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24 nips-2011-Active learning of neural response functions with Gaussian processes

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Author: Mijung Park, Greg Horwitz, Jonathan W. Pillow

Abstract: A sizeable literature has focused on the problem of estimating a low-dimensional feature space for a neuron’s stimulus sensitivity. However, comparatively little work has addressed the problem of estimating the nonlinear function from feature space to spike rate. Here, we use a Gaussian process (GP) prior over the inﬁnitedimensional space of nonlinear functions to obtain Bayesian estimates of the “nonlinearity” in the linear-nonlinear-Poisson (LNP) encoding model. This approach offers increased ﬂexibility, robustness, and computational tractability compared to traditional methods (e.g., parametric forms, histograms, cubic splines). We then develop a framework for optimal experimental design under the GP-Poisson model using uncertainty sampling. This involves adaptively selecting stimuli according to an information-theoretic criterion, with the goal of characterizing the nonlinearity with as little experimental data as possible. Our framework relies on a method for rapidly updating hyperparameters under a Gaussian approximation to the posterior. We apply these methods to neural data from a color-tuned simple cell in macaque V1, characterizing its nonlinear response function in the 3D space of cone contrasts. We ﬁnd that it combines cone inputs in a highly nonlinear manner. With simulated experiments, we show that optimal design substantially reduces the amount of data required to estimate these nonlinear combination rules. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 However, comparatively little work has addressed the problem of estimating the nonlinear function from feature space to spike rate. [sent-8, score-0.388]

2 Here, we use a Gaussian process (GP) prior over the inﬁnitedimensional space of nonlinear functions to obtain Bayesian estimates of the “nonlinearity” in the linear-nonlinear-Poisson (LNP) encoding model. [sent-9, score-0.215]

3 We then develop a framework for optimal experimental design under the GP-Poisson model using uncertainty sampling. [sent-13, score-0.289]

4 This involves adaptively selecting stimuli according to an information-theoretic criterion, with the goal of characterizing the nonlinearity with as little experimental data as possible. [sent-14, score-0.4]

5 We apply these methods to neural data from a color-tuned simple cell in macaque V1, characterizing its nonlinear response function in the 3D space of cone contrasts. [sent-16, score-0.394]

6 We ﬁnd that it combines cone inputs in a highly nonlinear manner. [sent-17, score-0.203]

7 With simulated experiments, we show that optimal design substantially reduces the amount of data required to estimate these nonlinear combination rules. [sent-18, score-0.29]

8 1 Introduction One of the central problems in systems neuroscience is to understand how neural spike responses convey information about environmental stimuli, which is often called the neural coding problem. [sent-19, score-0.405]

9 One approach to this problem is to build an explicit encoding model of the stimulus-conditional response distribution p(r|x), where r is a (scalar) spike count elicited in response to a (vector) stimulus x. [sent-20, score-0.71]

10 Most prior work has focused on: simple parametric forms [6, 9, 11]; non-parametric methods that do not scale easily to high 1 input inverse-link nonlinearity Poisson spiking response history filter Figure 1: Encoding model schematic. [sent-23, score-0.361]

11 The nonlinear function f converts an input vector x to a scalar, which g then transforms to a non-negative spike rate λ = g(f (x)). [sent-24, score-0.373]

12 The spike response r is a Poisson random variable with mean λ. [sent-25, score-0.391]

13 In this paper, we use a Gaussian process (GP) to provide a ﬂexible, computationally tractable model of the multi-dimensional neural response nonlinearity f (x), where x is a vector in feature space. [sent-29, score-0.298]

14 We use a ﬁxed inverse link function g to transform f (x) to a non-negative spike rate, which ensures the posterior over f is log-concave [6, 20]. [sent-36, score-0.408]

15 The method relies on uncertainty sampling [31], which involves selecting the stimulus x for which g(f (x)) is maximally uncertain given the data collected in the experiment so far. [sent-39, score-0.478]

16 We show that the GP-Poisson model provides a ﬂexible, tractable model for these responses, and that optimal design can substantially reduce the number of stimuli required to characterize them. [sent-41, score-0.39]

17 1 GP-Poisson neural encoding model Encoding model (likelihood) We begin by deﬁning a probabilistic encoding model for the neural response. [sent-43, score-0.232]

18 Let ri be an observed neural response (the spike count in some time interval T ) at the i’th trial given the input stimulus xi . [sent-44, score-0.708]

19 1), an input vector xi passes through a nonlinear function f , whose real-valued output is transformed to a positive spike rate through a (ﬁxed) function g. [sent-47, score-0.373]

20 The spike response is a Poisson random variable with mean g(f (x)), so the conditional probability of a stimulusresponse pair is Poisson: p(ri |xi , f ) = 1 ri −λi , ri ! [sent-48, score-0.49]

21 , rN ) is a vector of spike responses, 1 is a vector of ones, and f = (f (x1 ), . [sent-53, score-0.269]

22 This allows us to place a Gaussian prior on f without allocating probability mass to negative spike rates, and obviates the need for constrained optimization of f (but see [22] for a highly efﬁcient solution). [sent-62, score-0.323]

23 The hyperparameters deﬁning this prior are a mean µf and a kernel function k(xi , xj ) that speciﬁes the covariance between function values f (xi ) and f (xj ) for any pair of input points xi and xj . [sent-69, score-0.346]

24 Here, we use a Gaussian kernel, since neural response nonlinearities are expected to be smooth in general: k(xi , xj ) = ρ exp −||xi − xj ||2 /(2τ ) , (4) where hyperparameters ρ and τ control the marginal variance and smoothness scale, respectively. [sent-73, score-0.395]

25 3 The maximum a posteriori (MAP) estimate can be obtained by numerically maximizing the posterior for f . [sent-76, score-0.265]

26 (5) As noted above, this posterior has a unique maximum fmap so long as g is convex and log-concave. [sent-80, score-0.841]

27 However, the solution vector fmap deﬁned this way contains only the function values at the points in the training set X. [sent-81, score-0.756]

28 The GP prior provides a simple analytic formula for the maximum of the joint marginal containing the training data and any new point f ∗ = f (x∗ ), for a new stimulus x∗ . [sent-83, score-0.273]

29 j i i In practice, the prior covariance matrix K is often ill-conditioned when datapoints in X are closely spaced and smoothing hyperparameter τ is large, making it impossible to numerically compute K −1 . [sent-96, score-0.283]

30 4 Efﬁcient evidence optimization for θ The hyperparameters θ = {µf , ρ, τ } that control the GP prior have a major inﬂuence on the shape of the inferred nonlinearity, particularly in high dimensions and when data is scarce. [sent-101, score-0.225]

31 2), evaluated at fmap , and K −1 is the inverse prior covariance (eq. [sent-107, score-0.788]

32 8) gives us a formula for evaluating approximate evidence, exp L(f ) N (f |µf , K) p(r|θ) ≈ , (10) N (f |fmap , Λ) which we evaluate at f = fmap , since the Laplace approximation is the most accurate there [20, 33]. [sent-111, score-0.729]

33 The hyperparameters θ directly affect the prior mean and covariance (µf , K), as well as the posterior mean and covariance (fmap , Λ), all of which are essential for evaluating the evidence. [sent-112, score-0.429]

34 Finding fmap and Λ given θ requires numerical optimization of log p(f |r, θ), which is computationally expensive to perform for each search step in θ. [sent-113, score-0.702]

35 The logic here is that a Gaussian posterior and prior imply a likelihood function proportional to a Gaussian, which in turn allows prior and posterior moments to be computed analytically for each θ. [sent-115, score-0.444]

36 9), and m = H −1 (Λ−1 fmap − K −1 µf ) for the likelihood mean, which comes from the standard formula for the product of two Gaussians. [sent-118, score-0.756]

37 fmap = Λ(Hi mi + K −1 µf ) and Λ = (Hi + K −1 )−1 . [sent-121, score-0.702]

38 Note that this signiﬁcantly expedites evidence optimization since we do not have to numerically optimize fmap for each θ. [sent-122, score-0.828]

39 The true nonlinear response function g(f (x)) is in gray, the posterior mean is in black solid, 95% conﬁdence interval is in black dotted, stimulus is in blue dots. [sent-124, score-0.53]

40 A (top): Random design: responses were measured with 20 (left) and 100 (right) additional stimuli, with stimuli sampled uniformly over the interval shown on the x axis. [sent-125, score-0.303]

41 A (bottom): Optimal design: responses were measured with same numbers of additional stimuli selected by uncertainty sampling (see text). [sent-126, score-0.455]

42 The optimal design achieved half the error rate of the random design experiment. [sent-128, score-0.303]

43 3 Optimal design: uncertainty sampling So far, we have introduced an efﬁcient algorithm for estimating the nonlinearity f and hyperparameters θ for an LNP encoding model under a GP prior. [sent-129, score-0.51]

44 Here we introduce a method for adaptively selecting stimuli during an experiment (often referred to as active learning or optimal experimental design) to minimize the amount of data required to estimate f [29]. [sent-130, score-0.364]

45 The basic idea is that we should select stimuli that maximize the expected information gained about the model parameters. [sent-131, score-0.21]

46 Uncertainty sampling [31] is an algorithm that is appropriate when the model parameters and stimulus space are in a 1-1 correspondence. [sent-133, score-0.267]

47 It involves selecting the stimulus x for which the posterior over parameter f (x) has highest entropy, which in the case of a Gaussian posterior corresponds to the highest posterior variance. [sent-134, score-0.606]

48 Here we alter the algorithm slightly to select stimuli for which we are most uncertain about the spike rate g(f (x)), not (as stated above) the stimuli where we are most uncertain about our underlying function f (x). [sent-135, score-0.79]

49 Our strategy therefore focuses on uncertainty in the expected spike-rate rather than uncertainty in f . [sent-138, score-0.226]

50 For each point, we j compute the posterior uncertainty γj about the spike rate g(f (x∗ )) using the delta method, i. [sent-141, score-0.56]

51 , j γj = g (f (x∗ ))σj , where σj is the posterior standard deviaton (square root of the posterior variance) j at f (xj ) and g is the derivative of g with respect to its argument. [sent-143, score-0.278]

52 The stimulus selected next on trial t + 1, given all data observed up to time t, is selected randomly from the set: xt+1 ∈ {x∗ | γj ≥ γi ∀i}, j (12) that is, the set of all stimuli for which uncertainty γ is maximal. [sent-144, score-0.524]

53 To ﬁnd {σj } at each candidate point, we must ﬁrst update θ and fmap . [sent-145, score-0.734]

54 After each trial, we update fmap by numerically optimizing the posterior, then update the hyperparameters using (eq. [sent-146, score-0.873]

55 11), and then numerically re-compute fmap and Λ given the new θ. [sent-147, score-0.765]

56 5 Algorithm 1 Optimal design for nonlinearity estimation under a GP-Poisson model 1. [sent-150, score-0.248]

57 , rt }, the posterior mode fmap t , and hyper∗ parameters θt , compute the posterior mean and standard deviation (fmap , σ ∗ ) at a grid of ∗ candidate stimulus locations {x }. [sent-157, score-1.251]

58 ﬁnd fmap t+1 |Dt+1 , θt ; update θi+1 by maximizing evidence; ﬁnd fmap t+1 |Dt+1 , θt+1 4 Simulations We tested our method in simulation using a 1-dimensional feature space, where it is easy to visualize the nonlinearity and the uncertainty of our estimates (Fig. [sent-161, score-1.677]

59 The stimulus space was taken to be the range [0, 100], the true f was a sinusoid, and spike responses were simulated as Poisson with rate g(f (x)). [sent-163, score-0.579]

60 We compared the estimate of g(f (x)) obtained using optimal design to the estimate obtained with “random sampling”, stimuli drawn uniformly from the stimulus range. [sent-164, score-0.59]

61 2 shows the estimates of g(f (x)) after 20 and 100 trials using each method, along with the marginal posterior standard deviation, which provides a ±2 SD Bayesian conﬁdence interval for the estimate. [sent-166, score-0.217]

62 The optimal design method effectively decreased the high variance in the middle (near 50) because it drew more samples where uncertainty about the spike rate was higher (due to the fact that variance increases with mean for Poisson neurons). [sent-167, score-0.641]

63 As shown in (B), uncertainty sampling achieved roughly half the error rate of the random sampling after 20 datapoints. [sent-171, score-0.32]

64 6 20 10 0 0 20 trial # 40 60 Figure 3: Raw experimental data: stimuli in 3D conecontrast space (above) and recorded spike counts (below) during the ﬁrst 60 experimental trials. [sent-174, score-0.599]

65 Several (3-6) stimulus staircases along different directions in color space were randomly interleaved to avoid the effects of adaptation; a color direction is deﬁned as the relative proportions of L, M, and S cone contrasts, with [0 0 0] corresponding to a neutral gray (zero-contrast) stimulus. [sent-175, score-0.515]

66 This sampling procedure permitted a broad survey of the stimulus space, with the objective that many stimuli evoked a statistically reliable but non-saturating response. [sent-177, score-0.453]

67 In all, 677 stimuli in 65 color directions were presented for this neuron. [sent-178, score-0.246]

68 Contrast was varied using multiple interleaved staircases along different axes in color space, and spikes were counted during a 557ms window beginning 100ms after stimulus appeared. [sent-181, score-0.292]

69 The staircase design was used because the experiments were carried out prior to formulating the optimal design methods described in this paper. [sent-182, score-0.318]

70 However, we will analyze them here for a “simulated optimal design experiment”, where we choose stimuli sequentially from the list of stimuli that were actually presented during the experiment, in an order determined by our information-theoretic criterion. [sent-183, score-0.569]

71 6 0 0 M spike rate 150 datapoints random sampling uncertainty sampling 0 L 30 0. [sent-211, score-0.723]

72 6 0 0 spike rate 150 datapoints random sampling uncertainty sampling B spike rate A all data posterior mean 95% conf. [sent-212, score-1.202]

73 4 S Figure 4: One and two-dimensional conditional “slices” through the 3D nonlinearity of a V1 simple cell in cone contrast space. [sent-217, score-0.297]

74 A: 1D conditionals showing spike rate as a function of L, M, and S cone contrast, respectively, with other cone contrasts ﬁxed to zero. [sent-218, score-0.747]

75 Note that even with only 1/4 of data, the optimal design estimate is nearly identical to the estimate obtained from all 677 datapoints. [sent-220, score-0.221]

76 B: 2D conditionals on M and L (ﬁrst row), S and L (second row), M and S (third row) cones, respectively, with the other cone contrast set to zero. [sent-221, score-0.264]

77 2D conditionals using optimal design sampling (middle column) with 150 data points are much closer to the 2D conditionals using all data (right column) than those from a random sub-sampling of 150 points (left column). [sent-222, score-0.541]

78 We ﬁrst used the entire dataset (677 stimulus-response pairs) to ﬁnd the posterior maximum fmap , with hyperparameters set by maximizing evidence (sequential optimization of fmap and θ (eq. [sent-223, score-1.741]

79 4 shows 1D and 2D conditional slices through the estimated 3D nonlinearity g(f (x)), with contour plots constructed using the MAP estimate of f on a ﬁne grid of points. [sent-226, score-0.238]

80 The contours for a neuron with linear summation of cone contrasts followed by an output nonlinearity (i. [sent-227, score-0.418]

81 4B) indicates that cone contrasts are summed together in a highly nonlinear fashion, especially for L and M cones (top). [sent-231, score-0.297]

82 We then performed a simulated optimal design experiment by selecting from the 677 stimulusresponse pairs collected during the experiment, and re-ordering them greedily according to the uncertainty sampling algorithm described above. [sent-232, score-0.518]

83 Using only 150 data points, the conditionals of the estimate using uncertainty sampling were almost identical to those using all data (677 points). [sent-235, score-0.333]

84 Although our software implementation of the optimal design method was crude (using Matlab’s fminunc twice to ﬁnd fmap and fmincon once to optimize the hyperparameters during each inter-trial interval), the speed was more than adequate for the experimental data collected (Fig. [sent-236, score-1.044]

85 1), to incorporate the effects of spike history on the neuron’s response, allowing us to account for the possible effects of adaptation on the spike counts obtained. [sent-242, score-0.633]

86 A: The run time for uncertainty sampling (including the posterior update and the evidence optimization) as a function of the number of data points observed. [sent-248, score-0.453]

87 (The grid of “candidate” stimuli {x∗ } was the subset of stimuli in the experimental dataset not yet selected, but the speed was not noticeably affected by scaling to much larger sets of candidate stimuli). [sent-249, score-0.527]

88 Note that the error of uncertainty sampling with 150 points is even lower than that from random sampling with 300 data points. [sent-252, score-0.335]

89 C: Estimated response-history ﬁlter h, which describes how recent spiking inﬂuences the neuron’s spike rate. [sent-253, score-0.306]

90 2 spikes per trial in predicted spike count, a 4 percent reduction in cross-validation error compared to the original model. [sent-260, score-0.341]

91 6 Discussion We have developed an algorithm for optimal experimental design, which allows the nonlinearity in a cascade neural encoding model to be characterized quickly and accurately from limited data. [sent-261, score-0.31]

92 The method relies on a fast method for updating the hyperparameters using a Gaussian factorization of the Laplace approximation to the posterior, which removes the need to numerically recompute the MAP estimate as we optimize the hyperparameters. [sent-262, score-0.207]

93 We described a method for optimal experimental design, based on uncertainty sampling, to reduce the number of stimuli required to estimate such response functions. [sent-263, score-0.51]

94 We applied our method to the nonlinear color-tuning properties of macaque V1 neurons and showed that the GP-Poisson model provides a ﬂexible, tractable model for these responses, and that optimal design can substantially reduce the number of stimuli required to characterize them. [sent-264, score-0.488]

95 One additional virtue of the GP-Poisson model is that conditionals and marginals of the high-dimensional nonlinearity are straightforward, making it easy to visualize their lowerdimensional slices and projections (as we have done in Fig. [sent-265, score-0.311]

96 We added a history term to the LNP model in order to incorporate the effects of recent spike history on the spike rate (Fig. [sent-267, score-0.695]

97 We expect that the ability to incorporate dependencies on spike history to be important for the success of optimal design experiments, especially with neurons that exhibit strong spike-rate adaptation [30]. [sent-269, score-0.465]

98 One potential criticism of our approach is that uncertainty sampling in unbounded spaces is known to “run away from the data”, repeatedly selecting stimuli that are far from previous measurements. [sent-270, score-0.437]

99 We wish to point out that in neural applications, the stimulus space is always bounded (e. [sent-271, score-0.227]

100 , by the gamut of the monitor), and in our case, stimuli at the corners of the space are actually helpful for initializing estimates the range and smoothness of the function. [sent-273, score-0.263]

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1 0.92476869 85 nips-2011-Emergence of Multiplication in a Biophysical Model of a Wide-Field Visual Neuron for Computing Object Approaches: Dynamics, Peaks, & Fits

Author: Matthias S. Keil

Abstract: Many species show avoidance reactions in response to looming object approaches. In locusts, the corresponding escape behavior correlates with the activity of the lobula giant movement detector (LGMD) neuron. During an object approach, its ﬁring rate was reported to gradually increase until a peak is reached, and then it declines quickly. The η-function predicts that the LGMD activity is a product ˙ between an exponential function of angular size exp(−Θ) and angular velocity Θ, and that peak activity is reached before time-to-contact (ttc). The η-function has become the prevailing LGMD model because it reproduces many experimental observations, and even experimental evidence for the multiplicative operation was reported. Several inconsistencies remain unresolved, though. Here we address ˙ these issues with a new model (ψ-model), which explicitly connects Θ and Θ to biophysical quantities. The ψ-model avoids biophysical problems associated with implementing exp(·), implements the multiplicative operation of η via divisive inhibition, and explains why activity peaks could occur after ttc. It consistently predicts response features of the LGMD, and provides excellent ﬁts to published experimental data, with goodness of ﬁt measures comparable to corresponding ﬁts with the η-function. 1 Introduction: τ and η Collision sensitive neurons were reported in species such different as monkeys [5, 4], pigeons [36, 34], frogs [16, 20], and insects [33, 26, 27, 10, 38]. This indicates a high ecological relevance, and raises the question about how neurons compute a signal that eventually triggers corresponding movement patterns (e.g. escape behavior or interceptive actions). Here, we will focus on visual stimulation. Consider, for simplicity, a circular object (diameter 2l), which approaches the eye at a collision course with constant velocity v. If we do not have any a priori knowledge about the object in question (e.g. its typical size or speed), then we will be able to access only two information sources. These information sources can be measured at the retina and are called optical variables (OVs). The ﬁrst is the visual angle Θ, which can be derived from the number of stimulated photore˙ ˙ ceptors (spatial contrast). The second is its rate of change dΘ(t)/dt ≡ Θ(t). Angular velocity Θ is related to temporal contrast. ˙ How should we combine Θ and Θ in order to track an imminent collision? The perhaps simplest ˙ combination is τ (t) ≡ Θ(t)/Θ(t) [13, 18]. If the object hit us at time tc , then τ (t) ≈ tc − t will ∗ Also: www.ir3c.ub.edu, Research Institute for Brain, Cognition, and Behaviour (IR3C) Ediﬁci de Ponent, Campus Mundet, Universitat de Barcelona, Passeig Vall d’Hebron, 171. E-08035 Barcelona 1 give us a running estimation of the time that is left until contact1 . Moreover, we do not need to know anything about the approaching object: The ttc estimation computed by τ is practically independent of object size and velocity. Neurons with τ -like responses were indeed identiﬁed in the nucleus retundus of the pigeon brain [34]. In humans, only fast interceptive actions seem to rely exclusively on τ [37, 35]. Accurate ttc estimation, however, seems to involve further mechanisms (rate of disparity change [31]). ˙ Another function of OVs with biological relevance is η ≡ Θ exp(−αΘ), with α = const. [10]. While η-type neurons were found again in pigeons [34] and bullfrogs [20], most data were gathered from the LGMD2 in locusts (e.g. [10, 9, 7, 23]). The η-function is a phenomenological model for the LGMD, and implies three principal hypothesis: (i) An implementation of an exponential function exp(·). Exponentation is thought to take place in the LGMD axon, via active membrane conductances [8]. Experimental data, though, seem to favor a third-power law rather than exp(·). (ii) The LGMD carries out biophysical computations for implementing the multiplicative operation. It has been suggested that multiplication is done within the LGMD itself, by subtracting the loga˙ rithmically encoded variables log Θ − αΘ [10, 8]. (iii) The peak of the η-function occurs before ˆ ttc, at visual angle Θ(t) = 2 arctan(1/α) [9]. It follows ttc for certain stimulus conﬁgurations (e.g. ˆ l/|v| 5ms). In principle, t > tc can be accounted for by η(t + δ) with a ﬁxed delay δ < 0 (e.g. −27ms). But other researchers observed that LGMD activity continuous to rise after ttc even for l/|v| 5ms [28]. These discrepancies remain unexplained so far [29], but stimulation dynamics perhaps plays a role. We we will address these three issues by comparing the novel function “ψ” with the η-function. LGMD computations with the ψ-function: No multiplication, no exponentiation 2 A circular object which starts its approach at distance x0 and with speed v projects a visual angle Θ(t) = 2 arctan[l/(x0 − vt)] on the retina [34, 9]. The kinematics is hence entirely speciﬁed by the ˙ half-size-to-velocity ratio l/|v|, and x0 . Furthermore, Θ(t) = 2lv/((x0 − vt)2 + l2 ). In order to deﬁne ψ, we consider at ﬁrst the LGMD neuron as an RC-circuit with membrane potential3 V [17] dV Cm = β (Vrest − V ) + gexc (Vexc − V ) + ginh (Vinh − V ) (1) dt 4 Cm = membrane capacity ; β ≡ 1/Rm denotes leakage conductance across the cell membrane (Rm : membrane resistance); gexc and ginh are excitatory and inhibitory inputs. Each conductance gi (i = exc, inh ) can drive the membrane potential to its associated reversal potential Vi (usually Vinh ≤ Vexc ). Shunting inhibition means Vinh = Vrest . Shunting inhibition lurks “silently” because it gets effective only if the neuron is driven away from its resting potential. With synaptic input, the neuron decays into its equilibrium state Vrest β + Vexc gexc + Vinh ginh V∞ ≡ (2) β + gexc + ginh according to V (t) = V∞ (1 − exp(−t/τm )). Without external input, V (t 1) → Vrest . The time scale is set by τm . Without synaptic input τm ≡ Cm /β. Slowly varying inputs gexc , ginh > 0 modify the time scale to approximately τm /(1 + (gexc + ginh )/β). For highly dynamic inputs, such as in late phase of the object approach, the time scale gets dynamical as well. The ψ-model assigns synaptic inputs5 ˙ ˙ ˙ ˙ gexc (t) = ϑ(t), ϑ(t) = ζ1 ϑ(t − ∆tstim ) + (1 − ζ1 )Θ(t) (3a) e ginh (t) = [γϑ(t)] , ϑ(t) = ζ0 ϑ(t − ∆tstim ) + (1 − ζ0 )Θ(t) 1 (3b) This linear approximation gets worse with increasing Θ, but turns out to work well until short before ttc (τ adopts a minimum at tc − 0.428978 · l/|v|). 2 LGMD activity is usually monitored via its postsynaptic neuron, the Descending Contralateral Movement Detector (DCMD) neuron. This represents no problem as LGMD spikes follow DCMD spikes 1:1 under visual stimulation [22] from 300Hz [21] to at least 400Hz [24]. 3 Here we assume that the membrane potential serves as a predictor for the LGMD’s mean ﬁring rate. 4 Set to unity for all simulations. 5 LGMD receives also inhibition from a laterally acting network [21]. The η-function considers only direct feedforward inhibition [22, 6], and so do we. 2 Θ ∈ [7.63Â°, 180.00Â°[ temporal resolution ∆ tstim=1.0ms l/|v|=20.00ms, β=1.00, γ=7.50, e=3.00, ζ0=0.90, ζ1=0.99, nrelax=25 0.04 scaled dΘ/dt continuous discretized 0.035 0.03 Θ(t) (input) ϑ(t) (filtered) voltage V(t) (output) t = 56ms max t =300ms c 0.025 0 10 2 η(t): α=3.29, R =1.00 n =10 → t =37ms log Θ(t) amplitude relax max 0.02 0.015 0.01 0.005 0 −0.005 0 50 100 150 200 250 300 −0.01 0 350 time [ms] 50 100 150 200 250 300 350 time [ms] (b) ψ versus η (a) discretized optical variables Figure 1: (a) The continuous visual angle of an approaching object is shown along with its discretized version. Discretization transforms angular velocity from a continuous variable into a series of “spikes” (rescaled). (b) The ψ function with the inputs shown in a, with nrelax = 25 relaxation time steps. Its peak occurs tmax = 56ms before ttc (tc = 300ms). An η function (α = 3.29) that was ﬁtted to ψ shows good agreement. For continuous optical variables, the peak would occur 4ms earlier, and η would have α = 4.44 with R2 = 1. For nrelax = 10, ψ is farther away from its equilibrium at V∞ , and its peak moves 19ms closer to ttc. t =500ms, dia=12.0cm, ∆t c =1.00ms, dt=10.00µs, discrete=1 stim 250 n relax = 50 2 200 α=4.66, R =0.99 [normal] n = 25 relax 2 α=3.91, R =1.00 [normal] n =0 relax tmax [ms] 150 2 α=1.15, R =0.99 [normal] 100 50 0 β=1.00, γ=7.50, e=3.00, V =−0.001, ζ =0.90, ζ =0.99 inh −50 5 10 15 20 25 30 0 35 1 40 45 50 l/|v| [ms] (a) different nrelax (b) different ∆tstim ˆ ˆ Figure 2: The ﬁgures plot the relative time tmax ≡ tc − t of the response peak of ψ, V (t), as a function of half-size-to-velocity ratio (points). Line ﬁts with slope α and intercept δ were added (lines). The predicted linear relationship in all cases is consistent with experimental evidence [9]. (a) The stimulus time scale is held constant at ∆tstim = 1ms, and several LGMD time scales are deﬁned by nrelax (= number of intercalated relaxation steps for each integration time step). Bigger values of nrelax move V (t) closer to its equilibrium V∞ (t), implying higher slopes α in turn. (b) LGMD time scale is ﬁxed at nrelax = 25, and ∆tstim is manipulated. Because of the discretization of optical variables (OVs) in our simulation, increasing ∆tstim translates to an overall smaller number of jumps in OVs, but each with higher amplitude. Thus, we say ψ(t) ≡ V (t) if and only if gexc and ginh are deﬁned with the last equation. The time ˙ scale of stimulation is deﬁned by ∆tstim (by default 1ms). The variables ϑ and ϑ are lowpass ﬁltered angular size and rate of expansion, respectively. The amount of ﬁltering is deﬁned by memory constants ζ0 and ζ1 (no ﬁltering if zero). The idea is to continue with generating synaptic input ˙ after ttc, where Θ(t > tc ) = const and thus Θ(t > tc ) = 0. Inhibition is ﬁrst weighted by γ, and then potentiated by the exponent e. Hodgkin-Huxley potentiates gating variables n, m ∈ [0, 1] instead (potassium ∝ n4 , sodium ∝ m3 , [12]) and multiplies them with conductances. Gabbiani and co-workers found that the function which transforms membrane potential to ﬁring rate is better described by a power function with e = 3 than by exp(·) (Figure 4d in [8]). 3 Dynamics of the ψ-function 3 Discretization. In a typical experiment, a monitor is placed a short distance away from the insect’s eye, and an approaching object is displayed. Computer screens have a ﬁxed spatial resolution, and as a consequence size increments of the displayed object proceed in discrete jumps. The locust retina is furthermore composed of a discrete array of ommatidia units. We therefore can expect a corresponding step-wise increment of Θ with time, although optical and neuronal ﬁltering may ˙ smooth Θ to some extent again, resulting in ϑ (ﬁgure 1). Discretization renders Θ discontinuous, ˙ For simulating the dynamics of ψ, we discretized angular size what again will be alleviated in ϑ. ˙ with ﬂoor(Θ), and Θ(t) ≈ [Θ(t + ∆tstim ) − Θ(t)]/∆tstim . Discretized optical variables (OVs) were re-normalized to match the range of original (i.e. continuous) OVs. To peak, or not to peak? Rind & Simmons reject the hypothesis that the activity peak signals impending collision on grounds of two arguments [28]: (i) If Θ(t + ∆tstim ) − Θ(t) 3o in consecutively displayed stimulus frames, the illusion of an object approach would be lost. Such stimulation would rather be perceived as a sequence of rapidly appearing (but static) objects, causing reduced responses. (ii) After the last stimulation frame has been displayed (that is Θ = const), LGMD responses keep on building up beyond ttc. This behavior clearly depends on l/|v|, also according to their own data (e.g. Figure 4 in [26]): Response build up after ttc is typically observed for sufﬁ˙ ciently small values of l/|v|. Input into ψ in situations where Θ = const and Θ = 0, respectively, ˙ is accommodated by ϑ and ϑ, respectively. We simulated (i) by setting ∆tstim = 5ms, thus producing larger and more infrequent jumps in discrete OVs than with ∆tstim = 1ms (default). As a consequence, ϑ(t) grows more slowly (deˆ layed build up of inhibition), and the peak occurs later (tmax ≡ tc − t = 10ms with everything else ˆ ˆ identical with ﬁgure 1b). The peak amplitude V = V (t) decreases nearly sixfold with respect to default. Our model thus predicts the reduced responses observed by Rind & Simmons [28]. Linearity. Time of peak ﬁring rate is linearly related to l/|v| [10, 9]. The η-function is consistent ˆ with this experimental evidence: t = tc − αl/|v| + δ (e.g. α = 4.7, δ = −27ms). The ψ-function reproduces this relationship as well (ﬁgure 2), where α depends critically on the time scale of biophysical processes in the LGMD. We studied the impact of this time scale by choosing 10µs for the numerical integration of equation 1 (algorithm: 4th order Runge-Kutta). Apart from improving the numerical stability of the integration algorithm, ψ is far from its equilibrium V∞ (t) in every moment ˙ t, given the stimulation time scale ∆tstim = 1ms 6 . Now, at each value of Θ(t) and Θ(t), respectively, we intercalated nrelax iterations for integrating ψ. Each iteration takes V (t) asymptotically closer to V∞ (t), and limnrelax 1 V (t) = V∞ (t). If the internal processes in the LGMD cannot keep up with stimulation (nrelax = 0), we obtain slopes values that underestimate experimentally found values (ﬁgure 2a). In contrast, for nrelax 25 we get an excellent agreement with the experimentally determined α. This means that – under the reported experimental stimulation conditions (e.g. [9]) – the LGMD would operate relatively close to its steady state7 . Now we ﬁx nrelax at 25 and manipulate ∆tstim instead (ﬁgure 2b). The default value ∆tstim = 1ms corresponds to α = 3.91. Slightly bigger values of ∆tstim (2.5ms and 5ms) underestimate the experimental α. In addition, the line ﬁts also return smaller intercept values then. We see tmax < 0 up to l/|v| ≈ 13.5ms – LGMD activity peaks after ttc! Or, in other words, LGMD activity continues to increase after ttc. In the limit, where stimulus dynamics is extremely fast, and LGMD processes are kept far from equilibrium at each instant of the approach, α gets very small. As a consequence, tmax gets largely independent of l/|v|: The activity peak would cling to tmax although we varied l/|v|. 4 Freeze! Experimental data versus steady state of “psi” In the previous section, experimentally plausible values for α were obtained if ψ is close to equilibrium at each instant of time during stimulation. In this section we will thus introduce a steady-state 6 Assuming one ∆tstim for each integration time step. This means that by default stimulation and biophysical dynamics will proceed at identical time scales. 7 Notice that in this moment we can only make relative statements - we do not have data at hand for deﬁning absolute time scales 4 tc=500ms, v=2.00m/s ψ∞ → (β varies), γ=3.50, e=3.00, Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, γ=3.50, (e varies), Vinh=−0.001 300 tc=500ms, v=2.00m/s ψ∞ → β=2.50, (γ varies), e=3.00, Vinh=−0.001 350 300 β=10.00 β=5.00 norm. rmse = 0.058...0.153 correlation (β,α)=−0.90 (n=4) ∞ β=1.00 e=4.00 norm. |η−ψ | = 0.009...0.114 e=3.00 300 norm. rmse = 0.014...0.160 correlation (e,α)=0.98 (n=4) ∞ e=2.50 250 250 norm. |η−ψ | = 0.043...0.241 ∞ norm. rmse = 0.085...0.315 correlation (γ,α)=1.00 (n=5) 150 tmax [ms] 200 tmax [ms] 200 tmax [ms] γ=5.00 γ=2.50 γ=1.00 γ=0.50 γ=0.25 e=5.00 norm. |η−ψ | = 0.020...0.128 β=2.50 250 200 150 100 150 100 100 50 50 50 0 5 10 15 20 25 30 35 40 45 0 5 50 10 15 20 l/|v| [ms] 25 30 35 40 45 0 5 50 10 15 20 l/|v| [ms] (a) β varies 25 30 35 40 45 50 l/|v| [ms] (b) e varies (c) γ varies ˆ ˆ Figure 3: Each curve shows how the peak ψ∞ ≡ ψ∞ (t) depends on the half-size-to-velocity ratio. In each display, one parameter of ψ∞ is varied (legend), while the others are held constant (ﬁgure title). Line slopes vary according to parameter values. Symbol sizes are scaled according to rmse (see also ﬁgure 4). Rmse was calculated between normalized ψ∞ (t) & normalized η(t) (i.e. both functions ∈ [0, 1] with original minimum and maximum indicated by the textbox). To this end, the ˆ peak of the η-function was placed at tc , by choosing, at each parameter value, α = |v| · (tc − t)/l (for determining correlation, the mean value of α was taken across l/|v|). tc=500ms, v=2.00m/s ψ∞ → (β varies), γ=3.50, e=3.00, Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, γ=3.50, (e varies), Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, (γ varies), e=3.00, Vinh=−0.001 0.25 β=5.00 0.12 β=2.50 β=1.00 0.1 0.08 (normalized η, ψ∞) 0.12 β=10.00 (normalized η, ψ∞) (normalized η, ψ∞) 0.14 0.1 0.08 γ=5.00 γ=2.50 0.2 γ=1.00 γ=0.50 γ=0.25 0.15 0.06 0.04 0.02 0 5 10 15 20 25 30 35 40 45 50 meant |η(t)−ψ∞(t)| meant |η(t)−ψ∞(t)| meant |η(t)−ψ∞(t)| 0.06 0.04 e=5.00 e=4.00 e=3.00 0.02 e=2.50 10 l/|v| [ms] 15 20 25 30 35 40 45 50 l/|v| [ms] (a) β varies (b) e varies 0.1 0.05 0 5 10 15 20 25 30 35 40 45 50 l/|v| [ms] (c) γ varies Figure 4: This ﬁgure complements ﬁgure 3. It visualizes the time averaged absolute difference between normalized ψ∞ (t) & normalized η(t). For η, its value of α was chosen such that the maxima of both functions coincide. Although not being a ﬁt, it gives a rough estimate on how the shape of both curves deviate from each other. The maximum possible difference would be one. version of ψ (i.e. equation 2 with Vrest = 0, Vexc = 1, and equations 3 plugged in), ψ∞ (t) ≡ e ˙ Θ(t) + Vinh [γΘ(t)] e ˙ β + Θ(t) + [γΘ(t)] (4) (Here we use continuous versions of angular size and rate of expansion). The ψ∞ -function makes life easier when it comes to ﬁtting experimental data. However, it has its limitations, because we brushed the whole dynamic of ψ under the carpet. Figure 3 illustrates how the linˆ ear relationship (=“linearity”) between tmax ≡ tc − t and l/|v| is inﬂuenced by changes in parameter values. Changing any of the values of e, β, γ predominantly causes variation in line slopes. The smallest slope changes are obtained by varying Vinh (data not shown; we checked Vinh = 0, −0.001, −0.01, −0.1). For Vinh −0.01, linearity is getting slightly compromised, as slope increases with l/|v| (e.g. Vinh = −1 α ∈ [4.2, 4.7]). In order to get a notion about how well the shape of ψ∞ (t) matches η(t), we computed timeaveraged difference measures between normalized versions of both functions (details: ﬁgure 3 & 4). Bigger values of β match η better at smaller, but worse at bigger values of l/|v| (ﬁgure 4a). Smaller β cause less variation across l/|v|. As to variation of e, overall curve shapes seem to be best aligned with e = 3 to e = 4 (ﬁgure 4b). Furthermore, better matches between ψ∞ (t) and η(t) correspond to bigger values of γ (ﬁgure 4c). And ﬁnally, Vinh marches again to a different tune (data not shown). Vinh = −0.1 leads to the best agreement (≈ 0.04 across l/|v|) of all Vinh , quite different from the other considered values. For the rest, ψ∞ (t) and η(t) align the same (all have maximum 0.094), 5 ˙ (a) Θ = 126o /s ˙ (b) Θ = 63o /s Figure 5: The original data (legend label “HaGaLa95”) were resampled from ref. [10] and show ˙ DCMD responses to an object approach with Θ = const. Thus, Θ increases linearly with time. The η-function (ﬁtting function: Aη(t+δ)+o) and ψ∞ (ﬁtting function: Aψ∞ (t)+o) were ﬁtted to these data: (a) (Figure 3 Di in [10]) Good ﬁts for ψ∞ are obtained with e = 5 or higher (e = 3 R2 = 0.35 and rmse = 0.644; e = 4 R2 = 0.45 and rmse = 0.592). “Psi” adopts a sigmoid-like curve form which (subjectively) appears to ﬁt the original data better than η. (b) (Figure 3 Dii in [10]) “Psi” yields an excellent ﬁt for e = 3. RoHaTo10 gregarious locust LV=0.03s Θ(t), lv=30ms e011pos014 sgolay with 100 t =107ms max ttc=5.00s ψ adj.R2 0.95 (LM:3) ∞ η(t) adj.R2 1 (TR::1) 2 ψ : R =0.95, rmse=0.004, 3 coefficients ∞ → β=2.22, γ=0.70, e=3.00, V =−0.001, A=0.07, o=0.02, δ=0.00ms inh η: R2=1.00, rmse=0.001 → α=3.30, A=0.08, o=0.0, δ=−10.5ms 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 time [s] (b) α versus β (a) spike trace Figure 6: (a) DCMD activity in response to a black square (l/|v| = 30ms, legend label “e011pos14”, ref. [30]) approaching to the eye center of a gregarious locust (ﬁnal visual angle 50o ). Data show the ﬁrst stimulation so habituation is minimal. The spike trace (sampled at 104 Hz) was full wave rectiﬁed, lowpass ﬁltered, and sub-sampled to 1ms resolution. Firing rate was estimated with Savitzky-Golay ﬁltering (“sgolay”). The ﬁts of the η-function (Aη(t + δ) + o; 4 coefﬁcients) and ψ∞ -function (Aψ∞ (t) with ﬁxed e, o, δ, Vinh ; 3 coefﬁcients) provide both excellent ﬁts to ﬁring rate. (b) Fitting coefﬁcient α (→ η-function) inversely correlates with β (→ ψ∞ ) when ﬁtting ﬁring rates of another 5 trials as just described (continuous line = line ﬁt to the data points). Similar correlation values would be obtained if e is ﬁxed at values e = 2.5, 4, 5 c = −0.95, −0.96, −0.91. If o was determined by the ﬁtting algorithm, then c = −0.70. No clear correlations with α were obtained for γ. despite of covering different orders of magnitude with Vinh = 0, −0.001, −0.01. Decelerating approach. Hatsopoulos et al. [10] recorded DCMD activity in response to an ap˙ proaching object which projected image edges on the retina moving at constant velocity: Θ = const. ˙ This “linear approach” is perceived as if the object is getting increasingly implies Θ(t) = Θ0 + Θt. slower. But what appears a relatively unnatural movement pattern serves as a test for the functions η & ψ∞ . Figure 5 illustrates that ψ∞ passes the test, and consistently predicts that activity sharply rises in the initial approach phase, and subsequently declines (η passed this test already in the year 1995). 6 Spike traces. We re-sampled about 30 curves obtained from LGMD recordings from a variety of publications, and ﬁtted η & ψ∞ -functions. We cannot show the results here, but in terms of goodness of ﬁt measures, both functions are in the same ballbark. Rather, ﬁgure 6a shows a representative example [30]. When α and β are plotted against each other for ﬁve trials, we see a strong inverse correlation (ﬁgure 6b). Although ﬁve data points are by no means a ﬁrm statistical sample, the strong correlation could indicate that β and α play similar roles in both functions. Biophysically, β is the leakage conductance, which determines the (passive) membrane time constant τm ∝ 1/β of the neuron. Voltage drops within τm to exp(−1) times its initial value. Bigger values of β mean shorter τm (i.e., “faster neurons”). Getting back to η, this would suggest α ∝ τm , such that higher (absolute) values for α would possibly indicate a slower dynamic of the underlying processes. 5 Discussion (“The Good, the Bad, and the Ugly”) Up to now, mainly two classes of LGMD models existed: The phenomenological η-function on the one hand, and computational models with neuronal layers presynaptic to the LGMD on the other (e.g. [25, 15]; real-world video sequences & robotics: e.g. [3, 14, 32, 2]). Computational models predict that LGMD response features originate from excitatory and inhibitory interactions in – and between – presynaptic neuronal layers. Put differently, non-linear operations are generated in the presynaptic network, and can be a function of many (model) parameters (e.g. synaptic weights, time constants, etc.). In contrast, the η-function assigns concrete nonlinear operations to the LGMD [7]. The η-function is accessible to mathematical analysis, whereas computational models have to be probed with videos or artiﬁcial stimulus sequences. The η-function is vague about biophysical parameters, whereas (good) computational models need to be precise at each (model) parameter value. The η-function establishes a clear link between physical stimulus attributes and LGMD activity: It postulates what is to be computed from the optical variables (OVs). But in computational models, such a clear understanding of LGMD inputs cannot always be expected: Presynaptic processing may strongly transform OVs. The ψ function thus represents an intermediate model class: It takes OVs as input, and connects them with biophysical parameters of the LGMD. For the neurophysiologist, the situation could hardly be any better. Psi implements the multiplicative operation of the η-function by shunting inhibition (equation 1: Vexc ≈ Vrest and Vinh ≈ Vrest ). The η-function ﬁts ψ very well according to our dynamical simulations (ﬁgure 1), and satisfactory by the approximate criterion of ﬁgure 4. We can conclude that ψ implements the η-function in a biophysically plausible way. However, ψ does neither explicitly specify η’s multiplicative operation, nor its exponential function exp(·). Instead we have an interaction between shunting inhibition and a power law (·)e , with e ≈ 3. So what about power laws in neurons? Because of e > 1, we have an expansive nonlinearity. Expansive power-law nonlinearities are well established in phenomenological models of simple cells of the primate visual cortex [1, 11]. Such models approximate a simple cell’s instantaneous ﬁring rate r from linear ﬁltering of a stimulus (say Y ) by r ∝ ([Y ]+ )e , where [·]+ sets all negative values to zero and lets all positive pass. Although experimental evidence favors linear thresholding operations like r ∝ [Y − Ythres ]+ , neuronal responses can behave according to power law functions if Y includes stimulus-independent noise [19]. Given this evidence, the power-law function of the inhibitory input into ψ could possibly be interpreted as a phenomenological description of presynaptic processes. The power law would also be the critical feature by means of which the neurophysiologist could distinguish between the η function and ψ. A study of Gabbiani et al. aimed to provide direct evidence for a neuronal implementation of the η-function [8]. Consequently, the study would be an evidence ˙ for a biophysical implementation of “direct” multiplication via log Θ − αΘ. Their experimental evidence fell somewhat short in the last part, where “exponentation through active membrane conductances” should invert logarithmic encoding. 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