Abstract

Experiments by Loffler and Orbach on the integration of motion signals across space [J. Opt. Soc. Am. A 20, 1461 (2003)] revealed that both three-dimensional analysis and object interpretation play a much smaller role than previously assumed. These results motivated the quantitative description of a low-level, bottom-up model presented here. Motion is computed in parallel at different spatial sites, and excitatory interactions operate between sites. The strength of these interactions is determined mainly by distance. Simulations correctly predict behavior for a variety of manipulations on multi-aperture stimuli: aligned and skewed lines, different presentation times, different inter-aperture gaps, and different spatial frequencies. However, strictly distance-dependent mechanisms are too simplistic to account for all experimental data. Mismatches for grossly misoriented lines suggest collinear facilitation as a promising extension. Once incorporated, collinear facilitation not only correctly predicts results for misoriented patterns but also accounts for the lack of motion integration between heterogeneous stimuli such as lines and dots.

© 2003 Optical Society of America

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2003 (1)

2002 (2)

Y. Weiss, E. P. Simoncelli, E. H. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598–604 (2002).
[CrossRef] [PubMed]

M. A. Georgeson, G. S. A. Barbieri-Hesse, T. C. A. Freeman, “The primal sketch revisited: locating and representing edges in human vision via Gaussian-derivative filtering,” Perception 31, 1 (2002).

2001 (7)

C. C. Pack, R. T. Born, “Temporal dynamics of a neural solution to the aperture problem in visual area MT of macaque brain,” Nature 409, 1040–1042 (2001).
[CrossRef] [PubMed]

D. Bradley, “MT signals: better with time,” Nat. Neurosci. 4, 346–348 (2001).
[CrossRef] [PubMed]

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

R. F. Hess, W. H. A. Beaudot, K. T. Mullen, “Dynamics of contour integration,” Vision Res. 41, 1023–1037 (2001).
[CrossRef] [PubMed]

P. J. Bex, A. J. Simmers, S. C. Dakin, “Snakes and ladders: the role of temporal modulation in visual contour integration,” Vision Res. 41, 3775–3782 (2001).
[CrossRef] [PubMed]

H. S. Orbach, G. Loffler, “Motion integration across apertures: theory and experiment,” Invest. Ophthalmol. Visual Sci. 42, 4685 (2001).

Z. Y. Yang, A. Shimpi, D. Purves, “A wholly empirical explanation of perceived motion,” Proc. Natl. Acad. Sci. USA 98, 5252–5257 (2001).

1999 (5)

E. Castet, V. Charton, A. Dufour, “The extrinsic/intrinsic classification of two-dimensional motion signals with barber-pole stimuli,” Vision Res. 39, 915–932 (1999).
[CrossRef] [PubMed]

G. Loffler, H. S. Orbach, “Computing feature motion without feature detectors: a model for terminator motion without end-stopped cells,” Vision Res. 39, 859–871 (1999).
[CrossRef] [PubMed]

J. Lorenceau, L. Zago, “Cooperative and competitive spatial interactions in motion integration,” Vision Res. 16, 755–770 (1999).

R. Hess, D. Field, “Integration of contours: new insights,” Trends Cogn. Sci. 3, 480–486 (1999).
[CrossRef] [PubMed]

For example, L. Liden, C. Pack, “The role of terminators and occlusion cues in motion integration and segmentation: a neural network model,” Vision Res. 39, 3301–3320 (1999).
[CrossRef]

1998 (1)

L. Liden, E. Mingolla, “Monocular occlusion cues alter the influence of terminator motion in the barber pole phenomenon,” Vision Res. 38, 3883–3898 (1998).
[CrossRef]

1997 (5)

E. Castet, S. Wuerger, “Perception of moving lines: in-teractions between local perpendicular signals and 2D motion signals,” Vision Res. 37, 705–720 (1997).
[CrossRef] [PubMed]

K. E. Schmidt, R. Goebel, S. Lowel, W. Singer, “The perceptual grouping criterion of collinearity is reflected by anisotropies of connections in the primary visual cortex,” Eur. J. Neurosci. 9, 1083–1089 (1997).
[CrossRef] [PubMed]

W. H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,” J. Neurosci. 17, 2112–2127 (1997).
[PubMed]

J. Kim, H. R. Wilson, “Motion integration over space: interaction of the center and surround motion,” Vision Res. 37, 991–1005 (1997).
[CrossRef] [PubMed]

J. Chey, S. Grossberg, E. Mingolla, “Neural dynamics of motion grouping: from aperture ambiguity to object speed and direction,” J. Opt. Soc. Am. A 14, 2570–2594 (1997).
[CrossRef]

1996 (2)

J. Kim, H. R. Wilson, “Direction repulsion between components in motion transparency,” Vision Res. 36, 1177–1187 (1996).
[CrossRef] [PubMed]

J. A. Movshon, W. T. Newsome, “Visual response properties of striate cortical neurons projecting to area MT in macaque monkeys,” J. Neurosci. 16, 7733–7741 (1996).
[PubMed]

1995 (3)

S. Raiguel, M. M. Van Hulle, D.-K. Xiao, V. L. Marcar, G. A. Orban, “Shape and spatial distribution of receptive fields and antagonistic motion surrounds in the middle temporal area (V5) of the macaque,” Eur. J. Neurosci. 7, 2064–2082 (1995).
[CrossRef] [PubMed]

S. J. Nowlan, T. J. Sejnowski, “A selection model for motion processing in area MT of primates,” J. Neurosci. 15, 1195–1214 (1995).
[PubMed]

M. B. Ben-Av, M. Shiffrar, “Disambiguating velocity estimates across image space,” Vision Res. 35, 2889–2895 (1995).
[CrossRef] [PubMed]

1994 (4)

C. D. Salzman, W. T. Newsome, “Neural mechanisms for forming a perceptual decision,” Science 264, 231–237 (1994).
[CrossRef] [PubMed]

H. S. Orbach, H. R. Wilson, “Fourier and non-Fourier terminators in motion perception,” Invest. Ophthalmol. Visual Sci. 35, 1827 (1994).

H. R. Wilson, J. Kim, “A model for motion coherence and transparency,” Visual Neurosci. 11, 1205–1220 (1994).
[CrossRef]

H. R. Wilson, J. Kim, “Perceived motion in the vector sum direction,” Vision Res. 34, 1835–1842 (1994).
[CrossRef] [PubMed]

1993 (11)

J. Kim, H. R. Wilson, “Dependence of plaid motion coherence on component grating directions,” Vision Res. 33, 2479–2489 (1993).
[CrossRef] [PubMed]

S. Grossberg, E. Mingolla, “Neural dynamics of motion perception: direction fields, apertures, and resonant grouping,” Percept. Psychophys. 53, 248–278 (1993).

M. B. Ben-Av, M. Shiffrar, “When ambiguous becomes unambiguous,” Invest. Ophthalmol. Visual Sci. 34, 1028–1028 (1993).

F. L. Kooi, “Local direction of edge motion causes and abolishes the barberpole illusion,” Vision Res. 33, 2347–2351 (1993).
[CrossRef] [PubMed]

N. Rubin, S. Hochstein, “Isolating the effect of one-dimensional motion signals on the perceived direction of moving 2-dimensional objects,” Vision Res. 33, 1385–1396 (1993).
[CrossRef] [PubMed]

J. Lorenceau, M. Shiffrar, N. Wells, E. Castet, “Different motion sensitive units are involved in recovering the direction of moving lines,” Vision Res. 33, 1207–1217 (1993).
[CrossRef] [PubMed]

A. P. Georgopoulos, M. Taira, A. Lukashin, “Cognitive neurophysiology of the motor cortex,” Science 260, 47–52 (1993).
[CrossRef] [PubMed]

R. A. Andersen, L. H. Snyder, C.-S. Li, B. Stricanne, “Coordinate transformations in the representation of spatial information,” Curr. Opin. Neurobiol. 3, 171–176 (1993).
[CrossRef] [PubMed]

R. Malach, Y. Amir, M. Harel, A. Grinvald, “Relationship between intrinsic connections and functional architecture revealed by optical imaging and in-vivo targeted biocytin injections in primate striate cortex,” Proc. Natl. Acad. Sci. USA 90, 10469–10473 (1993).
[CrossRef]

U. Polat, D. Sagi, “Lateral interactions between spatial channels—suppression and facilitation revealed by lateral masking experiments,” Vision Res. 33, 993–999 (1993).
[CrossRef] [PubMed]

D. J. Field, A. Hayes, R. F. Hess, “Contour integration by the human visual-system—evidence for a local association field,” Vision Res. 33, 173–193 (1993).
[CrossRef] [PubMed]

1992 (8)

R. T. Born, R. B. H. Tootell, “Segregation of global and local motion processing in primate middle temporal visual area,” Nature 357, 497–499 (1992).
[CrossRef] [PubMed]

E. Mingolla, J. T. Todd, J. F. Norman, “The perception of globally coherent motion,” Vision Res. 32, 1015–1031 (1992).
[CrossRef] [PubMed]

J. Lorenceau, M. Shiffrar, “The influence of terminators on motion integration across space,” Vision Res. 32, 263–273 (1992).
[CrossRef] [PubMed]

H. R. Wilson, W. A. Richards, “Curvature and separation discrimination at texture boundaries,” J. Opt. Soc. Am. A 9, 1653–1662 (1992).
[CrossRef] [PubMed]

H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
[CrossRef]

C. Yo, H. R. Wilson, “Perceived direction of moving two-dimensional patterns depends on duration, contrast and eccentricity,” Vision Res. 32, 135–147 (1992).
[CrossRef] [PubMed]

C. D. Salzman, C. M. Murasugi, K. H. Britten, W. T. Newsome, “Microstimulation in visual area MT—effects on direction discrimination performance,” J. Neurosci. 12, 2331–2355 (1992).
[PubMed]

K. H. Britten, M. N. Shadlen, W. T. Newsome, J. A. Movshon, “The analysis of visual-motion—a comparison of neuronal and psychophysical performance,” J. Neurosci. 12, 4745–4765 (1992).
[PubMed]

1990 (3)

G. Sclar, J. R. Maunsell, P. Lennie, “Coding of image contrast in central visual pathways of the macaque monkey,” Vision Res. 30, 1–10 (1990).
[CrossRef] [PubMed]

L. S. Stone, A. B. Watson, J. B. Mulligan, “Effect of contrast on the perceived direction of a moving plaid,” Vision Res. 30, 1049–1067 (1990).
[CrossRef] [PubMed]

M. Nawrot, R. Sekuler, “Assimilation and contrast in motion perception—explorations in cooperativity,” Vision Res. 30, 1439–1451 (1990).
[CrossRef]

1989 (3)

R. J. Snowden, “Motions in orthogonal directions are mutually suppressive,” J. Opt. Soc. Am. A 6, 1096–1101 (1989).
[CrossRef]

H. R. Rodman, T. D. Albright, “Single unit analysis of patter-motion selective properties in the middle temporal area (MT),” Exp. Brain Res. 75, 53–64 (1989).
[CrossRef]

S. Shimojo, G. H. Silverman, K. Nakayama, “Occlusion and the solution to the aperture problem for motion,” Vision Res. 29, 619–626 (1989).
[CrossRef] [PubMed]

1988 (1)

K. Nakayama, G. H. Silverman, “The aperture problem I. Perception of nonrigidity and motion direction in translating sinusoidal lines,” Vision Res. 28, 739–746 (1988).
[CrossRef]

1986 (1)

K. Tanaka, K. Hikosaka, H.-A. Saito, M. Yukie, Y. Fukada, E. Iwai, “Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey,” J. Neurosci. 6, 134–144 (1986).
[PubMed]

1985 (3)

J. Allman, F. Miezin, E. McGuinness, “Stimulus specific responses from beyond the classical receptive-field: neurophysiological mechanisms for local–global comparisons in visual neurons,” Annu. Rev. Neurosci. 8, 407–430 (1985).
[CrossRef]

H. R. Wilson, “A model for direction selectivity in threshold motion perception,” Biol. Cybern. 51, 213–222 (1985).
[CrossRef] [PubMed]

J. P. H. van Santen, G. Sperling, “Elaborated Reichardt detectors,” J. Opt. Soc. Am. A 2, 300–321 (1985).
[CrossRef] [PubMed]

1984 (4)

H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A 1, 124–131 (1984).
[CrossRef] [PubMed]

G. G. Blasdel, D. Fitzpatrick, “Physiological organisation of layer-4 in macaque striate cortex,” J. Neurosci. 4, 880–895 (1984).
[PubMed]

D. C. Van Essen, “The visual field representation in striate cortex of the macaque monkey: asymmetries, anisotropies, and individual variability,” Vision Res. 24, 429–448 (1984).
[CrossRef] [PubMed]

T. D. Albright, “Direction and orientation selectivity of neurons in visual area MT of the macaque,” J. Neurophysiol. 52, 1106–1130 (1984).
[PubMed]

1983 (1)

J. H. R. Maunsell, D. C. Van Essen, “Functional properties of neurons in middle temporal visual area of the macaque monkey. I. Selectivity for stimulus direction, speed, and orientation,” J. Neurophysiol. 49, 1127–1147 (1983).
[PubMed]

1982 (2)

D. G. Albrecht, D. B. Hamilton, “Striate cortex of monkey and cat: contrast response functions,” J. Neurophysiol. 48, 217–237 (1982).
[PubMed]

K. S. Rockland, J. S. Lund, “Widespread periodic intrinsic connections in the tree shrew visual-cortex,” Science 215, 1532–1534 (1982).
[CrossRef] [PubMed]

1980 (1)

G. Mather, B. Moulden, “A simultaneous shift in apparent direction: further evidence for a ‘distributional-shift’ model of direction coding,” Q. J. Exp. Psychol. 32, 325–333 (1980).
[CrossRef] [PubMed]

1979 (1)

W. Marshak, R. Sekuler, “Mutual repulsion between moving visual targets,” Science 205, 1399–1401 (1979).
[CrossRef] [PubMed]

1977 (1)

H. R. Wilson, “Hysterisis in binocular grating perception: contrast effects,” Vision Res. 17, 843–851 (1977).
[CrossRef]

1973 (2)

H. R. Wilson, J. D. Cowan, “A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue,” Kybernetik 13, 55–80 (1973).
[CrossRef] [PubMed]

S. Grossberg, “Contour enhancement, short-term memory and constances in reverberating neural networks,” Stud. Appl. Math. 52, 217–257 (1973).

1968 (1)

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of the monkey striate cortex,” J. Physiol. 195, 215–243 (1968).

1966 (1)

K. I. Naka, W. A. Rushton, “S-potentials from colour units in the retina of the fish,” J. Physiol. 185, 584–599 (1966).

1935 (1)

H. Wallach, “Über visuell wahrgenommene Bewegungsrichtung,” Psychol. Forsch. 20, 325–380 (1935).
[CrossRef]

Adelson, E. H.

Y. Weiss, E. P. Simoncelli, E. H. Adelson, “Motion illusions as optimal percepts,” Nat. Neurosci. 5, 598–604 (2002).
[CrossRef] [PubMed]

Albrecht, D. G.

D. G. Albrecht, D. B. Hamilton, “Striate cortex of monkey and cat: contrast response functions,” J. Neurophysiol. 48, 217–237 (1982).
[PubMed]

Albright, T. D.

H. R. Rodman, T. D. Albright, “Single unit analysis of patter-motion selective properties in the middle temporal area (MT),” Exp. Brain Res. 75, 53–64 (1989).
[CrossRef]

T. D. Albright, “Direction and orientation selectivity of neurons in visual area MT of the macaque,” J. Neurophysiol. 52, 1106–1130 (1984).
[PubMed]

Allman, J.

J. Allman, F. Miezin, E. McGuinness, “Stimulus specific responses from beyond the classical receptive-field: neurophysiological mechanisms for local–global comparisons in visual neurons,” Annu. Rev. Neurosci. 8, 407–430 (1985).
[CrossRef]

Amir, Y.

R. Malach, Y. Amir, M. Harel, A. Grinvald, “Relationship between intrinsic connections and functional architecture revealed by optical imaging and in-vivo targeted biocytin injections in primate striate cortex,” Proc. Natl. Acad. Sci. USA 90, 10469–10473 (1993).
[CrossRef]

Andersen, R. A.

R. A. Andersen, L. H. Snyder, C.-S. Li, B. Stricanne, “Coordinate transformations in the representation of spatial information,” Curr. Opin. Neurobiol. 3, 171–176 (1993).
[CrossRef] [PubMed]

Barbieri-Hesse, G. S. A.

M. A. Georgeson, G. S. A. Barbieri-Hesse, T. C. A. Freeman, “The primal sketch revisited: locating and representing edges in human vision via Gaussian-derivative filtering,” Perception 31, 1 (2002).

Beaudot, W. H. A.

R. F. Hess, W. H. A. Beaudot, K. T. Mullen, “Dynamics of contour integration,” Vision Res. 41, 1023–1037 (2001).
[CrossRef] [PubMed]

Ben-Av, M. B.

M. B. Ben-Av, M. Shiffrar, “Disambiguating velocity estimates across image space,” Vision Res. 35, 2889–2895 (1995).
[CrossRef] [PubMed]

M. B. Ben-Av, M. Shiffrar, “When ambiguous becomes unambiguous,” Invest. Ophthalmol. Visual Sci. 34, 1028–1028 (1993).

Bex, P. J.

P. J. Bex, A. J. Simmers, S. C. Dakin, “Snakes and ladders: the role of temporal modulation in visual contour integration,” Vision Res. 41, 3775–3782 (2001).
[CrossRef] [PubMed]

Blasdel, G. G.

G. G. Blasdel, D. Fitzpatrick, “Physiological organisation of layer-4 in macaque striate cortex,” J. Neurosci. 4, 880–895 (1984).
[PubMed]

Born, R. T.

C. C. Pack, R. T. Born, “Temporal dynamics of a neural solution to the aperture problem in visual area MT of macaque brain,” Nature 409, 1040–1042 (2001).
[CrossRef] [PubMed]

R. T. Born, R. B. H. Tootell, “Segregation of global and local motion processing in primate middle temporal visual area,” Nature 357, 497–499 (1992).
[CrossRef] [PubMed]

Bosking, W. H.

W. H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,” J. Neurosci. 17, 2112–2127 (1997).
[PubMed]

Bradley, D.

D. Bradley, “MT signals: better with time,” Nat. Neurosci. 4, 346–348 (2001).
[CrossRef] [PubMed]

Britten, K. H.

K. H. Britten, M. N. Shadlen, W. T. Newsome, J. A. Movshon, “The analysis of visual-motion—a comparison of neuronal and psychophysical performance,” J. Neurosci. 12, 4745–4765 (1992).
[PubMed]

C. D. Salzman, C. M. Murasugi, K. H. Britten, W. T. Newsome, “Microstimulation in visual area MT—effects on direction discrimination performance,” J. Neurosci. 12, 2331–2355 (1992).
[PubMed]

Castet, E.

E. Castet, V. Charton, A. Dufour, “The extrinsic/intrinsic classification of two-dimensional motion signals with barber-pole stimuli,” Vision Res. 39, 915–932 (1999).
[CrossRef] [PubMed]

E. Castet, S. Wuerger, “Perception of moving lines: in-teractions between local perpendicular signals and 2D motion signals,” Vision Res. 37, 705–720 (1997).
[CrossRef] [PubMed]

J. Lorenceau, M. Shiffrar, N. Wells, E. Castet, “Different motion sensitive units are involved in recovering the direction of moving lines,” Vision Res. 33, 1207–1217 (1993).
[CrossRef] [PubMed]

Charton, V.

E. Castet, V. Charton, A. Dufour, “The extrinsic/intrinsic classification of two-dimensional motion signals with barber-pole stimuli,” Vision Res. 39, 915–932 (1999).
[CrossRef] [PubMed]

Chey, J.

Cowan, J. D.

H. R. Wilson, J. D. Cowan, “A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue,” Kybernetik 13, 55–80 (1973).
[CrossRef] [PubMed]

Dakin, S. C.

P. J. Bex, A. J. Simmers, S. C. Dakin, “Snakes and ladders: the role of temporal modulation in visual contour integration,” Vision Res. 41, 3775–3782 (2001).
[CrossRef] [PubMed]

Dufour, A.

E. Castet, V. Charton, A. Dufour, “The extrinsic/intrinsic classification of two-dimensional motion signals with barber-pole stimuli,” Vision Res. 39, 915–932 (1999).
[CrossRef] [PubMed]

Ferrera, V. P.

H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
[CrossRef]

Field, D.

R. Hess, D. Field, “Integration of contours: new insights,” Trends Cogn. Sci. 3, 480–486 (1999).
[CrossRef] [PubMed]

Field, D. J.

D. J. Field, A. Hayes, R. F. Hess, “Contour integration by the human visual-system—evidence for a local association field,” Vision Res. 33, 173–193 (1993).
[CrossRef] [PubMed]

Fitzpatrick, D.

W. H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,” J. Neurosci. 17, 2112–2127 (1997).
[PubMed]

G. G. Blasdel, D. Fitzpatrick, “Physiological organisation of layer-4 in macaque striate cortex,” J. Neurosci. 4, 880–895 (1984).
[PubMed]

Freeman, T. C. A.

M. A. Georgeson, G. S. A. Barbieri-Hesse, T. C. A. Freeman, “The primal sketch revisited: locating and representing edges in human vision via Gaussian-derivative filtering,” Perception 31, 1 (2002).

Fukada, Y.

K. Tanaka, K. Hikosaka, H.-A. Saito, M. Yukie, Y. Fukada, E. Iwai, “Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey,” J. Neurosci. 6, 134–144 (1986).
[PubMed]

Gallogly, D. P.

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

Geisler, W. S.

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

Gelb, D. J.

Georgeson, M. A.

M. A. Georgeson, G. S. A. Barbieri-Hesse, T. C. A. Freeman, “The primal sketch revisited: locating and representing edges in human vision via Gaussian-derivative filtering,” Perception 31, 1 (2002).

Georgopoulos, A. P.

A. P. Georgopoulos, M. Taira, A. Lukashin, “Cognitive neurophysiology of the motor cortex,” Science 260, 47–52 (1993).
[CrossRef] [PubMed]

Goebel, R.

K. E. Schmidt, R. Goebel, S. Lowel, W. Singer, “The perceptual grouping criterion of collinearity is reflected by anisotropies of connections in the primary visual cortex,” Eur. J. Neurosci. 9, 1083–1089 (1997).
[CrossRef] [PubMed]

Grinvald, A.

R. Malach, Y. Amir, M. Harel, A. Grinvald, “Relationship between intrinsic connections and functional architecture revealed by optical imaging and in-vivo targeted biocytin injections in primate striate cortex,” Proc. Natl. Acad. Sci. USA 90, 10469–10473 (1993).
[CrossRef]

Grossberg, S.

J. Chey, S. Grossberg, E. Mingolla, “Neural dynamics of motion grouping: from aperture ambiguity to object speed and direction,” J. Opt. Soc. Am. A 14, 2570–2594 (1997).
[CrossRef]

S. Grossberg, E. Mingolla, “Neural dynamics of motion perception: direction fields, apertures, and resonant grouping,” Percept. Psychophys. 53, 248–278 (1993).

S. Grossberg, “Contour enhancement, short-term memory and constances in reverberating neural networks,” Stud. Appl. Math. 52, 217–257 (1973).

Grzywacz, N. M.

N. M. Grzywacz, A. L. Yuille, “Theories for the visual perception of local velocity and coherent motion,” in Computional Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 231–252.

Hamilton, D. B.

D. G. Albrecht, D. B. Hamilton, “Striate cortex of monkey and cat: contrast response functions,” J. Neurophysiol. 48, 217–237 (1982).
[PubMed]

Harel, M.

R. Malach, Y. Amir, M. Harel, A. Grinvald, “Relationship between intrinsic connections and functional architecture revealed by optical imaging and in-vivo targeted biocytin injections in primate striate cortex,” Proc. Natl. Acad. Sci. USA 90, 10469–10473 (1993).
[CrossRef]

Hayes, A.

D. J. Field, A. Hayes, R. F. Hess, “Contour integration by the human visual-system—evidence for a local association field,” Vision Res. 33, 173–193 (1993).
[CrossRef] [PubMed]

Hess, R.

R. Hess, D. Field, “Integration of contours: new insights,” Trends Cogn. Sci. 3, 480–486 (1999).
[CrossRef] [PubMed]

Hess, R. F.

R. F. Hess, W. H. A. Beaudot, K. T. Mullen, “Dynamics of contour integration,” Vision Res. 41, 1023–1037 (2001).
[CrossRef] [PubMed]

D. J. Field, A. Hayes, R. F. Hess, “Contour integration by the human visual-system—evidence for a local association field,” Vision Res. 33, 173–193 (1993).
[CrossRef] [PubMed]

Hikosaka, K.

K. Tanaka, K. Hikosaka, H.-A. Saito, M. Yukie, Y. Fukada, E. Iwai, “Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey,” J. Neurosci. 6, 134–144 (1986).
[PubMed]

Hildreth, E. C.

E. C. Hildreth, The Measurement of Visual Motion (MIT Press, Cambridge, Mass., 1984).

Hochstein, S.

N. Rubin, S. Hochstein, “Isolating the effect of one-dimensional motion signals on the perceived direction of moving 2-dimensional objects,” Vision Res. 33, 1385–1396 (1993).
[CrossRef] [PubMed]

Hubel, D. H.

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of the monkey striate cortex,” J. Physiol. 195, 215–243 (1968).

Iwai, E.

K. Tanaka, K. Hikosaka, H.-A. Saito, M. Yukie, Y. Fukada, E. Iwai, “Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey,” J. Neurosci. 6, 134–144 (1986).
[PubMed]

Kim, J.

J. Kim, H. R. Wilson, “Motion integration over space: interaction of the center and surround motion,” Vision Res. 37, 991–1005 (1997).
[CrossRef] [PubMed]

J. Kim, H. R. Wilson, “Direction repulsion between components in motion transparency,” Vision Res. 36, 1177–1187 (1996).
[CrossRef] [PubMed]

H. R. Wilson, J. Kim, “Perceived motion in the vector sum direction,” Vision Res. 34, 1835–1842 (1994).
[CrossRef] [PubMed]

H. R. Wilson, J. Kim, “A model for motion coherence and transparency,” Visual Neurosci. 11, 1205–1220 (1994).
[CrossRef]

J. Kim, H. R. Wilson, “Dependence of plaid motion coherence on component grating directions,” Vision Res. 33, 2479–2489 (1993).
[CrossRef] [PubMed]

Kooi, F. L.

F. L. Kooi, “Local direction of edge motion causes and abolishes the barberpole illusion,” Vision Res. 33, 2347–2351 (1993).
[CrossRef] [PubMed]

Lennie, P.

G. Sclar, J. R. Maunsell, P. Lennie, “Coding of image contrast in central visual pathways of the macaque monkey,” Vision Res. 30, 1–10 (1990).
[CrossRef] [PubMed]

Li, C.-S.

R. A. Andersen, L. H. Snyder, C.-S. Li, B. Stricanne, “Coordinate transformations in the representation of spatial information,” Curr. Opin. Neurobiol. 3, 171–176 (1993).
[CrossRef] [PubMed]

Liden, L.

For example, L. Liden, C. Pack, “The role of terminators and occlusion cues in motion integration and segmentation: a neural network model,” Vision Res. 39, 3301–3320 (1999).
[CrossRef]

L. Liden, E. Mingolla, “Monocular occlusion cues alter the influence of terminator motion in the barber pole phenomenon,” Vision Res. 38, 3883–3898 (1998).
[CrossRef]

Loffler, G.

G. Loffler, H. S. Orbach, “Factors affecting motion integration,” J. Opt. Soc. Am. A 20, 1461–1471 (2003).
[CrossRef]

H. S. Orbach, G. Loffler, “Motion integration across apertures: theory and experiment,” Invest. Ophthalmol. Visual Sci. 42, 4685 (2001).

G. Loffler, H. S. Orbach, “Computing feature motion without feature detectors: a model for terminator motion without end-stopped cells,” Vision Res. 39, 859–871 (1999).
[CrossRef] [PubMed]

G. Loffler, “The integration of motion signals across space,” Ph.D. thesis (Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, UK, 1999).

Lorenceau, J.

J. Lorenceau, L. Zago, “Cooperative and competitive spatial interactions in motion integration,” Vision Res. 16, 755–770 (1999).

J. Lorenceau, M. Shiffrar, N. Wells, E. Castet, “Different motion sensitive units are involved in recovering the direction of moving lines,” Vision Res. 33, 1207–1217 (1993).
[CrossRef] [PubMed]

J. Lorenceau, M. Shiffrar, “The influence of terminators on motion integration across space,” Vision Res. 32, 263–273 (1992).
[CrossRef] [PubMed]

Lowel, S.

K. E. Schmidt, R. Goebel, S. Lowel, W. Singer, “The perceptual grouping criterion of collinearity is reflected by anisotropies of connections in the primary visual cortex,” Eur. J. Neurosci. 9, 1083–1089 (1997).
[CrossRef] [PubMed]

Lukashin, A.

A. P. Georgopoulos, M. Taira, A. Lukashin, “Cognitive neurophysiology of the motor cortex,” Science 260, 47–52 (1993).
[CrossRef] [PubMed]

Lund, J. S.

K. S. Rockland, J. S. Lund, “Widespread periodic intrinsic connections in the tree shrew visual-cortex,” Science 215, 1532–1534 (1982).
[CrossRef] [PubMed]

Malach, R.

R. Malach, Y. Amir, M. Harel, A. Grinvald, “Relationship between intrinsic connections and functional architecture revealed by optical imaging and in-vivo targeted biocytin injections in primate striate cortex,” Proc. Natl. Acad. Sci. USA 90, 10469–10473 (1993).
[CrossRef]

Marcar, V. L.

S. Raiguel, M. M. Van Hulle, D.-K. Xiao, V. L. Marcar, G. A. Orban, “Shape and spatial distribution of receptive fields and antagonistic motion surrounds in the middle temporal area (V5) of the macaque,” Eur. J. Neurosci. 7, 2064–2082 (1995).
[CrossRef] [PubMed]

Marshak, W.

W. Marshak, R. Sekuler, “Mutual repulsion between moving visual targets,” Science 205, 1399–1401 (1979).
[CrossRef] [PubMed]

Mather, G.

G. Mather, B. Moulden, “A simultaneous shift in apparent direction: further evidence for a ‘distributional-shift’ model of direction coding,” Q. J. Exp. Psychol. 32, 325–333 (1980).
[CrossRef] [PubMed]

Maunsell, J. H. R.

J. H. R. Maunsell, D. C. Van Essen, “Functional properties of neurons in middle temporal visual area of the macaque monkey. I. Selectivity for stimulus direction, speed, and orientation,” J. Neurophysiol. 49, 1127–1147 (1983).
[PubMed]

Maunsell, J. R.

G. Sclar, J. R. Maunsell, P. Lennie, “Coding of image contrast in central visual pathways of the macaque monkey,” Vision Res. 30, 1–10 (1990).
[CrossRef] [PubMed]

McGuinness, E.

J. Allman, F. Miezin, E. McGuinness, “Stimulus specific responses from beyond the classical receptive-field: neurophysiological mechanisms for local–global comparisons in visual neurons,” Annu. Rev. Neurosci. 8, 407–430 (1985).
[CrossRef]

Miezin, F.

J. Allman, F. Miezin, E. McGuinness, “Stimulus specific responses from beyond the classical receptive-field: neurophysiological mechanisms for local–global comparisons in visual neurons,” Annu. Rev. Neurosci. 8, 407–430 (1985).
[CrossRef]

Mingolla, E.

L. Liden, E. Mingolla, “Monocular occlusion cues alter the influence of terminator motion in the barber pole phenomenon,” Vision Res. 38, 3883–3898 (1998).
[CrossRef]

J. Chey, S. Grossberg, E. Mingolla, “Neural dynamics of motion grouping: from aperture ambiguity to object speed and direction,” J. Opt. Soc. Am. A 14, 2570–2594 (1997).
[CrossRef]

S. Grossberg, E. Mingolla, “Neural dynamics of motion perception: direction fields, apertures, and resonant grouping,” Percept. Psychophys. 53, 248–278 (1993).

E. Mingolla, J. T. Todd, J. F. Norman, “The perception of globally coherent motion,” Vision Res. 32, 1015–1031 (1992).
[CrossRef] [PubMed]

Moulden, B.

G. Mather, B. Moulden, “A simultaneous shift in apparent direction: further evidence for a ‘distributional-shift’ model of direction coding,” Q. J. Exp. Psychol. 32, 325–333 (1980).
[CrossRef] [PubMed]

Movshon, J. A.

J. A. Movshon, W. T. Newsome, “Visual response properties of striate cortical neurons projecting to area MT in macaque monkeys,” J. Neurosci. 16, 7733–7741 (1996).
[PubMed]

K. H. Britten, M. N. Shadlen, W. T. Newsome, J. A. Movshon, “The analysis of visual-motion—a comparison of neuronal and psychophysical performance,” J. Neurosci. 12, 4745–4765 (1992).
[PubMed]

Mullen, K. T.

R. F. Hess, W. H. A. Beaudot, K. T. Mullen, “Dynamics of contour integration,” Vision Res. 41, 1023–1037 (2001).
[CrossRef] [PubMed]

Mulligan, J. B.

L. S. Stone, A. B. Watson, J. B. Mulligan, “Effect of contrast on the perceived direction of a moving plaid,” Vision Res. 30, 1049–1067 (1990).
[CrossRef] [PubMed]

Murasugi, C. M.

C. D. Salzman, C. M. Murasugi, K. H. Britten, W. T. Newsome, “Microstimulation in visual area MT—effects on direction discrimination performance,” J. Neurosci. 12, 2331–2355 (1992).
[PubMed]

Naka, K. I.

K. I. Naka, W. A. Rushton, “S-potentials from colour units in the retina of the fish,” J. Physiol. 185, 584–599 (1966).

Nakayama, K.

S. Shimojo, G. H. Silverman, K. Nakayama, “Occlusion and the solution to the aperture problem for motion,” Vision Res. 29, 619–626 (1989).
[CrossRef] [PubMed]

K. Nakayama, G. H. Silverman, “The aperture problem I. Perception of nonrigidity and motion direction in translating sinusoidal lines,” Vision Res. 28, 739–746 (1988).
[CrossRef]

Nawrot, M.

M. Nawrot, R. Sekuler, “Assimilation and contrast in motion perception—explorations in cooperativity,” Vision Res. 30, 1439–1451 (1990).
[CrossRef]

Newsome, W. T.

J. A. Movshon, W. T. Newsome, “Visual response properties of striate cortical neurons projecting to area MT in macaque monkeys,” J. Neurosci. 16, 7733–7741 (1996).
[PubMed]

C. D. Salzman, W. T. Newsome, “Neural mechanisms for forming a perceptual decision,” Science 264, 231–237 (1994).
[CrossRef] [PubMed]

C. D. Salzman, C. M. Murasugi, K. H. Britten, W. T. Newsome, “Microstimulation in visual area MT—effects on direction discrimination performance,” J. Neurosci. 12, 2331–2355 (1992).
[PubMed]

K. H. Britten, M. N. Shadlen, W. T. Newsome, J. A. Movshon, “The analysis of visual-motion—a comparison of neuronal and psychophysical performance,” J. Neurosci. 12, 4745–4765 (1992).
[PubMed]

Norman, J. F.

E. Mingolla, J. T. Todd, J. F. Norman, “The perception of globally coherent motion,” Vision Res. 32, 1015–1031 (1992).
[CrossRef] [PubMed]

Nowlan, S. J.

S. J. Nowlan, T. J. Sejnowski, “A selection model for motion processing in area MT of primates,” J. Neurosci. 15, 1195–1214 (1995).
[PubMed]

Orbach, H. S.

G. Loffler, H. S. Orbach, “Factors affecting motion integration,” J. Opt. Soc. Am. A 20, 1461–1471 (2003).
[CrossRef]

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Z. Y. Yang, A. Shimpi, D. Purves, “A wholly empirical explanation of perceived motion,” Proc. Natl. Acad. Sci. USA 98, 5252–5257 (2001).

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K. Tanaka, K. Hikosaka, H.-A. Saito, M. Yukie, Y. Fukada, E. Iwai, “Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey,” J. Neurosci. 6, 134–144 (1986).
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[CrossRef]

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[CrossRef]

J. Kim, H. R. Wilson, “Motion integration over space: interaction of the center and surround motion,” Vision Res. 37, 991–1005 (1997).
[CrossRef] [PubMed]

E. Mingolla, J. T. Todd, J. F. Norman, “The perception of globally coherent motion,” Vision Res. 32, 1015–1031 (1992).
[CrossRef] [PubMed]

J. Lorenceau, L. Zago, “Cooperative and competitive spatial interactions in motion integration,” Vision Res. 16, 755–770 (1999).

J. Lorenceau, M. Shiffrar, “The influence of terminators on motion integration across space,” Vision Res. 32, 263–273 (1992).
[CrossRef] [PubMed]

W. S. Geisler, J. S. Perry, B. J. Super, D. P. Gallogly, “Edge co-occurrence in natural images predicts contour grouping performance,” Vision Res. 41, 711–724 (2001).
[CrossRef] [PubMed]

U. Polat, D. Sagi, “Lateral interactions between spatial channels—suppression and facilitation revealed by lateral masking experiments,” Vision Res. 33, 993–999 (1993).
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K. Nakayama, G. H. Silverman, “The aperture problem I. Perception of nonrigidity and motion direction in translating sinusoidal lines,” Vision Res. 28, 739–746 (1988).
[CrossRef]

S. Shimojo, G. H. Silverman, K. Nakayama, “Occlusion and the solution to the aperture problem for motion,” Vision Res. 29, 619–626 (1989).
[CrossRef] [PubMed]

E. Castet, V. Charton, A. Dufour, “The extrinsic/intrinsic classification of two-dimensional motion signals with barber-pole stimuli,” Vision Res. 39, 915–932 (1999).
[CrossRef] [PubMed]

E. Castet, S. Wuerger, “Perception of moving lines: in-teractions between local perpendicular signals and 2D motion signals,” Vision Res. 37, 705–720 (1997).
[CrossRef] [PubMed]

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Other (13)

W. Reichardt, “Autocorrelation, a principle for the evaluation of sensory information by the central nervous system,” in Sensory Communication, W. A. Rosenblith, ed. (Wiley, New York, 1961), pp. 303–317.

Note that neither the inputs, Ix,y[Eq. (1)], nor the recurrent inhibition [Eq. (2)] extends over the entire range of directions (±180°) but instead are restricted to relative angles of ±120°. The only reason for this limitation is to allow the local network to signal more than one direction of motion in circumstances of transparency.27Following this argument, such a network signals transparency by a bimodality in MT pattern unit responses. However, in agreement with psychophysical data,7all global simulations presented here resulted in a single direction of motion, and the simulations never predicted transparency.

To simulate the temporal dynamics of the coupled differential equations, the fast Euler method has been employed.To verify this approach, initial sample simulations were undertaken in which the Euler method was compared with the considerably slower but more stable fourth-order Runge–Kutta method. These sample simulations proved that for a sufficiently low step size of 1/4τ, the two methods give indistinguishable results, with the Euler method being faster by a factor of ∼3.

N. M. Grzywacz, A. L. Yuille, “Theories for the visual perception of local velocity and coherent motion,” in Computional Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 231–252.

E. C. Hildreth, The Measurement of Visual Motion (MIT Press, Cambridge, Mass., 1984).

G. Loffler, “The integration of motion signals across space,” Ph.D. thesis (Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, UK, 1999).

Note the difference between this term, which depends on line-segment orientation, and the motion-discontinuity term in Eq. (4), which depends on direction of motion.

To simulate this condition, a small amount of (random) noise was added to the initial V1 simple cell responses. Without this noise, it would be impossible to extract a maximally excited cell, as all cells would have the same firing rate. The randomly added noise (unique to this condition) is the reason for the small capturing behavior of the network for the smallest gap which represents the average over 20 model simulations.

H. R. Wilson, “Psychophysical models of spatial vision and hyperacuity,” in Spatial Vision, D. Regan, ed. (MacMillan, New York, 1991), pp. 64–86.

There are two conceptually different ways to treat neural sites corresponding to locations of a scene without stimulation (e.g., aperture gaps). Grzywacz and Yuille14proposed a model in which lateral interactions result in motion signals at every point of the visual field regardless of whether the field was initially stimulated by a moving object. It is unclear whether this correctly reflects neurophysiology. The approach taken here is different. Spatial sites that are not initially activated by motion in their receptive field are not activated by lateral interactions; rather, they stay silent. This is consistent with the approach of relating the activity of MT pattern neurons directly to behavior48,49and noting that, behaviorally, parts of the visual field that have not been stimulated do not appear to have motion associated with them. Mathematically, this approach is achieved by modulating lateral interactions by a site’s activity: If a site does not receive any bottom-up input through its local computations, lateral excitation stays silent.

H. R. Wilson, Spikes, Decisions, and Actions (Oxford U. Press, Oxford, UK, 1999).

The nature and stability of the steady state was estimated by considering the linear terms of a Taylor expansion of the nonlinear dynamics at the equilibrium points (for details of this approach see Ref. 85). The corresponding exponentials have negative real parts, and the equilibrium is consequently stable.

It is important to emphasize that this kind of undamped propagation does not necessarily result in a single direction of motion in the more complicated case of a dynamic multiobject environment. Signal propagation is strong only for adjacent or overlapping sites. As our experiments show, any gap between sites weakens propagation and allows different directions of motion for nearby objects in a scene.

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Figures (17)

Fig. 1
Fig. 1

Rigid moving diamond. The large arrow represents the veridical velocity. The circles show areas where local motion estimation could take place, and the small arrows give the local-motion estimation at the location of the corners (terminator) or the featureless edges.

Fig. 2
Fig. 2

Outline of the global model. Three identical copies of a local-motion model are applied in parallel to different parts of the stimulus (in this case, a line behind a three-aperture mask). Each of the local-motion models contains two parallel pathways (Fourier and non-Fourier) which extract the motion of luminance boundaries and texture boundaries, respectively. The convolution of the stimulus (A) with differently oriented V1 simple-cell filters (B) defines the filters’ sensitivity function. (For clarity, the figure shows only one orientation.) A power-law function (C) models the nonlinear simple-cell contrast response. Subsequently, the signal is processed in parallel. The Fourier pathway extracts motion by using directionally tuned Reichardt detectors (GF). The output is normalized (HF) by a feedforward divisive term that is calculated in terms of the sum over differently oriented V1 simple cells’ responses. MT component cells (firing rates shown in IF) sum the outputs of motion units located at different spatial positions (omitted for clarity) but tuned to the same direction of motion. In the non-Fourier pathway, V1 simple cells’ responses are squared (D) and second-stage filtered (assumed to be carried out by cells in area V2) (E). These filters are tuned to a lower spatial frequency and are oriented orthogonal to the initial filters to extract texture boundaries. Qualitatively, the same steps follow as described for the Fourier pathway: power-law nonlinearity (F), texture boundary motion (GNF), feedforward normalization (HNF), and MT component cell (INF) pooling. Finally, the signals of Fourier and non-Fourier pathways are combined at the level of MT pattern units (K). The output for each local model in the absence of any lateral interconnections (i.e., a multi-local model) would give a different direction of motion for each site. In that case, the central site would signal an incorrect direction of motion. The global model differs from the multilocal version by the lateral interactions at the level of model MT pattern cells indicated at the bottom of the picture.

Fig. 3
Fig. 3

Lateral interactions at the level of model MT pattern units. Each pattern unit is sensitive to a different direction of motion (arrows); 24 such units span the whole range of motions in 15° increments. The excitatory signal (+) between units tuned to similar directions of motion at different locations depends on the distance between locations (weighting defined by a Gaussian function of distance) and is modulated (*) by hypothesized motion-discontinuity detectors, which calculate the difference in motion energy between pattern units at different locations (dashed lines); the detectors are necessary to avoid mutually excitatory feedback when two locations signal the same direction of motion (see Appendix A and text for details). A Naka–Rushton (NR) nonlinearity is included for physiological plausibility and, in addition, avoids signal explosion in such an excitatory network.

Fig. 4
Fig. 4

Simulation methods. Each circular area depicts the receptive fields of a set of model MT pattern units (each tuned to a different direction of motion). The left and center panels show the three sites that are active for the line stimuli with small and large inter-aperture gaps, respectively. The icon on the right shows the situation in condition 4C where the diameter of the central aperture was manipulated. It can be seen that there is some overlap of receptive fields in this condition and the number of active units increases with the size of the central aperture (dashed circle).

Fig. 5
Fig. 5

Results of the local model applied independently to each of three sites (upper and lower terminator and central featureless line segment). The left panel shows the stimulus and its physical direction of motion. The line orientation is 45° relative to the horizontal, its width is 0.25°, and its motion is upward (arrow) at 5°/s, matching one of the psychophysical conditions. Contrast was 1.0, and the diameter of each of the circular apertures was 1.6°. The central panels plot the responses of model MT component cells as a function of preferred motion direction for both pathways (Fourier and non-Fourier). The bold numbers next to the polar plots indicate signal strength. The arrow in the right panel depicts the final model output, the direction of motion, at the level of MT pattern units. For terminators, the shape of the polar plots for the Fourier pathway exhibit a broad bandwidth with a bias (>15°) perpendicular to the line orientation. In the case of the non-Fourier units, the response curve is sharply tuned with a peak always biased toward the line orientation. Neither of the pathways alone signals the veridical direction of motion. The responses in the case of isolated Fourier (F) and non-Fourier (NF) pathways are shown inside the polar plots. However, the combined network response (Σ, right column) is very close to the true physical direction of motion.13 For central line segment, as would be expected on the basis of the aperture problem, the computed direction of motion is perpendicular to the line orientation (note that there is no contribution from the non-Fourier pathway for line segments). For subsequent simulations it is of note that the signal strength of the line segment’s Fourier pathway slightly exceeds that of the terminator sites.

Fig. 6
Fig. 6

Results for the effect of the size of the gap between apertures. The plots show perceived and predicted direction of motion of the central, featureless segment as a function of the gap between adjacent apertures. As indicated by the icons on the right, the direction of motion is shown relative to the physical direction of the line. The model predictions, given by the squares connected by the solid lines, are in close agreement with psychophysical data7 for all inter-aperture gaps and line tilts (45°, center; 30°, lower left; and 60°, lower right). Error bars for the data are standard errors of the mean. Regardless of line tilt, the model correctly predicts motion capture of the central segment by the adjacent terminators for small gaps, a direction orthogonal to the line orientation for large gaps, and an intermediate direction in between.

Fig. 7
Fig. 7

Model predictions with strong and weak space-constant parameter [σ, Eq. (5)]. Solid circles give model predictions for a high space constant (2.5, strong lateral interactions), and solid squares show results for a low space constant (1.5, weak interactions). As can be seen, a small change of a single model parameter can account for the variability across individual subjects. Note that the space constant was not used as a free parameter for subsequent simulations, but instead the same constant (σ=2) was employed throughout the simulations presented here.

Fig. 8
Fig. 8

Results for the effect of presentation time. Data are for a 105-ms presentation time compared with 200-ms time in the previous experiment. The only difference between this condition and the longer presentation (Fig. 6) is the somewhat reduced capturing effect for an intermediate gap (2°), which is convincingly matched by the model.

Fig. 9
Fig. 9

Time course for a small, intermediate, and large gap in the case of a 45° oriented line. Different conditions give rise to different asymptotic steady states. Moreover, the time taken to reach this asymptotic level differs: It is shorter for a small inter-aperture gap (solid line) than for an intermediate gap (dashed line). For large gaps (dotted line), lateral interactions show no effect.

Fig. 10
Fig. 10

Effect of changing the diameter of the central aperture and therefore altering the gap between apertures while keeping the physical distance between terminators and central line segment fixed. The model predicts experimental observations: The crucial variable for motion capture is the gap and not the physical distance.

Fig. 11
Fig. 11

Effect of misorientation (0°, top left; 10°, top right; 20°, bottom left; 45°, bottom right) between central line segment and line terminators (icons). The model incorporating collinear facilitation correctly predicts motion capture for small gaps and up to moderate skews (10° and 20°) but lack of capture for grossly misoriented contours (45°). Without collinear facilitation, the model predicts motion capture for small gaps regardless of skew, at variance with experimental data.

Fig. 12
Fig. 12

Results for stimuli defined by a cross-sectional profile of a D6 pattern. The spatial frequency of the central patch is fixed (1.7 cpd), but that of the truncations varies (1.7 cpd, top left; 2.8 cpd, top right; 5.0 cpd, bottom left). The single-spatial-frequency model is in qualitative agreement with experimental data but slightly underestimates the effect of capture for dissimilar spatial frequencies. This suggests that lateral interactions between motion signals arising in different parts of the visual field are restricted to a small range of spatial frequencies but are not completely isolated to a single channel.

Fig. 13
Fig. 13

Effect of dots on line segments. Model predictions are in agreement with psychophysics. There is little effective interaction between dotlike features and line segments even if they share the same spatial-frequency components. For the model simulation, this behavior is due to the lack of collinear facilitation, because dots do not exhibit an orientational preference. The small capturing effect for the smallest gap is a result of adding random noise to the V1 simple-cell outputs feeding into the collinear facilitation mechanism. Occasionally, the noise supports the physical direction of motion, and the averaged simulation shows a small bias toward the real direction of motion.

Fig. 14
Fig. 14

Lateral excitation and multiplicative modulation. Hypothesized motion-discontinuity detectors are shown below the line stimulus. The dashed lines indicate the sites from which they receive inputs. The output of the differential operators is used to modulate (*) the excitatory (+) lateral interactions. In the model, discontinuity detectors are driven by directionally selective units at different sites. Hence they correspond to neurons that respond to directionally specific motion discontinuities.

Fig. 15
Fig. 15

Collinear facilitation where lateral interactions depend on the stimulus configuration. The thick lines with solid arrowheads indicate strong interactions between sites that share the same orientational preference. There are no interactions (“0,” open arrowheads) when orientations are sufficiently different. It is not yet clear what kinds of interactions there are in cases where orientations are widely different (e.g., perpendicular). It is conceivable that the interactions, rather than being absent, should be negative, i.e., inhibitory. Further work is required to determine these interactions (indicated by “?”).

Fig. 16
Fig. 16

Left: Four-neuron, two-site network representing the interactions between model MT pattern units. For simplicity, each site (A and B) contains only two neurons (A1 and A2, B1 and B2) selective for one of two directions of motion (e.g., up or down). Neurons at the same site mutually inhibit each other in a winner-take-all fashion (dashed lines with minus signs). In addition to this intra-site winner-take-all computation, there are inter-site excitatory connections (solid lines with plus signs) between sites signaling the same direction of motion (A1 and B1, A2 and B2). These inter-site interactions are responsible for the capturing effect. Right, example of the temporal dynamics of a four-neuron, two-site network. Capture is evident by the fact that at one site (B) the neuron (B1) wins the competition over its partner (B2) despite receiving a much weaker input. This is due to the lateral excitation between units signaling the same direction of motion (A1 and B1) and a strong input to A1 (see text for details).

Fig. 17
Fig. 17

Propagation of an unambiguous terminator signal. Initially and in parallel (top row) many local computations (circles) occur at different sites along the line. Each circle represents the area over which a local-motion model receives its input. Only at the terminator location does the local-model output match the physical direction of motion of the line (in this case upward and to the right). At all the other positions the computed directions (shown by the bold arrows) are perpendicular to the line’s orientation. The arrows under each circle show the lateral excitatory interactions between sites. A successful model should exhibit the behavior portrayed on the figure: The terminator signal first captures the site next to it (second row), and the captured signal in turn affects its neighboring site (third row) and so on until all sites signal the same, veridical motion (bottom row). At each step, the most active lateral connection is highlighted.  

Tables (2)

Tables Icon

Table 1 Inhibitory Intra-Site Weighting, i , and Excitatory Inter-Site Weighting, e , among Pattern Units as a Function of Difference in Directional Tuning (Angle)

Tables Icon

Table 2 Parameters for the Two Naka–Rushton Functions Describing the Nonlinearities of the Postsynaptic Potentials of MT Pattern Units (NRP) and Motion Discontinuity Detectors (NRE)

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Iϕ,x,y=θ=ϕ-120°θ=ϕ+120°{[MTcomponentF(θ)+αMTcomponentNF(θ)]cos(ϕ-θ)},
dPϕ,x,ydt=1τ-Pϕ,x,y+NRIϕ,x,y-θ=ϕ-120θ=ϕ+120iθ Pθ,x,y,
dPϕ,x,ydt=1τ-Pϕ,x,y+NRPIϕ,x,y-θ=ϕ-120θ=ϕ+120iθ Pθ,x,y+Eϕ,x,y.
Eϕ,x,y=x,yd(x-x, y-y)θ=ϕ-15ϕ+15(eθ Pθ,x,y)×NRE(|Pϕ,x,y-Pϕ,x,y|).
d(x, y)=a exp[-(x2+y2)/σ2].
d(x, x)=a exp[-(x-x)2/σ2]×|cos[b2π(orimax,x-orimax,x)]|>0.
dA1dt=1τ [-A1+NR(SA1-iA2+eB1)],
dA2dt=1τ [-A2+NR(SA2-iA1+eB2)],
dB1dt=1τ [-B1+NR(SB1-iB2+eA1)],
dB2dt=1τ [-B2+NR(SB2-iB1+eA2)],
NR(x)=RmaxxNμN+xN.
SA1=10,SB1=0,
SA2=5,SB2=8.
SA1=10,SB1=0,SC1=0,
SD1=0,SE1=0,SF1=0,
SA2<1h>=8,SB2=8,SC2=8,
SD2=8,SE2=8,SF2=8.
dA1dt=1τ {-A1+NR[SA1-iA2+eB1(|A1-B1|)]}.

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