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119 nips-2010-Implicit encoding of prior probabilities in optimal neural populations

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Author: Deep Ganguli, Eero P. Simoncelli

Abstract: unkown-abstract

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sentIndex sentText sentNum sentScore

1 Here we consider the inﬂuence of a prior probability distribution over sensory variables on the optimal allocation of neurons and spikes in a population. [sent-5, score-0.469]

2 We model the spikes of each cell as samples from an independent Poisson process with rate governed by an associated tuning curve. [sent-6, score-0.503]

3 For this response model, we approximate the Fisher information in terms of the density and amplitude of the tuning curves, under the assumption that tuning width varies inversely with cell density. [sent-7, score-1.064]

4 This family includes lower bounds on mutual information and perceptual discriminability as special cases. [sent-9, score-0.396]

5 In all cases, we ﬁnd a closed form expression for the optimum, in which the density and gain of the cells in the population are power law functions of the stimulus prior. [sent-10, score-0.968]

6 We show preliminary evidence that the theory successfully predicts the relationship between empirically measured stimulus priors, physiologically measured neural response properties (cell density, tuning widths, and ﬁring rates), and psychophysically measured discrimination thresholds. [sent-12, score-0.933]

7 1 Introduction Many bottom up theories of neural encoding posit that sensory systems are optimized to represent sensory information, subject to limitations of noise and resources (e. [sent-13, score-0.498]

8 A substantial literature has considered population models in which each neuron’s mean response to a scalar variable is characterized by a tuning curve [e. [sent-17, score-0.802]

9 In these results, the distribution of sensory variables is assumed to be uniform and the populations are assumed to be homogeneous with regard to tuning curve shape, spacing, and amplitude. [sent-21, score-0.618]

10 It would seem natural that a neural system should devote more resources to regions of sensory space that occur with higher probability, analogous to results in coding theory [11]. [sent-23, score-0.328]

11 At the population level, non-uniform allocations of neurons with identical tuning curves have been shown to be optimal for non-uniform stimulus distributions [16, 17]. [sent-25, score-1.216]

12 Here, we examine the inﬂuence of a sensory prior on the optimal allocation of neurons and spikes in a population, and the implications of this optimal allocation for subsequent perception. [sent-26, score-0.568]

13 Given a prior distribution over a scalar stimulus parameter, and a resource budget of N neurons with an average of R spikes/sec for the entire population, we seek the optimal shapes, positions, and amplitudes of tuning curves. [sent-27, score-0.86]

14 We assume a population with independent Poisson spiking, and consider a family of objective functions based on Fisher information. [sent-28, score-0.415]

15 We then approximate the Fisher information in terms of two continuous resource variables, the density and gain of the tuning curves. [sent-29, score-0.668]

16 For all objective functions, we ﬁnd that the optimal tuning curve properties (cell density, tuning width, and gain) are power-law functions of the stimulus prior, with exponents dependent on the speciﬁc choice of objective function. [sent-31, score-1.14]

17 Through the Fisher information, we also derive a bound on perceptual discriminability, again in the form a power-law of the stimulus prior. [sent-32, score-0.362]

18 Thus, our framework provides direct and experimentally testable links between sensory priors, properties of the neural representation, and perceptual discriminability. [sent-33, score-0.396]

19 2 Encoding model We assume a conventional model for a population of N neurons responding to a single scalar variable, s [1–6]. [sent-35, score-0.483]

20 The number of spikes emitted (per unit time) by the nth neuron is a sample from an independent Poisson process, with mean rate determined by its tuning function, hn (s). [sent-36, score-0.573]

21 The probability density of the population response can be written as N p(r|s) = hn (s)rn e−hn (s) . [sent-37, score-0.645]

22 n=1 We also assume the total expected spike rate, R, of the population is ﬁxed, which places a constraint on the tuning curves: N hn (s) ds = R, p(s) (1) n=1 where p(s) is the probability distribution of stimuli in the environment. [sent-39, score-0.99]

23 We refer to this as a sensory prior, in anticipation of its future use in Bayesian decoding of the population response. [sent-40, score-0.567]

24 To formulate a family of objective functions which depend on both p(s), and the tuning curves, we ﬁrst rely on Fisher information, If (s), which can be written as a function of the tuning curves [1, 18]: If (s) = − p(r|s) ∂2 log p(r|s) dr ∂s2 N = h′2 (s) n . [sent-42, score-0.894]

25 hn (s) n=1 The Fisher information can be used to express lower bounds on mutual information [16], the variance of an unbiased estimator [18], and perceptual discriminability [19]. [sent-43, score-0.524]

26 The Cramer-Rao inequality allows us to express the minimum expected squared 2 stimulus discriminability achievable by any decoder1 : p(s) ds. [sent-45, score-0.43]

27 We formulate a generalized objective function that includes the Fisher bounds on information and discriminability as special cases: N N arg max hn (s) p(s) f h′2 (s) n hn (s) n=1 ds, s. [sent-47, score-0.496]

28 To make the problem tractable, we ﬁrst introduce a parametrization of the population in terms of cell density and gain. [sent-62, score-0.593]

29 The cell density controls both the spacing and width of the tuning curves, and the gain controls their maximum average ﬁring rates. [sent-63, score-0.844]

30 Finally, re-writing the objective function and constraints in these terms allows us to obtain closed-form solutions for the optimal tuning curves. [sent-65, score-0.463]

31 1 Density and gain for a homogeneous population If p(s) is uniform, then by symmetry, the Fisher information for an optimal neural population should also be uniform. [sent-67, score-0.932]

32 We assume a convolutional population of tuning curves, evenly spaced on the unit lattice, such that they approximately “tile” the space: N h(s − n) ≈ 1. [sent-68, score-0.869]

33 n=1 We also assume that this population has an approximately constant Fisher information: N If (s) = h′2 (s − n) h(s − n) n=1 N φ(s − n) ≈ Iconv . [sent-69, score-0.397]

34 = (5) n=1 That is, we assume that the Fisher information curves for the individual neurons, φ(s − n), also tile the stimulus space. [sent-70, score-0.438]

35 The value of the constant, Iconv , is dependent on the details of the tuning curve shape, h(s), which we leave unspeciﬁed. [sent-71, score-0.408]

36 1(a-b) shows that the Fisher information for a convolutional population of Gaussian tuning curves, with appropriate width, is approximately constant. [sent-73, score-0.836]

37 Now we introduce two scalar values, a gain (g), and a density (d), that affect the convolutional population as follows: n (6) hn (s) = g h d(s − ) . [sent-74, score-0.808]

38 Here, we use it to bound the squared discriminability of the estimator, as expressed in the stimulus space, which is independent of bias [19]. [sent-76, score-0.43]

39 (a) Homogeneous population with Gaussian tuning curves on the unit lattice. [sent-80, score-0.858]

40 55 is chosen so that the curves approximately tile the stimulus space. [sent-82, score-0.471]

41 (b) The Fisher information of the convolutional population (green) is approximately constant. [sent-83, score-0.487]

42 The cumulative integral of this density, D(s), alters the positions and widths of the tuning curves in the convolutional population. [sent-85, score-0.658]

43 (d) The warped population, with tuning curve peaks (aligned with tick marks, at locations sn = D−1 (n)), is scaled by the gain function, g(s) (blue). [sent-86, score-0.641]

44 A single tuning curve is highlighted (red) to illustrate the effect of the warping and scaling operations. [sent-87, score-0.408]

45 (e) The Fisher information of the inhomogeneous population is approximately proportional to d2 (s)g(s). [sent-88, score-0.397]

46 The density controls both the spacing and width of the tuning curves: as the density increases, the tuning curves become narrower, and are spaced closer together so as to maintain their tiling of stimulus space. [sent-90, score-1.49]

47 (5), that the Fisher information of the convolutional population is approximately constant with respect to s. [sent-93, score-0.487]

48 If the original (unit-spacing) convolutional population is supported on the interval (0, Q) of the stimulus space, then the number of neurons in the modulated population must be N (d) = Qd to cover the same interval. [sent-95, score-1.146]

49 Under the assumption that the tuning curves tile the stimulus space, Eq. [sent-96, score-0.787]

50 2 Density and gain for a heterogeneous population Intuitively, if p(s) is non-uniform, the optimal Fisher information should also be non-uniform. [sent-99, score-0.557]

51 This can be achieved through inhomogeneities in either the tuning curve density or gain. [sent-100, score-0.531]

52 We thus generalize density and gain to be continuous functions of the stimulus, d(s) and g(s), that warp and scale the convolutional population: hn (s) = g(sn ) h(D(s) − n). [sent-101, score-0.444]

53 Optimal heterogeneous population properties, for objective functions speciﬁed by Eq. [sent-103, score-0.475]

54 s Here, D(s) = −∞ d(t)dt, the cumulative integral of d(s), warps the shape of the prototype tuning curve. [sent-105, score-0.349]

55 The value sn = D−1 (n) represents the preferred stimulus value of the (warped) nth tuning curve (Fig. [sent-106, score-0.696]

56 Note that the warped population retains the tiling properties of the original convolutional population. [sent-108, score-0.534]

57 As in the uniform case, the density controls both the spacing and width of the tuning curves. [sent-109, score-0.635]

58 We can now write the Fisher information of the heterogeneous population of neurons in Eq. [sent-113, score-0.543]

59 As earlier, the constant Iconv is determined by the precise shape of the tuning curves. [sent-118, score-0.349]

60 To attain the proper rate, we use the fact that the warped tuning curves sum to unity (before multiplication by the gain function) and use Eq. [sent-121, score-0.648]

61 3 Objective function and solution for a heterogeneous population Approximating Fisher information as proportional to squared density and gain allows us to re-write the objective function and resource constraints of Eq. [sent-124, score-0.826]

62 In all cases, the solution speciﬁes a power-law relationship between the prior, and the density and gain of the tuning curves. [sent-130, score-0.575]

63 In general, all solutions allocate more neurons, with correspondingly narrower tuning curves, to higher-probability stimuli. [sent-131, score-0.426]

64 The shape of the optimal gain function depends on the objective function: for α < 0, neurons with lower ﬁring rates are used to represent stimuli with higher probabilities, and for α > 0, neurons with higher ﬁring rates are used for stimuli with higher probabilities. [sent-133, score-0.506]

65 (c) Orientation discrimination thresholds averaged across four human subjects [24]. [sent-138, score-0.369]

66 (d & e) Infomax and discrimax predictions of orientation distribution. [sent-139, score-0.424]

67 In addition to power-law relationships between tuning properties and sensory priors, our formulation offers a direct relationship between the sensory prior and perceptual discriminability. [sent-144, score-0.948]

68 5 Experimental evidence Our framework predicts a quantitative link between the sensory prior, physiological parameters (the density, tuning widths, and gain of cells), and psychophysically measured discrimination thresholds. [sent-148, score-0.905]

69 We obtained subsets of these quantities for two visual stimulus variables, orientation and spatial frequency, both of believed to be encoded by cells in primary visual cortex (area V1). [sent-149, score-0.502]

70 For each variable, we use the infomax and discrimax solutions to convert the physiological and perceptual measurements, using the appropriate exponents from Table 1, into predictions of the stimulus prior p(s). [sent-150, score-1.055]

71 1 Orientation We estimated the prior distribution of orientations in the environment by averaging orientation statistics across three natural image databases. [sent-153, score-0.316]

72 The average distribution of orientations exhibits higher probability at the cardinal orientations (vertical and horizontal) than at the oblique orientations (Fig. [sent-156, score-0.33]

73 Measurements of cell density for a population of 79 orientation-tuned V1 cells in Macaque [23] show more cells tuned to the cardinal orientations than the oblique orientations (Fig. [sent-158, score-1.075]

74 Finally, perceptual discrimination thresholds, averaged across four human subjects [24] show a similar bias (Fig. [sent-160, score-0.426]

75 If a neural population is designed to maximize information, then the cell density and inverse discrimination thresholds should match the stimulus prior, as expressed in infomax column of Table 1. [sent-163, score-1.275]

76 (b) Cell density as a function of preferred spatial frequency for a population of 317 V1 cells [25, 28] Dark blue: average number of cells tuned to each spatial frequency. [sent-167, score-0.953]

77 (d & e) Infomax and discrimax predictions of spatial frequency distribution. [sent-172, score-0.464]

78 We see that the predictions arising from cell density and discrimination thresholds are consistent with one another, and both are consistent with the stimulus prior. [sent-177, score-0.771]

79 For the discrimax objective function, the exponents in the power-law relationships (expressed in Table 1) are too small, resulting in poor qualitative agreement between the stimulus prior and predictions from the physiology and perception (Fig. [sent-179, score-0.81]

80 For example, predicting the prior from perceptual data, under the discrimax objective function, requires exponentiating discrimination thresholds to the fourth power, resulting in an over exaggeration of the cardinal bias. [sent-181, score-0.803]

81 2 Spatial frequency We obtained a prior distribution over spatial frequencies averaged across two natural image databases [20, 21]. [sent-183, score-0.344]

82 We also obtained spatial frequency tuning properties for a population of 317 V1 cells [25]. [sent-188, score-0.958]

83 On average, we see there are more cells, with correspondingly narrower tuning widths, tuned to low spatial frequencies (Fig. [sent-189, score-0.548]

84 These data support the model assumption that tuning width is inversely proportional to cell density. [sent-191, score-0.562]

85 We also obtained average discrimination thresholds for sinusoidal gratings of different spatial frequencies from two studies (Fig. [sent-192, score-0.401]

86 We again test the infomax and discrimax solutions by comparing predicted distributions obtained from the physiological and perceptual data, to the measured prior. [sent-196, score-0.701]

87 The infomax case shows striking agreement between the measured stimulus prior, and predictions based on the physiological and perceptual measurements (Fig. [sent-198, score-0.727]

88 However, as in the orientation case, discrimax predictions are poor (Fig. [sent-200, score-0.424]

89 3(e)), suggesting that information maximization provides a better optimality principle for explaining the neural and perceptual encoding of spatial frequency than discrimination maximization. [sent-201, score-0.536]

90 7 6 Discussion We have examined the inﬂuence sensory priors on the optimal allocation of neural resources, as well as the inﬂuence of these optimized resources on subsequent perception. [sent-202, score-0.343]

91 Fisher information is known to provide a poor bound on mutual information when there are a small number of neurons, a short decoding time, or non-smooth tuning curves [16, 29]. [sent-207, score-0.577]

92 These assumptions allow us to approximate Fisher information in terms of cell density and gain (Fig. [sent-211, score-0.332]

93 Our framework offers an important generalization of the population coding literature, allowing for non-uniformity of sensory priors, and corresponding heterogeneity in tuning and gain properties. [sent-213, score-1.043]

94 Second, tuning curve encoding models only specify neural responses to single stimulus values. [sent-216, score-0.701]

95 Previous studies assume that prior probabilities are either uniform [6], represented in the spiking activity of a separate population of neurons [5], or represented (in sample form) in the spontaneous activity [35]. [sent-224, score-0.543]

96 Our encoding formulation provides a mechanism whereby the prior is implicitly encoded in the density and gains of tuning curves, which presumably arise from the strength of synaptic connections. [sent-225, score-0.581]

97 Narrow versus wide tuning curves: What’s best for a population code? [sent-258, score-0.713]

98 Optimal tuning widths in population coding of periodic variables. [sent-264, score-0.84]

99 Maximally informative stimuli and tuning curves for sigmoidal ratecoding neurons and populations. [sent-280, score-0.655]

100 Optimal neural population coding of an auditory spatial cue. [sent-286, score-0.54]

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