We have recently shown that fixational eye movements improve discrimination of the orientation of a high spatial frequency grating masked by low-frequency noise, but do not help with a low-frequency grating masked by high-frequency noise (M. Rucci, R. Iovin, M. Poletti, & F. Santini, 2007). In this study, we explored the neural mechanisms responsible for this phenomenon. Models of parvocellular ganglion cells were stimulated by the same visual input experienced by subjects in our psychophysical experiments, i.e., the spatiotemporal signals resulting from viewing stimuli during eye movements. We show that the spatial organization of correlated activity in the model predicts the subjects' performance in the experiments. During viewing of high-frequency gratings, fixational eye movements modulated the responses of modeled neurons in a way that depended on the relative alignment of cell receptive fields. Responses covaried strongly only when receptive fields were aligned parallel to the grating's orientation. Such a dependence on the axis of receptive-field alignment did not occur during viewing of low-frequency gratings. In this case, the responses of cells on the parallel and orthogonal axes were similarly affected by eye movements. These results support a role for oculomotor synchronization of neural activity in the representation of visual information in the retina.

*f*

_{c}= 5 cycles/deg. In Experiment 2, the frequencies of the grating and the pattern of noise were inverted so that the frequency of the grating (4 cycles/deg) was lower than the frequency band of noise (high-pass cut-off frequency of

*f*

_{c}= 10 cycles/deg). In both cases, the power of the noise decreased proportionally to the square of the spatial frequency, as occurs in the power spectrum of natural images (Field, 1987). Stimuli were either flashed at a fixed location on the display (unstabilized condition) or were moved with the eye, under real-time computer control, in order to cancel the retinal motion resulting from fixational eye movements (stabilized condition). In this latter condition, the stimulus always appeared to the observer to be immobile at the center of the fovea. Eye position was continuously recorded by means of a Generation 6 DPI eyetracker (Fourward Technologies, Inc.).

*k*-th trial of the psychophysical experiments, model neurons received as input the movie

*I*

_{ k}(

**x**,

*t*), which reconstructed the spatiotemporal signal resulting from the specific combination of the stimulus

*S*

_{ k}(

**x**) presented in the considered trial and (in the unstabilized condition) the recorded trace of eye movements,

*ξ*_{ k}(

*t*). In the simulation of an unstabilized trial, the position of the stimulus changed at each frame of the movie so as to be centered at the current location of gaze:

*I*

_{ k}(

**x**,

*t*) =

*S*

_{ k}(

**x**+

*ξ*_{ k}(

*t*)). In this way, the receptive fields of modeled neurons effectively scanned the stimulus following the recorded scanpath. In a stabilized trial, instead, since retinal image motion had been canceled out by the procedure of retinal stabilization, each frame of

*I*

_{ k}(

**x**,

*t*) consisted of the same image:

*I*

_{ k}(

**x**,

*t*) =

*S*

_{ k}(

**x**). In the simulations, to enable analysis of correlated activity for pairs of cells with receptive fields up to 1° apart, stimuli were enlarged by removing the Gaussian window used in the experiments. As in the experiments, stimuli were displayed for 1 s and followed by a high-energy mask.

*K*(

**x**,

*t*), which convolved the visual input

*I*(

**x**,

*t*):

**x**is the location of the center of a cell's receptive field, and the operator [·]

_{ γ}indicates rectification with threshold

*γ*: [

*z*]

_{ γ}=

*z*−

*γ*if

*z*>

*γ,*and [

*z*]

_{ γ}= 0 if

*z*≤

*γ*. In the context of this paper, since the instantaneous firing rate already provides statistical averaging, it is a more computationally efficient signal to simulate than the actual train of spikes.

*F*(

**x**), was modeled as the standard difference of two-dimensional Gaussian functions, so that its contrast sensitivity function

*F*(

**u**) varied with spatial frequency

**u**as:

*S*

_{c},

*S*

_{s},

*r*

_{c}, and

*r*

_{s}represent the sensitivities and radii of center and surround Gaussians, respectively. Since our experimental stimuli contained gratings at two different frequencies, we modeled two populations of parvocellular cells with different peaks in their spatial sensitivity. As shown in Figure 2, one set of neurons was highly sensitive to low spatial frequencies. These neurons had maximal sensitivity at 4.3 cycles/deg and responded strongly to the gratings of Experiment 2. Cells in the second group were sensitive to a higher range of spatial frequencies. Their contrast sensitivity function peaked at 11.4 cycles/deg, so that their responses were severely in influenced by the gratings of Experiment 1. For both neuronal populations, parameters were adjusted on the basis of neurophysiological data from Derrington and Lennie (1984) to model the contrast sensitivities of individual cells with receptive fields within the central 10° of visual field. Neurons in the first population modeled cell A in Figure 3 of Derrington and Lennie (1984). For these neurons, the values of the parameters in Equation 3 were set to

*S*

_{c}= 15.03 impulses s

^{−1},

*r*

_{c}= 0.015°,

*r*

_{s}= 0.072°,

*S*

_{s}= 0.580 impulses s

^{−1}. Units in the population sensitive to high spatial frequencies modeled cell C in Figure 3 of Derrington and Lennie (1984). In this case, the parameters of Equation 3 were set to the following values:

*S*

_{c}= 10.74 impulses s

^{−1},

*r*

_{c}= 0.03°,

*r*

_{s}= 0.202°,

*S*

_{s}= 0.158 impulses s

^{−1}.

*H*(

*t*) was computed as the inverse Fourier transform of the temporal-frequency response

*H*(

*ω*) to a sine grating modulated at temporal frequency

*ω*.

*H*(

*ω*) was modeled by a set of low-pass filters and a high-pass stage as:

*A*represents the overall gain;

*H*

_{S}, the strength of the subtractive stage;

*τ*

_{S}and

*τ*

_{L}, the time constants of the high- and low-pass stages;

*N*

_{L}, the number of low-pass stages; and

*D,*the brief transmission delay along the RGC axon, which was included to model the output of the retina. In this model, the cascade of filters is a mathematical commodity with the purpose of data fitting; these filters are not meant to describe individual biological components. This model has been successfully applied to fit data from macaque's ganglion cells (Benardete & Kaplan, 1997, 1999a, 1999b; Benardete, Kaplan, & Knight, 1992). Temporal parameters of the model were set to the median values measured in P cells in the macaque's retina (Benardete & Kaplan, 1999b):

*A*= 601.48 impulses s

^{−1},

*N*

_{L}= 51,

*τ*

_{S}= 0.87 ms,

*H*

_{S}= 0.77,

*τ*

_{L}= 31.73 ms,

*D*= 4 ms.

*k,*the degree of covariance between the responses of two model cells,

*η*

_{ x}(

*t*) and

*η*

_{ y}(

*t*), was quantified by means of their correlation coefficient:

*T*= 1 s duration of the trial. Correlation coefficients were then averaged (a) over the ensemble

*F*of all pairs of simulated cells with receptive field centers at a fixed distance

*d*from each other and aligned at the same angle

*φ*with respect to the grating, and (b) over all experimental trials, yielding the spatial correlation function

*r*(

*d, φ*) (see Figure 3):

*φ*represents the angle subtended by the line intersecting the centers of the two receptive fields and the grating's axis (

*θ*

_{k}= ±45°). Each value of

*φ*identifies a specific orientation axis relative to the grating. The function

*r*(

*d, φ*) gives the average correlation coefficient between the responses of two cells with receptive field at various distances on a selected orientation

*φ*.

*φ*= 0°) or orthogonal (

*φ*= 90°) to the grating's orientation. These two angles define the two possible orientations that a grating could assume in a trial. The responses of retinal ganglion cells aligned on these two axes are likely to play a critical role in the neural processes underlying the forced-choice decision of our experiments. These responses converge onto neurons selective for the two possible orientations in the primary visual cortex (Reid & Alonso, 1995). In this paper, we use the symbols

*r*

_{⊥}(

*d*) and

*r*

_{∥}(

*d*) to represent the spatial correlation functions on these two orthogonal axes:

*r*

_{∥}(

*d*) and

*r*

_{⊥}(

*d*) as the parallel and orthogonal correlation functions. Levels of covariance in the simulations were averaged over a total of

*N*= 200 trials, each with its unique pattern of eye movements. In each trial, means were evaluated over at least 10 pairs of cells for every value of separation

*d*and angle

*φ*.

*R*(

**u**,

*ω*), where

**u**and

*ω*indicate spatial and temporal frequencies, respectively. Under the assumption of linearity,

*r*(

**x**,

*t*) (the correlation between the responses of pairs of cells at separation

**x**measured at a time lag

*t*) can be evaluated as

*F*

^{−1}{∣

*K*(

**u**,

*ω*)∣

^{2}

*R*(

**u**,

*ω*)}, where

*K*is the cell linear kernel, and

*F*

^{−1}indicates the inverse Fourier Transform operator (see for example Bendat & Piersol, 1986). Input spectra in Figure 8 were estimated by means of Welch's method over all the trials available from one subject (see Rucci et al., 2007).

*r*

_{∥}(

*d*) and

*r*

_{⊥}(

*d*), shown in Figures 3d– 3e. The averages of these two functions over cells at different separations,

_{∥}and

_{⊥}, were used as indices of the global degrees of synchronization in neural activity on the two axes.

*r*

_{∥}(

*d*) and

*r*

_{⊥}(

*d*), in Equation 7) for both populations of simulated retinal units (cells sensitive to either low or high spatial frequencies). Both results obtained in the presence of the normal fixational motion of the retinal image and in simulations of retinal stabilization are shown in Figure 4.

*r*

_{∥}(

*d*) varied little with the distance

*d*between receptive fields. In contrast, on the orthogonal axis, cell responses were synchronously modulated only when the separations between their receptive fields matched the period of the grating, as shown by the oscillations in the orthogonal correlation function,

*r*

_{⊥}(

*d*). Although these modulations were visible in both neuronal populations, they were more pronounced for neurons sensitive to high spatial frequencies. Because of their spatial characteristics, these neurons responded strongly to the grating and were relatively unaffected by the low-band pattern of noise present in the stimulus.

_{∥}and

_{⊥}, was larger during normal retinal image motion than under retinal stabilization. This effect was particularly evident for neurons sensitive to high spatial frequencies, which responded strongly to the changes in luminance introduced by fixational instability. For this neuronal population, the difference between mean levels of covariance on the parallel and orthogonal axes was significant (

*z*= 5.33,

*p*< 0.01; two-tailed

*z*-test).

*r*

_{∥}(

*d*). Neurons with receptive fields aligned orthogonally to the grating were simultaneously active only if the separation between their receptive fields was a multiple of the period of the grating, yielding an orthogonal correlation function which oscillated at the grating's frequency. This pattern of correlated activity occurred for both neuronal populations. In this case, however, the influence of the grating was most visible in the responses of neurons sensitive to low spatial frequencies. Because of their contrast sensitivity functions, these neurons were driven strongly by the grating and largely unaffected by the pattern of noise.

_{∥}and

_{⊥}, also changed little between the two conditions of presence and absence of retinal image motion.

*z*= −6.22,

*p*< 0.01; two-tailed

*z*-test). In contrast, the parallel and orthogonal correlation functions measured during the initial 100 ms were almost identical to those obtained in simulations of retinal stabilization. The reasons underlying the differential impact of the two sources of retinal image motion, modulations of stimulus contrast and oculomotor activity, are explained in Discussion.

*r*(

**x**,

*t*) between the responses of pairs of cells at separation

**x**measured at a time lag

*t*is, by definition, the inverse Fourier Transform of the power spectrum of neural activity. This spectral density function can be obtained by multiplying the power spectrum of visual input by the squared amplitude of the cell linear kernel (see for example Bendat & Piersol, 1986). This analysis gives insights into the mechanisms by which fixational eye movements modulate neuronal responses, as it enables quantification of the contributions from different spatial and temporal frequencies to the patterns of correlated activity.

*I*(

**x**,

*t*) is a 3D function of space and time, its power spectrum is a 3D function of spatial and temporal frequency. The left side of Figure 8b shows the power spectra of the visual input signals in the various cases considered in this study. Each panel represents a space–time section of the corresponding 3D spectrum taken along plane

*α*in Figure 8a. Choice of this plane enabled display of both the grating and the pattern of noise present in the stimulus. As shown by these sections, the two conditions of normal retinal image motion and retinal stabilization produced visual input signals with significantly different spectral distributions. Under retinal stabilization, the input power was confined to the spatial frequency plane at zero temporal frequency because the retinal stimulus did not change. In the normal viewing condition, instead, the motion of the eye spread the spatial power of the stimulus across temporal frequencies. Interestingly, the extent of this temporal spreading increased with the spatial frequency. It was this dependence on spatial frequency of the temporal power generated by eye movements which was ultimately responsible for the results presented in this paper.

*f*

_{d}(i.e., temporal contrast sensitivity function different from zero only at

*f*

_{d}). In Experiment 1, the SNR of this ideal neuron is 243 times larger if the neuron responds to

*f*

_{d}= 15 Hz than if it only responds to static stimuli (

*f*

_{d}= 0 Hz). The opposite effect occurs in Experiment 2, in which sensitivity to

*f*

_{d}= 15 Hz lowers the SNR by a factor of 26 with respect to sensitivity to

*f*

_{d}= 0 Hz. Thus, the enhancement of high spatial frequencies caused by fixational eye movements tends to be more pronounced in neurons that prefer high temporal frequencies.