatial frequency. Since the function -360~~~ is an odd function of spatial frequency, its addition to a point-symmetric phase curve will not disrupt the point symmetry. The preservation of point symmetry is demonstrated in Figs 4(a) and 7(a) with the mod DOG model. Figure 4(a) shows the spatial transfer

function for a spatiotemporally separable and spatially asymmetric receptive field; Fig. 7(a) shows the spatial transfer function for the same receptive field but transfated +O.l deg. In both cases, the spatial phase curve is point-symmetric. One can still test for spatiotemporal separability even if the eye position changes between measurements of spatial transfer functions. Item 2b of the

S. DAWIS rr rri SPATIAL

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FIN. 5. Test for even symmetry in spatial amplitude curves measured at temporal frequencies of (a) ? Hz and (b) II Hz. The amplitude data were taken from Fig. 3(c). The data at negative spatial frequencies (open triangles) were reflected onto the data at positive spatial frequencies (open circles). The 2 Hz spatial amplitude curve is even symmetric; the I2Hz spatial amplitude curve is slightly asymmetric. Referring back to Fig. 4. one may conclude that at I2 Hz the receptive field is spatially asymmetric and the phase difference between center and surround responses is not 180‘. and that at 2 Hz either the receptive field becomes spatially symmetric, the phase difference between center and surround responses becomes 180‘. or both.

test will be invalid if the eye moves between measurements of these functions, but the correction for item 2b is clear. One has to allow for an additional component in the spatial phase curve which is proportional to spatial frequency. Furthermore, one can distinguish between spatial transfer functions of separable and even symmetric tields after spatial translation such as is caused by eye movements. The addition of the odd function -360.~~~ to a spatial phase curve does not create a point symmetry where none existed previously. This is illustrated in Figs 4(d) and 7(b) which demonstrate that a spatially symmetric but coupled field fails the SPATIAL

PHASE

Fig. 7. The effect of spatial position on the spatial transfer function illustrated with the mod DOG model. (Note that the spatial frequency scales are linear.) (a) The receptive field is the spatiotemporally separable, spatially asymmetric receptive tield of Fig. 4(a) but translated +O.i v deg. The translation does not affect the spatial amplitude curve. and although the spatial phase curve is modified by the addition of a spatial phase component due to translation. its point symmetry is retained. Point symmetry is retained because the additional component is a linear, and therefore odd, function of spatial frequency. (b) The receptive field is the spatiotemporally coupled. spatially symmetric receptive tield of Fig. 4(d) but translated iO.l vdeg. Again. translation does not affect the spatial amplitude curve. However. in this case, the even symmetry of the spatial phase curve is disrupted by the addition of the spatial phase component due to translation.

test for spatiotemporal separability whether it is placed at or away from the spatial origin. The -360.~“~ formula for spatial translation will be used in following sections. Since the constant of CURVE I2 HERTZ

2 HERTZ

0 PHASE (deq) 1111hlllllll .50 -1.0

-0.1

0.1 SPATIAL

IO

-180

50 FREOUENCY

(cycles/degl

Fig. 6. Test for point symmetry in spatial phase curves measured at temporal frequencies of (a) 2 Hz and (b) 12 Hz. The phase data were collected from the X-cell s-potential of Fig. 3(c). For each panel, the original data (open circles) were rotated 180’ about the extrapolated value for the phase at zero spatial frequency (indicated by the x ) and replotted (as open triangles). If the rotated data superimpose on the original data, then the spatial phase curve is point-symmetric around the phase at zero spatial frequency. satisfying one of the requirements for spatiotemporal separability. The 12 Hz phase data in panel (b) fail the test: therefore, the X-cell response is spatiotemporally coupled.

X-cell receptive fields

proportionality is -360.x0, the position .r, of the receptive field can be determined from phase measurements. This could be important for visual information processing. As Lee ef ui. ( 198 1) have sugposition information may be gested. spatial transmitted in the visual system by response phases. Receptive j&z/d s_wnmetr~

We now consider the test for spatial symmetry. Figures 4(c) and (d) show that a receptive field which has even symmetry in space and is centered at the spatial origin will produce spatial amplitude and phase curves which are even functions of spatial frequency (see also the Appendix). If the receptive held is centered at a point .r, visual degrees from the spatial origin, an odd symmetric component -360.~~~ must be added to the even symmetric spatial phase curve. Therefore, the test for receptive field symmetry cannot be simply to look for even spatial amplitude and phase curves. One solution would be to center very accurately the receptive fietd at the origin before the spatial transfer function is measured. A more convenient solution is, after the measurement, to decompose the spatial phase curve into even and odd components. As with any real function, the spatial phase curve can be decomposed into the sum of an even function and an odd function: if the phase at spatial frequency v is denoted by 0 (v), then the even component is given by [O(v) + 0 (-~)]/2 and the odd component by [O(v) - 0 f- v)]/2. Once the spatial phase curve is so decomposed, one can then examine whether the odd component was due only to a spatial o&et from the origin. Specifically, the test for receptive field symmetry is: the receptive field has even symmetry in space if and only if at each temporal frequency (1) the spatial amplitude of spatial frequency (2) the odd component can be described -360.x, \‘.

curve is an even function and of the spatial phase curve by the linear function

Let us apply the second part of the test for spatial symmetry to the retinal X-cell data shown in Fig. 8. phe first part of the test is identical to item la of the test for spatiotemporal separability and can be illustrated by reconsidering Fig. 5.1 Figure 8 shows the even and odd components of the 2 and I2 Hz spatial phase curves shown previously in Fig. 6. In Fig. 8 the odd components of the 2 and 12 Hz spatial phase curves deviate at low spatial frequencies from a line passing through the origin; therefore, the receptive field is slightly asymmetric. Another look at spatiotemporal separability. De-

composing the spatial phase curve into even and odd components also is useful in analyzing separability. Constancy of the even component of the spatiai phase curve is equivalent to point symmetry of the entire spatial phase curve, one of the requirements

557

for spatiotemporal separability. It was noted earlier that if a response is spatiotemporally separable, then the spatial phase curve is point symmetric around the phase point at zero spatial frequency (item lb). According to equation (2). the point symmetry is given by e(v)-e(o)= -[B(-r)-e(O)]. Rearranging the equation, we get [e(v) + e( -v)]/ 2 = B (0). The function [6’(v) + B (- v)]jt is the even component of the function 8(v). Therefore. if a response is spatiotemporally separable, then at any given temporal frequency the even component of the spatial phase curve must be equal to a constant given by the value of the phase measured at zero spatial (01

24

SPATIAL

I

2 HERTZ

t

12 HERTZ

FREOUENCY

cl E

(bl \

300

1.0 SPATIAL

I

I

20

30

FREOUENCY

Fig. 8. The even and odd components of the spatial phase curves of the X-cell s-potential which were presented in Fig 6. (Note that the spatial frequency scales are linear.) A test based on phase data can distinguish spatiotemporal separability from spatial symmetry. If the odd component of the phase versus spatial frequency is a linear function of spatial frequency, then the receptive field is spatially symmetric. The amount by which the odd component deviates from linearity will indicate the magnitude of any existing spatial asymmetry; thus, the odd component of the 2 Hz data (a) indicates a larger asymmetry than the slight asymmetry indicated by the odd component of the 12 Hz data (b). If the even component of the spatial phase curve is a constant function of spatial frequency, then the spatial phase curve is point-symmetric around the phase at zero spatial frequency, satisfying one of the requirements for spatiotemporal separability. Spatiotemporal coupling is indicated by the inconstant even component of the I2Hz data, reiterating the results of the point symmetry test shown in Fig. 6. Phase data also contain position information. The slope of the odd component of the spatial phase curve at high spatial frequencies is a measure of the position of the center mechanism. In panels (a) and (b), sectors subtending the odd phase components at 1.6, 2.4 and 3.2c/deg were drawn to give an idea of the limitations imposed by the data on the precision of the center position. Both sectors indicate that the precision is on the order of 1 visuaI minute.

558

(a)

fbl

SP&T’

LL

TRANSFER

“DIFFERENCE

OF

FUNCTION GPUSSI1NS

WITH ”

MODiFifO

MODEL

?2 HERTZ

PHASE (deq 1

--5.0

wblllll

1.0 5.0-5.0 -1.0

-0.1 0.1

-1.0

SPATIAL

FREOUENCY

-0.1

0.1

1.0

5.0

(cycles/deg)

(d)

(c) SPATIAL OF

TRANSFER

GAUSSIANS”

FUNCTION

WITH

“DIFFERENCE

MODEL

2 HERTZ

PHASE

tdegl

-5.0

-1.0

-0.1

0.1

SPATIAL

1.0

5.0

-5.0

FREQUENCY

-1.0

[cycles

-0.1

0.1

1.0

5.0

[degl

Fig. 9. A comparison of fits by the mod DOG model (a and b) and the original DOG model (c and d) to spatial transfer functions. The 2 Hz spatial transfer function of an X-cell s-potential is reproduced twice in (a) and (c); the 12 Hz spatial transfer function from the same ceil is reproduced in (b) and (d). (The amplitude and phase data were taken from Figs 5 and 6.) For the mod DOG model, equation (3) was used to fit the data. To fit the data with the original DOG model, equation (3) was used with two constraints: 0, - ep = 180’ and x, = x,. The parameter values for the continuous curves are given in Table 1; corresponding receptive field plots are drawn in the upper right comer of panels (a)-(d).

X-cell receptive fields Table I. Parameter values used to fit the spatial transfer functmns m Fig 9 lb)

(a) h-_ (imp set) K, (imp sect rz (visual deg) I, (visual deg) 0, (deg) 8, (deg) c,. (visual deg) x, (visual dcg) SSE

15.3 7.52 0.126 1.129 -0.9 173.2 0.0s 0.63 63.73

16.35 8.39 O.i3i 1.614 136.0 - 76.6 0251 0.342 88.89

ICI

15.13 8.21 0.125 1.732 - I.4 17s 6 0.076 0.076 137.10

(d) IS.59 6.2 0.1% 2.234 I.U.0 - 36.0 0.279 0.279 149.26

Also listed is the sum-squared-error (SSE) betueen the model and the data. Equation (3). the formula for the spatial transfer function of the mod DOG model. was used with these values to generate the continuous curves in Fig. 9(a>-(d). Note that in columns fc) and (d) If, - f?C= 180’ and x, XX,; therefore, the continuous curves in panels IC) and (d) +.+eregenerated by the original DOG model. See text for a discussion of the difference in the values for 0,. f3,$I, and I, given by the two (modified and original) models.

spatial transfer functions of the s-potential responses analyzed previously. A gradient method (Fletcher and Powell, 1963) was used to search for the values of the eight parameters that minimized the sum of the squared error between the data and the model in the complex plane (the data, given as amplitudes with corresponding phases, can be viewed as complex numbers and the range of the function in equation (3) is the complex plane). The best-fitting parameter values and the sum-squared-error are listed in columns (a) and (b) of Table I. As Fig. 9(a) and (b) illustrate, the model describes amplitude and phase data quite well. The mud DOG model When the original DOG model is used to fit Our findings of spatiotemporal coupling and occaamplitude data, only four parameters are necessary: sional spatial asymmetries mean that an adequate the strengths and extents of the center and surround model of the receptive field must deal with amplitude mechanism. The two other parameters, the temporal and phase data simultaneously. One candidate is the phase and position of the center mechanism, are mod DOG model. In the Models section, it was noted determined by phase data which are usually ignored that the mod DOG model is characterized by eight when using the DOG model. Therefore, in practice, parameters: the strengths, extents, temporal phases, the mod DOG model uses twice as many parameters and positions of the center and surround mechas the DOG model. However, there is no increase in anisms. The eight parameters of the modified model the “degree of freedom” in fitting the model to the appear in its spatial transfer function (see the Appendata because the modified model is applied to amplidix for a derivation) tude and phase pairs whereas the DOG model is applied only to amplitudes. Y~,$2nv) = K,(w,) exp[-(ar,v)’ Our previous analysis of phase data demonstrated + i(n/lSO)(f?,(c!&) - 360X
3!

105

Fig. 10. The logarithm of the strength ratio and phase difference between center and surround as functions of temporal frequency for (a) the LGS X-cell of Fig. 3(a) and (b) the X-cell s-potential of Fig. 3(c). Measurements were obtained at stimulus contrasts of 0. I4 (open triangles) and 0.28 (solid triangles). Values for the strengths and the phases were obtained by fitting the data with the mod DOG model.

fit was improved for the 2 Hz data by introducing an offset between center (.Y.,)and surround (s,) positions and for the 12 Hz data by increasing the phase lag between center and surround (0, - 0,) from - 180’ to -212.6‘. The parameters of the surround mechanism tend to be more variable than those of the center mechanism, This is due. in part. to the problem of isolating the surround mechanism. At high spatial frequencies the center mechanism is manifested in relative isolation while at low spatial frequencies the center and surround mechanisms appear together. In Fig. 9 the center mechanism is so well isolated at high spatial frequencies that the DOG model and the modified model gave essentially the same parameter values for the center mechanism (compare Fig. 9(a) with (c) and Fig. 9(b) with (d)]. The spatial amplitude data from Fig. 3(a) and (c) were combined with their corresponding spatial phase data [not shown except for the 2 and 12 Hz spatial phase data in Fig. 6(a) and (b) and reanalyzed using the mod DOG model. In Figs 10. I1 and 12, the results for the LGN X-cell of Fig. 3(a) are given in panel (a), and those for the X-cell s-potential of Fig. 3(c) are given in panel (b). Spuriolemporal coupling. Let us reconsider spatiotemporal coupling with the aid of the mod DOG model. The KS/Kc. ratios and the 0, - 0, differences determined at various temporal frequencies are plotted in Fig. IO as open triangles (0.14 contrast) and solid triangles (0.28 contrast). The results in Fig. IO(a) show two forms of spatiotemporal coupling: (1) the weakening and (2) the lagging of the surround response relative to the center response at high temporal frequencies. The results in Fig. IO(b) exhibit only the latter form of coupling. A third form of spatiotemporal coupling is shown in Fig. I I, In Fig. I I the model parameters rc and ri (the extent of the center and surround) are plotted as circles and squares. respectively, as a function of

:

_A

;

-5

Fig. I I. The extents of the center mechanism (circles1 and surround mechanism (squares) as functions of temporal frequency for (a) the LGK X-cell of Fig, 3(a) and (b) the X-ceil s-potential of Fig. 3(c). Measurements were obtained at stimulus contrasts of 0.14 (open symbols) and 0.28 (solid symbols). Values for the extents were obtained by fitting the data with the mod DOG model.

temporal frequency for 0.14 contrast (open symbols) and 0.28 contrast (solid symbols). In Fig. I I(a), the spatial extent of the center remains constant while that of the surround decreases with temporal frequency. This means that the center mechanism is spatiotemporally separable while the surround mechanism itself is spatjotemporally coupled. In Fig. 1I(b), let us concentrate on the results obtained at the lower contrast (open symbols). The extent of the center mechanism is constant with temporal frequency, so the center mechanism is spatiotemporally separable. The extent of the surround is constant from 3 to 16 Hz. so in this range of temporal frequencies the surround mechanism will appear to be spatiotemporally separable. However. at iow temporal frequencies the extent of the surround decreases, an indication that the surround mechanism is spatiotemporally coupled.

The mod DOG model handles receptive field asymmetry by introducing an ohset betueen .Y, and I,. In Fig. 12(a) the value for s, (squares) are always more negative than for .Y>(circles) except at I Hz and 0.28 contrast. Discarding the 1 and 24 Hz data (the 1 HZ measurements were limited to positivs spatial frequencies and the 24 Hz data uere noisy), the average

Oar

(bl

06

04 02 0

I TEMPORAL

FAEOUENCY

2 3 4 6 8 I2 16 ( HII

Fig. 12. The positions of the center mechanism (circles) and surround mechanism (squares) as functions of temporal frequency for (a) the LGN X-cell of Fig. 3(a) and (b) the X-cell s-potential of Fig. 3(c). Measurements were obtained at stimulus contrasts of 0.14 (open symbols) and 0.78 (solid symbols). Values for the positions were obtained by fitting the data with the mod DOG model.

X-cell receptive fields

offset in visual degrees is -0.183 k 0.083 (SD of an observation). The average offset is about -VYOof the average value for rl. shown in Fig. 1I(a) (n = 8. rr = 0.453 + 0.0 I5 visual degree). The average offset is significantly different from zero at the l?, level for a two-tailed l-test. In Fig. 12(b) the value for .v, is more positive than for .Y,.in 14 out of 16 cases. The average offset is 0.122 f. 0.159 deg: this also gives a r-value which is significant at the 19~ level for a two-tailed t-test. There were 26 retinal (s-potential) X-cells and 22 geniculate (action potential) X-cells in which at least 6 spatial transfer functions were measured. Fitting these data with the mod DOG, we calculated offsets between center and surround positions that were significantly different from zero (jg, level for a two-tailed r-test) in 16 out of the 26 retinal X-cells and in 8 out of the 22 geniculate X-ceils. The offsets, though detectable, were slight-on the order of 0.12 deg. The offsets could be larger since we did not select grating orientations to maximize offsets. SUMMARY

AND DISCUSSION

Phase measurements of responses to gratings of many spatial frequencies drifted in two opposite directions and at many velocities gave information about spatiotemporal coupling, receptive field position, and receptive field asymmetry. The effects on phase of spatiotemporal coupling and receptive field asymmetry appeared at low spatial frequencies. This indicates the involvement of the surround mechanism in producing

these

effects.

Receptive

field

position

could be determined best from phases of responses to high spatial frequencies. This underscores the importance of the center mechanism in relaying positional information. The mod DOG model, given by equation (3). was introduced to aid in the interpretation of the measured spatial transfer functions. The model handles: (I) spatiotemporal coupling, by allowing KJK,., the ratio of the surround to center strength, and 0, - O,, the phase difference between surround and center responses, to vary with the temporal frequency of the stimulus: (2) position, by resolving a -360~~~ component in the spatial phase curve, where .vr is the position of the peak of the center Gaussian and v is spatial frequency; and (3) asymmetry, by allowing an offset between center and surround positions. Analysis of the data with the model demonstrated the occurrence of two previously proposed forms of spatiotemporal coupling-a decrease in the surround to center strength with increasing temporal frequency (Maffei ef al., 1971: Derrington and Lennie, 1982) and an increase in the amount by which the surround response lagged the center response with increasing temporal frequency (Kaplan ef al., 1979; SchweitzerTong et al., 1981; Derrington and Lennie, 1982: Enroth-Cugell ef al., 1983). In addition, the model indicated that the spatial extent of the surround was a function of temporal frequency. These findings

561

profoundly affect our concept of receptive field organization. Since the entire receptive field is spatiotemporally coupled, a natural hypothesis is that the receptive field is the sum of the center and surround mechanisms which are each spatiotemporally separable f(.v,j.. I)=5,(.r.)‘)h,(1)+g,(.T.~)h,(f)

(4)

where g<(_r._t,)and /i,(1) are the spatial and temporal functions. respectively. of the center mechanism and similarly for the surround mechanism. SchweitzerTong ef al. (1981) and Enroth-Cugell ef (11.(1983) have used this premise, and we began with it too. Support for this hypothesis comes from the fact that the extent of the center is remarkably constant with changes in temporal frequency. However, since the extent of the surround depends on the temporal frequency of the stimulus, the surround mechanism is irself spatiotemporally coupled. Therefore, we conclude that the receptive field can be approximated by the sum of a center mechanism that is spatiotemporally separable and a surround mechanism that is spatiotemporally coupled I(s,~,f)=g,.(.u,~)I~,(f)+~;(.\-,~.f)

(5)

wherej;(s,_r, f) is the spatiotemporal function of the surround mechanism. Equation (5) states that the center mechanism is spatially homogeneous in its dynamics whereas the surround mechanism varies from point to point in its dynamics. We will offer further evidence for this new concept of a spatially inhomogeneous surround mechanism in a separate paper. Possible explanations for spatiotemporal coupling of the surround mechanism include membrane cable properties and feedback. For example, if the surround were formed by summation of electrotonic signals in the horizontal cell. cable properties of the cell membrane would suggest that the extent of the surround would decrease as the temporal frequency is increased. In any case, the spatiotemporal coupling of the surround mechanism in X-cells is a significant feature of retinal and geniculate function. Further investigation of this phenomenon is required for an adequate understanding of vision and visual mechanisms. Arkno~~lrclgrmmrs-We thank Robert Soodak for his help. This investigation was supported by National Institutes of Health grants EY 05483. EY 1472, EY 188. EY 1428, MH 15125, and a Research Career Development Award to R. Shapley. REFERENCES Bishop P. O., Burke W. and Davis R. (1962) The interpretations of the extracellular response of single lateral

geniculate cells. J. Physiol. 162, 45 l-472. Brodie S. E., Knight 9. W. and RatlifT F. (1978) The response of the Limulus retina to moving stimuli: a prediction by Fourier synthesis, J. gen. Phpiol. 72,

129-166. Cooper G.

R. and

and Svkm York.

(1967)Merhods o/Signal Holt. Rinehart & Winston, New

;McGillem C. D.

Analvsis.

Derrington A. M?. and Lenme P. (1982) The influence of temporal frequency and adaptation level on receptive field organization of retinal gmglion cells in cat. J. Ph,bxiol. 333, 343-366. Enroth-Cugeli C. and Robson J. G. (1966) The contrast sensitivity of retinal ganglion cells of the cat. J. Ph,r&i. 187, 517-552. Enroth-Cugell C., Robson J. G.. Schwettzer-Tong D. E. and Watson A. B. (1983) Spatio-temporal interactions in cat retinal ganglion cells showing linear spatial summation. J. Physiol. 341, 279-307. Fletcher R. and Powell M. J. D. (1963) X rapid descent method for minimization. Comprrrer 1. 6, 163-168. Hochstein S. and Shapley R. M. (1976) Quantttative analysis of retinal ganglion cell classifications. J. Physid. 262, 233-264. Kaplan E.. IMarcus S. and So I’. T. (1979) Effects of dark adaptation on spatial and temporal properties of receptive fields in cat lateral geniculate nucleus. J. Physial. 294, 561-5SO. huffier S.

W. (1953) Discharge patterns and functional organization of mammalian retina. J. ~V~c~~5ph~~~~~. 16, 37-68. Lee B. 3.. Elepfandt A. and Virsu V. (1981) Phase of responses to moving sinusoidal gratings in cells of cat retina and lateral geniculate nucleus. J. Neurophysiol, 45, 807-817. Lehmkuhle S., Kratz K. E., Mangel S. C. and Sherman S. M. (1980) Spatial and temporal sensitivity of X- and Y-cells in dorsal lateral geniculare nucleus of the cat. J. ,~‘euro~h~sio~. 43, SIC54 t . Maffei L., Fiorentini A. and Cervetto L. (1971) Homeostasis in retinal receptive fields. J. IVeuraplr~sioi. 34, 579-587. Milkman N., Schick G.. Rossetto M., Ratliff F.. Shapley R. and Victor J. (1980) AMethods and designs: a twodimensional computer-controlled visual stimulator. BeJirrr‘.Rev. Merh. Instrut. 12, 283-792. Rodieck R. W. (1965) Quantita&ive analysis of cat retinal ganglion cell response to visual stimuli. t&ion Res. 5, ;83-60 I. Rodieck R. W. and Stone J. (1965) Analysts of receptive fields of cat retinal ganglion cells. J. ~Veurophysiol. 28,

different positions in the receptive field and recording the response at each position. The line spread function. L(X), is measured by sinusoidally modulating the intensity of a line at different positions in the receptive field and recording the amplitude and phase of the response at each position. The subscript 0 refers to the temporal frequency of the sinusoidal stimulus. The line spread function is related to the spatiotemporal impulse function by

In other words. the value of the line spread function at s is given by the value of the (temporal) Fourier transform of the spatiotemporal impulse function. at the temporal frequency cl,,. For a linear system. the spatial transfer function _zZ’~~(<) is defined to be the (spatial) Fourier transform of the tine spread function

where < is the spatial frequency in radians:(visual degree). The spatial transfer function is a complex function of the signed quantity <. Therefore, a complete plot of the spatial transfer function comprises (I) a spatial amplitude curve and (2) a corresponding spatial phase curve over positive and negative <, In Brodie e{ nl. (1978). the spatiotemporal transfer function was developed: it was shown to be the spatiotemporal Fourier transform of the spatiotemporal impulse function

832-849.

Schweitzer-Tong, Robson J. G., Watson A. B.. Shou T. and Enroth-Cugell C. (f981) S~tio-tempo~~ properties of one class of cat retinal ganglion ceils. fnl-esr. Opktkuf. rkwl Sci., Suppl. 20, l4.So Y. T. and Shaolev

R. 119791 Soatial orouerties

of X and

Y cells in the &&al geniculate nucieus’of the cat and conduction velocities of their inputs, Expl Brain Res. 36, 533-550. So Y. T. and Shapley R. (1981) Spatial tuning ofcelis in and around lateral geniculate nucleus of the cat: X and Y relay

cells and perigeniculate interneurons. J. ?v’etrroph_~,~jol. 45, 107.120. Tolhurst D. J. and Movshon J. A. (1975) Spatial and temporal contrast sensitivity of striate cortical neurones. IVuture. Lund. 257, 674-675. Wilson H. (1980) Spatiotemporai characterization of a transient mechanism in the human visual system. Vision RG.

20, 443-42.

APPENDIX

The following treatment deals only with visual stimuli which can be expressed as a function of one spatial variable, X, and time, 1. It is assumed that the visual system is linear, stationary, and continuous, Relationships terkations

among

ofthe

various

spat&temporal

churac-

visual response

The spatiotemporai impulse function, Z(X, t), of a single neuron is measured by flashing a line at

[Brodie et al. (1975) used a different notation. Y(<, w), to denote the spatiotemporal transfer function. Also, they defined the spatiotemporal impulse response as the response of a homogeneous population to a line impulsed at .Y= 0; whereas, we view the spatiotemporal impulse response in terms of the single unit. Consequently, a sign change must be made: for example, in the present notation the velocity of the drifting grating is given by c+>~~:&, whereas in the notation of Brodie et al. (1978) the velocity is given by -w,;&. ] Substituting (Al) into (AZ) and comparing with (A3). we obtain Y ‘,,,,(< 1= ,r,‘. m,f.

(A?)

Equation (A4) states that the spatial transfer function is obtained from the spatiotemporal transfer function by holding the temporal frequency constant; in other words, the spatial transfer function is a cross-section of the spatiotemporal transfer function. S_vmmetric

receptice jieieki

Assume that the receptive field has even symmetry. If the axis of symmetry is placed at s = 0. then I (.Y,I J = I f - .Y.I).

(As)

X-cell receptive

From symmetry properties of the Fourier transform (see Cooper and iMcGillem, 1967) we get 44,(C) = YJ-<).

(A6)

From (A6). one sees that if the receptive field has even symmetry and is centered at x = 0, the spatial transfer function has the same value at -< as it has at <. This means that the amplitude and phase spectra are even functions of < as shown in Fig. 4(d).

in Fig. 4(a) retained its point symmetry while that in Fig. 4(d) lost its even symmetry. A modified diffkrence of Gawians model A simple model that incorporates the possibility that the ratio between center and surround strengths and the difference between center and surround phases may depend on the temporal frequency w, is an impulse function given by

Spatiotemporal separability

I(.~,t)=g,(.~)h,(t)+g~(~~*)h,(f)

Assume that the spatiotemporal impulse function can be factored into spatial and temporal terms I(& t)=g(x)h(t).

(A7)

Combining equations (A3), (A4) and (A?) and rearranging, we obtain the spatial transfer function $400(<) = G (5) ff (wJ

(A@

where

H (4

X0(<) = G,(;‘) H,(%) + G,(Z) If,(%) e-““g (x) d.u

(A8.1)

edi**‘h (t) dr.

(A8.2)

J --z x =

J0

(A9)

where g,(x) and g,(r) are the spatial profiles of the center and surround mechanisms, respectively. and h,(r) and h,(t) are the time courses of center and surround impulses, respectively. The model of Enroth-Cugell et al. (1983) is also based on the premise given by (A9). Combining (A3), (A4) and (A9) and rearranging, we obtain the spatial transfer function of (A9).

Lz G(5)=

563

fields

(AIO)

where G,(C), G,(r), Hc(wO).and IY,(w,) are defined by equations similar to (AS.!) and (A8.2). Defining A,(o)J expIi(x/lSO)

O,(w,)] = H,(w,)

(Al 1)

In other words, the spatial transfer function is and similarly for subscript s, we can rewrite (AIO) as equal to the Fourier transform of g (x) multiplied by -K%(t) = G,(s’) A,(Q+J expli(nl~80) 9,(&f the complex number N(o,). Since g(x) is a real + G,(S) A,(w,)exp[i(n;180)8.~(wo)l6412) function of x, its Fourier transform has a spatial amplitude curve which is an even function of < and where 6,(w,) and Q,$(@ are the phases (in degrees) of a spatial phase curve which is an odd function of ?j the center and surround responses, respectively, at (Cooper and McGillem, 1967). The multiplication of temporal frequency w,. Therefore, the impulse functhe Fourier transform of g (s) by a complex number tion (A9) gives rise to spatial transfer function (A 12) means that the spatial amplitude curve is translated which consists of a center spatial term G,(t) and a vertically along a logarithmic amplitude axis and the surround spatial term G,(c) having a relative weightspatial phase curve is translated vertically along the ing and a phase difference which is dependent on aO. phase axis; the result is that the spatial amplitude We assume that the center and surround spatial curve is an even function of < and the spatial phase profiles are Gaussians which are not necessarily curve has a point of symmetry around the phase centered at the same point point at zero spatial frequency even if the receptive g,(x) = k, exp ( --KY - .~,)/r,]~) (A 13.1) field is asymmetric as shown in Fig. 4(a). g,(r) = k,exp { -[(.Y - .r,)/r,]‘~

Translation in space

If a spatiotemporal impulse function I(.Y, f) is shifted +.Y,, units, the spatial transfer function of the shifted impulse function 1(x -x0, t) is given by e-‘cY P%(r) (Cooper and McGillem, 1967). In other words, the spatial transfer function of f(s -x0, t) is obtained by multiplying the spatial transfer function of 1(x, t) by e-IEro. Since e-‘:Q has unit amplitude, translating the spatiotemporal impulse function in space has no effect on the spatial amplitude curve of the spatial transfer function. On the other hand, a translation of the impulse function by _r, units means that -<.r,, must be added to the spatial phase curve. The effects of imp&se function translations on the spatial transfer functions of Fig. 4 are illustrated in Fig. 7. Since -<-Q is an odd function of <, the spatial phase curve

(A 13.2)

where r, and rr are the spatial extents of the center and surround mechanisms, respectively, and .r, and X, are the locations of the peaks of the center and surround Gaussians, respectively. The resulting spatial transfer function is .Y%(Zxv) = K,(w,) exp[-(nr,v)’ f i(7~/180)(0~(w,,)- 36Ox,~)] f K&0,) exp[ - (nr,v)’ + i(n/lgO)(0,(o,)

- 36Ox,v)]

(A14)

where (A14.1) (A14.2) (A 14.3)

Units of measurement

are

[:I = radians (visual degree) (.415.l) (A15.1) 11.1= cycles (visual degree) [wfl]= radians second (.415.3) Is,] = dimensionless (Alj.4) [X-J= dimensionless (Al5.5) [Ec,(c&] = (response units)(stimuius unit) (A15.6) (A15.7) [r,l = visual degrees [8,(w,)] = degrees (Alj.8) [X,]= visual degrees (Alj.9) [A,(w,)] = (response units)~~~stimulus unit) x (visual degree)) (A15.10)

and similarly for subscript s. At a given temporal frequency (c?~"fixed). equation (Al-l) has eight parameters: KC(w,,). K,(w,,). t)~(w)). ~),(cJ~), r, . r,, .y and .r,. The original DOG model is obtained b;, setting .‘I,= .r and ~~c(c~J,JieXp[i~n lXO)U,(c~~~)] = - rl,(w,,) exp[i’(rr 180) 0; (w,,)]: equation (Al-t) becomes -ii:,,~(2n~) =

(k,r,

X/i

k >r,\ I: exp[-(nr>v):])

exp[-(ilr.v)l] &(~,,)exp[i(n

180)0‘((!~,,)]. (A16)