{"title": "Extending position/phase-shift tuning to motion energy neurons improves velocity discrimination", "book": "Advances in Neural Information Processing Systems", "page_first": 809, "page_last": 816, "abstract": null, "full_text": "Extending position/phase-shift tuning to motion \nenergy neurons improves velocity discrimination\n\nStanley Yiu Man Lam and Bertram E. Shi\n\nDepartment of Electronic and Computer Engineering\nHong Kong Univeristy of Science and Technology\n\nClear Water Bay, Kowloon, Hong Kong\n\n{eelym,eebert}@ee.ust.hk\n\nAbstract\n\nWe extend position and phase-shift tuning, concepts already well established in\nthe disparity energy neuron literature, to motion energy neurons. We show that\nReichardt-like detectors can be considered examples of position tuning, and that\nmotion energy filters whose complex valued spatio-temporal receptive fields are\nspace-time separable can be considered examples of phase tuning. By combining\nthese two types of detectors, we obtain an architecture for constructing motion\nenergy neurons whose center frequencies can be adjusted by both phase and posi-\ntion shifts. Similar to recently described neurons in the primary visual cortex,\nthese new motion energy neurons exhibit tuning that is between purely space-\ntime separable and purely speed tuned. We propose a functional role for this\nintermediate level of tuning by demonstrating that comparisons between pairs of\nthese motion energy neurons can reliably discriminate between inputs whose\nvelocities lie above or below a given reference velocity.\n\n1 Introduction\nImage motion is an important cue used by both biological and artificial visual systems to extract\ninformation about the environment. Although image motion is commonly used, there are different\nmodels for image motion processing in different systems. The Reichardt model is a dominant\nmodel for motion detection in insects, where image motion analysis occurs at a very early stage [1].\nFor mammals, the bulk of visual processing for motion is thought to occur in the cortex, and the\nmotion energy model is one of the dominant models [2][3]. However, despite the differences in\ncomplexity between these two models, they are mathematically equivalent given appropriate\nchoices of the spatial and temporal filters [4].\nThe motion energy model is very closely related to the disparity energy model, which has been\nused to model the outputs of disparity selective neurons in the visual cortex [5]. The disparity tun-\ning of neurons in this model can be adjusted via two mechanisms: a position shift between the cen-\nter locations of the receptive fields in the left and right eyes or a phase shift between the receptive\nfield organization in the left and right eyes [6][7]. It appears that biological systems use a combina-\ntion of these two mechanisms. \nIn Section 2, we extend the concepts of position and phase tuning to the construction of motion\nenergy neurons. We combine the Reichardt model and the motion energy model to obtain an archi-\ntecture for constructing motion energy neurons whose tuning can be adjusted by the analogs of\nposition and phase shifts. In Section 3, we investigate the functional advantages of position and\nphase shifts, inspired by a similar comparison from the disparity energy literature. We show that by\nsimply comparing the outputs of pair of motion energy cells with combined position/phase shift\ntuning enables us to discriminate reliably between stimuli moving above and below a reference\n\n\fvelocity. Finally, in Section 4, we place this work in the context of recent results on speed tuning in\nV1 neurons.\n\n)\n\n(\n\n\u2013\n\n). \n\n2ab=\n\n)2 a2 b2+\n\n2 Extending Position/Phase Tuning to Motion Energy Models\nFigure 1(a) shows a 1D array of three Reichardt detectors[1] tuned to motion from left to right.\nEach detector computes the correlation between its photosensor input and the delayed input from\nthe photosensor to the left. The delay could be implemented by a low pass filter. Usually, the corre-\nlation is assumed to be computed by a multiplication between the current and delayed signals. For\nconsistency with the following discussion, we show the output as a summation followed by a\nsquaring. Squaring the sum is essentially equivalent to the product, since the product could be\nrecovered by subtracting the sum of the squared inputs from the squared sum (e.g.\na b+(\nDelbruck proposed a modification of the Reichardt detector (Figure 1(b)), which switches the order\nof the delay and the sum, resulting in a delay-line architecture [8]. The output of a detector is the\nsum of its photosensor input and the delayed output of the detector to the left. This recurrent con-\nnection extends the spatio-temporal receptive field of the detector, since the input from the second-\nnearest-neighboring photosensor to the left is now connected to the detector through two delays,\nwhereas the Reichardt detector never sees the output of its second-nearest-neighboring photosen-\nsor.\nThe velocity tuning of these detectors is determined by the combination of the temporal delay and\nthe position shift between the neighboring detectors. As the delay increases, the tuned velocity\ndecreases. As the position shift increases, the tuned velocity also increases. This position-tuning of\nvelocity is reminiscent of the position-tuning of disparity energy neurons, where the larger the posi-\ntion shift between the spatial receptive fields being combined from the left and right eyes, the larger\nthe disparity tuning [9].\nFigure 1(c) shows a 1D array of three motion energy detectors[2][3]. At each spatial location, the\noutputs of the photosensors in a neighborhood around each spatial location are combined through\neven and odd symmetric linear spatial receptive fields, which are here modelled by spatial Gabor\nfunctions. In 1D, the even and odd symmetric Gabor receptive field profiles are the real and imagi-\nnary parts of the function\n\ngs x( )\n\n=\n\n1\n-----------------\n2\u03c0\u03c3x\n\nexp\n\n\u2013\n\n\u239b\n\u239c\n\u239d\n\n\u239e\nx2\n---------\n\u239f\n\u23a0\n2\n2\u03c3x\n\nexp\n\n(\n\njx\u03a9x\n)\n\n=\n\n1\n-----------------\n2\u03c0\u03c3x\n\nexp\n\n\u2013\n\n\u239b\n\u239c\n\u239d\n\n\u239e\nx2\n---------\n\u239f\n\u23a0\n2\n2\u03c3x\n\n(\n\ncos\n\n\u03a9xx(\n)\n\n+\n\nj\n\nsin\n\n\u03a9xx(\n)\n\n)\n\n(1)\n\n\u03a9x\n\n determines the preferred spatial frequency of the receptive field, and \n\nwhere \n determines its\nspatial extent. The even and odd spatial filter outputs are then combined through temporal filters to\nproduce two outputs which are then squared and summed to produce the motion energy. In many\ncases, the temporal receptive field profiles are also Gabor functions. The combined spatial and tem-\nporal receptive fields of the two neurons are separable when considered as a single complex valued\nfunction:\n\n\u03c3x\n\ng x t,(\n)\n\n=\n\n1\n-----------------\n2\u03c0\u03c3x\n\nexp\n\n\u2013\n\n\u239b\n\u239c\n\u239d\n\n\u239e\nx2\n---------\n\u239f\n\u23a0\n2\n2\u03c3x\n\nexp\n\n(\n\nj\u03a9xx\n)\n\n\u22c5\n\n1\n-----------------\n2\u03c0\u03c3t\n\nexp\n\n\u2013\n\n\u239b\n\u239c\n\u239d\n\n\u239e\nt2\n---------\n\u239f\n\u23a0\n2\n2\u03c3t\n\nexp\n\n(\n\nj\u03a9tt\n)\n\n(2)\n\n\u03c3t\n\n\u03a9t\n\n and \n\n determine the preferred temporal frequency and temporal extent of the temporal\nwhere \nreceptive fields. Strictly speaking, these spatio-temporal filters are not velocity tuned, since the\nvelocity at which a moving sine-wave grating stimulus produces maximum response varies with\nthe spatial frequency of the sine-wave grating. However, since spatial frequencies of \n lead to the\nthought of as having a preferred velocity\nlargest responses, \n.\n\u03a9t \u03a9x\u2044\nvpref\n\nis sometimes \n\nthe filter \n\n\u2013=\n\n\u03a9x\n\n\f2\n\n2\n\n2\n\n\u03c4\n\n\u03c4\n\n(a)\n\n2\n\n2\n\n2\n\n\u03c4\n\n\u03c4\n\n2\n\n\u03c4\n\n2\n\n\u03c4\n\na\nej\n\u03a9\n\n2\n\n\u03c4\n\n2\n\n\u03c4\n\na\nej\n\u03a9\n\n2\n\n\u03c4\n\n2\n\n\u03c4\n\na\nej\n\u03a9\n\n(b)\n\n(c)\n\nim\nre\n\naej\u03a9\n\nim\nre\n\nim\n\nre\n\na cos\u03a9\n-a sin\u03a9\na sin\u03a9\na cos\u03a9\n\nim\n\nre\n\n2\n\n\u03c4\n\n2\n\n\u03c4\n\n2\n\n\u03c4\n\n2\n\n\u03c4\n\naej\u03a9\n\naej\u03a9\n\n2\n\n\u03c4\n\n2\n\n\u03c4\n\n(d)\n\nFigure 1. (a) 1D array of three Reichardt detectors tuned to motion from left to right. The \n block\nrepresents a temporal delay. The semi-circles represent photosensors. (b) Delbruck delay-line\ndetector. (c) 1D array of three motion energy detectors. The bottom blocks represent even and odd\nsymmetric spatial receptive fields modelled by Gabor functions. (d) The proposed motion detector\nby combining the position and phase tuning mechanisms of (b) and (c).\n\n\u03c4\n\nOne problem with using spatio-temporal Gabor functions is that they are non-causal in time. In this\nwork, we consider the use of a causal recurrently implemented temporal filter. If we let the real and\n denote the even and odd spatial filter outputs, then the two temporal fil-\nimaginary parts of \n\nux t,(\n)\n\n\fter outputs of the temporal filter are given by the real and imaginary parts of \nfies\n\nvx t,(\n)\n\n, which satis-\n\n\u22c5\n\n(\n\n=\n\nexp\n\na 1<\n\nj\u03a9t\n)\n\n) a\n\nv x t 1\u2013\n,(\n\n(3)\nv x t,(\n are real valued constants. We derive this equation from Fig. 1(c) by consider-\nwhere \n and \ning the time delay \n as a unit sample discrete time delay. We consider discrete time operation here\nfor consistency with our experimental results, however, a corresponding continuous time temporal\nfilter can be obtained by replacing the time delay by a first order continuous-time recurrent filter\nwith time constant \n\n. The frequency response of this complex-valued filter is\n\n) u x t,(\n\u22c5\n)\n\n1 a\u2013(\n\n\u03a9t\n\u03c4\n\n+\n\n\u03c4\n\n)\n\n(\n)\nV\u03c9x \u03c9t,\n----------------------\n(\n)\nU\u03c9x \u03c9t,\n\n=\n\n1 a\u2013\n--------------------------------------------------------------\nj \u03c9t \u03a9t\u2013\n\u2013(\n\u22c5\u2013\n)\n(\n)\n1 a\n\nexp\n\n(4)\n\n\u03c9x\n\nwhere \n and \nimum value at \nthe combined spatio-temporal receptive field can be approximated by the continuous function:\n\n are spatial and temporal frequency variables. This function achieves unity max-\n\u03a9t\u2013=\n. Assuming the same Gabor spatial receptive field,\n\n, independently of \n\n\u03c9t\n\u03c9t\n\n\u03c9x\n\ng x t,(\n)\n\n=\n\n1\n-----------------\n2\u03c0\u03c3x\n\nexp\n\n\u2013\n\n\u239b\n\u239c\n\u239d\n\n\u239e\nx2\n---------\n\u239f\n\u23a0\n2\n2\u03c3x\n\nexp\n\n(\n\nj\u03a9xx\n)\n\n\u22c5\n\n\u03c4 1\u2013\n\nexp\n\nt \u03c4\u2044\u2013(\n\n)\n\nexp\n\n(\n\nj\u03a9tt\n\n)ht( )\n\n(5)\n\n.\n\n)\n\n\u2248\n\n\u03c4 1\u2013\n\n\u03a9x\n\n\u03a9t\n\n\u2013=\n\nht( )\n\nvpref\n\nux t,(\n)\n\n1 a\u2013(\n)\n\n\u03a9t \u03a9x\u2044\n\n and the output \n\n between the input \n\n is the unit step function, and \n\n. Again, strictly speaking, the filter is not\n, the composite\n\nwhere \nvelocity tuned, but for input sine-wave gratings with a spatial frequency near \nspatio-temporal filter has a preferred velocity near \nThe velocity tuning of this filter is determined by the combination of the time delay and a phase\nshift \n. The longer the time delay, the\nslower the preferred velocity. However, the larger the phase-shift, the higher the preferred velocity.\nThis phase-tuning of velocity is reminiscent of the phase-tuning of disparity tuned neurons, where\nthe larger the phase shift between the left and right receptive fields, the larger the preferred dispar-\nity.\nThe possibility to adjust velocity tuning using two complementary mechanisms, suggests that it\nshould be possible to combine these two methods, as observed in disparity neurons. Figure 1(d)\nshows how the position and phase tuning mechanisms of Figures 1(b) and 1(c) can be combined.\n will be determined by the sum of the preferred\nThe preferred velocity for spatial frequencies \nvelocities determined by the position and phase-shift mechanisms, i.e. \n,\n1 \u03a9t \u03a9x\u2044\nassuming a unit spatial displacement between adjacent photosensors.\n\nvx t 1\u2013\n,(\n\nvpref\n\n\u03a9x\n\n=\n\n\u2013\n\n3 Motion energy pairs for velocity discrimination\nGiven the possibility of combining the position and phase tuning mechanisms, an interesting ques-\ntion is how these two mechanisms might be exploited when constructing populations of motion\nenergy neurons. Velocity can be estimated using a population of neurons tuned to different spatio-\ntemporal frequencies [10][11]. However, the output of a single motion energy neuron is an ambigu-\nous indicator of velocity, since its output depends upon other stimulus dimensions in addition to\nmotion, (e.g. orientation, contrast).\nGiven the long history of position/phase shifts in disparity tuning, it is natural to start with an inspi-\nration taken from the context of binocular vision. It has been shown that the responses from a pop-\nulation of phase-tuned disparity energy are more comparable than the responses from a population\nof position-tuned disparity energy neurons [12]. In particular, the preferred disparity of the neuron\nwith maximum response in a population of phase tuned neurons is a more reliable indicator of the\nstimulus disparity than the preferred disparity of the neuron with maximum response in a popula-\ntion of position tuned neurons, especially for neurons with small phase shifts. The disadvantage of\npurely phase tuned neurons is that their preferred disparities can be tuned only over a limited range\ndue to phase-wraparound in the sinusoidal modulation of the spatial Gabor. However, there is no\n\n\frestriction on the range of preferred disparities when using position shifts. Thus, it has been sug-\ngested that position shifts can be used to \u201cbias\u201d the preferred disparity of a population around a\nrough estimate of the stimulus disparity, and then use a population of neurons tuned by phase shifts\nto obtain a more accurate estimation of the actual disparity.\nIn this section, we demonstrate that a similar phenomenon holds for motion energy neurons. In par-\nticular, we show that we can use position shifts to place the tuned velocity (for a spatial frequency\nof \n, and then use phase\nshifts with equal magnitude but opposite sign to place the preferred velocities symmetrically\naround this bias velocity. We then show that by comparing the outputs of these two neurons, we can\naccurately discriminate between velocities above and below \nThe equation describing the complex valued output of the spatio-temporal filtering stage \nfor the detector shown in Figure 1(d) is \n\n) in a population of two neurons around a desired bias velocity, \n\nwx t,(\n)\n\nvbias\n\nvbias\n\n\u03a9x\n\n. \n\nwx t,(\nThe frequency response is\n\n) a\n\n=\n\nexp\n\n(\n\nj\u03a9t\n)\n\n\u22c5\n\nw x 1\u2013 t 1\u2013\n(\n\n,\n\n)\n\n+\n\n1 a\u2013(\n\n) u x t,(\n\u22c5\n)\n\n(6)\n\n(7)\n\n. \n\n=\n\n\u03a9t\n\n=\n\n\u03c9x\n\n and \n\n\u03c9t\n\n\u00b1=\n\n2\u03c0 20\u2044\n\n\u03a9t\n(\n\nexp\n=\n\n(\n\n\u03c9x \u03c9t,\n)\n\n\u03c9x \u03c9t\u2013\n\n2\u03c0 20\u2044\n\u03a9t\n\nW \u03c9x \u03c9t,\n(\n)\n------------------------\n(\n)\nU\u03c9x \u03c9t,\n\n\u2013(\n\u03c9x \u03a9t+\n\n but opposite temporal frequencies \n\n independently of \n\u03c9x\n\n1 a\u2013\n---------------------------------------------------------------------------\n\u2013\n+\nj \u03c9t \u03c9x \u03a9t\u2013\n)\n(\n)\n\u22c5\n1 a\nand achieves its maximum along the line \n, as seen in the contour plot of the spatio-\n\u03c9t\ntemporal frequency response magnitude of the cascade of (1) and (7) in Fig. 2(a). In comparison,\nthe spatio-temporal frequency response of the cascade of (1) and (4) shown in Fig. 2(e), achieves its\nmaximum at \n. For a moving sine wave grating input with spatial and tem-\n\u03a9t\nporal frequencies \n, the steady state motion energy outputs will be proportional to the\nsquared magnitudes of the spatio-temporal frequency response evaluated at \nAssume that we have two such motion cells with the same preferred spatial frequency\n. The motion energy cell with\n\u03a9x\n is tuned to\npositive \n is tuned to fast velocities, while the motion energy cell with negative \nslow velocities. If we compare the frequency response magnitudes at frequency \n, the\n\u03c9x \u03c9t,\n)\nboundary between the regions in the \n plane where the magnitude of one is larger than the\nother is a line passing thorough the origin with slope equal to 1, as shown in Fig. 2(c). This suggests\nthat we can determine whether the velocity of the grating is faster or slower than 1 pixel per frame\nby checking the relative magnitude of the motion energy outputs, at least for sine-wave gratings.\nAlthough the sine-wave grating is a particularly simple input, this property is not shared by other\npairs of motion energy neurons. For example, Fig. 2(f) shows the spatio-temporal frequency\nresponses two motion energy neurons that have the same spatio-temporal center frequencies as con-\nsidered above, but are constructed by phase tuning (the cascade of (1) and (4)). In this case, the\nboundary is a horizontal line. Thus, the velocity boundary depends upon the spatial frequency. For\nlower spatial frequencies, the relative magnitudes will switch at higher velocities. Another com-\nmonly considered arrangement of Gabor-filters is to place the center frequencies around a circle.\nFor two neurons, this corresponds to displacing the two center frequencies by an equal amount per-\npendicularly to the line \n (Fig. 2(k)). For motion energy filters built from non-causal\nGabor filters, the spatio-temporal frequency responses exhibit perfect circular symmetry, and the\ndecision boundary also coincides with the diagonal line \n (see Figure 9 in [13]). However,\nnon-causal filters are not physically realizable. If we consider motion energy neurons constructed\nfrom temporally causal functions (e.g. the cascade (1) and (4)), the boundary only matches the\ndiagonal line in a small neighborhood of \nWe have characterized the performance of the three motion pairs on the fast/slow velocity discrim-\nination task for a variety of inputs, including sine-wave gratings, square wave gratings, and drifting\nrandom dot stimuli with varying coherence.\nWe first consider drifting sinusoidal gratings with spatial frequencies \nties \n\n and veloci-\n. For each spatial frequency and velocity, we compare the two motion energy\n\n, as shown in Fig. 2(i).\n\n0 2\u03c0 10\u2044\n,[\n\n\u03c9x \u03a9x\n\n=\n\n\u2208\n\n0 2,[\n\n]\n\n\u03c9x\n\n=\n\n\u03c9x\n\n=\n\n\u03c9x\n\n\u2208\n\n]\n\n\u03c9t\n\n\u03c9t\n\nvinput\n\n\ffast cell\n\nslow cell\n\nmotion pair\n\ntuning curve\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n\n0.5\n\n0\n\n-0.5\n-1\n\n-1.5\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n0.5\n\n0\n-0.5\n-1\n\n-1.5\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n\n0.5\n\n0\n\n-0.5\n-1\n\n-1.5\n\nd\ne\nn\nu\nt\n-\nn\no\ni\nt\ni\ns\no\np\n\n/\ne\ns\na\nh\np\n\n)\nl\na\nc\ni\nt\nr\ne\nv\n(\n \nd\ne\nn\nu\nt\n-\ne\ns\na\nh\np\n\n)\nl\na\nn\no\ng\no\nh\nt\nr\no\n(\n \nd\ne\nn\nu\nt\n-\ne\ns\na\nh\np\n\n-1\n\n1\nspatial frequency\n\n0\n\n(a)\n\n1\n-1\nspatial frequency\n\n0\n\n(e)\n\n1\n-1\nspatial frequency\n\n0\n\n(i)\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n\n0.5\n\n0\n\n-0.5\n-1\n\n-1.5\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n0.5\n\n0\n-0.5\n-1\n\n-1.5\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n\n0.5\n\n0\n\n-0.5\n-1\n\n-1.5\n\n-1\n1\nspatial frequency\n\n0\n\n(b)\n\n-1\n1\nspatial frequency\n\n0\n\n(f)\n\n-1\n1\nspatial frequency\n\n0\n\n(j)\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n\n0.5\n\n0\n\n-0.5\n-1\n\n-1.5\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n0.5\n\n0\n-0.5\n-1\n\n-1.5\n\ny\nc\nn\ne\nu\nq\ne\nr\nf\n \nl\na\nr\no\np\nm\ne\nt\n\n1.5\n\n1\n0.5\n\n0\n-0.5\n-1\n\n-1.5\n\n-1\n1\nspatial frequency\n\n0\n\n(c)\n\n-1\n\n1\nspatial frequency\n\n0\n\n(g)\n\n-1\n1\nspatial frequency\n\n0\n\n(k)\n\ne\nd\nu\nt\ni\nl\np\nm\na\n\ne\nd\nu\nt\ni\nl\np\nm\na\n\ne\nd\nu\nt\ni\nl\np\nm\na\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n\n-0.4\n\n-0.2\n\n0\n\n0.2\n\n0.4\n\n0.6\n\ndistance\n\n(d)\n\n-0.4\n\n-0.2\n\n0\n\n0.2\n\n0.4\n\n0.6\n\ndistance\n\n(h)\n\n-0.4\n\n-0.2\n\n0\n\n0.2\n\n0.4\n\n0.6\n\ndistance\n\n(l)\n\n,\n\n)\n\n)\n\n,\n\n(\n\n(\n\n(\n\n(\n\n)\n\n and \n\n=\n\n(\n\n0.314 0,\n\n)\n\n\u03c9x \u03c9t,\n)\n\n0.314 0.628\n\nFigure 2. Frequency response amplitudes of the motion pairs formed by types of motion cells. First\nrow: Phase and position tuned motion cells. The center frequencies of the fast (a) and slow (b) cells\nare \n respectively. Second row: Vertically displaced\nphase-tuned motion energy cells. The center frequencies of the fast (e) and slow (f) cells are\n respectively. Third row: Orthogonally displaced phase-tuned\n(\n0.314 0.628\nmotion energy cells. The center frequencies of the fast (i) and slow (j) cells are \n)\n0.092 0.536\n respectively. The third column shows the contour plot of difference between\nand \nthe frequency response amplitudes of the fast cell from the slow cell. The dashed line shows the\ndecision boundary at zero. The fourth column shows the cross sections of the frequency response\namplitudes along the line connecting the two center frequencies (fast = solid, slow = dashed). Zero\ndenotes the point on the line that crosses \n\n0.536 0.092\n\n0.314 0,\n\n and \n\n(\n\n,\n\n)\n\n,\n\n\u03c9t\n\n=\n\n\u03c9x\n\n.\n\noutputs at different phase shifts of the input grating, and calculate the percentage where the\nresponse of the fast cell is larger than that of the slow cell. Fig. 3(a)-(c) show the percentages as the\ngrey scale value for each combination of input spatial frequency and velocity. Ideally, the top half\nshould be white (i.e. the fast cell\u2019s response is larger for all inputs whose velocity is greater than\none), and the bottom half should be black. For the phase-shifted motion cells with unit position-\ntuned velocity bias, the responses are correct over a wide range of spatial frequencies. On the other\nhand, for the motion pairs with the same center frequencies but tuned by pure phase shifts\n(Fig. 3(c)), the velocity at which the relative responses switch decreases with spatial frequency.\nThis is consistent with the horizontal decision boundary computed by comparing the frequency\nresponse magnitudes. For the phase-tuned motion-energy cells with orthogonally displaced center\nfrequencies, the boundary rapidly diverges from the horizontal as the spatial frequency moves away\nfrom \n. Fig. 3(d) shows the overall accuracy by combining the responses over all velocities. The\ndetector utilizing the phase-tuned cells with position bias have the highest accuracy over the widest\nrange of spatial frequencies.\nFig. 3(e)-(h) show the responses of the motion pairs to square wave gratings. The results are similar\nto the case of sinusoidal gratings, except that the performance at low spatial frequencies is worse.\n\n\u03a9x\n\n\fposition/phase tuned\n\nphase-tuned (vertical)\n\nphase-tuned (orthogonal)\n\naverage accuracy\n\ns\ng\nn\ni\nt\na\nr\ng\n\n \ne\nv\na\nw\n \ne\nn\ni\ns\n\ns\ng\nn\ni\nt\na\nr\ng\n \ne\nv\na\nw\n \ne\nr\na\nu\nq\ns\n\n \n\ns\nt\no\nd\nm\no\nd\nn\na\nr\n \n\ng\nn\ni\nt\nf\ni\nr\nd\n\n2\n\n1.5\n\n2\n\n1.5\n\n2\n\n1.5\n\ny\nt\ni\nc\no\nl\ne\nv\n\n1\n\n0.5\n\ny\nt\ni\nc\no\nl\ne\nv\n\n1\n\n0.5\n\ny\nt\ni\nc\no\nl\ne\nv\n\n1\n\n0.5\n\n0.3\n\n0.2\n\n0.1\n0.6\ninput spatial frequency\n\n0.5\n\n0.4\n\n(a)\n\n0\n\n0\n\n2\n\n1.5\n\n0.3\n\n0.2\n\n0.1\n0.6\ninput spatial frequency\n\n0.4\n\n0.5\n\n(b)\n\n0\n\n0\n\n2\n\n1.5\n\n0\n\n0\n\n2\n\n1.5\n\n0.3\n\n0.2\n\n0.1\n0.6\ninput spatial frequency\n\n0.4\n\n0.5\n\n(c)\n\ny\nt\ni\nc\no\nl\ne\nv\n\n1\n\n0.5\n\n1\n\ny\nt\ni\nc\no\nl\ne\nv\n\n0.5\n\ny\nt\ni\nc\no\nl\ne\nv\n\n1\n\n0.5\n\n0.3\n\n0.2\n\n0.1\n0.6\ninput spatial frequency\n\n0.5\n\n0.4\n\n(e)\n\n0\n\n0\n\n2\n\n1.5\n\n0.3\n\n0.2\n\n0.1\n0.6\ninput spatial frequency\n\n0.5\n\n0.4\n\n(f)\n\n0\n\n0\n\n2\n\n1.5\n\n0\n\n0\n\n2\n\n1.5\n\n0.3\n\n0.2\n\n0.1\n0.6\ninput spatial frequency\n\n0.5\n\n0.4\n\n(g)\n\n1\n\ny\nt\ni\nc\no\nl\ne\nv\n\n0.5\n\n1\n\ny\nt\ni\nc\no\nl\ne\nv\n\n0.5\n\n1\n\ny\nt\ni\nc\no\nl\ne\nv\n\n0.5\n\n1\n\n0.9\n\n0.8\n\n0.7\n\n0.6\n\n0.5\n0\n\n1\n\n0.9\n\n0.8\n\n0.7\n\n0.6\n\n0.5\n0\n\n1\n\n0.9\n\n0.8\n\n0.7\n\n0.6\n\ny\nc\na\nr\nu\nc\nc\na\n\ny\nc\na\nr\nu\nc\nc\na\n\n0.6\n\n0.6\n\n0.3\n\n0.2\n\n0.1\n\n0.5\ninput spatial frequency\n\n0.4\n\n(d)\n\n0.1\n\n0.2\n\n0.5\ninput spatial frequency\n\n0.4\n\n0.3\n\n(h)\n\n0\n0.5\n\n0.6\n\n0.8\n\n0.7\n\n0.9\ncoherence level\n\n1\n\n0\n0.5\n\n0.6\n\n0.8\n\n0.7\n\n0.9\ncoherence level\n\n1\n\n0\n0.5\n\n0.6\n\n0.7\n\n0.8\n\n0.9\ncoherence level\n\n1\n\n0.5\n\n0.5\n\n0.7\n\n0.8\n\n0.6\ncoherence level\n\n0.9\n\n1\n\n(i)\n\n(j)\n\n(k)\n\n(l)\n\nFigure 3. Performance on the velocity discrimination task for different stimuli. First row: sine\nwave gratings; second row: square wave gratings; third row: drifting random dots. The first three\ncolumns show the percentage of stimuli where the fast motion energy cell\u2019s response is larger than\nthe slow cell\u2019s response. First column: motion cells with position-tuned velocity bias; second\ncolumn: phase tuned motion cells with the same center frequencies; third column: phase-tuned\nmotion cells with orthogonal offset. The fourth column shows the average accuracy over all input\nvelocities. Solid line: motion cells with position-tuned velocity bias; dashed line: phase tuned\nmotion cells with the same center frequencies; dash-dot line: phase-tuned motion cells with\northogonal offset.\n\nthis is expected, since for low spatial frequencies, the square wave gratings have large constant\nintensity areas that convey no motion information.\nFig. 3(i)-(l) show the responses for drifting random dot stimuli at different velocities and coherence\nlevels. The dots were one pixel wide. The motion pair using the phase-shifted cells with position\ntuned bias velocity maintain a consistently higher accuracy over all coherence levels tested.\n\n4 Discussion\nWe described a new architecture for motion energy filters obtained by combining the position tun-\ning mechanism of the Reichardt-like detectors and the phase tuning mechanism of motion energy\ndetectors based on complex-valued spatio-temporal separable filters. Motivated by results with dis-\nparity energy neurons indicating that the responses of phase-tuned neurons with small phase shifts\nare more comparable, we have examined the ability of the proposed velocity detectors to discrimi-\nnate between input stimuli above and below a fixed velocity. Our experimental and analytical\nresults confirm that comparisons between pairs constructed by using a position shift to center the\ntuned velocities around the border and using phase shifts to offset the tuned velocity of the pair to\nopposite sides of the boundary is consistently better than previously proposed architectures that\nwere based on pure phase tuning.\n\n\fRecent experimental evidence has cast doubt upon the belief that the motion neurons in V1 and MT\nhave very distinct properties. Traditionally, the tuning of V1 motion sensitive neurons is thought to\nbe separable along the spatial and temporal frequency dimensions, while the frequency tuning MT\nneurons is inseparable, consistent with constant speed tuning. However, it now seems that both V1\nand MT neurons actually show a continuum in the degree to which preferred velocity changes with\nspatial frequency [14][15][16]. Our proposed neurons constructed by position and phase shifts also\nshow an intermediate behavior between speed tuning and space-time separable tuning. With pure\nphase shifts, the tuning is space-time separable. With position shifts, the neurons become speed\ntuned. An intermediate tuning is obtained by combining position and phase tuning. Our results on a\nsimple velocity discrimination task suggest a functional role for this intermediate level of tuning in\ncreating motion energy pairs whose relative responses truly indicate changes in velocity around a\nreference level for stimuli with a broad band of spatial frequency content. Pair-wise comparisons\nhave been previously proposed as a potential method for coding image speed [17][18]. Here, we\nhave demonstrated a systematic way of constructing reliably comparable pairs of neurons using\nsimple neurally plausible circuits.\n\nAcknowledgements\nThis work was supported in part by the Hong Kong Research Grants Council under Grant\nHKUST6300/04E.\n\nReferences\n[1]\n\n[2]\n\n[3]\n\n[4]\n\n[5]\n\n[6]\n\n[7]\n\n[8]\n\n[9]\n\n[10]\n\n[11]\n\n[12]\n\n[14]\n\n[15]\n\n[16]\n\n[17]\n\n[18]\n\nW. Reichardt, \u201cAutocorrelation, a principle for the evaluation of sensory information by the central\nnervous system,\u201d in Sensory Communication, W. A. Rosenblith, ed. 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