Ординатура / Офтальмология / Английские материалы / Seeing_De Valois_2000

54 Larry N. Thibos

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56 Robert Shapley

that measurements of the receptive field provide a clear picture of the spatial mapping of receptor inputs to a sensory neuron. Once one knows the receptive field, so it has been thought, one can predict how the neuron will react to an infinite variety of stimuli. It is important now to realize that this is an implicit commitment to a particular view of how the nervous system works.

The concept of a visual receptive field originally came with the implication that the neuron combines signals by linear summation, and that its inputs are coming via a feedforward neural network. As I aim to show, departures from this linear feedforward conceptual framework require stretching the idea of “receptive field” until it hardly means anything definite at all. In the application to cortical neurophysiology, where nonlinear mechanisms abound, the receptive field concept begins to break down until there is nothing left of the original idea, though the term is still used. In the description and analysis of many cortical cells, it is my belief that one needs to change the analytical framework and abandon receptive fields as a conceptual tool in favor of nonlinear dynamical systems theory. My aim in this chapter is to make the case for a new organizing principle—the neural network as a spatiotemporal, dynamical system. Some illustrations will come later, when we discuss the major findings of cortical neurophysiology.

One should also ask, before embarking on a detailed review of the visual receptive fields of single neurons, what is the use of this concept in understanding perception? This is a very di cult question to answer at this point in time. The main positive outcome one can identify is that the nature of receptive fields can be related to limits of perception. For example, the spatial scale of the smallest receptive fields is believed to be a most important factor in limiting spatial resolution (see for example, Hawken & Parker, 1990; Parker & Newsome, 1998). Second, if it were true that the spatial layout of the receptive fields of simple cells in the cortex determined their orientation tuning selectivity, then the limits of angular discrimination could be explained in terms of receptive field structure (cf. Vogels & Orban, 1991). This second example is problematic, however, as discussed below. Another possibly important consequence of understanding neuronal receptive fields is that it could give insights into how the brain works, how it represents the sensory world, how it decides to act based on evidence. That is, in studying and learning to understand how receptive fields work, we might learn about fundamental rules of structure– function correlation, or synaptic physiology, that generalize to all aspects of brain function. However, if it is true that the analytical framework for understanding the neural network of the cerebral cortex requires a new organizing principle, then the receptive field concept will turn out to have been only a first step in the direction of understanding the true complexity of the brain. Before we can judge, we must go through the evidence and the explanations together.

II. RECEPTIVE FIELDS OF RETINAL GANGLION CELLS

The original idea of visual receptive fields came from H. K. Hartline in his studies of the visual properties of retinal ganglion cells in the frog retina (Hartline, 1940).

This was an idea Hartline derived from E. D. Adrian, who in turn had used a term invented by C. S. Sherrington (cf. the scholarly review of the history of retinal ganglion cell physiology by Enroth-Cugell, 1993). The frog ganglion cells from which Hartline recorded had no maintained discharge of nerve impulses in the absence of stimulation, so the only nerve impulses they fired were either responses to spots of light that Hartline flashed on the frog’s retina, or responses to the turning o of the light. The area on the retina from which he could evoke nerve impulses from a ganglion cell Hartline termed the receptive field of that ganglion cell. Figure 1, a reproduction of Hartline’s classic figure, shows the compact, approximately elliptical receptive field of a frog’s retinal ganglion cell.

Later work by Stephen Ku er on cat retinal ganglion cells (Ku er, 1953) required a modification of Hartline’s definition of receptive field because cat ganglion cells have a maintained impulse rate in the absence of stimulation. The modi- fied definition was that the receptive field was the region on the retina where stimulation could cause a modulation of the impulse rate around the average level of firing in the absence of stimulation.

It is also important that because of the maintained activity, Ku er could observe stimulus-dependent modulations in spike rate above and below the maintained activity. For this reason, Ku er was the first to recognize center-surround antagonism in the cat retinal ganglion cell: stimuli in the periphery of the receptive field caused a response modulation of the spike rate opposite in sign to that evoked by central stimuli (Ku er, 1953).

FIGURE 1 The original visual receptive fields (Hartline, 1940). These are maps of the threshold response contours from two di erent retinal ganglion cells of a bullfrog, R. catesbiana. (a) The data are from an “on–o ”retinal ganglion optic nerve fiber, the activity of which was recorded with a wick electrode after microdissection of the nerve fiber on the surface of the retina in an excised eye. The log units of attenuation of the light source are drawn on the threshold contour. The unattenuated light source had a luminance of 2 * 104 cd/m2. The exploring spot had a diameter of 50 m on the retina. It was flashed on from darkness for several seconds. (b) Similar experiment on an “on” type frog ganglion cell. The central shaded region is where a sustained “on” response was obtained.

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From the beginning, it was clear that the receptive field was not an invariant characterization of the spatial, visual properties of a neuron. This is because, as defined, it varies in size with stimulus size, intensity, color, and any other stimulus variable that determines the e ectiveness of the stimulus in exciting the neuron. Hartline stated explicitly that the receptive field varied in size with size or intensity of stimulus spot (Hartline, 1940). For example, if one uses a stimulus spot with light intensity 1 unit, and then a second stimulus spot with light intensity 100, the receptive field mapped out with the second spot may be ten times bigger (or perhaps more!) than the field mapped out with the first spot. However, from his work on spatial summation, it is reasonable to suppose that already in the 1930s, Hartline had the idea of spatial invariants of the visual properties of the ganglion cells he studied. These are the spatial distributions of sensitivity. For instance, in Figure 1, the closed curves connect points in the visual field of equal sensitivity because there are the boundaries at which a threshold response was elicited with the stimulus intensity indicated in the figure. If one defines sensitivity as 1/threshold intensity, these are thus equal sensitivity contours. A collection of equal sensitivity contours, one for each intensity of stimulus spot, will trace out the 2-dimensional (2-D) distribution, a surface, of sensitivity for the neuron studied. Hartline further predicted the sensitivity for a compound stimulus, which was the sum of two simpler stimuli. He found reasonably good agreement with linear summation of sensitivity (Hartline, 1940), the first result consistent with the concept of linear spatial summation weighted by the spatial sensitivity distribution.

A. The Two-Mechanisms Model: Center and Surround

Rodieck (1965) was the first to state explicitly the idea of spatial distributions of sensitivity and the linear combination of signals weighted by the sensitivity distributions. Rodieck and Stone (1965) demonstrated that the way in which to extract the spatial distributions of sensitivity was to map sensitivity with small spots, just as Hartline did originally.They presented their data as 1-D slices, termed sensitivity profiles, through the 2-D sensitivity distributions. But then they went further to account for the measured sensitivity profiles with a two-mechanism model, the Di erence of Gaussians (DOG) model (Rodieck, 1965). Figure 2 (Rodieck, 1965) is a drawing of the DOG model for a cat retinal ganglion cell’s sensitivity profiles. The two mechanisms were the receptive field Center mechanism, and the receptive field Surround mechanism. Let us call Sc(r) the Center’s sensitivity distribution as a function of position relative to the receptive field middle, and Ss(r) the Surround’s sensitivity distribution. Each of these mechanisms was assumed to be well fit by Gaussian function of position, so that

Because total sensitivity, S(r), is the di erence between Sc(r) and Ss(r), this is a DOG model. In these formulae, kc is the peak local sensitivity of the Center; ks is the peak local sensitivity of the Surround; c is the spatial spread of the Center, the distance from the receptive field middle at which sensitivity declines by 1/e; and s is the spatial spread of the Surround.

Rodieck (1965) also included a temporal response component, making his DOG model truly a spatiotemporal model. He assumed that the step response of the Center mechanism was a delayed (delay tdc) sharp transient followed by an exponential relaxation to a steady state value, and that the Surround temporal response was similar with a similar time constant of relaxation but with a longer delay to response onset, tds.

Ss(r,t) 0; when t tds

FIGURE 2 The DOG model of the receptive field of cat ganglion cells (after Rodieck, 1965). Two mechanisms overlap in space: a Center Mechanism and an antagonistic Surround Mechanism (both indicated with solid lines). Each mechanism has a sensitivity as a function of position that is a Gaussian function (see text). Their summed response (the DOG or Di erence of Gaussians) produces the response of the retinal ganglion cell, and it is drawn as a dotted line. Each mechanism has a spatial sensitivity distribution that is a Gaussian function of space. The spread of the center is smaller than that of the Surround. In this drawing, the line-weighting function of the model is depicted: the response as a function of position for a thin line stimulus. The Center and Surround are set equal in summed e ect on the cell, and the surround is drawn as three times wider than the Center.

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Rodieck (1965) was explicit that his model was a linear model of the receptive field of cat retinal ganglion cells. The point of making such a model was to synthesize (and therefore to predict) the response to arbitrary stimuli by means of convolution, a standard technique of linear systems analysis. He applied this conceptual tool to synthesize the ganglion cell responses to drifting bars. Convolution means, basically, linear summation of the e ects of the moving bars, point by point within the receptive field. The qualitative agreement between theory and experiment supports the notion of approximate linear summation within the retina (at least for the neurons Rodieck studied).

One fundamental idea of the DOG model for receptive fields is that the Center and Surround mechanisms are feed-forward, converging with their separately computed signals onto the ganglion cell in parallel.This idea is consistent with many of the data on signal summation of center and surround signals in the cat retina (Enroth-Cugell & Pinto, 19792; Enroth-Cugell et al., 1977).

The work of Enroth-Cugell and Robson (1966) is a more elaborate study of linearity and receptive field properties of retinal ganglion cells. These authors introduced the idea of spatial frequency analysis of the sensitivity distributions of a retinal ganglion cell’s receptive field. Using Rodieck’s DOG model, they were able to account for the shape (both qualitatively and quantitatively) of the spatial frequency response of many cat retinal ganglion cells (the X cells). Figure 3, a summary figure from their Friedenwald lecture (Enroth-Cugell & Robson, 1984), illustrates (in b) the fact that the spatial spread of the receptive field Center is much smaller than

FIGURE 3 Spatial frequency analysis of the receptive field (Enroth-Cugell & Robson, 1984). (a) Spatial frequency response functions for Center, Surround, and their sum (the ganglion cell’s net response). The Center’s spatial frequency response is drawn with small dashes, the Surround’s with broader dashes, and the sum is the solid curve. Three di erent illustrative spatial frequencies are indicated as a,b,c; respectively, these are in the Center-Surround antagonistic regime, at the peak of the cell’s spatial frequency response and at a high spatial frequency at which the Center’s response is unopposed by Surround response. (b) The DOG model in the space domain, with spatial sinusoids at spatial frequencies a, b, c also drawn.

that of the Surround. This (as illustrated in a) causes the spatial frequency range of the Center to be much broader than the spatial frequency range of the Surround. In other words, the spatial frequency resolution of the Center exceeds greatly the resolution of the Surround, so that in these cells at high spatial frequency, the response comes from one mechanism only: the Center mechanism. At lower spatial frequencies, the response is a linear subtraction of the Surround’s response from the Center’s. The typical bandpass shape of the spatial frequency response is thus accounted for by the Center–Surround model and linear spatial summation.

An important technical simplification that enables one to “see” the spatial frequency consequence of the DOG model is the following: the Fourier Transform of a Gaussian function of spatial location is a Gaussian function of spatial frequency. The standard deviation of the Gaussian of spatial frequency is the reciprocal of the standard deviation of the spatial Gaussian.Thus the defining equations for the DOG model become, in the dimension of spatial frequency, v:

This set of equations {C} are what is illustrated graphically in Figure 3. Although the results of Rodieck and Stone (1965), and Enroth-Cugell and Rob-

son (1966), are impressive in their rigor and predictive power, it is worth being critical and worth asking how many of their results are crucial for understanding receptive fields. For instance, is it important that the DOG model uses Gaussian functions to account for receptive field mechanisms? What is the significance functionally to the nervous system of linear summation within the receptive fields?

Later work has revealed that the Gaussian shape of a receptive field mechanism in the DOG model is only an approximation to the true spatial profile. In the case of the receptive field center of X cells in the cat retina and lateral geniculate nucleus (LGN), the overall sensitivity profile of the center mechanism appears to be the envelope of a number of smaller subprofiles, each of which is smaller than the total center (Soodak, Shapley, & Kaplan, 1991). Furthermore, the receptive field surround mechanism seems to be constructed out of multiple subregions, each of them bigger in spatial extent than the receptive field center (Shapley and Kaplan, unpublished results). However, the essence of the DOG model seems to be the idea that you will obtain a concentric Center–Surround organization even if the antagonistic Center and Surround mechanisms overlap in space, as long as the Surround is larger in spatial extent than is the Center mechanism. And this basic idea has been confirmed unequivocally in studies of Center–Surround organization of coloropponent retinal ganglion cells and LGN cells of the monkey visual system (Reid & Shapley, 1992). In monkey ganglion cells of the P-type, the M(530 nm) cones and the L(560 nm) cones are almost always connected with opposite signature to

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the ganglion cell, so that, for example, a cell may be M , L , or M , L . The M cone input may be to the smaller Center mechanism, or to the broader Surround mechanism. Reid and Shapley (1992) used spatial stimuli for receptive field mapping that were also cone-isolating (so that only, say, M cones were mapped out in one experimental run; only L cones in a subsequent run). As shown in Figure 4, they found that, for example, an M cone Surround always overlapped in space an L cone Center; in general, Surround and Center always overlapped in space. This is strong confirmation of the basic idea of the DOG model of Center–Surround receptive fields.

The question of the functional significance of linearity is much more general, and there is no definite answer to this question, though candidate answers have been proposed. I believe the biological evidence is very convincing that there must be some important advantage to linear summation because the retina goes to some trouble to produce it. It is known that until one reaches the retinal ganglion cells, retinal intercellular signaling is by slow potentials not spike trains. The slow potentials in bipolar cells must be a linear function of the receptors’intracellular potentials in order for linear signal summation to work. This implies that the synapses between photoreceptors and bipolar cells must be linear transducers, and this makes them special synapses. How this is done is still an outstanding problem in synaptic biophysics. Nevertheless, that it is done seems to imply that there must be an important functional reason for the nervous system to depart from its default synapses (which are quite nonlinear). Speculations for the perceptual reason for preserving linearity are concerned with the accuracy of the computation of brightness, color, and direction of motion, all of which require precise computations across spatial locations.

B. A Third Mechanism: Nonlinear Subunits

That the nervous system does not always preserve linear signal summation in the visual pathway is indicated by another major finding in the Enroth-Cugell and Rob-

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FIGURE 4 Receptive field map of a macaque Parvocellular lateral geniculate nucleus (LGN) cell with cone-isolating stimuli (Reid & Shapley, 1992). The response was measured by reverse correlation with a spatiotemporal m-sequence. This gives a map of locations that excited the neuron under study, as a function of time. Here the time of peak response was chosen (at 48 ms after a stimulus) and the spatial sensitivity distributions plotted for this peak time. When an increment of intensity causes excitation of the neuron, the picture element is made brighter than the mean gray; when a decrement causes excitation, the pixel is made darker than mean gray. Strength of excitation is encoded as amount of lightness or blackness of the pixel. Three di erent spatial distributions are shown. (a) The spatial sensitivity map for black/white stimuli. This corresponds to the Center-Surround maps of Ku er (1953). This cell was excited by decrements in its Center, so it would classically be called an “o ” center neuron. (b) The spatial sensitivity map for L cone-isolating stimuli. The pixels in the stimuli were shades of red and green that were equally e ective for M and S cones, so only L cones responded. This is the map of L cone input to the neuron. (c) The spatial sensitivity map for M cone-isolating stimuli. The pixels in the stimuli were shades of red and green that were equally e ective for L and S cones, so only M cones responded. This is the map of M cone input to the neuron.