nips nips2003 nips2003-140 knowledge-graph by maker-knowledge-mining

140 nips-2003-Nonlinear Processing in LGN Neurons

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Author: Vincent Bonin, Valerio Mante, Matteo Carandini

Abstract: According to a widely held view, neurons in lateral geniculate nucleus (LGN) operate on visual stimuli in a linear fashion. There is ample evidence, however, that LGN responses are not entirely linear. To account for nonlinearities we propose a model that synthesizes more than 30 years of research in the field. Model neurons have a linear receptive field, and a nonlinear, divisive suppressive field. The suppressive field computes local root-meansquare contrast. To test this model we recorded responses from LGN of anesthetized paralyzed cats. We estimate model parameters from a basic set of measurements and show that the model can accurately predict responses to novel stimuli. The model might serve as the new standard model of LGN responses. It specifies how visual processing in LGN involves both linear filtering and divisive gain control. 1 In t rod u ct i on According to a widely held view, neurons in lateral geniculate nucleus (LGN) operate linearly (Cai et al., 1997; Dan et al., 1996). Their response L(t) is the convolution of the map of stimulus contrast S(x,t) with a receptive field F(x,t): L (t ) = [ S∗ F ] ( 0, t ) The receptive field F(x,t) is typically taken to be a difference of Gaussians in space (Rodieck, 1965) and a difference of Gamma functions in time (Cai et al., 1997). This linear model accurately predicts the selectivity of responses for spatiotemporal frequency as measured with gratings (Cai et al., 1997; Enroth-Cugell and Robson, 1966). It also predicts the main features of responses to complex dynamic video sequences (Dan et al., 1996). 150 spikes/s Data Model Figure 1. Response of an LGN neuron to a dynamic video sequence along with the prediction made by the linear model. Stimuli were sequences from Walt Disney’s “Tarzan”. From Mante et al. (2002). The linear model, however, suffers from limitations. For example, consider the response of an LGN neuron to a complex dynamic video sequences (Figure 1). The response is characterized by long periods of relative silence interspersed with brief events of high firing rate (Figure 1, thick traces). The linear model (Figure 1, thin traces) successfully predicts the timing of these firing events but fails to account for their magnitude (Mante et al., 2002). The limitations of the linear model are not surprising since there is ample evidence that LGN responses are nonlinear. For instance, responses to drifting gratings saturate as contrast is increased (Sclar et al., 1990) and are reduced, or masked, by superposition of a second grating (Bonin et al., 2002). Moreover, responses are selective for stimulus size (Cleland et al., 1983; Hubel and Wiesel, 1961; Jones and Sillito, 1991) in a nonlinear manner (Solomon et al., 2002). We propose that these and other nonlinearities can be explained by a nonlinear model incorporating a nonlinear suppressive field. The qualitative notion of a suppressive field was proposed three decades ago by Levick and collaborators (1972). We propose that the suppressive field computes local root-mean-square contrast, and operates divisively on the receptive field output. Basic elements of this model appeared in studies of contrast gain control in retina (Shapley and Victor, 1978) and in primary visual cortex (Cavanaugh et al., 2002; Heeger, 1992; Schwartz and Simoncelli, 2001). Some of these notions have been applied to LGN (Solomon et al., 2002), to fit responses to a limited set of stimuli with tailored parameter sets. Here we show that a single model with fixed parameters predicts responses to a broad range of stimuli. 2 Mod el In the model (Figure 2), the linear response of the receptive field L(t) is divided by the output of the suppressive field. The latter is a measure of local root-mean-square contrast c local. The result of the division is a generator potential V (t ) = Vmax L (t ) c50 + clocal , where c 50 is a constant. F(x,t) Stimulus S(x,t) V0 L(t) Receptive Field R(t) Firing rate Rectification H(x,t) S*(x,t) c50 clocal Suppressive Field Filter Figure 2. Nonlinear model of LGN responses. The suppressive field operates on a filtered version of the stimulus, S*=S*H, where H is a linear filter and * denotes convolution. The squared output of the suppressive field is the local mean square (the local variance) of the filtered stimulus: clocal = S ( x, t ) G ( x ) dx dt , 2 * 2 ∫∫ where G(x) is a 2-dimensional Gaussian. Firing rate is a rectified version of generator potential, with threshold V thresh:   R(t ) = V (t ) − Vthresh + .   To test the nonlinear model, we recorded responses from neurons in the LGN of anesthetized paralyzed cats. Methods for these recordings were described elsewhere (Freeman et al., 2002). 3 Resu l t s We proceed in two steps: first we estimate model parameters by fitting the model to a large set of canonical data; second we fix model parameters and evaluate the model by predicting responses to a novel set of stimuli. B 60 40 20 0 0.01 0.2 4 Spatial Frequency (cpd) Response (spikes/s) Response (spikes/s) A 80 60 40 20 0 0.5 5 50 Temporal Frequency (Hz) Figure 3. Estimating the receptive field in an example LGN cell. Stimuli are gratings varying in spatial (A) and temporal (B) frequency. Responses are the harmonic component of spike trains at the grating temporal frequency. Error bars represent standard deviation of responses. Curves indicate model fit. Response (spikes/s) 0.25 0.50 0.75 Test contrast 100 80 60 40 20 0 1.00 100 D 80 60 40 20 0 0.5 Response (spikes/s) 50 0 C B 100 Response (spikes/s) Response (spikes/s) A 0.25 0.50 0.75 1.00 Mask contrast 100 80 60 40 20 4.0 32.0 Mask diameter (deg) 0 0.01 0.20 4.00 Mask spatial frequency (cpd) Figure 4. Estimating the suppressive field in the example LGN cell. Stimuli are sums of a test grating and a mask grating. Responses are the harmonic component of spike trains at the temporal frequency of test. A: Responses to test alone. B-D: Responses to test+mask as function of three mask attributes: contrast (B), diameter (C) and spatial frequency (D). Gray areas indicate baseline response (test alone, 50% contrast). Dashed curves are predictions of linear model. Solid curves indicate fit of nonlinear model. 3.1 C h a r a c te r i z i n g t he r e c e p ti v e fi e l d We obtain the parameters of the receptive field F(x,t) from responses to large drifting gratings (Figure 3). These stimuli elicit approximately constant output in the suppressive field, so they allow us to characterize the receptive field. Responses to different spatial frequencies constrain F(x,t) in space (Figure 3A). Responses to different temporal frequencies constrain F(x,t) in time (Figure 3B). 3.2 C h a r a c te r i z i n g t he s u p p r e s s i v e f i e l d To characterize the divisive stage, we start by measuring how responses saturate at high contrast (Figure 4A). A linear model cannot account for this contrast saturation (Figure 4A, dashed curve). The nonlinear model (Figure 4A, solid curve) captures saturation because increases in receptive field output are attenuated by increases in suppressive field output. At low contrast, no saturation is observed because the output of the suppressive field is dominated by the constant c 50. From these data we estimate the value of c50. To obtain the parameters of the suppressive field, we recorded responses to sums of two drifting gratings (Figure 4B-D): an optimal test grating at 50% contrast, which elicits a large baseline response, and a mask grating that modulates this response. Test and mask temporal frequencies are incommensurate so that they temporally label a test response (at the frequency of the test) and a mask response (at the frequency of the mask) (Bonds, 1989). We vary mask attributes and study how they affect the test responses. Increasing mask contrast progressively suppresses responses (Figure 4B). The linear model fails to account for this suppression (Figure 4B, dashed curve). The nonlinear model (Figure 4B, solid curve) captures it because increasing mask contrast increases the suppressive field output while the receptive field output (at the temporal frequency of the test) remains constant. With masks of low contrast there is little suppression because the output of the suppressive field is dominated by the constant c 50 . Similar effects are seen if we increase mask diameter. Responses decrease until they reach a plateau (Figure 4C). A linear model predicts no decrease (Figure 4C, dashed curve). The nonlinear model (Figure 4C, solid curve) captures it because increasing mask diameter increases the suppressive field output while it does not affect the receptive field output. A plateau is reached once masks extend beyond the suppressive field. From these data we estimate the size of the Gaussian envelope G(x) of the suppressive field. Finally, the strength of suppression depends on mask spatial frequency (Figure 4D). At high frequencies, no suppression is elicited. Reducing spatial frequency increases suppression. This dependence of suppression on spatial frequency is captured in the nonlinear model by the filter H(x,t). From these data we estimate the spatial characteristics of the filter. From similar experiments involving different temporal frequencies (not shown), we estimate the filter’s selectivity for temporal frequency. 3.3 P r e d i c ti n g r e s p o n s e s t o n o v e l s ti m u l i We have seen that with a fixed set of parameters the model provides a good fit to a large set of measurements (Figure 3 and Figure 4). We now test whether the model predicts responses to a set of novel stimuli: drifting gratings varying in contrast and diameter. Responses to high contrast stimuli exhibit size tuning (Figure 5A, squares): they grow with size for small diameters, reach a maximum value at intermediate diameter and are reduced for large diameters (Jones and Sillito, 1991). Size tuning , however, strongly depends on stimulus contrast (Solomon et al., 2002): no size tuning is observed at low contrast (Figure 5A, circles). The model predicts these effects (Figure 5A, curves). For large, high contrast stimuli the output of the suppressive field is dominated by c local, resulting in suppression of responses. At low contrast, c local is much smaller than c50, and the suppressive field does not affect responses. Similar considerations can be made by plotting these data as a function of contrast (Figure 5B). As predicted by the nonlinear model (Figure 5B, curves), the effect of increasing contrast depends on stimulus size: responses to large stimuli show strong saturation (Figure 5B, squares), whereas responses to small stimuli grow linearly (Figure 5B, circles). The model predicts these effects because only large, high contrast stimuli elicit large enough responses from the suppressive field to cause suppression. For small, low contrast stimuli, instead, the linear model is a good approximation. B 100 Response (spikes/s) A 80 60 40 20 0 0.50 4.00 32.00 Diameter (deg) 0.00 0.25 0.50 0.75 1.00 Contrast Figure 5. Predicting responses to novel stimuli in the example LGN cell. Stimuli are gratings varying in diameter and contrast, and responses are harmonic component of spike trains at grating temporal frequency. Curves show model predictions based on parameters as estimated in previous figures, not fitted to these data. A: Responses as function of diameter for different contrasts. B: Responses as function of contrast for different diameters. 3.4 M o d e l pe r f or m a nc e To assess model performance across neurons we calculate the percentage of variance in the data that is explained by the model (see Freeman et al., 2002 for methods). The model provides good fits to the data used to characterize the suppressive field (Figure 4), explaining more than 90% of the variance in the data for 9/13 cells (Figure 6A). Model parameters are then held fixed, and the model is used to predict responses to gratings of different contrast and diameter (Figure 5). The model performs well, explaining in 10/13 neurons above 90% of the variance in these novel data (Figure 6B, shaded histogram). The agreement between the quality of the fits and the quality of the predictions suggests that model parameters are well constrained and rules out a role of overfitting in determining the quality of the fits. To further confirm the performance of the model, in an additional 54 cells we ran a subset of the whole protocol, involving only the experiment for characterizing the receptive field (Figure 3), and the experiment involving gratings of different contrast and diameter (Figure 5). For these cells we estimate the suppressive field by fitting the model directly to the latter measurements. The model explains above 90% of the variance in these data in 20/54 neurons and more than 70% in 39/54 neurons (Figure 6B, white histogram). Considering the large size of the data set (more than 100 stimuli, requiring several hours of recordings per neuron) and the small number of free parameters (only 6 for the purpose of this work), the overall, quality of the model predictions is remarkable. Estimating the suppressive field A # cells 6 n=13 4 2 0 Size tuning at different contrasts 15 n=54 10 # cells B 5 0 0 50 100 Explained variance (%) Figure 6. Percentage of variance in data explained by model. A: Experiments to estimate the suppressive field. B: Experiments to test the model. Gray histogram shows quality of predictions. White histogram shows quality of fits. 4 Co n cl u si o n s The nonlinear model provides a unified description of visual processing in LGN neurons. Based on a fixed set of parameters, it can predict both linear properties (Figure 3), as well as nonlinear properties such as contrast saturation (Figure 4A) and masking (Figure 4B-D). Moreover, once the parameters are fixed, it predicts responses to novel stimuli (Figure 5). The model explains why responses are tuned for stimulus size at high contrast but not at low contrast, and it correctly predicts that only responses to large stimuli saturate with contrast, while responses to small stimuli grow linearly. The model implements a form of contrast gain control. A possible purpose for this gain control is to increase the range of contrast that can be transmitted given the limited dynamic range of single neurons. Divisive gain control may also play a role in population coding: a similar model applied to responses of primary visual cortex was shown to maximize independence of the responses across neurons (Schwartz and Simoncelli, 2001). We are working towards improving the model in two ways. First, we are characterizing the dynamics of the suppressive field, e.g. to predict how it responds to transient stimuli. Second, we are testing the assumption that the suppressive field computes root-mean-square contrast, a measure that solely depends on the secondorder moments of the light distribution. Our ultimate goal is to predict responses to complex stimuli such as those shown in Figure 1 and quantify to what degree the nonlinear model improves on the predictions of the linear model. Determining the role of visual nonlinearities under more natural stimulation conditions is also critical to understanding their function. The nonlinear model synthesizes more than 30 years of research. It is robust, tractable and generalizes to arbitrary stimuli. As a result it might serve as the new standard model of LGN responses. Because the nonlinearities we discussed are already present in the retina (Shapley and Victor, 1978), and tend to get stronger as one ascends the visual hierarchy (Sclar et al., 1990), it may also be used to study how responses take shape from one stage to another in the visual system. A c k n o w l e d g me n t s This work was supported by the Swiss National Science Foundation and by the James S McDonnell Foundation 21st Century Research Award in Bridging Brain, Mind & Behavior. References Bonds, A. B. (1989). Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex. Vis Neurosci 2, 41-55. Bonin, V., Mante, V., and Carandini, M. (2002). The contrast integration field of cat LGN neurons. Program No. 352.16. In Abstract Viewer/Itinerary Planner (Washington, DC, Society for Neuroscience). Cai, D., DeAngelis, G. C., and Freeman, R. D. (1997). Spatiotemporal receptive field organization in the lateral geniculate nucleus of cats and kittens. J Neurophysiol 78, 10451061. Cavanaugh, J. R., Bair, W., and Movshon, J. A. (2002). Selectivity and spatial distribution of signals from the receptive field surround in macaque v1 neurons. J Neurophysiol 88, 25472556. Cleland, B. G., Lee, B. B., and Vidyasagar, T. R. (1983). Response of neurons in the cat's lateral geniculate nucleus to moving bars of different length. J Neurosci 3, 108-116. Dan, Y., Atick, J. J., and Reid, R. C. (1996). Efficient coding of natural scenes in the lateral geniculate nucleus: experimental test of a computational theory. J Neurosci 16, 3351-3362. Enroth-Cugell, C., and Robson, J. G. (1966). The contrast sensitivity of retinal ganglion cells of the cat. J Physiol (Lond) 187, 517-552. Freeman, T., Durand, S., Kiper, D., and Carandini, M. (2002). Suppression without Inhibition in Visual Cortex. Neuron 35, 759. Heeger, D. J. (1992). Normalization of cell responses in cat striate cortex. Vis Neurosci 9, 181-197. Hubel, D., and Wiesel, T. N. (1961). Integrative action in the cat's lateral geniculate body. J Physiol (Lond) 155, 385-398. Jones, H. E., and Sillito, A. M. (1991). The length-response properties of cells in the feline dorsal lateral geniculate nucleus. J Physiol (Lond) 444, 329-348. Levick, W. R., Cleland, B. G., and Dubin, M. W. (1972). Lateral geniculate neurons of cat: retinal inputs and physiology. Invest Ophthalmol 11, 302-311. Mante, V., Bonin, V., and Carandini, M. (2002). Responses of cat LGN neurons to plaids and movies. Program No. 352.15. In Abstract Viewer/Itinerary Planner (Washington, DC, Society for Neuroscience). Rodieck, R. W. (1965). Quantitative analysis of cat retina ganglion cell response to visual stimuli. Vision Res 5, 583-601. Schwartz, O., and Simoncelli, E. P. (2001). Natural signal statistics and sensory gain control. Nat Neurosci 4, 819-825. Sclar, G., Maunsell, J. H. R., and Lennie, P. (1990). Coding of image contrast in central visual pathways of the macaque monkey. Vision Res 30, 1-10. Shapley, R. M., and Victor, J. D. (1978). The effect of contrast on the transfer properties of cat retinal ganglion cells. J Physiol 285, 275-298. Solomon, S. G., White, A. J., and Martin, P. R. (2002). Extraclassical receptive field properties of parvocellular, magnocellular, and koniocellular cells in the primate lateral geniculate nucleus. J Neurosci 22, 338-349.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 org Abstract According to a widely held view, neurons in lateral geniculate nucleus (LGN) operate on visual stimuli in a linear fashion. [sent-2, score-0.595]

2 There is ample evidence, however, that LGN responses are not entirely linear. [sent-3, score-0.343]

3 Model neurons have a linear receptive field, and a nonlinear, divisive suppressive field. [sent-5, score-0.755]

4 To test this model we recorded responses from LGN of anesthetized paralyzed cats. [sent-7, score-0.455]

5 We estimate model parameters from a basic set of measurements and show that the model can accurately predict responses to novel stimuli. [sent-8, score-0.416]

6 It specifies how visual processing in LGN involves both linear filtering and divisive gain control. [sent-10, score-0.156]

7 1 In t rod u ct i on According to a widely held view, neurons in lateral geniculate nucleus (LGN) operate linearly (Cai et al. [sent-11, score-0.443]

8 Their response L(t) is the convolution of the map of stimulus contrast S(x,t) with a receptive field F(x,t): L (t ) = [ S∗ F ] ( 0, t ) The receptive field F(x,t) is typically taken to be a difference of Gaussians in space (Rodieck, 1965) and a difference of Gamma functions in time (Cai et al. [sent-14, score-1.4]

9 This linear model accurately predicts the selectivity of responses for spatiotemporal frequency as measured with gratings (Cai et al. [sent-16, score-0.741]

10 It also predicts the main features of responses to complex dynamic video sequences (Dan et al. [sent-18, score-0.45]

11 The linear model (Figure 1, thin traces) successfully predicts the timing of these firing events but fails to account for their magnitude (Mante et al. [sent-28, score-0.225]

12 The limitations of the linear model are not surprising since there is ample evidence that LGN responses are nonlinear. [sent-30, score-0.381]

13 For instance, responses to drifting gratings saturate as contrast is increased (Sclar et al. [sent-31, score-0.763]

14 , 1990) and are reduced, or masked, by superposition of a second grating (Bonin et al. [sent-32, score-0.143]

15 Moreover, responses are selective for stimulus size (Cleland et al. [sent-34, score-0.429]

16 We propose that these and other nonlinearities can be explained by a nonlinear model incorporating a nonlinear suppressive field. [sent-37, score-0.746]

17 The qualitative notion of a suppressive field was proposed three decades ago by Levick and collaborators (1972). [sent-38, score-0.857]

18 We propose that the suppressive field computes local root-mean-square contrast, and operates divisively on the receptive field output. [sent-39, score-1.388]

19 Basic elements of this model appeared in studies of contrast gain control in retina (Shapley and Victor, 1978) and in primary visual cortex (Cavanaugh et al. [sent-40, score-0.371]

20 , 2002), to fit responses to a limited set of stimuli with tailored parameter sets. [sent-43, score-0.503]

21 Here we show that a single model with fixed parameters predicts responses to a broad range of stimuli. [sent-44, score-0.474]

22 2 Mod el In the model (Figure 2), the linear response of the receptive field L(t) is divided by the output of the suppressive field. [sent-45, score-1.148]

23 The latter is a measure of local root-mean-square contrast c local. [sent-46, score-0.139]

24 The suppressive field operates on a filtered version of the stimulus, S*=S*H, where H is a linear filter and * denotes convolution. [sent-50, score-0.927]

25 The squared output of the suppressive field is the local mean square (the local variance) of the filtered stimulus: clocal = S ( x, t ) G ( x ) dx dt , 2 * 2 ∫∫ where G(x) is a 2-dimensional Gaussian. [sent-51, score-0.977]

26   To test the nonlinear model, we recorded responses from neurons in the LGN of anesthetized paralyzed cats. [sent-53, score-0.583]

27 3 Resu l t s We proceed in two steps: first we estimate model parameters by fitting the model to a large set of canonical data; second we fix model parameters and evaluate the model by predicting responses to a novel set of stimuli. [sent-56, score-0.492]

28 Estimating the receptive field in an example LGN cell. [sent-61, score-0.531]

29 Stimuli are gratings varying in spatial (A) and temporal (B) frequency. [sent-62, score-0.253]

30 Responses are the harmonic component of spike trains at the grating temporal frequency. [sent-63, score-0.211]

31 Estimating the suppressive field in the example LGN cell. [sent-81, score-0.857]

32 Stimuli are sums of a test grating and a mask grating. [sent-82, score-0.345]

33 Responses are the harmonic component of spike trains at the temporal frequency of test. [sent-83, score-0.191]

34 B-D: Responses to test+mask as function of three mask attributes: contrast (B), diameter (C) and spatial frequency (D). [sent-85, score-0.607]

35 1 C h a r a c te r i z i n g t he r e c e p ti v e fi e l d We obtain the parameters of the receptive field F(x,t) from responses to large drifting gratings (Figure 3). [sent-90, score-1.078]

36 These stimuli elicit approximately constant output in the suppressive field, so they allow us to characterize the receptive field. [sent-91, score-0.819]

37 2 C h a r a c te r i z i n g t he s u p p r e s s i v e f i e l d To characterize the divisive stage, we start by measuring how responses saturate at high contrast (Figure 4A). [sent-95, score-0.589]

38 A linear model cannot account for this contrast saturation (Figure 4A, dashed curve). [sent-96, score-0.235]

39 The nonlinear model (Figure 4A, solid curve) captures saturation because increases in receptive field output are attenuated by increases in suppressive field output. [sent-97, score-1.592]

40 At low contrast, no saturation is observed because the output of the suppressive field is dominated by the constant c 50. [sent-98, score-0.974]

41 To obtain the parameters of the suppressive field, we recorded responses to sums of two drifting gratings (Figure 4B-D): an optimal test grating at 50% contrast, which elicits a large baseline response, and a mask grating that modulates this response. [sent-100, score-1.415]

42 Test and mask temporal frequencies are incommensurate so that they temporally label a test response (at the frequency of the test) and a mask response (at the frequency of the mask) (Bonds, 1989). [sent-101, score-0.875]

43 We vary mask attributes and study how they affect the test responses. [sent-102, score-0.26]

44 The nonlinear model (Figure 4B, solid curve) captures it because increasing mask contrast increases the suppressive field output while the receptive field output (at the temporal frequency of the test) remains constant. [sent-105, score-2.05]

45 With masks of low contrast there is little suppression because the output of the suppressive field is dominated by the constant c 50 . [sent-106, score-1.18]

46 Similar effects are seen if we increase mask diameter. [sent-107, score-0.228]

47 The nonlinear model (Figure 4C, solid curve) captures it because increasing mask diameter increases the suppressive field output while it does not affect the receptive field output. [sent-110, score-1.877]

48 A plateau is reached once masks extend beyond the suppressive field. [sent-111, score-0.534]

49 From these data we estimate the size of the Gaussian envelope G(x) of the suppressive field. [sent-112, score-0.469]

50 Finally, the strength of suppression depends on mask spatial frequency (Figure 4D). [sent-113, score-0.447]

51 This dependence of suppression on spatial frequency is captured in the nonlinear model by the filter H(x,t). [sent-116, score-0.373]

52 From similar experiments involving different temporal frequencies (not shown), we estimate the filter’s selectivity for temporal frequency. [sent-118, score-0.2]

53 We now test whether the model predicts responses to a set of novel stimuli: drifting gratings varying in contrast and diameter. [sent-121, score-0.833]

54 Responses to high contrast stimuli exhibit size tuning (Figure 5A, squares): they grow with size for small diameters, reach a maximum value at intermediate diameter and are reduced for large diameters (Jones and Sillito, 1991). [sent-122, score-0.475]

55 Size tuning , however, strongly depends on stimulus contrast (Solomon et al. [sent-123, score-0.287]

56 , 2002): no size tuning is observed at low contrast (Figure 5A, circles). [sent-124, score-0.17]

57 For large, high contrast stimuli the output of the suppressive field is dominated by c local, resulting in suppression of responses. [sent-126, score-1.3]

58 At low contrast, c local is much smaller than c50, and the suppressive field does not affect responses. [sent-127, score-0.857]

59 Similar considerations can be made by plotting these data as a function of contrast (Figure 5B). [sent-128, score-0.139]

60 As predicted by the nonlinear model (Figure 5B, curves), the effect of increasing contrast depends on stimulus size: responses to large stimuli show strong saturation (Figure 5B, squares), whereas responses to small stimuli grow linearly (Figure 5B, circles). [sent-129, score-1.3]

61 The model predicts these effects because only large, high contrast stimuli elicit large enough responses from the suppressive field to cause suppression. [sent-130, score-1.605]

62 For small, low contrast stimuli, instead, the linear model is a good approximation. [sent-131, score-0.177]

63 Predicting responses to novel stimuli in the example LGN cell. [sent-141, score-0.491]

64 Stimuli are gratings varying in diameter and contrast, and responses are harmonic component of spike trains at grating temporal frequency. [sent-142, score-0.775]

65 B: Responses as function of contrast for different diameters. [sent-145, score-0.139]

66 4 M o d e l pe r f or m a nc e To assess model performance across neurons we calculate the percentage of variance in the data that is explained by the model (see Freeman et al. [sent-147, score-0.252]

67 The model provides good fits to the data used to characterize the suppressive field (Figure 4), explaining more than 90% of the variance in the data for 9/13 cells (Figure 6A). [sent-149, score-0.991]

68 Model parameters are then held fixed, and the model is used to predict responses to gratings of different contrast and diameter (Figure 5). [sent-150, score-0.773]

69 The model performs well, explaining in 10/13 neurons above 90% of the variance in these novel data (Figure 6B, shaded histogram). [sent-151, score-0.152]

70 The agreement between the quality of the fits and the quality of the predictions suggests that model parameters are well constrained and rules out a role of overfitting in determining the quality of the fits. [sent-152, score-0.199]

71 To further confirm the performance of the model, in an additional 54 cells we ran a subset of the whole protocol, involving only the experiment for characterizing the receptive field (Figure 3), and the experiment involving gratings of different contrast and diameter (Figure 5). [sent-153, score-0.987]

72 For these cells we estimate the suppressive field by fitting the model directly to the latter measurements. [sent-154, score-0.96]

73 The model explains above 90% of the variance in these data in 20/54 neurons and more than 70% in 39/54 neurons (Figure 6B, white histogram). [sent-155, score-0.21]

74 Estimating the suppressive field A # cells 6 n=13 4 2 0 Size tuning at different contrasts 15 n=54 10 # cells B 5 0 0 50 100 Explained variance (%) Figure 6. [sent-157, score-1.018]

75 4 Co n cl u si o n s The nonlinear model provides a unified description of visual processing in LGN neurons. [sent-163, score-0.177]

76 Based on a fixed set of parameters, it can predict both linear properties (Figure 3), as well as nonlinear properties such as contrast saturation (Figure 4A) and masking (Figure 4B-D). [sent-164, score-0.321]

77 Moreover, once the parameters are fixed, it predicts responses to novel stimuli (Figure 5). [sent-165, score-0.571]

78 The model explains why responses are tuned for stimulus size at high contrast but not at low contrast, and it correctly predicts that only responses to large stimuli saturate with contrast, while responses to small stimuli grow linearly. [sent-166, score-1.604]

79 The model implements a form of contrast gain control. [sent-167, score-0.217]

80 A possible purpose for this gain control is to increase the range of contrast that can be transmitted given the limited dynamic range of single neurons. [sent-168, score-0.179]

81 Divisive gain control may also play a role in population coding: a similar model applied to responses of primary visual cortex was shown to maximize independence of the responses across neurons (Schwartz and Simoncelli, 2001). [sent-169, score-0.847]

82 First, we are characterizing the dynamics of the suppressive field, e. [sent-171, score-0.469]

83 Second, we are testing the assumption that the suppressive field computes root-mean-square contrast, a measure that solely depends on the secondorder moments of the light distribution. [sent-174, score-0.857]

84 Our ultimate goal is to predict responses to complex stimuli such as those shown in Figure 1 and quantify to what degree the nonlinear model improves on the predictions of the linear model. [sent-175, score-0.615]

85 The nonlinear model synthesizes more than 30 years of research. [sent-177, score-0.157]

86 Because the nonlinearities we discussed are already present in the retina (Shapley and Victor, 1978), and tend to get stronger as one ascends the visual hierarchy (Sclar et al. [sent-180, score-0.201]

87 , 1990), it may also be used to study how responses take shape from one stage to another in the visual system. [sent-181, score-0.371]

88 Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex. [sent-186, score-0.217]

89 Spatiotemporal receptive field organization in the lateral geniculate nucleus of cats and kittens. [sent-203, score-0.798]

90 Selectivity and spatial distribution of signals from the receptive field surround in macaque v1 neurons. [sent-211, score-0.591]

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92 Efficient coding of natural scenes in the lateral geniculate nucleus: experimental test of a computational theory. [sent-228, score-0.235]

93 The contrast sensitivity of retinal ganglion cells of the cat. [sent-234, score-0.285]

94 The length-response properties of cells in the feline dorsal lateral geniculate nucleus. [sent-259, score-0.268]

95 Lateral geniculate neurons of cat: retinal inputs and physiology. [sent-268, score-0.245]

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99 The effect of contrast on the transfer properties of cat retinal ganglion cells. [sent-303, score-0.321]

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Abstract: We describe a neuromorphic chip that utilizes transistor heterogeneity, introduced by the fabrication process, to generate orientation maps similar to those imaged in vivo. Our model consists of a recurrent network of excitatory and inhibitory cells in parallel with a push-pull stage. Similar to a previous model the recurrent network displays hotspots of activity that give rise to visual feature maps. Unlike previous work, however, the map for orientation does not depend on the sign of contrast. Instead, signindependent cells driven by both ON and OFF channels anchor the map, while push-pull interactions give rise to sign-preserving cells. These two groups of orientation-selective cells are similar to complex and simple cells observed in V1. 1 Orientation Maps Neurons in visual areas 1 and 2 (V1 and V2) are selectively tuned for a number of visual features, the most pronounced feature being orientation. Orientation preference of individual cells varies across the two-dimensional surface of the cortex in a stereotyped manner, as revealed by electrophysiology [1] and optical imaging studies [2]. The origin of these preferred orientation (PO) maps is debated, but experiments demonstrate that they exist in the absence of visual experience [3]. To the dismay of advocates of Hebbian learning, these results suggest that the initial appearance of PO maps rely on neural mechanisms oblivious to input correlations. Here, we propose a model that accounts for observed PO maps based on innate noise in neuron thresholds and synaptic currents. The network is implemented in silicon where heterogeneity is as ubiquitous as it is in biology. 2 Patterned Activity Model Ernst et al. have previously described a 2D rate model that can account for the origin of visual maps [4]. Individual units in their network receive isotropic feedforward input from the geniculate and recurrent connections from neighboring units in a Mexican hat profile, described by short-range excitation and long-range inhibition. If the recurrent connections are sufficiently strong, hotspots of activity (or ‘bumps’) form periodically across space. In a homogeneous network, these bumps of activity are equally stable at any position in the network and are free to wander. Introducing random jitter to the Mexican hat connectivity profiles breaks the symmetry and reduces the number of stable states for the bumps. Subsequently, the bumps are pinned down at the locations that maximize their net local recurrent feedback. In this regime, moving gratings are able to shift the bumps away from their stability points such that the responses of the network resemble PO maps. Therefore, the recurrent network, given an ample amount of noise, can innately generate its own orientation specificity without the need for specific hardwired connections or visually driven learning rules. 2.1 Criticisms of the Bump model We might posit that the brain uses a similar opportunistic model to derive and organize its feature maps – but the parallels between the primary visual cortex and the Ernst et al. bump model are unconvincing. For instance, the units in their model represent the collective activity of a column, reducing the network dynamics to a firing-rate approximation. But this simplification ignores the rich temporal dynamics of spiking networks, which are known to affect bump stability. More fundamentally, there is no role for functionally distinct neuron types. The primary criticism of the Ernst et al.’s bump model is that its input only consists of a luminance channel, and it is not obvious how to replace this channel with ON and OFF rectified channels to account for simple and complex cells. One possibility would be to segregate ON-driven and OFF-driven cells (referred to as simple cells in this paper) into two distinct recurrent networks. Because each network would have its own innate noise profile, bumps would form independently. Consequently, there is no guarantee that ON-driven maps would line up with OFF-driven maps, which would result in conflicting orientation signals when these simple cells converge onto sign-independent (complex) cells. 2.2 Simple Cells Solve a Complex Problem To ensure that both ON-driven and OFF-driven simple cells have the same orientation maps, both ON and OFF bumps must be computed in the same recurrent network so that they are subjected to the same noise profile. We achieve this by building our recurrent network out of cells that are sign-independent; that is both ON and OFF channels drive the network. These cells exhibit complex cell-like behavior (and are referred to as complex cells in this paper) because they are modulated at double the spatial frequency of a sinusoidal grating input. The simple cells subsequently derive their responses from two separate signals: an orientation selective feedback signal from the complex cells indicating the presence of either an ON or an OFF bump, and an ON–OFF selection signal that chooses the appropriate response flavor. Figure 1 left illustrates the formation of bumps (highlighted cells) by a recurrent network with a Mexican hat connectivity profile. Extending the Ernst et al. model, these complex bumps seed simple bumps when driven by a grating. Simple bumps that match the sign of the input survive, whereas out-of-phase bumps are extinguished (faded cells) by push-pull inhibition. Figure 1 right shows the local connections within a microcircuit. An EXC (excitatory) cell receives excitatory input from both ON and OFF channels, and projects to other EXC (not shown) and INH (inhibitory) cells. The INH cell projects back in a reciprocal configuration to EXC cells. The divergence is indicated in left. ON-driven and OFF-driven simple cells receive input in a push-pull configuration (i.e., ON cells are excited by ON inputs and inhibited by OFF inputs, and vise-versa), while additionally receiving input from the EXC–INH recurrent network. In this model, we implement our push-pull circuit using monosynaptic inhibitory connections, despite the fact that geniculate input is strictly excitatory. This simplification, while anatomically incorrect, yields a more efficient implementation that is functionally equivalent. ON Input Luminance OFF Input left right EXC EXC Divergence INH INH Simple Cells Complex Cells ON & OFF Input ON OFF OFF Space Figure 1: left, Complex and simple cell responses to a sinusoidal grating input. Luminance is transformed into ON (green) and OFF (red) pathways by retinal processing. Complex cells form a recurrent network through excitatory and inhibitory projections (yellow and blue lines, respectively), and clusters of activity occur at twice the spatial frequency of the grating. ON input activates ON-driven simple cells (bright green) and suppresses OFF-driven simple cells (faded red), and vise-versa. right, The bump model’s local microcircuit: circles represent neurons, curved lines represent axon arbors that end in excitatory synapses (v shape) or inhibitory synapses (open circles). For simplicity, inhibitory interneurons were omitted in our push-pull circuit. 2.3 Mathematical Description • The neurons in our network follow the equation CV = −∑ ∂(t − tn) + I syn − I KCa − I leak , • n where C is membrane capacitance, V is the temporal derivative of the membrane voltage, δ(·) is the Dirac delta function, which resets the membrane at the times tn when it crosses threshold, Isyn is synaptic current from the network, and Ileak is a constant leak current. Neurons receive synaptic current of the form: ON I syn = w+ I ON − w− I OFF + wEE I EXC − wEI I INH , EXC I syn = w+ ( I ON + I OFF ) + wEE I EXC − wEI I INH + I back , OFF INH I syn = w+ I OFF − w− I ON + wEE I EXC − wEI I INH , I syn = wIE I EXC where w+ is the excitatory synaptic strength for ON and OFF input synapses, w- is the strength of the push-pull inhibition, wEE is the synaptic strength for EXC cell projections to other EXC cells, wEI is the strength of INH cell projections to EXC cells, wIE is the strength of EXC cell projections to INH cells, Iback is a constant input current, and I{ON,OFF,EXC,INH} account for all impinging synapses from each of the four cell types. These terms are calculated for cell i using an arbor function that consists of a spatial weighting J(r) and a post-synaptic current waveform α(t): k ∑ J (i − k ) ⋅ α (t − t n ) , where k spans all cells of a given type and n indexes their spike k ,n times. The spatial weighting function is described by J (i − k ) = exp( − i − k σ ) , with σ as the space constant. The current waveform, which is non-zero for t>0, convolves a 1 t function with a decaying exponential: α (t ) = (t τ c + α 0 ) −1 ∗ exp(− t τ e ) , where τc is the decay-rate, and τe is the time constant of the exponential. Finally, we model spike-rate adaptation with a calcium-dependent potassium-channel (KCa), which integrates Ca triggered by spikes at times tn with a gain K and a time constant τk, as described by I KCa = ∑ K exp(tn − t τ k ) . n 3 Silicon Implementation We implemented our model in silicon using the TSMC (Taiwan Semiconductor Manufacturing Company) 0.25µm 5-metal layer CMOS process. The final chip consists of a 2-D core of 48x48 pixels, surrounded by asynchronous digital circuitry that transmits and receives spikes in real-time. Neurons that reach threshold within the array are encoded as address-events and sent off-chip, and concurrently, incoming address-events are sent to their appropriate synapse locations. This interface is compatible with other spike-based chips that use address-events [5]. The fabricated bump chip has close to 460,000 transistors packed in 10 mm2 of silicon area for a total of 9,216 neurons. 3.1 Circuit Design Our neural circuit was morphed into hardware using four building blocks. Figure 2 shows the transistor implementation for synapses, axonal arbors (diffuser), KCa analogs, and neurons. The circuits are designed to operate in the subthreshold region (except for the spiking mechanism of the neuron). Noise is not purposely designed into the circuits. Instead, random variations from the fabrication process introduce significant deviations in I-V curves of theoretically identical MOS transistors. The function of the synapse circuit is to convert a brief voltage pulse (neuron spike) into a postsynaptic current with biologically realistic temporal dynamics. Our synapse achieves this by cascading a current-mirror integrator with a log-domain low-pass filter. The current-mirror integrator has a current impulse response that decays as 1 t (with a decay rate set by the voltage τc and an amplitude set by A). This time-extended current pulse is fed into a log-domain low-pass filter (equivalent to a current-domain RC circuit) that imposes a rise-time on the post-synaptic current set by τe. ON and OFF input synapses receive presynaptic spikes from the off-chip link, whereas EXC and INH synapses receive presynaptic spikes from local on-chip neurons. Synapse Je Diffuser Ir A Ig Jc KCa Analog Neuron Jk Vmem Vspk K Figure 2: Transistor implementations are shown for a synapse, diffuser, KCa analog, and neuron (simplified), with circuit insignias in the top-left of each box. The circuits they interact with are indicated (e.g. the neuron receives synaptic current from the diffuser as well as adaptation current from the KCa analog; the neuron in turn drives the KCa analog). The far right shows layout for one pixel of the bump chip (vertical dimension is 83µm, horizontal is 30 µm). The diffuser circuit models axonal arbors that project to a local region of space with an exponential weighting. Analogous to resistive divider networks, diffusers [6] efficiently distribute synaptic currents to multiple targets. We use four diffusers to implement axonal projections for: the ON pathway, which excites ON and EXC cells and inhibits OFF cells; the OFF pathway, which excites OFF and EXC cells and inhibits ON cells; the EXC cells, which excite all cell types; and the INH cells, which inhibits EXC, ON, and OFF cells. Each diffuser node connects to its six neighbors through transistors that have a pseudo-conductance set by σr, and to its target site through a pseudo-conductance set by σg; the space-constant of the exponential synaptic decay is set by σr and σg’s relative levels. The neuron circuit integrates diffuser currents on its membrane capacitance. Diffusers either directly inject current (excitatory), or siphon off current (inhibitory) through a current-mirror. Spikes are generated by an inverter with positive feedback (modified from [7]), and the membrane is subsequently reset by the spike signal. We model a calcium concentration in the cell with a KCa analog. K controls the amount of calcium that enters the cell per spike; the concentration decays exponentially with a time constant set by τk. Elevated charge levels activate a KCa-like current that throttles the spike-rate of the neuron. 3.2 Experimental Setup Our setup uses either a silicon retina [8] or a National Instruments DIO (digital input–output) card as input to the bump chip. This allows us to test our V1 model with real-time visual stimuli, similar to the experimental paradigm of electrophysiologists. More specifically, the setup uses an address-event link [5] to establish virtual point-to-point connectivity between ON or OFF ganglion cells from the retina chip (or DIO card) with ON or OFF synapses on the bump chip. Both the input activity and the output activity of the bump chip is displayed in real-time using receiver chips, which integrate incoming spikes and displays their rates as pixel intensities on a monitor. A logic analyzer is used to capture spike output from the bump chip so it can be further analyzed. We investigated responses of the bump chip to gratings moving in sixteen different directions, both qualitatively and quantitatively. For the qualitative aspect, we created a PO map by taking each cell’s average activity for each stimulus direction and computing the vector sum. To obtain a quantitative measure, we looked at the normalized vector magnitude (NVM), which reveals the sharpness of a cell’s tuning. The NVM is calculated by dividing the vector sum by the magnitude sum for each cell. The NVM is 0 if a cell responds equally to all orientations, and 1 if a cell’s orientation selectivity is perfect such that it only responds at a single orientation. 4 Results We presented sixteen moving gratings to the network, with directions ranging from 0 to 360 degrees. The spatial frequency of the grating is tuned to match the size of the average bump, and the temporal frequency is 1 Hz. Figure 3a shows a resulting PO map for directions from 180 to 360 degrees, looking at the inhibitory cell population (the data looks similar for other cell types). Black contours represent stable bump regions, or equivalently, the regions that exceed a prescribed threshold (90 spikes) for all directions. The PO map from the bump chip reveals structure that resembles data from real cortex. Nearby cells tend to prefer similar orientations except at fractures. There are even regions that are similar to pinwheels (delimited by a white rectangle). A PO is a useful tool to describe a network’s selectivity, but it only paints part of the picture. So we have additionally computed a NVM map and a NVM histogram, shown in Figure 3b and 3c respectively. The NVM map shows that cells with sharp selectivity tend to cluster, particularly around the edge of the bumps. The histogram also reveals that the distribution of cell selectivity across the network varies considerably, skewed towards broadly tuned cells. We also looked at spike rasters from different cell-types to gain insight into their phase relationship with the stimulus. In particular, we present recordings for the site indicated by the arrow (see Figure 3a) for gratings moving in eight directions ranging from 0 to 360 degrees in 45-degree increments (this location was chosen because it is in the vicinity of a pinwheel, is reasonably selective, and shows considerable modulation in its firing rate). Figure 4 shows the luminance of the stimulus (bottom sinusoids), ON- (cyan) and OFF-input (magenta) spike trains, and the resulting spike trains from EXC (yellow), INH (blue), ON- (green), and OFFdriven (red) cell types for each of the eight directions. The center polar plot summarizes the orientation selectivity for each cell-type by showing the normalized number of spikes for each stimulus. Data is shown for one period. Even though all cells-types are selective for the same orientation (regardless of grating direction), complex cell responses tend to be phase-insensitive while the simple cell responses are modulated at the fundamental frequency. It is worth noting that the simple cells have sharper orientation selectivity compared to the complex cells. This trend is characteristic of our data. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 300 250 200 150 100 50 20 40 60 80 100 120 140 160 180 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3: (a) PO map for the inhibitory cell population stimulated with eight different directions from 180 to 360 degrees (black represents no activity, contours delineate regions that exceed 90 spikes for all stimuli). Normalized vector magnitude (NVM) data is presented as (b) a map and (c) a histogram. Figure 4: Spike rasters and polar plot for 8 directions ranging from 0 to 360 degrees. Each set of spike rasters represent from bottom to top, ON- (cyan) and OFF-input (magenta), INH (yellow), EXC (blue), and ON- (green) and OFF-driven (red). The stimulus period is 1 sec. 5 Discussion We have implemented a large-scale network of spiking neurons in a silicon chip that is based on layer 4 of the visual cortex. The initial testing of the network reveals a PO map, inherited from innate chip heterogeneities, resembling cortical maps. Our microcircuit proposes a novel function for complex-like cells; that is they create a sign-independent orientation selective signal, which through a push-pull circuit creates sharply tuned simple cells with the same orientation preference. Recently, Ringach et al. surveyed orientation selectivity in the macaque [9]. They observed that, in a population of V1 neurons (N=308) the distribution of orientation selectivity is quite broad, having a median NVM of 0.39. We have measured median NVM’s ranging from 0.25 to 0.32. Additionally, Ringach et al. found a negative correlation between spontaneous firing rate and NVM. This is consistent with our model because cells closer to the center of the bump have higher firing rates and broader tuning. While the results from the bump chip are promising, our maps are less consistent and noisier than the maps Ernst et al. have reported. We believe this is because our network is tuned to operate in a fluid state where bumps come on, travel a short distance and disappear (motivated by cortical imaging studies). But excessive fluidity can cause non-dominant bumps to briefly appear and adversely shift the PO maps. We are currently investigating the role of lateral connections between bumps as a means to suppress these spontaneous shifts. The neural mechanisms that underlie the orientation selectivity of V1 neurons are still highly debated. This may be because neuron responses are not only shaped by feedforward inputs, but are also influenced at the network level. If modeling is going to be a useful guide for electrophysiologists, we must model at the network level while retaining cell level detail. Our results demonstrate that a spike-based neuromorphic system is well suited to model layer 4 of the visual cortex. The same approach may be used to build large-scale models of other cortical regions. References 1. Hubel, D. and T. Wiesel, Receptive firelds, binocular interaction and functional architecture in the cat's visual cortex. J. Physiol, 1962. 160: p. 106-154. 2. Blasdel, G.G., Orientation selectivity, preference, and continuity in monkey striate cortex. J Neurosci, 1992. 12(8): p. 3139-61. 3. Crair, M.C., D.C. Gillespie, and M.P. Stryker, The role of visual experience in the development of columns in cat visual cortex. Science, 1998. 279(5350): p. 566-70. 4. Ernst, U.A., et al., Intracortical origin of visual maps. Nat Neurosci, 2001. 4(4): p. 431-6. 5. Boahen, K., Point-to-Point Connectivity. IEEE Transactions on Circuits & Systems II, 2000. vol 47 no 5: p. 416-434. 6. Boahen, K. and Andreou. A contrast sensitive silicon retina with reciprocal synapses. in NIPS91. 1992: IEEE. 7. Culurciello, E., R. Etienne-Cummings, and K. Boahen, A Biomorphic Digital Image Sensor. IEEE Journal of Solid State Circuits, 2003. vol 38 no 2: p. 281-294. 8. Zaghloul, K., A silicon implementation of a novel model for retinal processing, in Neuroscience. 2002, UPENN: Philadelphia. 9. Ringach, D.L., R.M. Shapley, and M.J. Hawken, Orientation selectivity in macaque V1: diversity and laminar dependence. J Neurosci, 2002. 22(13): p. 5639-51.

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