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TOPICAL REVIEW |
Departments of Psychology and Neurobiology, Jules Stein Eye Institute and Brain Research Institute, David Geffen School of Medicine, University of California, Los Angeles, Los Angeles, CA 90095, USA
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(Received 1 April 2004;
accepted after revision 17 May 2004;
first published online 21 May 2004)
Correspondence D. L. Ringach: Departments of Psychology, Neurobiology, Jules Stein Eye Institute and Brain Research Institute, David Geffen School of Medicine, University of California, Los Angeles, Los Angeles, CA 90095, USA. Email: dario{at}ucla.edu
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Introduction |
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Hubel and Wiesel pursued a similar line of research when they first started recording from cat visual cortex. In one of their early experiments, they noticed that a cell would discharge in response to a moving line shadow cast by the edge of a slide as it was inserted into the ophthalmoscope (Hubel & Wiesel, 1998). They soon realized that the cell would fire only when the line was oriented within a narrow range. Further measurements confirmed that many other cells were also selective to the orientation of boundaries. Thus, the discovery of orientation tuning – perhaps the major transformation in the organization of receptive fields in the early visual pathway – was accidental. Since that discovery much research has focused on whether there are principled ways of determining those ‘features’ of the environment to which sensory neurones are selective.
By flashing orientated lines at various locations along the receptive field, Hubel and Wiesel classified cortical neurones into two distinct groups: simple and complex. Simple cells were defined as those whose receptive fields could be divided into separate ‘on’ and ‘off ’ subregions (Fig. 1A), while complex cells were defined by exclusion. They also introduced the concept of a hierarchical organization of receptive fields. According to this proposal, simple cells are constructed first from the convergence of geniculate receptive fields aligned in space to produce the observed elongation of ‘on’ and ‘off ’ subfields (Fig. 1B), and complex cells are subsequently constructed from the convergence of simple-cell receptive fields with similar orientation tuning but varying positions (or spatial phases) to generate a field of ‘on–off ’ responses (Fig. 1C). As we discuss below, this initial description of two neuronal classes in primary visual cortex and their associated hierarchical circuitry have shaped both experimental and theoretical studies of cortical function.
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The linear–non-linear model and the reverse correlation method
Arguably, one of the first influential concepts was the idea that some visual receptive fields could be viewed as a spatio-temporal filter acting on the stimulus followed by rectification (the so-called linear–non-linear model), which originated in the study of retinal ganglion cells (Rodieck & Stone, 1965a,b; Enroth-Cugell & Robson, 1966). Rodieck and Stone successfully used the model to explain the responses of retinal ganglion cells to a variety of static and moving stimuli from the measurements of neural responses to flashing dots. This work yielded an understanding of ganglion cell receptive fields as a linear combination of a centre and surround mechanism that was delayed in time. Similar ideas were, in fact, implicit in the earlier work of Kuffler, Hartline, Barlow and coworkers (Barlow, 1953; Kuffler, 1953; Hartline et al. 1956; Hartline & Ratliff, 1958). For example, Kuffler discussed the origin of transient responses as a combination of ‘on–off ’ delayed mechanisms, so he was well aware of the importance of spatio-temporal inseparability of the receptive field (Kuffler, 1953).
Mathematically, the model can be described as follows. The visual stimulus, defined as the spatio-temporal distribution of luminance across the receptive field, is summarized in a vector x(t) representing the stimulus within the time interval (t – T, t). The assumption is that the neurone has finite memory, so that the response at time t is not influenced by stimuli at times t' < t – T. Normally, in primary visual cortex, a selection of T
300 ms ensures that all the relevant history of the stimulus is encoded in the vector x(t). The linear–non-linear model postulates that the spike train is an inhomogeneous Poisson process with instantaneous mean rate given by 
(t) =
(wTx) (Hunter & Korenberg, 1986; Chichilnisky, 2001; Nykamp & Ringach, 2002; Marmarelis & Marmarelis 1978). Here, wTx describes the linear operation of the filter, and (•) is a static non-linearity (typically monotonically increasing).
One efficient way to measure the ‘kernel’, w, is by cross-correlation with spatio-temporal white noise (see Chichilnisky, 2001 for a recent review). To the best of my knowledge, the first recording of the full spatio-temporal kernel of a simple cell in cat visual cortex was reported by Erich Sutter in 1975 using a rather elegant apparatus (Fig. 2A and B) (Sutter, 1975). Because this work appears to be relatively unknown, it is appropriate to briefly describe it here. The stimulus was produced by feeding white noise to the position, (x,y), and intensity, z, inputs of an oscilloscope. These signals were generated by playing the different tracks of a single tape (Fig. 2A, left recorder). Sutter arranged a second tape recorder (Fig. 2A, right recorder) so its head was a distance d apart from the first tape recorder. This second recorder received the tape from the first one and recorded the signals arriving from a microelectrode on another track of the tape. Thus, the temporal relationship between the stimulus and the response was preserved. At any one location on the tape, one can find recorded the neural response and the stimulus that preceded it by d/v seconds. Here, d is the distance between the heads of the tape recorders and v is the linear velocity of the tape.
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d, which would represent a time lag of 
=d/v. Once a time lag is selected, the stimulus is re-played on the oscilloscope such that the location signals (x,y) were unchanged, but the intensity signal was multiplied by a brief pulse sequence indicating the presence or absence of a spike in the response audio track. The stimulus was therefore multiplied by the response, and the resulting images were added photographically by having a camera with an open shutter pointed at the oscilloscope screen. One picture was taken for each t and thus the entire spatio-temporal receptive field was computed. This initial study showed the feasibility of applying such methods in visual receptive fields. A careful analysis of their shapes in a large population of cells came at a later stage. Simple receptive fields have Gabor-like shapes, but not all Gabor-like shapes are receptive fields
Movshon et al. (1978) first demonstrated that simple-cortical cells exhibit spatio-temporal summation within their receptive fields. They observed that the spatial profile of the receptive field compared well with synthesized profiles resulting from measurements in response to drifting sinusoidal gratings, indicating that simple-cells showed linear summation up to the spike rectification stage. Jones and Palmer used sparse stimulation with bright and dark ‘dots’ to measure receptive fields of simple cells in cat area 17 (Jones & Palmer, 1987b). By cross-correlating the evoked spikes and with the positions and times of occurrence of the stimuli, they estimated the spatial impulse response at a fixed time lag (50 ms). Some of their original measurements are shown in Fig. 2C. This study provided the first measurements of simple-cell receptive field measurements over a population of neurones and established that their shapes were well approximated by two-dimensional Gabor functions (Jones & Palmer, 1987a). Using a similar method, the full spatio-temporal kernel in cat area 17 was measured by DeAngelis et al., who also studied their development in kittens (DeAngelis et al. 1993b).
In macaque primary visual cortex, I used a modification of the reverse correlation method where the receptive field is probed with a fast sequence of gratings and found that the receptive field shapes were similar to those seen in cat (Ringach, 2002). Figure 2D shows that the distribution of these shapes in monkey (open circles) and cat (crosses; data from Jones & Palmer, 1987a). The parameters nx and ny provide a measure proportional to the width and length of the receptive field, respectively, in units of the period of the grating in a two-dimensional Gabor fit to the kernel. Blob-like receptive fields are mapped to points near the origin. Receptive fields with a number of elongated subfields are mapped to points away from the origin. The distribution of (nx, ny) appears to lie, approximately, on a one-dimensional curve and cat and monkey data are comparable. The results indicate that a particular family of filter shapes is present in primary visual cortex. Furthermore, current theories of simple-cell receptive fields (Olshausen & Field, 1996; Bell & Sejnowski, 1997; van Hateren & Ruderman, 1998), based on ‘optimal’ linear representations of the image, fail to account for this distribution (Ringach, 2002). Finding out what is special about this family from a computational point of view may yield clues about their function.
The importance of the output non-linearity and its measurement
Reverse correlation provides a measure of the front-end filter (Chichilnisky, 2001). However, it should be clear that the tuning properties of a linear-nonlinear (LN) system (such as its orientation or spatial frequency bandwidth) do not depend solely on its linear kernel; they will also be influenced by the static non-linearity in the spike generation mechanism, represented by (•). For example, it has been suggested that both direction and orientation tuning can be sharpened significantly by thresholding or by an accelerating non-linearity (Reid et al. 1987, 1991; Jagadeesh et al. 1993; DeAngelis et al. 1993a,b; Anzai et al. 1999). Thus, estimating (•) is clearly important when comparing the tuning properties of the model to tuning curves obtained with drifting sinusoidal gratings.
One strategy to measure the output non-linearity in the LN system is to perform a two-step analysis (Anzai et al. 1999; Chichilnisky, 2001; see also Hunter & Korenberg, 1986). First, measure the linear kernel using reverse correlation, then generate a scatter plot of the linear prediction wTx against the actual response and smooth it to obtain an estimate of the static non-linearity. If we know that the non-linearity can be approximated by some functional family, such as a half power-law rectifier, then a more efficient method can be used where the parameters of the non-linearity are found by matching the input–output moments of the model to the data (Nykamp & Ringach, 2002).
Spatio-temporal inseparable receptive fields and direction selectivity
The concept of the receptive field as a spatio-temporal entity was instrumental in advancing our knowledge of receptive field function (Adelson & Bergen, 1985; Reid et al. 1987; Emerson et al. 1992; DeAngelis et al. 1995). First, it clarified that ‘on’ and ‘off’ responses at various locations of the receptive field could be interpreted within the framework of the LN model (DeAngelis et al. 1995). Second, it demonstrated how non-trivial computations, such as direction selectivity, may arise from linear mechanisms if the receptive field is inseparable in space and time (Reid et al. 1987, 1991).
A receptive field is spatio-temporal separable if it its structure can be written as a product of a spatial and temporal function, w(x,y,t) =h(x,y)g(t). Spatio-temporal inseparability of receptive fields was first shown to be involved in the generation of direction selectivity by measuring the response of the cells with contrast reversing gratings as a function of spatial phase (Enroth-Cugell & Robson, 1966; Hochstein & Shapley, 1976a,b; Reid et al. 1987). Full measurements of the spatio-temporal receptive fields in both cat and monkey confirmed these early reports and showed a characteristic tilt of ‘on’ and ‘off’ subregions in the space–time plane (DeAngelis et al. 1993a,b). Such tilt in the space–time plane endows the neurone with an asymmetric receptive field that causes the response to be larger when the stimulus moves in one direction than the other. Experimentally, the linear prediction of direction selectivity matches very well the preferred direction of the neurone but underestimates its magnitude by about 1/3 (Reid et al. 1991). Intracellular measurements (Jagadeesh et al. 1993) have demonstrated that linearity holds very well when one considers the membrane voltage of the neurones, but that thresholding contributes significantly to making the spike responses better tuned than the intracellular voltage demonstrating, once again, the importance of the output non-linearity in the LN system when predicting the tuning properties of spike responses (but see Baker, 2001).
The gain control model
One of the recent advances in the field has been the realization that the LN model of simple cells fails to account for a number of phenomena (Albrecht & Geisler, 1991; Geisler & Albrecht, 1991; Robson, 1991; Heeger, 1992; Carandini et al. 1997; Tolhurst & Heeger, 1997). First, measurements of response as a function of orientation show saturation at different levels. This would not be expected from a system where the maximum spike rate saturates due to a fixed output non-linearity. Second, the responses of neurones are suppressed by stimuli that, by themselves, do not cause the cell to fire. A well-documented example is the suppression caused by adding a stimulus orientated orthogonally to the preferred orientation of the cell, a phenomenon referred to as cross-orientation inhibition (Morrone et al. 1982, 1987; Bonds, 1989). Third, in response to drifting gratings, the response to increasing contrast does not simply scale but advances in time. In other words, the responses become faster, while a LN model predicts no change in the temporal structure of the response. Fourth, the spatial summation of cells changes with the contrast of the stimulus (Polat et al. 1998; Kapadia et al. 1999; Sceniak et al. 1999); the higher the contrast, the smaller the degree of spatial summation. A LN model would predict no change in spatial summation with contrast. Albrecht et al. (2003) provide a recent review of these and other non-linear response properties of cortical neurones.
A number of investigators proposed to extend the LN model by adding a gain control mechanism (Albrecht & Geisler, 1991; Bonds, 1991; Heeger, 1992; Carandini & Heeger, 1994; Tolhurst & Heeger, 1997; Carandini et al. 1997). The idea is that the output of the linear filter is divided (or ‘normalized’) by the overall activity in a population of cortical cells that represent the ‘normalization pool’ (Fig. 3A). The model attributes the selectivity for orientation entirely to the linear filter; it is only the gain that is determined by the normalization signal. The gain control model explains saturation because the activity of the local population, and therefore the normalization signal, increases with contrast. It also explains cross-orientation inhibition because the normalization signal includes signals that are also tuned to the orthogonal orientation. It has also been recently extended to include the surround from the classical receptive field to explain changes in spatial summation as a function of stimulus contrast (Cavanaugh et al. 2002). However, it is likely that a single gain control mechanism may be insufficient to explain the change of contrast–response curves in a variety of suppression phenomena (Sengpiel et al. 1998; Carandini et al. 2002).
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0 for all x (in which case the matrix H is called a positive semidefinite matrix), 
is the semisaturation constant, and (•) is a static non-linearity usually selected to represent a power-law rectifier: (x) =xß if x > 0 and zero otherwise. A full identification of this model requires that we estimate the linear filter w, the semisaturation constant , the exponent ß, and the matrix H. The gain control model is attractive because it explains a set of interesting phenomena in a parsimonious way (Carandini et al. 1997; Cavanaugh et al. 2002). Furthermore, Simoncelli and colleagues have put forward an interesting theoretical framework for gain control (Simoncelli & Olshausen, 2001; Schwartz & Simoncelli, 2001). Within this framework gain control works to increase the degree of independence between neural responses when the system is stimulated with natural signals. It is also worth noting that mechanisms for gain control were also described in classes of retinal (Shapley et al. 1972; Shapley & Victor, 1978, 1979) and geniculate neurones (Kaplan et al. 1987; Purpura et al. 1988; Benardete et al. 1992; Benardete & Kaplan, 1999). One should consider that at least part of the cortical effects may originate from non-linearities in the LGN inputs (see discussion in Carandini et al. 1997).
Gain control and intracortical sharpening of tuning
The gain control signal appears to be broadly tuned for orientation, spatial frequency and temporal frequency (DeAngelis et al. 1992). It has often been assumed that, because of its broad tuning, the gain control signal cannot sharpen the tuning conferred by the linear filter. This is not entirely correct, and Fig. 3B provides an example of how a gain control signal that is untuned in orientation and low-pass in spatial frequency can enhance the tuning of the neurone in the Fourier domain (Ringach et al. 2002). The leftmost panel illustrates the spatial kernel of a Gabor receptive field, the two panels on the right show the tuning of the neurone in the Fourier domain in two conditions: with and without gain control. Clearly, the tuning of both spatial frequency and orientation can be enhanced because the gain control signal is ‘carving away’ the responses near the origin in the Fourier plane (Ringach et al. 2002). The result is that the most responsive region in the Fourier domain is shifted away from the origin relative to the response of the linear filter alone. The angular extent of the response enhancement region is reduced. Thus, in cases where the original filter has significant power at low spatial frequencies, gain control can have the net effect of enhancing both spatial frequency and orientation selectivity.
Examples of tuning in the Fourier domain for three macaque V1 cells are shown in Fig. 3C. Over the population, such measurements show a correlation between the degree of suppression and tuning in both orientation and spatial frequency, suggesting the same circuitry involved in gain control could be responsible for enhancing tuning selectivity. In some instances, when large stimulus patches are used so both the classical receptive field and its surround are stimulated, it is sometimes possible to see a suppressive signal that is tuned in orientation (Fig. 3C, right panel), consistent with a role of suppression in enhancing tuning selectivity (Ringach et al. 1997, 2002, 2003).
Complex cells and the energy model
Most quantitative models of complex cells derive from the original formulation of Hubel and Wiesel who proposed that they result from the convergence of inputs from a number of simple cells sharing the same preference for orientation (Spitzer & Hochstein, 1985, 1988; see Martinez & Alonso, 2003 for a recent review). One instantiation of this circuit, known as the energy model, considers a pair of linear filters, tuned for orientation and spatial frequency, arranged in quadrature (Adelson & Bergen, 1985). The outputs of the filters are squared and added together to produce a response. The response can be considered a measure of the local signal energy (within a frequency band), therefore the term ‘energy model’. Clearly, a LN model applied to a complex cell would not work.
A recent approach used to model complex cells is to learn the input–output function by a two-layer neural network (Lau et al. 2002). This method, models the instantaneous rate of firing as
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(•) is a sigmoidal non-linearity (a hyperbolic tangent was used in this case). The model is, in effect, an instantiation of the Hubel–Wiesel feed-forward model, as each term can be considered the response of a simple cell. The back-propagation algorithm was used by Lau et al. (2002) to minimize the mean-square error on a training dataset and the performance of the model evaluated on a different dataset. The stimulus was a sequence of black/white bars orientated optimally for the cell. Some of the disadvantages of back-propagation are well known: it can settle into local minima and convergence can be rather slow. Nevertheless, the models estimated using this approach, which involves recording the response to flashed bars, were reasonably good at predicting other properties of the neurones, such as the direction selectivity index to a drifting sinusoidal gratings (Lau et al. 2002). A more efficient approach that is yielding interesting results is to study the spike-triggered covariance of the stimulus in response to spatio-temporal Gaussian white-noise. Here, one estimates the covariance matrix Cspike=E{x xT|spike} and compares it to the prior Cprior=E{x xT}. Because this matrix is supposed to represent the central second order moment, the result obtained from the spike-triggered average must be subtracted from all the stimuli first (Simoncelli et al. 2004). To select directions in stimulus space that appear relevant to establishing the cell's response one computes the eigenvalues of Cspike and determines which of these are significantly different from the null distribution of eigenvalues of Cprior. Both bootstrap methods (Touryan et al. 2002) and analytical results (Everson & Roberts, 2000) can been used to determine the statistical significance of the eigenvalues. Once this is done, the associated eigenvectors provide a subspace of interest that may be further studied by modelling how the neural response depends on the projection of the stimulus onto the ‘relevant subspace’. If an eigenvalue of Cspike is significantly larger than expected by chance, the associated eigenvalue is said to lie within the ‘excitatory’ subspace. Similarly, if an eigenvalue of Cspike is significantly lower than expected its eigenvector denotes a direction in stimulus space that suppresses the cell's response. Therefore, the eigenvector is said to lie within the ‘inhibitory’ subspace.
An example of this method applied to complex cells in cat area 17, from the work of Dan and colleagues, is shown in Fig. 4A. In this example, the eigenvectors associated with the two significant (excitatory) eigenvalues are tilted in space–time, as expected from one of the opponent pathway in the energy model of a directional complex cell (Adelson & Bergen, 1985). However, the eigenvectors only provide a basis for the ‘relevant subspace’ and should not be assigned a particular physical significance, such as that they represent simple-cell inputs to the cell. Both excitatory and inhibitory subspaces were observed in a similar study in macaque V1 by Rust et al. (2004) (Fig. 4B). In this directional cell, the eigenvectors for the excitatory and inhibitory subspaces preferred opposite directions of motion, suggesting a role for active suppression in the generation of direction selectivity. The results of Rust et al. (2004) in complex cells suggest that the quadrature-pair model can be refined in two ways. First, more than a pair of filters may be required to characterize the excitatory subspace. Second, a suppressive subspace appears to be required to appropriately model the responses of complex cells.
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If both the prior and spike-triggered distributions are Gaussian, the spike-triggered covariance has a nice interpretation in terms of the average information provided by the response about the stimulus (de Ruyter van Steveninck & Bialek, 1988; Chechik et al. 2004). Sharpee et al. (2003) has proposed a method to extend these ideas to non-Gaussian signals, such as naturalistic image sequences. The model put forward is a Markov chain, x
PSxP(spike), where the probability of spiking depends solely on the projection of the input onto a ‘relevant subspace’, denoted here by PSx. The subspace is identified by maximizing the mutual information between PSx and the neural response. The method may be considered a special case of the information-bottleneck technique (Tishby et al. 1999), where the coding of the stimulus is constrained to be linear. The scheme involves the optimization of a function with a large number of parameters, which can be a slow process and it is not guaranteed to converge. The main advantage of the technique is that it can be applied in situations where the signals are non-Gaussian.
Simple/complex cells, the hierarchical model, and theories of cortical function
The original description of simple and complex cells and the associated hierarchical model proposed by Hubel and Wiesel have had a strong impact in shaping theories of V1 function. This framework led many investigators to first develop theories of how simple cells represent the image, deferring the question about the function of complex cells (Maffei & Fiorentini, 1973; De Valois et al. 1979; Kulikowski & Bishop, 1981; Olshausen & Field, 1996; Bell & Sejnowski, 1997; Olshausen, 2001; Simoncelli & Olshausen, 2001; Hurri & Hyvarinen, 2003). The hierarchical model has also encouraged the search for coding principles that, when applied layer after layer in a hierarchy, will develop simple and complex-like behaviour (Rao & Ballard, 1997, 1999; Hyvarinen & Hoyer, 2001; Hoyer & Hyvarinen, 2002).
It has been recently suggested, however, that ‘simple’ and ‘complex’ cells may represent the ends of a continuum instead of two-discrete classes of neurones (Chance et al. 1999; Abbott & Chance, 2002; Mechler & Ringach, 2002). First, it has been demonstrated that the bimodality of the spike modulation ratio (or the F1/F0 ratio), taken to validate the existence of discrete classes of neurones (Skottun et al. 1991), is likely to be a consequence of the output rectification in what appears to be an otherwise unimodal populations of cells (Mechler & Ringach, 2002; Priebe N, Ferster D, Carandini M & Mechler F, unpublished observations). Second, the distribution of the F1/F0 ratio does not show a strong dependence with laminar location as predicted by the hierarchical model. Simple and complex cells (defined according to the F1/F0 ratio) are found in all cortical layers, both in monkey (Ringach et al. 2002b) and cat (Jacob et al. 2003). Because the F1/F0 ratio is not a direct measure of subfield overlap, it remains possible that a laminar segregation could be determined based on the relationship between ‘on/off ’ subregions (Conway & Livingstone, 2003; Martinez & Alonso, 2003; Hirsch, 2003; Kagan et al. 2003), but this remains to be determined.
The discreteness of simple/complex cells is more than a mere technical discussion about how to define these classes of neurones. The question is if the cortex can be considered to be composed of a hierarchy of distinct classes of receptive fields or not. The alternative is that receptive fields lie along a continuum, with simple and complex cells at the ends. A continuum of characteristics appears to hold for other receptive field attributes, such as colour tuning, length summation, spontaneous firing rate, etc. This is not to say that receptive field properties do not correlate across the population – they clearly do. As an example, ‘simple’ cells tend to have lower spontaneous rates than ‘complex’ cells (see discussion in Mechler & Ringach, 2002). However, the fact that several receptive field properties correlate does not constitute evidence of discreteness.
If we accept the view that receptive field properties appear to lie on a continuum, it would make sense to seek theoretical models that explain the distribution of receptive field properties and their correlations across the entire population, as well as trends in receptive field properties with laminar location. Such theories would have a quite different flavor from the ones that assume a ‘building-block’ cortex with simple and complex cells organized in strict hierarchy. Thus, the discreteness of neural populations in the cortex is something we must consider seriously, as the outcome may have a strong impact on how one views cortical organization and function. These are questions that, thanks to advances in receptive field mapping, can now be addressed in a rigorous manner.
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