Neuronal Population Codes and the Perception of Object Distance in Weakly Electric Fish

Estimating object distance

From a 2D electric image on their body surface, electric fish are able to determine the 3D location (x*, y*,z*) of the object producing the image (von der Emde et al., 1998). The object location in the body plane (x–y plane) can be estimated from the location at which the image has its peak amplitude. However, the peak amplitude of the electric image provides ambiguous information about the third dimension, lateral distance away from the fish (z*). In Figure 1a, two spherical objects of different sizes (and otherwise identical) are located at the same (x*, y*) location, but the larger object is farther away. For this, and many other combinations of object size and lateral distance, the peak amplitude of the image is the same and thus cannot be used to unambiguously determine the lateral distance of each object (see Materials and Methods; Eq. 3) (Rasnow, 1996). The image produced by the larger object is wider than the other (Fig.1a). This is shown more clearly by a one-dimensional slice through the image (Fig. 1b). When the image is normalized to its peak amplitude, its width can then be used to estimate lateral distance, z* (Rasnow, 1996; Assad et al., 1999).

To enable our analyses, we used a simplified description of the electric image. We assume the electric image has a 2D Gaussian shape, with its peak amplitude and half-width given by the parametersA_o and θ (see Materials and Methods) (Eqs. 1-3). Because θ provides a measure of normalized width and varies linearly with lateral distance, z* (Rasnow, 1996), it then can be used to estimate lateral distance (we use θ and image width interchangeably, although θ is actually the half-width).

Another image feature proposed as an indicator of object distance is the maximum slope of the image normalized to its peak amplitude (von der Emde et al., 1998; von der Emde, 1999). For a Gaussian image, this quantity varies as 1/θ and also fits the published maximum slope data (von der Emde et al., 1998) very well (Lewis and Maler, unpublished observations). Because of the direct relationship between θ, maximum slope, and previously reported data, we have discussed our results in terms of θ alone.

Estimation accuracy and tuning curve width

In the present context, downstream electrosensory networks must extract information about object location given a noisy profile of activity in the ELL pyramidal neuron population. We have formulated a simple model of the ELL population response to a stereotyped electric image (see Materials and Methods). Figure 2a shows the electric image produced by a small sphere (Eq. 1), which provides the input to the 2D grid of model neurons that constitute the ELL network (Fig. 2b). Each neuron on the grid integrates input from the electric image over a restricted range or receptive field (shown schematically by the shaded region in Fig. 2b; Eq. 4), such that for a point stimulus each neuron has a 2D Gaussian-shaped tuning curve (in the x–y plane). The tuning width is given by 2ς (measured at a height corresponding toe^−1/2 of the tuning curve peak). We ignore any contributions that dynamics may provide, with the response of each neuron given by a spike count over an integration time of 1 sec. After the addition of noise the ELL population response profile resembles a noisy replication of the electric image (Fig. 2c). Because the electric image is not a point stimulus, the actual response profile of the ELL population is wider than the image, to an extent that depends on the relative values of θ and ς (Fig. 2d) (see next section).

Given the noisy response profile of the ELL population, the typical population decoding problem is to determine the features of the object (x*, y*, z*,r_o ) that produced the response (Abbott, 1994; Salinas and Abbott, 1994; Deneve et al., 1999; Zhang and Sejnowski, 1999). As discussed before in a functional context, to unambiguously determine z* andr_o , two features of the electric image produced by the object must be estimated: the image width and the amplitude of the image peak (θ and A_o , respectively). The accuracy of estimation is limited by the accuracy with which the ELL neurons jointly encode these different features. Using a common approach from statistical estimation theory (Kay, 1993), we can determine an upper limit on this accuracy by computing the Cramer-Rao lower bound for estimating each object featurex*, y*, z*, andr_o , as well as the image features θ andA_o (Eq. 6) (see Materials and Methods). Accuracy in this context is given by the mean-squared estimation error, or equivalently, the variance of the estimate. We investigated the influence of two parameters on estimation accuracy: the lateral distance of the object z* and the tuning curve width ς. The error bound for estimating all features increases withz* (data not shown). This is not surprising because the image amplitude (and thus the effective signal-to-noise ratio) decreases fairly quickly with distance (Eq. 3). More interestingly, the effects of changing ς differ between the features (Fig. 3). There is no effect on estimating the x*–y* location (Fig.3a); the same result has been found previously for point stimuli (Snippe and Koenderink, 1992; Abbott and Dayan, 1999; Zhang and Sejnowski, 1999). On the other hand, increasing tuning width ς results in worse estimation of r_o and z* (Fig. 3a). For estimating electric image features (Fig.3b), increasing ς results in worse estimation of θ (larger error), but better estimation ofA_o (smaller error). Intuitively, this makes sense, wider tuning curves allow more neurons to accurately contribute to the estimation of A_o , and by averaging across neurons, a better estimate results. Estimating image width is different although, because the ELL neurons distort the image through a convolution with their tuning curves (Eq. 4, Fig.2d). This distortion increases with tuning width, resulting in more neurons that do not accurately represent image width, nonetheless influencing the θ estimate.

Shown also in Figure 3 are the results of network simulations. Using a network grid consisting of a physiological number and density of neurons (41 × 41 neuronal grid, density ρ = 46.7 neurons/cm²), we estimated the image features from the noisy neural responses using an ML approach (see Materials and Methods). The estimation error for this method is very close to the corresponding lower bound (Fig. 3, compare open symbols with solid lines). Note that for larger ς however, there is a slight deviation from the theoretical bound attributable mainly to edge effects (i.e., the neuronal image has above baseline values beyond the limits of the grid edges).

The relationship between the accuracy of image width estimation and tuning width can be made explicit by expressing the Fisher information for θ, I_F (θ), in terms of the grid spacing, Δ, the distance between tuning curve centers (Eq. 8). Differentiating I_F (θ) with respect to ς reveals that I_F (θ) decreases with ς (i.e., the derivative is negative and thus the estimation error increases) as long as θ² + ς² > (Δ/2)². This condition will hold as long as the tuning curve width is greater than the grid spacing (i.e., if 2ς > Δ). A similar calculation shows that the Fisher information for A_o ,I_F (A_o ), increases with ς for all ς > 0.

A simple neural algorithm for determining object distance

Estimating θ, in the present context, is equivalent to estimating the half-width of the image at a level of A_oe^−1/2. To provide an unambiguous estimate of z*, a measure of image width must be calculated from an image normalized by A_o . So in such a practical situation, peak amplitudeA_o must be estimated first, before image width.

One simple algorithm to calculate image width is to first normalize the neural responses to the maximal response and then count the number of neurons that are active above a certain threshold (Assad et al., 1999). One way to formalize this two-step algorithm is to first compute the average activity E_ave of all the neurons firing above a threshold, φ_a (Fig.4a). This step (step 1) provides an estimate of the peak response in the population, which can be used to normalize all neural activity. Then in step 2, the fraction of neurons (N_w) firing above a different threshold (φ_w) can be determined (Fig. 4b). These two thresholds are distinct in that φ_a is fixed and not relative to any neural response, whereas φ_w comes after the normalization step and is relative to the maximum response in the network. Figure 4, c and d shows how these measures vary with the features they are supposed to estimate. The actual peak neural activity (g_oA_o ) differs fromA_o by a constant factor and thus varies withz* in parallel with A_o (Fig.4c). However, E_ave underestimatesg_oA_o but still varies linearly withA_o (Fig. 4c, inset). Similarly,N_w varies in a near linear manner with θ (Fig. 4d).

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Fig. 4.

Two-step algorithm for estimating electric image width. a and b show representative response profiles E_i of a one-dimensional slice through the neuronal grid. In athe peak amplitude of the image is estimated by the average activityE_ave of all neurons firing above a threshold level of E_baseline+φ_a(E_baseline = 20). In bthe response profile is normalized byE_ave , and the image width is estimated by the number of neurons N_w firing above a threshold level of (E_baseline/E_ave )+φ_w.c, The average spike count of the neurons above threshold (E_ave ), as well as the decay in actual peak activity (g_oA_o ;g_o = 100) with increasing object distancez* (φ_a = 2η = 14; ς = 0.6). Over this range of z*,E_ave varies linearly withA_o (inset).d, The fraction of total neurons activated above a threshold level of φ_w = e^−1/2 (i.e., the number of neurons with a preferred location within a radius θ of the object location) plotted versus θ. For different values of ς, this measure increases in an almost linear manner with θ. The solid lines are the theoretical curves derived for a continuous distribution of neurons (see Results), and the open symbols show the measure for an actual model network (41 × 41 neuronal grid; ρ = 46.7).

We consider two specific neural implementations of this algorithm (Fig.5). Model 1 uses the same map (i.e., network), with tuning width ς₁, for estimating both θ and A_o. Model 2 uses two maps, one with a relatively large tuning width (ς₂ = 1) for estimating A_o and another with narrower tuning widths (ς₁ ≤ 1) for estimating θ. Model 1 is analogous to a single sensory map for all computations, and Model 2 is analogous to having two specialized sensory maps, one for estimating peak amplitude A_o , with larger tuning widths, and the other for estimating width θ, with smaller tuning widths. It is critical to note that both models use the same number of neurons in each processing step (each map is a 41 × 41 neuronal grid). The critical difference is that model 2 has two different tuning widths for each processing step.

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Fig. 5.

Schematic representation of two models for implementing the two-step algorithm for estimating electric image width. In model 1 (left), bothA_o and θ are estimated using the same map, map 1 with tuning widths ς₁. In model 2 (right), separate maps are used for each estimation step: map 2, with wide tuning curves (ς₂ = 1), is used to obtain an estimate of A_o , which is then used to normalize the activity in map 1, with narrow tuning curves (ς₁ < ς₂), from which an estimate of θ is obtained. BecauseA_o and θ are estimated separately in this two-step algorithm, both models use the same number of neurons to estimate each feature, although model 2 has two maps, and model 1 has only one.

To compare the performance of these models, we computeN_w for many simulated presentations of an object over a range of values of ς₁ andz* (φ_a = 2η; φ_w = e^−1/2). In this situation, the true value of N_w is given by the number of neurons with preferred locations within a circle of radius θ (πθ²ρ). The estimate of N_w from the present neural algorithm, however, is biased. This is in part because of its dependence on E_ave and also because it is determined from a neural profile that has an effective width of √(θ²+ς²) caused by the tuning curve convolution (Eq. 4, Fig. 2d). Because in the context of these models, downstream networks would have to use N_w to estimate object distance, and N_w is directly related to θ and object distance (Fig. 4d), we evaluate model performance from the bias and variance in N_w. In all cases tested (ς₁ = 0.15–1.0; z*= 1.0–1.4), model 2 outperforms model 1. For z* = 1.2, the estimation variance for both models is shown in Figure6a. The biases inN_w estimation are nearly identical for both models (data not shown), but the variance for model 2 is substantially less than that for model 1. Because model 2 is better at estimating peak amplitude (by virtue of its wide tuning curves for this step, ς₂ = 1), the variance is dominated by the width estimator, and thus increases with ς₁ in the same manner as the Cramer-Rao bound for θ (Fig. 3b). Model 1 must use a network with the same ς₁ for all steps, so there is a trade-off between accuracy of peak amplitude estimation and accuracy of width estimation. Peak amplitude estimation is better for larger ς₁ when more neurons are activated close to peak levels, but width estimation is better for smaller ς₁. In the case shown (Fig.6a), the amplitude estimate dominates even for small ς₁ and thus the overall variance decreases with ς₁, similar to the Cramer-Rao bound forA_o (Fig. 3b). When ς₁ = 1, both models have similar overall accuracy. Although it would seem that model 2 effectively has twice as many neurons as model 1, as stated earlier, it really uses the same number of neurons as model 1 for each processing step. The slightly better performance of model 2 for ς₁ = 1 is caused by the independence of the responses between the different maps. In other words, if the noise in the neuronal responses was exactly correlated between the two maps of model 2, the accuracy would be identical to that of model 1.

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Fig. 6.

Performance of the two models in implementing the two-step algorithm for estimating electric image width. a, This panel shows the variance in the estimate of N_w for each model. Model 1 (open symbols) results in higher variance than model 2 (ς₂ = 1; closed symbols) for all tuning widths ς₁. N_w is the fraction of neurons above threshold (i.e., the actual number normalized by the total number of neurons N²; parameter values are N = 41, φ_w= e^−1/2, φ_a = 2η, r_o = 0.5, (x*, y*, z*) = (0, 0, 1.2), ρ = 46.7,E_baseline = 20,g_o = 100, η = 7. The true value of N_w = πθ²ρ/N² ∼0.09.b, The variance of the θ estimate for a generalized decoding scheme in both models (see Results). All parameter values are the same in a and b, except that in b two independent maps are used for both models (see Results). Note that when ς₁= 1 in this case, both models are identical so the variances are necessarily the same. Each point in a andb represents the variance calculated from 3000 simulated trials.

The general trends shown in Figure 6a are similar for neuronal densities within ∼50–150% of that used in the simulations shown. We also tested several combinations of values for φ_a (range, 2η–4η) and φ_w (range, 0.25–0.75) for z* = 1.2 and ς₁ = 0.3. Similar trends resulted, so the increased accuracy of model 2 over model 1 does not depend critically on these threshold values. In addition, we also considered conditions in which the noise term E_noise was such that the SD of the ELL neuron responses was equal to their mean response F_ij (Eq. 4), rather than constant (η = 7) and independent of F_ij . This type of noise resulted in similar results (data not shown) and does not change our conclusions.

Although the previous analysis demonstrates a clear difference between the two models, it is important to prove that this difference is fundamental and is not simply attributable to the details of the algorithm implementation or the fact that model 2 uses two independent networks. We now consider two independent maps composed of 41 × 41 neuronal grids with tuning widths of ς₁ and ς₂, respectively. Map 2 provides an estimate ofA_o using ML estimation; this estimate is used to normalize the activity in map 1. Then map 1 is used to find the ML estimate of θ. For model 1, both maps have the same tuning width ς₁ = ς₂. Model 2 is identical to model 1 except that map 2 has a fixed tuning width ς₂ = 1. This constitutes a test of the two models in a general decoding framework in which the only difference is in the tuning width of map 2. The results are similar to those previous, with model 2 providing a better estimate of θ (Fig.6b). In this case however, the error increases with ς₁ for both model 1 and 2, in the same manner as the Cramer-Rao bound for θ (Fig. 3b), suggesting that θ estimation dominates the overall estimation error. This provides a theoretical validation of our conclusions, but it is certainly not an option for the fish. The ELL does not have multiple maps with the same tuning widths, and thus the fish does not have access to identical information from two identical maps. Our initial analysis (Fig.6a) shows how the specialized use of an additional map can improve the computation performed by a single map.

The two-step algorithm we have considered is based on previous ideas (Assad et al., 1999) and practical constraints (i.e., peak amplitude must be estimated before normalization can occur). But it is also interesting to ask how maps can be combined in the context of optimal estimation as defined by the Cramer-Rao bound. We again consider the two-map configuration analyzed in Figure 6b. We calculated the Cramer-Rao bounds for θ andA_o for two maps (see Materials and Methods) and compared them to that for a single map (Fig.7a,b). Two maps with identical tuning widths are twice as good as one map with that tuning width (i.e., the error decreases by half for two maps). The neuronal density is a critical factor in determining population coding accuracy (Zhang and Sejnowski, 1999). Having two identical maps is the same, in terms of accuracy, as having a single map with twice the density, not necessarily twice the number of neurons. Also shown in Figure 7is that having one of the maps with a fixed tuning width (ς₂ = 1) is better than two identical maps for estimating A_o , but worse for estimating θ. Thus, the relative importance of these parameters will influence the optimal configuration of the two maps; if a premium is placed on estimating θ independently ofA_o , then narrow tuning in both maps is better. In the two-step algorithm considered in this paper, accurately estimating A_o is critical for the overall accuracy of estimating θ, so a combination of tuning widths is best.

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Fig. 7.

Performance of two maps in the context of the Cramer-Rao bound. For two different combinations of two maps (similar to those in Fig. 6b), the analytically calculated Cramer-Rao bounds for estimating A_o(a), and θ (b) (see Materials and Methods), are shown by the thick solid anddotted lines, respectively. In one configuration (ς₂ = ς₁), both maps have the same tuning width, and in the other configuration (ς₂ = 1), one map has a tuning width of ς₁, and the other is fixed at ς₂ = 1. Also shown (thin solid lines) are the analytically calculated Cramer-Rao bounds for a single map (same as those in Fig. 3b).

Multiple maps and population coding

Weakly electric fish can accurately electrolocate objects in their surroundings using sensory information contained in a 2D electric image (Bastian, 1987; Heiligenberg, 1991; Nelson and MacIver, 1999; von der Emde, 1999). To unambiguously extract 3D object location, the fish must compute the width and location of the peak of the electric image that is normalized to its peak amplitude (Rasnow, 1996). The electric image is initially encoded in four populations of pyramidal neurons that comprise the four parallel maps in ELL. One map (ampullary system) is specialized for low-frequency signals. The neurons within the three remaining maps (tuberous system) can be distinguished, among other characteristics, by their distinct spatial response properties: the lateral map with large receptive fields, centromedial map with small receptive fields, and the centrolateral map with intermediate-sized receptive fields (Shumway, 1989a,b; Metzner, 1999; Turner and Maler, 1999). Our results suggest a novel function for the parallel sensory maps in ELL, as well as the occurrence of parallel maps in other sensory systems. Namely, in addition to coarse coding stimulus features on different scales, parallel sensory maps may also be optimized to encode features of a sensory stimulus to which the component neurons are not tuned in the same manner. Specifically, in addition to encoding the 2D electric image at different spatial scales, different ELL maps can also be specialized to accurately represent the sensory features required to compute the third dimension, i.e., object distance.

Previous theoretical studies have found that neuronal tuning width (or spatial resolution) should not affect encoding accuracy in 2D (Snippe and Koenderink, 1992; Abbott and Dayan, 1999; Zhang and Sejnowski, 1999). This is also the case for the coarse-coded features of a spatially extended stimulus (i.e., the electric image) (Fig.3a). This result does not apply when multiple 2D stimuli are given simultaneously, as in two-point discrimination, where narrower tuning curves are better (Snippe and Koenderink, 1992). We show that depending on the encoding strategy for a particular stimulus feature, either wider or narrower tuning curves improve encoding accuracy. We illustrate the impact of tuning width on the accuracy of determining object distance from the electric image using two simple models. To encode the peak amplitude of the electric image, wider tuning curves in the two coarse-coded dimensions (x and y) result in higher accuracy; whereas, to encode image width, narrower tuning curves are more accurate. This suggests that the lateral map in ELL (in addition to its other functions, such as processing high-frequency signals like chirps) (Shumway, 1989a; Metzner and Juranek, 1997), may provide information about image amplitude that can be used to normalize the activity in the centromedial map, which is then used to compute image width and object distance. Normalization could be mediated by the extensive cerebellar-like feedback that projects to ELL, through shunting inhibition or synaptic depression (Maler and Mugnaini, 1994;Bastian, 1996; Berman and Maler, 1999). This simple hypothesis can be readily tested with established experimental techniques (Bastian, 1987;Metzner and Juranek, 1997; Nelson and MacIver, 1999). For example, ablating the lateral map of ELL (wide tuning) should disrupt the accurate estimation of image amplitude and the subsequent normalization step, resulting in an ambiguous estimation of object distance. So, predictable behavioral errors should occur when animals attempt to distinguish objects with certain combinations of size and distance.

The locus of computation of object distance is not known, and it need not be the centromedial map itself, because information from all ELL maps could be combined in higher brain regions (i.e., torus semicircularis or optic tectum) (Heiligenberg, 1991). Indeed, it is not necessary that there be a locus of computation, or explicit neural map, of object distance in electric fish. Such information could remain in a combined population code throughout its processing stream. However, there is some evidence of neurons in both the tectum and cerebellum that are tuned to object distance (Bastian, 1986a). Similar “distance-tuned” neurons exist in the optic tectum of frogs and toads (House, 1989). There is also evidence that information from the different maps is treated very differently in the torus (Metzner and Juranek, 1997). So apparently the same information from different maps is not simply being averaged. Other constraints could lead to the formation of differently sized maps, such as specialized roles in temporal processing and communication (Metzner, 1999), as well as those proposed in the present paper.

The present analyses have primarily considered the location of an object. However, with an estimate of object distance (z*), the approximate size of the object (r_o ) can then be decoded from the amplitude estimate (Eq. 3). The accuracy of this estimate will be constrained by the Cramer-Rao bound shown in Figure 3a. Thus, extensive cross-talk between ELL maps (either within ELL or in their projections to higher centers) may be required to identify the complete array of necessary object properties (Assad et al., 1999).

Combined strategies in population coding

The population coding literature has primarily dealt with how neuronal populations encode features to which its component neurons exhibit bell-shaped tuning curves. These studies often focus on how a single value of the feature in question can be extracted from the neuronal population response. There can be more information in the population response than that one value; for example, the entire probability distribution of a stimulus feature can be decoded from the population response (Zemel et al., 1998). There is recent evidence, in the case of visual motion perception, that such information is actually used to form a specific percept (Treue et al. 2000). This information is still related to the coarse-coded stimulus features. To our knowledge, extracting information from a combined population code in a functional context, has not been previously considered.

Cues for electrosensory depth perception

Our study of electrosensory depth perception has considered only static cues of object distance. In the context of visual processing, the problem is analogous to judging the depth of a stationary object using only monocular information. Electric images resulting from near objects are narrower and of greater peak amplitude than those of far objects, and thus can be considered as having less blur and higher contrast. Blur and contrast can have a significant influence on visual depth perception and are commonly used by artists in the pictorial depiction of depth (O'Shea et al., 1994; Mather, 1997). In normal visual processing, such cues are usually effective only in the absence of others such as those resulting from stereopsis and motion. Although there is no binocular analog in electrosensory processing, electric fish certainly have many motion cues available. Indeed, some species of electric fish exhibit a back-and-forth hovering motion that could be used to generate specific cues. Also, looming cues, such as those resulting from a changing electric image as an object approaches, could also be used for computing a parameter such as the time-to-collision, often discussed in the context of visual looming (Sun and Frost, 1998;Gabbiani et al., 1999; Rind and Simmons, 1999). As yet, there is little known about electrosensory motion processing and how electric fish might use such information for electrolocation.