A motion sequence may be represented as a single pattern in x–y–t space; a velocity of motion corresponds to a three-dimensional orientation in this space. Motion sinformation can be extracted by a system that responds to the oriented spatiotemporal energy. We discuss a class of models for human motion mechanisms in which the first stage consists of linear filters that are oriented in space-time and tuned in spatial frequency. The outputs of quadrature pairs of such filters are squared and summed to give a measure of motion energy. These responses are then fed into an opponent stage. Energy models can be built from elements that are consistent with known physiology and psychophysics, and they permit a qualitative understanding of a variety of motion phenomena.

/pdf/spatiotemporal-energy-models-for-the-perception-of-motion-43mzze32mc.pdf

Spatiotemporal energy models for the perception of motion

Theoretical neuroscience provides a quantitative basis for describing what nervous systems do, determining how they function, and uncovering the general principles by which they operate This text introduces the basic mathematical and computational methods of theoretical neuroscience and presents applications in a variety of areas including vision, sensory-motor integration, development, learning, and memory The book is divided into three parts Part I discusses the relationship between sensory stimuli and neural responses, focusing on the representation of information by the spiking activity of neurons Part II discusses the modeling of neurons and neural circuits on the basis of cellular and synaptic biophysics Part III analyzes the role of plasticity in development and learning An appendix covers the mathematical methods used, and exercises are available on the book's Web site

Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems

Two-dimensional spatial linear filters are constrained by general uncertainty relations that limit their attainable information resolution for orientation, spatial frequency, and two-dimensional (2D) spatial position. The theoretical lower limit for the joint entropy, or uncertainty, of these variables is achieved by an optimal 2D filter family whose spatial weighting functions are generated by exponentiated bivariate second-order polynomials with complex coefficients, the elliptic generalization of the one-dimensional elementary functions proposed in Gabor’s famous theory of communication [ J. Inst. Electr. Eng.93, 429 ( 1946)]. The set includes filters with various orientation bandwidths, spatial-frequency bandwidths, and spatial dimensions, favoring the extraction of various kinds of information from an image. Each such filter occupies an irreducible quantal volume (corresponding to an independent datum) in a four-dimensional information hyperspace whose axes are interpretable as 2D visual space, orientation, and spatial frequency, and thus such a filter set could subserve an optimally efficient sampling of these variables. Evidence is presented that the 2D receptive-field profiles of simple cells in mammalian visual cortex are well described by members of this optimal 2D filter family, and thus such visual neurons could be said to optimize the general uncertainty relations for joint 2D-spatial–2D-spectral information resolution. The variety of their receptive-field dimensions and orientation and spatial-frequency bandwidths, and the correlations among these, reveal several underlying constraints, particularly in width/length aspect ratio and principal axis organization, suggesting a polar division of labor in occupying the quantal volumes of information hyperspace. Such an ensemble of 2D neural receptive fields in visual cortex could locally embed coarse polar mappings of the orientation–frequency plane piecewise within the global retinotopic mapping of visual space, thus efficiently representing 2D spatial visual information by localized 2D spectral signatures.

Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters.

The relative efficiency of any particular image-coding scheme should be defined only in relation to the class of images that the code is likely to encounter. To understand the representation of images by the mammalian visual system, it might therefore be useful to consider the statistics of images from the natural environment (i.e., images with trees, rocks, bushes, etc). In this study, various coding schemes are compared in relation to how they represent the information in such natural images. The coefficients of such codes are represented by arrays of mechanisms that respond to local regions of space, spatial frequency, and orientation (Gabor-like transforms). For many classes of image, such codes will not be an efficient means of representing information. However, the results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy (e.g., correlation between the intensities of neighboring pixels) into first-order redundancy (i.e., the response distribution of the coefficients). Such coding produces a relatively high signal-to-noise ratio and permits information to be transmitted with only a subset of the total number of cells. These results support Barlow's theory that the goal of natural vision is to represent the information in the natural environment with minimal redundancy.

/pdf/relations-between-the-statistics-of-natural-images-and-the-12zwc2ox1f.pdf

Relations between the statistics of natural images and the response properties of cortical cells.

Simple cells in the striate cortex have been depicted as half-wave-rectified linear operators. Complex cells have been depicted as energy mechanisms, constructed from the squared sum of the outputs of quadrature pairs of linear operators. However, the linear/energy model falls short of a complete explanation of striate cell responses. In this paper, a modified version of the linear/energy model is presented in which striate cells mutually inhibit one another, effectively normalizing their responses with respect to stimulus contrast. This paper reviews experimental measurements of striate cell responses, and shows that the new model explains a significantly larger body of physiological data.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0952523800009640

Normalization of cell responses in cat striate cortex

1. We have examined the responses of simple cells in the cat's atriate cortex to visual patterns that were designed to reveal the extent to which these cells may be considered to sum light-evoked influences linearly across their receptive fields. We used one-dimensional luminance-modulated bars and grating as stimuli; their orientation was always the same as the preferred orientation of the neurone under study. The stimuli were presented on an oscilloscope screen by a digital computer, which also accumulated neuronal responses and controlled a randomized sequence of stimulus presentations. 2. The majority of simple cells respond to sinusoidal gratings that are moving or whose contrast is modulated in time in a manner consistent with the hypothesis that they have linear spatial summation. Their responses to moving gratings of all spatial frequencies are modulated in synchrony with the passage of the gratings' bars across their receptive fields, and they do not produce unmodulated responses even at the highest spatial frequencies. Many of these cells respond to temporally modulated stationary gratings simply by changing their response amplitude sinusoidally as the spatial phase of the grating the grating is varied. Nonetheless, their behavior appears to indicate linear spatial summation, since we show in an Appendix that the absence of a 'null' phase in a visual neurone need not indicate non-linear spatial summation, and further that a linear neurone lacking a 'null' phase should give responses of the form that we have observed in this type of simple cell. 3. A minority of simple cells appears to have significant non-linearities of spatial summation. These neurones respond to moving gratings of high spatial frequency with a partially or totally unmodulated elevation of firing rate. They have no 'null' phases when tested with stationary gratings, and reveal their non-linearity by giving responses to gratings of some spatial phases that are composed partly or wholly of even harmonics of the stimulus frequency ('on-off' responses). 4. We compared simple receptive fields with their sensitivity to sinusoidal gratings of different spatial frequencies. Qualitatively, the most sensitive subregions of simple cells' receptive fields are roughly the same width as the individual bars of the gratings to which they are most sensitive. Quantitatively, their receptive field profiles measured with thin stationary lines, agree well with predicted profiles derived by Fourier synthesis of their spatial frequency tuning curves.

Spatial summation in the receptive fields of simple cells in the cat's striate cortex.

1. We have examined the spatial and temporal tuning properties of 238 cortical neurones, recorded using conventional techniques from acutely prepared anaesthetized cats. We determined spatial and temporal frequency tuning curves using sinusoidal grating stimuli presented to each neurone's receptive field by a digital computer on a cathode ray tube. 2. We measured tuning curves either by determining response amplitude as a function of spatial or temporal frequency, or by measuring contrast sensitivity (the inverse of the contrast of the grating that just elicited a detectable response). The two measures give very similar tuning curves in all cases. 3. We recorded from 184 neurones in area 17; of these 156 had receptive fields within 5 degrees of the area centralis. The range of preferred spatial frequency for these neurones was 0.3--3 c/deg, and their spatial frequency tuning band widths varied from 0.7 to 3.2 octaves at half-amplitude. The most common band width was roughly 1.3 octaves. Simple and complex cells in area 17 did not differ in their distributions of preferred spatial frequency, although complex cells were, on average, slightly less selective for spatial frequency than simple cells. 4. We recorded from fifty-four neurones from area 18, and performed several experiments in which we recorded from corresponding portions of both area 17 and area 18 in the same electrode penetration. Neurones in area 18 preferred spatial frequencies that were, on average, one third as high as those preferred by area 17 neurones at the same retinal eccentricity. Thus the range of preferred spatial frequency in area eighteen cells having receptive fields within 5 deg of the area centralis was between less than 0.1 and 0.5 c/deg. The distributions of optimum spatial frequency in the two areas were practically non-overlapping at eccentricities as high as 15 deg, the greatest eccentricity we examined. Neurones in area 18 were about as selective for spatial frequency as were neurones in area 17. 5. We determined temporal frequency tuning characteristics for some neurones from each area, using gratings that moved steadily across the screen. Neurones from area 17 all responded well to low temporal frequencies, and less well to higher frequencies (in excess of, usually, 2 or 4 Hz). In contrast, neurones recorded from area 18 sometimes had similar tuning properties, but more commonly showed a pronounced reduction in response as the temporal frequency was moved either above or below some optimum value (usually 2--8 Hz). 6. We conclude from these results that areas 17 and 18 act in parallel to process different aspects of the visual information relayed from the retina via the lateral geniculate complex. Some or all of the differences between the areas may be attributable to the predominance of Y cell input to area 18 and the predominance of X cell input to area 17...

Spatial and temporal contrast sensitivity of neurones in areas 17 and 18 of the cat's visual cortex.

1. All complex cells in the cat's striate cortex exhibit gross non-linearities of spatial summation when tested with sinusoidal grating stimuli. Their responses to moving gratings of all but the lowest spatial frequencies are usually dominated by a component that is not modulated by the passage of the bars of the grating across the receptive field. They give responses to temporally modulated stationary gratings that consist mostly of even harmonics of the stimulus frequency and that vary little in amplitude or wave form as the spatial phase of the grating is varied. 2. We compared complex cells' receptive fields with their sensitivity to sinusoidal gratings of different spatial frequencies. Qualitatively, the receptive fields are usually two to five times wider than the bars of the gratings that stimulate them most effectively. Quantitatively, the receptive field profiles of complex cells are invariably broader than those predicted by Fourier synthesis of their spatial frequency tuning curves, and in particular lack predicted spatially antagonistic regions. 3. We further examined the receptive field organization of these cells, using pairs of stationary lines flashed synchronously on their receptive fields. If both lines are of the same polarity (bright or dark), complex cells respond to the paired stimulus much less well than they do to either of its component bars, unless the bars are separated by less than about one quarter of the width of the receptive field. If the lines are of opposite polarity, one bright and one dark, the opposite situation obtains: closely spaced bars elicit small responses, while paired bars of larger separation are much more effective. In either case, the results are independent in general character of the absolute positions of the stimuli within the receptive field; rather, they depend in a manner characteristic of each cell on the relative positions of the two bars. 4. The two-line interaction profile that plots the change in a complex cell's response to one bar as a function of the position of a second added bar corresponds closely to the receptive field profile predicted from Fourier synthesis of the cell's spatial frequency tuning curve. These profiles may thus reveal the spatial characteristics of subunits within complex cell-receptive fields. We examined the nature of the interaction between these subunits by performing several two-line interaction experiments in which the onset of the second bar was delayed some time after the onset of the first. The results suggest that neighbouring subunits interact in a facilitatory fashion: for an interval after the presentation of one bar, responses to neighbouring bars are enhanced. 5. The subunits of a complex receptive field may, by their spatial properties, determine the spatial selectivities of complex cells, while the nature of the interaction among the subunits may determine these cells' sensitivity and selectivity for moving visual stimuli...

Receptive field organization of complex cells in the cat's striate cortex.

For neurones in the cat's striate cortex, we examined the dependence of response on the contrast of moving sinusoidal gratings. Most neurones showed a clear threshold contrast below which no response was elicited. Such thresholds presumably contribute to the animal's behavioural threshold, which should not be accounted for solely in terms of the detection of a signal in the presence of spontaneous “noise”. Above threshold, the response amplitude usually increased linearly with contrast until it began to saturate at the highest contrasts. The variance of the response increased with its amplitude; this finding perhaps underlies the Weber-Fechner relation for psychophysical contrast discrimination.

The Dependence of Response Amplitude and Variance of Cat Visual Cortical Neurones on Stimulus Contrast

https://www.cell.com/neuron/pdf/S0896-6273(03)00789-X.pdf

Ian D. Thompson

Papers

Spatial summation in the receptive fields of simple cells in the cat's striate cortex.

Spatial and temporal contrast sensitivity of neurones in areas 17 and 18 of the cat's visual cortex.

Receptive field organization of complex cells in the cat's striate cortex.

The Dependence of Response Amplitude and Variance of Cat Visual Cortical Neurones on Stimulus Contrast

Abnormal Functional Organization in the Dorsal Lateral Geniculate Nucleus of Mice Lacking the β2 Subunit of the Nicotinic Acetylcholine Receptor