Biol Cybern (2006) 94: 276–287 DOI 10.1007/s00422-005-0046-4

O R I G I NA L PA P E R

Alireza S. Mahani · Reza Khanbabaie · Harald Luksch Ralf Wessel

Sparse spatial sampling for the computation of motion in multiple stages

Received: 03 August 2005 / Accepted: 28 November 2005 / Published online: 10 January 2006 © Springer-Verlag 2006

Abstract The avian retino-tecto-rotundal pathway plays a central role in motion analysis and features complex connectivity. Yet, the relation between the pathway’s structural arrangement and motion computation has remained elusive. For an important type of tectal wide-field neuron, the stratum griseum centrale type I (SGC-I) neuron, we quantified its structure and found a spatially sparse but extensive sampling of the retinal projection. A computational investigation revealed that these structural properties enhance the neuron’s sensitivity to change, a behaviorally important stimulus attribute, while preserving information about the stimulus location in the SGC-I population activity. Furthermore, the SGC-I neurons project with an interdigitating topography to the nucleus rotundus, where the direction of motion is computed. We showed that, for accurate direction-of-motion estimation, the interdigitating projection of tectal wide-field neurons requires a two-stage rotundal algorithm, where the second rotundal stage estimates the direction of motion from the change in the relative stimulus position represented in the first stage. Keywords Anatomy · Computation · Motion · Population coding · Vision · Tectum

1 Introduction Neurons along the avian retino-tecto-rotundal pathway (Fig. 1a) represent increasingly abstract parameters of visual stimuli. While (at least certain classes of) the retinal ganglion cells (RGCs) simply report local image intensity A.S. Mahani · R. Khanbabaie · R. Wessel (B) Department of Physics, Washington University, St. Louis, MO 63130, USA Tel.: +1-314-9357976 Fax: +1-314-9356219 E-mail: [email protected] H. Luksch Institute of Biology II, RWTH, 52074 Aachen, Germany

or contrast, the tectal stratum griseum centrale type I (SGCI) neurons fire more vigorously when facing small moving stimuli (Frost 1993; Luksch et al. 2004), and neurons in the thalamic nucleus rotundus (Rt) are selective for the direction of motion of an object as well as looming patterns (Revzin 1981; Wang and Frost 1992; Chaves et al. 1993; Sun and Frost 1998; Laverghetta and Shimizu 1999). These differences in representations of visual stimuli are paralleled by qualitative structural differences of the neuronal elements in the three stages (Fig. 1b). A dense array of RGCs, each with a small dendritic field, projects to tectal layer 5 where their axon terminals form a topographic map and synapse onto dendritic endings of tectal SGC-I neurons (Karten et al. 1997; Luksch et al. 1998; Tömböl and Németh 1999; Hellmann and Güntürkün 2001). In contrast, tectal SGC-I neurons with somata in layer 13 and dendritic endings in layer 5 have large circular dendritic fields (Luksch et al. 1998). Finally, the point-to-point topography of the retinotectal projection is replaced by a more abstract topography in the tecto-rotundal projection (Benowitz and Karten 1976; Ngo et al. 1994; Karten et al. 1997; Deng and Rogers 1998; Hellmann and Güntürkün 2001) that appears to be interdigitating (Marin et al. 2003), i.e., a specific locus in the nucleus rotundus receives input from a sparse population of SGC neurons throughout the entire tectum. It has long been hypothesized that structure and function are correlated in neural systems (Stuart et al. 1999; Jan and Jan 2003; Poggio and Bizzi 2004; Chklovskii 2004). To explore this hypothesis in the retino-tecto-rotundal pathway, we first used anatomical techniques to quantify the structural features of SGC-I neurons. Based on these anatomical data, we then used physiologically-constrained (Luksch et al. 2004) computer simulations to examine the functional roles of the structural features for the population coding of visual stimuli in the retino-tecto-rotundal pathway. The structure of the paper is as follows. In Sect. 2, we present the anatomical data about the pathway. In Sect. 2.1 we estimate the number of SGC-I cells in the avian tectum. This estimate will be used later in computer simulations of the SGC-I population’s response to visual stimuli. In Sect. 2.2,

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Fig. 1 The avian retino-tecto-rotundal visual pathway. a Schematics of the avian brain with retinal ganglion cells (RGC), optic tectum (TO), and the thalamic nucleus rotundus (Rt). b Schematics of the connectivity in the avian retino-tecto-rotundal pathway. Retinal ganglion cell axons project with a topographic map to tectal layer 5. Tectal stratum griseum centrale type I (SGC-I) neurons form spatially distributed sets of dendritic endings in avian tectum layer 5, where they receive monosynaptic retinal inputs. SGC-I neurons project to the thalamic nucleus rotundus

we characterize the distribution of SGC-I dendritic endings, using two anatomical measures: number of dendritic endings per area of tectal surface, and distance between nearest dendritic endings in thin tectal slices. We then present a simple process for generating the dendritic endings of SGC-I cells that reproduces the above two experimental measures. This generative process will subsequently be used in our computer simulations. In Sect. 3, we use the anatomical data outlined in Sect. 2 to study how well the population of model SGC-I cells can discriminate stationary from moving stimuli. Specifically, we examine how this ability depends on the sparseness of each cell’s spatial distribution of dendritic endings. A similar analysis is done for the problem of estimating the location of a small moving object. Along with the anatomical information presented in this paper, we also incorporate the physiological properties of SGC-I cells into our model. These properties were previously revealed through in vitro studies of SGC-I cells in avian tectal slices (Luksch et al. 2004). In Sect. 4, we study the problem of estimating the direction of motion of a small moving object from the population response of model SGC-I neurons. We start with a one-stage algorithm, and then expand it to two stages. We study how the connectivity of the first and second stages affects the performance of the algorithm. The result is then compared with the anatomical facts about the tecto-rotundal projection. Section 5 summarizes the activity pattern generated across the population of SGC-I cells — according to our model — in response to small objects. It offers a wiring efficiency argument for our proposed two-stage algorithm for direction-of-motion estimation in the tecto-rotundal pathway, discusses the limitations of our approach, and finally offers concluding remarks. The histological, as well as computer simulation, procedures are detailed in Sect. 6. 2 Anatomical results 2.1 Number of SGC-I cells To estimate the number of SGC-I neurons, we took advantage of the fact that SGC subtypes innervate restricted rotundal subdivisions (Benowitz and Karten 1976; Karten et al. 1997; Luksch et al. 1998; Hellmann and Güntürkün 2001). Specifically, SGC-I neurons with dendritic endings in the retinorecipient layer 5, project to the anterior and centralis

rotundal divisions, whereas the SGC-II neurons with dendritic endings below the retinorecipient tectal layers, project to the posterior and triangularis rotundal divisions (Benowitz and Karten 1976; Luksch et al. 1998). Selective staining of the anterior and centralis rotundal divisions and subsequent counting of stained SGC-I neurons (Fig. 2a) revealed an estimated 85,000 SGC-I neurons per tectum.

2.2 SGC-I dendritic field Stratum griseum centrale type I neurons receive retinal synaptic inputs exclusively at their dendritic endings in layer 5. To quantify the spatial distribution of SGC-I dendritic endings (Fig. 2b) we filled SGC-I neurons in 450 µm-thick tectum slices and drew the two-dimensional projection of labeled neurons (Fig. 2c, d). In the following, we refer to the spatial distribution of dendritic endings as the “dendritic field”. From the center of the dendritic field in layer 5, as determined by the vertical projection of the soma, we binned the linear space in layer 5 in 100-µm intervals, counted the number of dendritic endings within each interval, and thus derived a measured density, ρ(r ), of dendritic endings as a function of the distance, r , in 100-µm increments, from the center of the dendritic field. SGC-I dendritic endings turned out to be cosine distributed about the center of the dendritic field: ρ(r ) = a + b cos(cr ), with fitted parameters a = 1.01# /100 × 450 µm2 , b = 1.04#/100 × 450 µm2 , and c = 0.0016 rad/µm (Fig. 2e). The total
number, N , of dendritic r endings per SGC-I cell is N = 0 max ρ(r )2πr dr ∼ = 160, where for simplicity we set a = b = 1#/100 × 450 µm2 and the upper integration limit is determined by [1+cos(crmax )] = 0. This number of dendritic endings per SGC-I neuron times the estimated number of SGC-I neurons per tectum yields a total of 14 million dendritic endings in layer 5 of one tectum. To further clarify the spatial distribution of dendritic endings of individual SGC-I neurons, we measured the distances between pairs of nearest neighbor dendritic endings from the same SGC-I neuron in 60 µm-thick sections irrespective of their absolute distance from the center of the dendritic field (Fig. 2b). On average, this nearest neighbor distance was < d >= 255 ± 276 µm (mean ± standard deviation; n = 373) with a range of 3–1, 440 µm. The distribution of

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Fig. 2 The anatomical organization of tectal SGC-I neurons. a Retrogradely labeled SGC-I somata (black) in tectal layer 13. Section thickness = 50 µm. Scale bar = 200 µm. b Three representative SGC-I dendritic endings in layer 5 in one 60 µm thick section. Scale bar = 100 µm. c, d Reconstruction of two representative SGC-I neurons in the tectum. The outlines of tectal layer 5 (top) and 13 (bottom) are shown in gray. Slice thickness = 450 µm. Scale bar = 200 µm. e Average density, ρ(r ), of “bottle brush” dendritic endings (BBE) per SGC-I neuron as a function of the distance from the center of the dendritic field in layer 5. The density was measured as the number of BBE in a 450-µm slice per 100 µm of distance from 51 SGC-I cells. Error bars are standard error. The line was obtained by fitting a cosine function of the form ρ(r ) = a + b cos (cr ) with parameters a = 1.01#/100 × 450 µm2 , b = 1.04#/100 × 450 µm2 , and c = 0.0016 rad/µm to the data points. f Histogram (thin lines) of measured nearest neighbor distances, d, between dendritic endings, binned in 5 µm bins. The thick black curve represents a fit of the exponential function, exp(−d/D), to the data points with fitting parameter D = 203 µm. g Simulated inhomogeneous Poisson cluster spatial distribution dendritic endings in a circular dendritic field with cosine density function as described in (e). The rectangle indicates the approximate outline of a typical tectal slice of 450-µm thickness and cut perpendicular to the surface of the tectum. h The 2D projection of dendritic endings within a slice of 450-µm thickness for three simulated dendritic fields (similar to (g)). Scale bar = 500 µm. i Histogram (thin lines) of measured distances, d, between dendritic endings of simulated dendritic fields (similar to (g)) and binned in 5 µm bins. The thick black curve represents a fit of the exponential function, exp(−d/D), to the data points with fitting parameter D = 230 µm

nearest neighbor distances was fit by an exponential function (Fig. 2f). Our next goal was to find a simple rule for generating the dendritic endings of an SGC-I cell, so that the above anatomical measures would be reproduced. The closest match was achieved with simulated dendritic fields (Fig. 2g) that were generated using an inhomogeneous “Poisson cluster” process (Diggle 2003) with the density function, ρ(r ), measured from anatomical work (see above). In the Poisson cluster process, locations were chosen with the density function, ρ(r )/2. (In a small area A, the probability of having a parent location is equal to ρ(r )/2×A) In the following, we refer to these locations as “parent locations” since these locations will be the origin of dendritic endings. Centered on each parent location, two dendritic endings were then generated out of a symmetric 2D Gaussian distribution with σ = 200 µm. The process is Poisson because the probability of having a parent location in a small area is independent of other small areas. It is inhomogeneous because this probability is not uniform across space for an SGC-I cell; rather, it depends on the distance, r , of

that small area from the center of the dendritic field of a cell. These simulated dendritic fields reproduced the measured average distance of 252 ± 263 µm (mean ± standard deviation; n = 374) with a range of 5–1,455 µm, the appearance of gaps and clusters in the 2D-projection of dendritic endings within 450 µm-thick slices (Fig. 2h) similar to reconstructions of real SGC-I dendritic endings, and the exponential distribution of nearest neighbor distances (Fig. 2i). While we used the cluster process, as described above, as a computational tool to reproduce the anatomical data, it is also biologically plausible as it can be loosely associated with the branching of the SGC-I dendrites before reaching the tectal layer 5 (Fig. 2c, d).

3 Change-sensitivity and location estimation with SGC-I neurons Since each dendritic ending has an RGC receptive field estimated as small as 0.5◦ radius (Karten et al. 1997), the total of

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Fig. 3 Simulated SGC-I population responses. a Schematic representation of a rightward moving stimulus and the retino-tecto connectivity for a population of SGC-I cells with overlapping dendritic fields. b A 5◦ long vertical stimulus bar moved with a speed of 10◦ /s from left to right across a 10◦ path that covers part of the receptive fields (circles) of three SGC-I cells as shown. The receptive fields had a diameter of D = 40◦ and their receptive field centers were separated by 10◦ . The corresponding cosine spatial distribution of SGC-I dendritic endings was simulated as described in the text. c Representative simulated spike trains for one trial for the moving stimulus and the three cells in (b), labeled A, B, and C. d Change detection as a function of receptive field diameter. Change detection is expressed as the signal-to-noise (SNR) of the difference between the population number of spikes during a temporal window in response to a static or a moving stimulus bar. The standard error of the SNR was derived by dividing the 1000 trials into 10 groups of 100 trials. e Average error in population encoding of stimulus location as a function of receptive field diameter

160 dendritic endings fills less than 1% of the area of an individual SGC-I receptive field of 40◦ -diameter. This anatomical organization raised the question why SGC-I neurons have such an extensive and ramified dendritic field corresponding to a spotty receptive field fine structure (Troje and Frost 1998; Letelier et al. 2002). To address this question, we examined to what extent the detection of spatiotemporal change in the visual scene, a behaviorally important stimulus, depends on the spatial distribution of dendritic endings. From the perspective of the SGC-I population response (Fig. 3a–c), one biologically plausible assumption, motivated by space and metabolic considerations, is to constrain the total number of SGC-I neurons (85,000) and the total number of dendritic endings (14 million) per tectum. As a result, the number of dendritic endings per cell is constrained to 160. We, therefore, investigated the functional effect of distributing the 160 dendritic endings per model cell over a variable dendritic field size. Because of the constant 160 dendritic endings per

model cell, the maximum spatial density of dendritic endings for one cell decreases with increasing receptive field diameter. An alternative assumption is evaluated at the end of this section. To examine the role of spatial density of dendritic endings in the detection of spatiotemporal stimulus change, we constructed a simplified model of the retinal inputs and the SGCI neuron (Fig. 3a). The model SGC-I neuron contained two important physiological elements: (i) a phasic retino-tectal synaptic signal transfer, described by the time-dependent response probability, P(t), where t is the RGC interspike interval, and (ii) the nonlinear interaction of dendritic spikes, which limits the SGC-I rate to the range between 0 and 33 Hz (Luksch et al. 2004). The stimulus classification task considered was to distinguish between a static and a moving stimulus bar (5 × 1◦ ), based on the SGC-I population activity. For the classification, the population number of spikes was measured during one

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simulation time step (10 ms) (i) in the steady-state response (100 ms after the onset of the static stimulus) to a static stimulus bar and (ii) in response to the same stimulus bar moving at the speed of 10◦ /s in the next simulation time step, i.e., when the stimulus moved by 0.1◦ . For each trial, the difference in the population number of spikes in response to the moving and the static stimulus was calculated. Because of the statistical nature of the spatial distribution of dendritic endings and the RGC and SGC-I spike generation, the repetition of trials yielded a distribution of differences in the population number of spikes. To quantify the stimulus classification performance, the signal-to-noise ratio (SNR), defined as the mean of the distribution of the difference of population number of spikes divided by the standard deviation of the distribution, was calculated. Starting from a receptive field diameter of 50◦ , the SNR decreased with decreasing receptive field diameter (Fig. 3d), i.e., with increasing maximum spatial density of dendritic endings for one cell. Similar curves with saturation happening at different sizes of dendritic fields are seen for other stimulus sizes and speeds, and stimulus brightness. The reduced stimulus classification performance with increasing maximum spatial density of dendritic endings for one cell has an intuitive biophysical explanation. The singleneuron response to a stationary stimulus is determined by the number of dendritic endings activated within the stimulus area and the time-dependent response probability, P(t), of the activated retino-tectal synapses. In contrast, the response to a moving stimulus receives an additional contribution from the fresh dendritic endings, P(t = ∞) = Pmax , that are activated by the leading moving stimulus edge. Both contributions increase with increasing maximum spatial density of dendritic endings for one cell and thus for a moving stimulus the SGC-I rate increase is always steeper, a fact that is independent of the parameter Pmax . However, the nonlinear interaction of dendritic spikes limits the SGC-I rate to 33 Hz. Therefore, the difference between the single-neuron SGCI response to a moving and a static object decreases with increasing maximum spatial density of dendritic endings for one cell. Our simulations show that this dependence persists in the SGC-I population response. In conclusion, numerous SGC-I neurons with extensive, overlapping, and spotty dendritic endings maximally utilize the limited dynamic range of their output firing rate to distinguish between static and moving objects. Does the population of SGC-I neurons with their spotty and large receptive fields contain information about the location of a moving stimulus? A classic problem in population coding with continuous tuning curves is the investigation of the effect of the tuning curve width and noise on the population encoding of a stimulus parameter (Pouget et al. 1999, 2003; Zhang and Sejnowski 1999; Eurich and Wilke 2000; Sanger 2003; Series et al. 2004). Here, the SGC-I receptive field consists of discrete units corresponding to its dendritic endings where they receive inputs from individual RGC units. We estimated the location, x(t), of a moving stimulus from the sampled SGC-I population activity. For each

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cell, the spikes within a sliding 30-ms window were counted. Since the stimulus was constrained to move in the x-direction, the average number of spikes, si (t), for all cells with the same x-coordinate, X i , (i.e., center of receptive field) was calculated, where i indexes the sampled x-coordinates. The sliding of time is x(t) ¯ =  location estimate as a function  (1/S(t)) i si (t)X i , where S(t) = i si (t) is a measure of the total activity of all sampled cells. The total RMS error was calculated from the location, x(t), of a moving stimulus and its estimate, x(t). ¯ Our analysis of this problem showed that the average error in population encoding of stimulus location increases with increasing receptive field diameter (Fig. 3e). Most importantly, however, for the receptive field diameters tested, the average error remains at less than 0.4◦ ; an error that is two orders of magnitude smaller than the typical neuron’s receptive field diameter of 40◦ . This observation is consistent with previous studies of population coding, which have shown that the location estimation error can be much smaller than the receptive field size (Dayan and Abbott 2001). In conclusion, the SGC-I sparse sampling enhances sensitivity to change while preserving stimulus location information in its population activity. Alternatively, to the above constraint of a constant 160 dendritic endings per cell, it is interesting to consider the constraint of holding the maximum spatial density of dendritic endings for one cell constant. Under this constraint, the total number of dendritic endings in the tectum increases with increasing receptive field diameter, which renders this constraint biologically less plausible. Enforcing this constraint leads to a change detection performance that shows a steady increase in SNR as a function of receptive field diameter, with no sign of saturation (data not shown). The location estimation error, on the other hand, becomes independent of the receptive field diameter (data not shown). Notably, the independence of estimation error from receptive field diameter under the assumption of constant maximum spatial density of dendritic endings for one cell (and thus constant maximum firing rate) is in agreement with previous theoretical investigations of population coding for 2-D problems (Zhang and Sejnowski 1999). 4 Tecto-rotundal projection Tectal SGC neurons project toward the thalamic nucleus rotundus (Karten et al. 1997; Luksch et al. 1998; Hellmann and Güntürkün 2001), where rotundal neurons compute the direction of motion and parameters of looming stimuli from the SGC population activity (Revzin 1981; Wang and Frost 1992; Sun and Frost 1998). The tecto-rotundal projection has long been thought to be diffuse without maintaining a retinotopic organization. However, small tracer injections into the nucleus rotundus labeled small fractions of SGC neurons throughout the entire optic tectum. As a result, each locus in the nucleus rotundus receives input from the entire visual field (Fig. 1b), with two neighboring loci receiving input from

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two populations of interdigitating SGC neurons (Marin et al. 2003). These anatomical findings raised the question to what extent the interdigitating topographic mapping between the optic tectum and the rotundus constrain computations, such as a direction-of-motion estimation algorithm. First, we used computer simulations to evaluate how using only a subset of an entire SGC-I population affects the accuracy of estimating the direction of motion of a stimulus. The subsampling process in the simulation corresponds to the anatomical observation that only a small fraction of SGC-I neurons throughout the tectum project to a locus in the nucleus rotundus (Fig. 1b). In our simulations, the distance between the closest neurons sampled in a subset is constant. We call this distance the “sampling distance”. We simulated SGC-I spike trains in response to moving stimuli for populations of model SGC-I neurons within the sampled subset, each with a spatial distribution of dendritic endings as described in Sect. 2.2. Dividing a total simulation time of 300 ms into six blocks of 50 ms each, we first estimated the average stimulus location for each block of time, using the population response during the corresponding 50 ms. The algorithm used for location estimation is the same as that described earlier (Sect. 3). We then applied a directionof-motion estimation algorithm (see below) to the resulting location estimates, and investigated the performance of this algorithm as a function of the sampling distance. The velocity of the stimulus was estimated by performing a standard linear regression on the six location estimates during our simulation time of 300 ms. The slope of the regressed line is the estimated velocity, and the sign of each velocity estimate indicates the direction of motion. Because of the statistical nature of the spatial distribution of dendritic endings and the process of spike generation for RGC and SGC-I neurons, repetition of trials yielded a distribution of velocity, as well as direction-of-motion, estimates. A natural measure of the performance in the direction-of-motion estimation is the signal-to-noise ratio (SNR1 ) defined as the mean of the distribution of velocity estimates divided by the standard deviation of the distribution. The SNR1 decreased steeply with increasing sampling distance (Fig. 4c). The above result is not surprising, because an increase in the sampling distance means a decrease in the number of SGC-I neurons in a subset. As fewer neurons are used in the estimation process, the error increases. Next, we formed several subsets of SGC-I cells, with two properties: first, they all have the same sampling distance. Second, each SGC-I cell is sampled by one and only one subset. We now postulate a rotundal two-stage scheme (Fig. 4a, b), in which the above-described subsets of SGC-I neurons project to intermediate rotundal units, which in turn converge on rotundal output neurons. Figure 4b illustrates a typical connectivity pattern between the SGC-I layer and the intermediate rotundal layer resulting from partitioning the tectal neurons according to the two properties described above. Physically, the intermediate rotundal units may be interneurons or dendritic branches. The entirety of the SGC-I population, as a result, projects onto a smaller rotundal population

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Fig. 4 Population decoding of motion in the retino-tecto-rotundal pathway. a Schematic representation of a rightward moving stimulus and the retino-tecto-rotundal connectivity for a population of SGC-I cells with overlapping dendritic fields and a two-stage rotundal analysis of the tectal population activity. b Schematic of the proposed projection of different subsets of tectal neurons (color-labeled) to intermediate rotundal units. All subsets have the same sampling distance, and each tectal neuron belongs to one and only one subset. c SNR of the directionof-motion estimation for one subset (open triangle), for the “estimate then sum” (open circle) and the “sum then estimate” (full circle) versions of the two-stage algorithm as a function of the sampling distance. The SGC-I receptive field diameter was 40◦ . The SNR values of the estimated velocities were extracted using 1,000 trials divided into 10 groups of 100 trials to estimate the standard error of the SNR

of intermediate units. The sampling distance and the spacing of the tectal neurons alone determine the number of intermediate rotundal units. In this scheme, to which we refer as “estimate then sum”, we applied the direction-of-motion estimation algorithm directly on one subset of the sampled SGC-I ensemble activity and√derived the total SNR including all subsets from S N R = k S N R1 , with S N R1 being the signal-to-noise ratio for each subset, and S N R being the total signal-to-noise ratio. Here k is the number of subsets, which increases with increasing sampling distance. The relationship

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is due to the fact that subsets have no SGC-I cells in common, and therefore, each subset estimate is statistically independent from others. Again, the total SNR decreased steeply with increasing sampling distance and vanished when the sampling distance equaled the receptive field diameter of 40◦ (Fig. 4c). This is, in spite of the fact that, for all sampling distances, all SGC-I cells are used in arriving at the final estimation. A qualitative argument allows for an intuitive understanding of this functional dependence. During the simulation time (300 ms) the stimulus travels a distance (3◦ for a speed of 10◦ /s) that is small compared to the diameter of the receptive field (40◦ ). Only sampled neurons with receptive fields within the stimulus path are activated. For a sampling distance equal to the receptive field diameter each subset of SGC-I neurons contains no more than one activated neuron. Motion sensing is impossible with only one activated neuron per subset: the symmetric shape of the underlying distribution of the dendritic endings for each neuron means that for any rightward motion, there is a corresponding leftward motion that would produce the same response in a single cell. The inability to distinguish opposite directions of motion means that SNR vanishes. Because of the mathematically well-behaved cosine spatial distribution of dendritic endings we expect a continuous decrease of the SNR for sampling distances from the minimum value of 0.5◦ to the receptive field diameter of 40◦ . The steep decline of the SNR with increasing sampling distance in the “estimate then sum” version appeared inconsistent with the anatomical evidence for a sparse tectorotundal projection (Marin et al. 2003), i.e., large sampling distance. We, therefore, investigated an alternative version of the direction-of-motion estimation algorithm to which we refer as “sum then estimate”. In this version, an intermediate rotundal unit simply sums its inputs and the motion estimation is deferred to the postsynaptic rotundal output neuron. Importantly, relative — but not absolute — position information can be preserved in this version if the sampling distance is sufficiently large and nearly constant. In other words, the population of intermediate rotundal units represents a compressed map of relative position. The motion-sensing algorithm, assumed to be implemented by rotundal neurons, is subsequently applied to this population of intermediate rotundal units. The output of an intermediate rotundal unit is the sum of all the SGC-I activity it is sampling. The population of intermediate rotundal units is treated in a cyclic way, using the vector method (Dayan and Abbott 2001) to estimate relative position. Linear regression is then applied to the relative position estimates as a function of time to estimate the stimulus velocity. The repetition of trials yielded a distribution of velocity estimates from which the SNR was calculated. Direction-of-motion estimates with this “sum then estimate” version of the algorithm revealed two interesting features (Fig. 4c). First, starting at half the receptive field diameter, the SNR increased with increasing sampling distance (i.e., sparse tecto-rotundal projection). Second, the SNR

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saturated when the sampling distance equaled the diameter (40◦ ) of the SGC-I receptive field. Interestingly, the SNR saturates at a sampling distance that is consistent in its order of magnitude with the anatomical evidence for a sparse tecto-rotundal projection (Marin et al. 2003). This decrease in the performance of the direction-ofmotion estimate with sampling distances decreasing from 40◦ has an intuitive qualitative explanation. If the sampling distance equals the SGC-I receptive field diameter, then a small moving object will activate only one of the tectal neurons projecting to each intermediate rotundal unit. In other words, at each point in time, one active intermediate rotundal unit corresponds to only one active tectal neuron. However, some hundred SGC-I neurons with overlapping receptive fields are activated by the small moving object and thus the same number of intermediate rotundal units are activated. The application of the motion-sensing algorithm to the population of intermediate rotundal units leads to a good estimation of the direction of motion. In contrast, for sampling distances smaller than the SGC-I receptive field diameter, at each point in time, one active intermediate rotundal unit corresponds to more than one active tectal neuron. Therefore, the relative position information is not represented in the population activity of the intermediate rotundal units. As a result, the direction of motion estimation worsens with decreasing sampling distance and the SNR finally vanishes at a sampling distance that is determined by the SGC-I receptive field diameter (Fig. 4c). Our simulations further showed that the “sum then estimate” version of the direction-of-motion estimation is robust with respect to small distortions of the tecto-rotundal topographic projection from a periodic pattern (data not shown). A qualitative argument accounts for this robustness. During the simulation time (0.3 s), the object moves 3◦ . In applying the linear regression to the position estimation data, the algorithm automatically emphasizes the contribution of the position estimation at the beginning and at the end of the motion. These estimates, in turn, have maximal contributions from cells whose density of dendritic endings have maximal slope at the location of the object. The location of such cells shifts by the same amount as the object moves. Thus, the motion estimation algorithm relies mainly on comparing the information from cells a few degrees apart from one another. As a result, the motion estimation algorithm is robust with respect to the projection disorder that reshuffles the order of projection for cells with distances below a few degrees. 5 Discussion It has long been hypothesized that the structure of neurons reflects their role in signal processing (Ramon and Cajal 1899); yet, one hundred years later, a quantitative theory of neuronal structure is still missing. Here, we have quantified the SGC-I dendritic structure, have shown that the measured sparse spatial distribution of SGC-I dendritic endings enhances stimulus classification while preserving location information, and have demonstrated that the subsequent

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interdigitating tectal projection to rotundal neurons allows the decoding of the direction of motion from the sparsely sampled population of SGC-I neurons. 5.1 Activity pattern generated by small objects Because of the sparse spatial sampling of retinal and tectal population activity, motion generates the following activation profile in the avian retino-tecto-rotundal motion pathway. A small moving bar sequentially activates thousands of small RGCs in its path; each RGC producing a spike train of brief duration (corresponding to its small receptive field size) containing one or just a few spikes. The sequential RGC spike trains in turn activate in rapid sequence (small temporal delay) hundreds of SGC-I neurons with overlapping receptive fields, each SGC-I producing a spike train of irregular pattern and long duration (corresponding to its large receptive field) overlapping significantly in time with the spike trains from the other SGC-I neurons. In other words, a cloud of SGC-I population activity spread over a tectal area of 4-mm diameter moves with the speed of the moving bar through the tectum. Because of the interdigitating topography, the population of SGC-I spikes activates a population of rotundal units, which together represent information about the direction of motion to be decoded by the rotundal neurons. When the bar stops moving, the cloud of tectal activity stops moving and reduces to a twinkle of a few sporadic spikes. 5.2 Why sparse, interdigitating tecto-rotundal projection? This study provided a potential explanation of how sparse tectal sampling and corresponding large dendritic fields can be reconciled with motion estimation and sparse interdigitating tecto-rotundal topography in a two-stage projection scheme. However, it could not address why the tecto-rotundal projection evolved towards sparse sampling, as opposed to a one-stage projection with small sampling distance, which apparently produces similar SNR values compared to the SNR values for large sampling distances for the two-stage “sum then estimate” version (Fig. 4c). Here, we discuss to what extent a wiring efficiency argument (Chklovskii 2004) may provide a plausible constraint to bias the evolution towards the sparse tecto-rotundal projection approach for motion estimation. In the one-stage scheme, Ni input tectal cells are connected to No motion-analyzing rotundal output neurons. Each of these rotundal neurons may be selective for a different direction of motion (and/or speed). Full connectivity between the input and output layers is required to maximize the signal-to-noise ratio, leading to a total of Ni × No connections. In the two-stage scheme, the same Ni tectal neurons project to Nu intermediate-layer rotundal units, which, in turn, are connected to the No motion-analyzing rotundal neurons. The interdigitating connectivity in the first stage implies that each tectal neuron projects to only one intermediate-layer unit. This results in Ni connections. The second stage involves full connectivity and thus leads to

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Nu × No connections. The ratio of the number of connections in the two-stage scheme to that of the one-stage scheme is (Nu × No + Ni )/(Ni × No ). For large values of No , such as Nu × No >> Ni , this ratio approaches Nu /Ni . The ratio can be much smaller than 1. For instance, if we assume that the size of the visual map on the surface of the tectum is 180◦ × 180◦ , then a sampling distance of 45◦ results in (180/45)2 = 16 SGC-I neurons projecting to each intermediate-layer unit, of which there must be Nu = Ni /16 to satisfy the condition that each tectal neuron projects to only one intermediate-layer unit. As a result, the ratio becomes Nu /Ni = 1/16 ∼ = 0.06 which is indeed much smaller than 1, thus providing a numerical estimate of the wiring efficiency.

5.3 Model limitations Since every model is based on a set of assumptions, several caveats exist in interpreting our modeling results. First, the electrophysiology experiments (Luksch et al. 2004) on which the present model of SGC-I cells is based, were done in brain slices and using single stimulus electrodes placed in the tectum (Luksch et al. 2001, 2004). The brain slice leaves out several visual centers that interact with the optic tectum, such as the nucleus isthmus and the pretectum. Under natural conditions, these centers can interact with the tectum and produce different response properties in SGC-I cells than what we see in brain slice experiments. On the other hand, in vivo experiments have documented the presence of cells in deep layers of the optic tectum that have large, spotty receptive fields, respond strongly to small moving objects, and show little directional selectivity (Troje and Frost 1998). The last two properties are consistent with our in vitro results, while the first one agrees with our anatomical observations. For the circuitry-level investigation of the tecto-rotundal projection, these properties may suffice. Therefore, the abovementioned limitations of our in vitro setting in characterizing SGC-I cells may not have a significant effect on the validity of our analysis of the tecto-rotundal pathway. On the other hand, our model cannot explain an observed property of neurons in the deep layers of the optic tectum, i.e., their selectivity for relative motion (Frost and Nakayama 1983). Second, we assumed that the retinal representation of visual stimuli consists of uncorrelated Poisson spike trains, representing local image intensity and with little processing happening in the retina. This is clearly a significant simplification; however, the phasic response of SGC-I cells to retinal stimulation means that the first spike is by far the most important feature of a retinal spike train arriving at the SGC-I synapse. In other words, the exact statistics of retinal response to visual stimulation may not be important to the subsequent SGC-I response. Third, because only a single stimulus electrode was used in the electrophysiology experiments reported in Luksch et al. (2004), we can assume that one, or only a few, dendritic endings of an SGC-I cell were excited during stimulation. As a result, our model simulations are limited to small-size objects

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that would activate few dendritic endings simultaneously. For large-size objects, any deviation from our assumption of independence of SGC-I synapses is expected to have significant impact and, indeed, the independence-based model fails to reproduce the observed SGC-I insensitivity to large objects and whole-field motion (Frost 1993). Finally, little is known about the physiological properties of the rotundal neurons which SGC-I cells project. While sensitivity to the direction-of-motion has been reported for some rotundal neurons (Revzin 1981), it is unclear whether these cells receive input from the tectal SGC-I cells. Furthermore, evidence for the existence of our proposed intermediate units in the rotundus remains to be found. Our theory about the two-stage estimation of the direction of motion in the tecto-rotundal pathway, therefore, is mainly of a predictive nature. Specifically, we have suggested a possible way to estimate the direction of motion of a small moving object from the population activity of SGC-I neurons that is consistent with the observed physiological and anatomical properties of SGC-I cells, as well as the known anatomical properties of the tecto-rotundal projection.

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whether the tectal projection out of the colliculus towards the thalamic homologue of the avian nucleus rotundus also follows an interdigitating topography. The demonstration in this study of the computational significance of the interdigitating topography (Fig. 4c) suggests an affirmative answer to the above question. More generally, it will be interesting to see whether a computational rationale for sparse connectivity similar to the one presented in this study for the retino-tecto-rotundal pathway will be found for other prominent computational pathways, such as the mammalian cerebral cortex, where only a small fraction of all possible connections between neurons physically exist, even within a local area (Braitenberg and Schuez 1991; Markram et al. 1997; Holmgren et al. 2003).

6 Methods 6.1 Histology

Eighteen White Leghorn chick hatchlings (Gallus gallus) of less than 5 days of age were used in this study. All procedures used in this study were approved by the local authorities and 5.4 Conclusion conform to the guidelines of the National Institutes of Health How universal is the sparse spatial distribution of single on the Care and Use of Laboratory Animals. cell dendritic endings and the sparse interdigitating topogInjections of the retrograde tracer, cholera toxin subunit raphy? Among birds, the morphological SGC cell types are B, into the anterior and centralis divisions of the ipsilateral not unique to the chicken tectum. Rather, comparable cell nucleus rotundus were performed following published protypes have been found in pigeon tectum (Hunt and Kün- cedures (Karten et al. 1997; Luksch et al. 1998). Animals zle 1976; Hardy et al. 1987; Karten et al. 1997; Hellmann were anesthetized, placed in a head holder on a sterotaxic and Güntürkün 2001) as well as in other avian species (barn frame, the skull was opened and the desired subdivision of owl, goose, duck, parrot; Luksch, unpublished observations). the rotundus was identified by extracellular recording techFurther, neurons with the characteristics of avian SGC cells niques or by using stereotaxic coordinates. Following the also exist in the superior colliculus of mammals (Major et al. localization of the desired structure, the recording electrode 2000). These “wide-field vertical cells” (Langer and Lund was removed and replaced with a micropipette filled with 1974) are situated in the Stratum griseum superficiale, SGS tracer. Approximately 100 nl of 1% (by weight) cholera toxin (Kanaseki and Sprague 1974). They have large dendritic fields, subunit B diluted in phosphate buffer (PB) was pressure inspecialized dendritic endings in upper layers, receive reti- jected. Following the application of tracer, the micropipette nal input, and project out of the colliculus towards the tha- was retracted, the wound was closed, and the animal was allamic homologue of the avian nucleus rotundus (Ogawa et lowed to recover. Animals survived for 2 days and were then al. 1985; Mooney et al. 1988; Lee and Hall 1995; Isa et al. anesthetized and perfused transcardially with 0.9% normal 1998; Major et al. 2000). On anatomical grounds, avian SGC saline followed by an ice-cold solution of 4% paraformalneurons and mammalian SGS neurons appear to be homolo- dehyde in PB. The brains were removed from the skull and gous (Major et al. 2000). Further, burst responses have been postfixed for at least 24 h. Then tissue was washed in PB, observed in vivo in wide-field neurons of primate superior immersed in 30% sucrose in PB for at least 24 h, and seccolliculus and the receptive field displayed a fine structure of tioned at 50 µm on a freezing, sliding microtome. Standard sparsely distributed spots, i.e., a reproducible discontinuous immunohistochemistry procedures were applied to localize response to a continuously moving spot of light (Humphrey the cholera toxin subunit B label (Luksch et al. 1998). Calcu1968). Sparse spatial distribution of single cell dendritic end- lations of SGC-I neuron number were based on soma counts ings thus seems common among wide-field tectal neurons of in several sections throughout the tectal extent, corrected for birds and mammals. The quantitative demonstration in this the systemic counting error (Guillery 2002), and multiplied study that the location of a stimulus is well represented in by the number of sections containing the SGC. the population activity of wide-field neurons (Fig. 3e) may Tectal slice preparation, SGC-I soma whole-cell recordbe particularly important for the interpretation of collicular ing, and SGC-I labeling were carried out as described previsensory-motor integration that underlies the animal’s ori- ously (Luksch et al. 1998, 2001). In brief, whole-cell patch entation behavior toward the location of a sensory stimu- recordings were obtained with glass micropipettes pulled lus (Stein and Meredith 1993). Finally, the question arises from borosilicate glass (1.5 mm OD, 0.86 mm ID, AM

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Systems, Carlsborg, WA, USA) on a horizontal puller (Sutter Instruments, San Rafael, CA or DMZ Universal Puller, Zeitz, Germany) and were filled with a solution containing 100 mM K-Gluconate, 40 mM KCl, 10 mM HEPES, 0,1 mM CaCl2 , 2 mM MgCl2 , 1.1 mM EGTA, 2 mM Mg-ATP, pH adjusted to 7.2 with KOH. Additionally, the solution contained 0.5% Biocytin (w/v) to label the recorded neurons. Individual SGCI cells were filled intracellularly in 450 µm-thick slices. After recording and labeling a maximum of two cells in one slice, slices were kept in oxygenated ACSF for an additional 30 min and subsequently fixed by immersion in 4% paraformaldehyde in PB for at least 4 h. Slices were then washed in phosphate buffer (PB, 0.1 M, pH 7.4) for at least 4 h, immersed in 30% sucrose in PB for at least 2 h and resectioned at 60 µm on a freezing microtome. The sections were collected in PB and the endogenous peroxidase blocked by a 15-min immersion in 0.6% hydrogen peroxide in methanol. The tissue was washed several times in PB, and then incubated in the avidinbiotin complex solution (ABC Elite kit, Vector Labs) and the reaction product visualized with a heavy-metal intensified DAB protocol. Following several washes in PB, the 60 µmthick sections were mounted on gelatin-coated slides, dried, dehydrated, and coverslipped. Sections were inspected for labeled neurons, and only data from cells that could unequivocally be classified according to published criteria (Luksch et al. 1998) were taken for further analysis. Thus classified, a total of 51 SGC-I neurons were analyzed. Cells were reconstructed at medium magnification (10×–20×) with a camera lucida on a Leica microscope and projected onto the 2D plane (Fig. 2c, d).

6.2 Simulations Computer simulations were performed to evaluate the response of model SGC-I cells to simple visual stimuli. In the bird, the azimuthal extent of the visual field of each eye is approximately 100◦ and projects onto a tectal circumference of approximately 10 mm. Assuming, for simplicity, a homogeneous spatial distribution of retinal ganglion cells in the retina and a homogeneous spatial distribution of RGC axon terminals on the tectal surface, we estimate that a typical SGC-I dendritic field of 4 mm diameter corresponds to a receptive field of 40◦ diameter. In our representation of visual stimuli the stimulus space was represented by an array of squares (or pixels) of 0.1◦ × 0.1◦ . A stimulus with uniform brightness was represented within this space. For most simulations, we considered a rectangular visual stimulus whose sides were 5◦ and 1◦ , perpendicular and parallel to the direction of motion, respectively. For such small objects, the assumption of independently responding model SGC-I cell synapses (see below) is valid and the resulting model reproduces a number of in vivo response properties (Luksch et al. 2004). This assumption, however, may not be valid for large visual stimuli, and the corresponding simulations do not reproduce the reduced in vivo response to large stimuli, such

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as whole-field motion (Frost 1993). Therefore, large visual stimuli were not used in this study. For the simulation of SGC-I population responses, we arranged SGC-I neurons in a regular lattice, with the centers of their receptive fields 0.5◦ apart in each direction, a value that was based on the SGC-I cell count and kept fixed throughout this study. For change-detection simulations, the stimulus size was 5 × 1◦ , and moved at 10◦ /s for 10 ms (after some initial period of rest). We assumed that the density of dendritic endings for each SGC-I cell is approximately constant over the area covered by the stimulus, and used the density corresponding to the location of the stimulus center at the beginning of the simulation to generate the dendritic endings of each SGC-I cell. Each dendritic ending received input from one RGC axon terminal. We assumed no spontaneous activity for the RGC, but when a stimulus covered its corresponding location, the RGC produced a Poisson spike train with a mean rate of 50 Hz. (We, therefore, ignored the size of the RGC receptive field.) Although it has been shown that retinal spike trains may not follow the Poisson statistics, for the purpose of our simulation this assumption suffices because mainly the first retinal spike decides the SGC-I response for moving stimuli. In a simulation time step of 10 ms, therefore, the probability of having an RGC spike is 50 × 0.01 = 0.5. Over a wide range of RGC firing rates, this parameter has little effect on simulation results (Luksch et al. 2004). An RGC spike arriving at an SGC-I dendritic ending caused a dendritic spike with a probability, P(t) = Pmax (1 − e−t/to ), where t is the time interval between two RGC spikes and Pmax = 0.87 and to = 2, 025 ms were the parameters determined from pulse train stimulation experiments (Luksch et al. 2004). For each dendritic spike, one SGC-I spike was generated if the previous SGC-I spike occurred more than 30 ms before (outside the interaction time). No SGC-I spike was generated if the previous SGC-I spike occurred within the interaction time of 30 ms. This limited the SGC rate to a range between 0 and 33 Hz. For moving stimuli used in location estimation and direction-of-motion estimation simulations, we adopted the following simplifying approximations. Since the stimulus (5 × 1◦ ) was moving at a speed of 10◦ /s, it spent approximately 100 ms within the receptive field of one RGC cell, which in our model equals the receptive field of one SGC dendritic ending. Therefore, each dendritic ending along the path of the stimulus received an RGC Poisson spike train of 50 Hz for 100 ms. The probability of this RGC spike train containing at least one spike is 1 − e−50×0.1 = 0.9933 ∼ = 1. Regardless of the exact arrival time of the first RGC spike, it has a chance of Pmax ∼ = 0.9 to cause a post-synaptic spike. The probability for additional RGC spikes to arrive within the 100 ms interval is small. Further, the probability for a dendritic ending to respond to one such additional RGC spike is smaller than P(t = 100 ms) ∼ = 0.04. Therefore, we adopted the approximation that there is a probability of approximately 0.9 to observe a post-synaptic spike for each dendritic ending along the path of the stimulus. Further, it can be shown that the uncertainty in the timing of this post-synaptic spike

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is equal to the average retinal interspike interval, which for a 50-Hz Poisson spike train is 20 ms. Since in our method (see below) for finding the direction of motion, we sampled the spike trains with a sliding 30-ms window, the 20 ms uncertainty was ignored. In summary, a dendritic spike was generated with a probability of 0.9 when the leading edge of the stimulus entered the dendritic ending’s receptive field. Within the valid parameter range for this approximation the SGC-I response is independent of the size of the object parallel to the motion and the RGC firing rate. During a simulation time step (10 ms), the leading edge advances 0.01 s × 10◦ /s = 0.1◦ and sweeps an area of A = 0.1◦ × 5◦ = 0.5◦ . For each dendritic ending in this area, there is a probability of Pmax ∼ = 0.9 to have a post-synaptic spike. If there are n dendritic endings, the probability of having at least one post-synaptic spike is 1 − (1 − Pmax )n . But the probability of having n dendritic endings is approximately ((ρ A)n /n!)e−ρ A , where ρ is the average local density of dendritic endings within the area A. Therefore, the probability of (at least) spike  to occur in this  one post-synaptic n −ρ A 1 − (1 − P n time-step equals +∞ max ) . n=1 ((ρ A) /n!)e In practice, only a few terms are needed. We used the first five terms in our simulations. As before, multiple dendritic spikes interacted in a mutually exclusive manner when they occurred within an interaction time of 30 ms. Because of the Poisson spatial distribution for the dendritic endings, the probability of finding a dendritic ending in a given small area is independent of the probability for its neighboring areas. This, in turn, means that the statistical uncertainties in the spike train of a given neuron for different times are uncorrelated and the responses of multiple neurons are uncorrelated. For all simulations of SGC-I population activity, the number of dendritic endings per cell was kept constant at an average of 160. Therefore, the maximum density parameter in ρ(r ) was adjusted for each receptive field diameter to maintain a constant number of dendritic endings per cell. Thus the maximum spatial density of dendritic endings for one cell decreases with increasing receptive field diameter (Fig. 3d,e). Since the total number of SGC-I dendritic endings (for all cells) is estimated to be about 14 million, and the total number of RGC axons may be smaller, there may be multiple dendritic endings corresponding to each RGC axon. Due to sparseness of individual SGC-I dendritic field, however, these dendritic endings are likely to belong to as many different SGC-I cells. Therefore, a given pair of SGC-I cells, even the adjacent ones, will have only a very small correlation in their responses to visual stimuli. Computer simulations incorporating the divergent input of RGC axons to SGC-I dendritic endings verified that the resulting correlation pattern in the SGC-I population response does not change the output of the decoding algorithms employed, and hence the arguments presented in this paper. Acknowledgements The authors thank K. Archie, D. Barbour, S. Brandt, A. Carlsson, J. Clark, B. Dellen, A. Eggebrecht, P. Stein, K. Thoroughman, and H. Wagner for critical reading of the manuscript. The work was supported by grants from DFG-Lu-622-2-2 to H.L. and the NIH (EY15678) to R.W..

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