Classification of Cat Ganglion Retinal Cells and Implications for Shape-Function Relationship

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2 Classification of Cat Ganglion Retinal Cells and Implications for Shape-Function Relationship Regina Célia Coelho 1,3, Cesare Valenti 2, Júlia Sawaki Tanaka 1,4 and Luciano da Fontoura Costa 1 1 Cybernetic Vision Research Group, IFSC University of São Paulo Caixa Postal 369, São Carlos, SP, Brazil, {reginac, julia, luciano}@if.sc.usp.br 2 Dipartimento di Matematica ed Applicazioni, University of Palermo Via Archirafi 34, 90123, Palermo, Italy cvalenti@math.unipa.it 3 State University of Maringá, Campus Universitário, Av. Colombo, 5790 Maringá, PR, Brazil reginac@din.uem.br 4 IQ UNESP São Paulo State University Caixa Postal 355, Araraquara, SP, Brazil, julia@iq.unesp.br Abstract This article presents a quantitative approach to ganglion cell classification by considering combinations of several geometrical features including fractal dimension, symmetry, diameter, eccentricity and convex hull. Special attention is given to moment and symmetrybased features. Several combinations of such features are fed to two clustering methods (Ward s hierarchical scheme and K-Means) and the respectively obtained classifications are compared. The results indicate the superiority of some features, also suggesting possible biological implications. 1. Introduction Although the morphological aspects of neurons and neural structures are potentially important, they have received relatively little attention from researchers in neuroscience. At the same time, an increasing amount of works have showed that the neural shape is directly related with the respective function [1 4], in the sense that the behavior exhibited by neural cells can be understood in terms of the biochemical processes in and outside the cells and their respective morphology. The morphology exhibited by neural cells is essential in the sense that it constrains the biochemical processes in the cells, defining the potential of interaction between each specific neuron and the rest of the neural system. Neural shape can be characterized quantitatively by obtaining morphometric measurements that allow us to have a reasonable representation of the physical constraints imparted by the cells [5]. Thus, the determination of a reasonable set of features expressing the neural behavior of different neuron classes can help to segregate cells into meaningful classes, and a cell classification scheme based on quantitative features can be used to this purpose [5,6]. These set of features should not just be related to the processes of the neurons, but also to the fields of influences affecting these processes, because these fields are essential in order to better understand the spatial cover and environment interaction exhibited by the cells. The primary cells used by morphological investigations are retinal ganglion cells, because their nearly planar dendritic arborization can be conveniently reduced to two dimensions. Most of such studies have been accomplished in cats because the central area of the retina of these animals is similar to the human fovea. An important evidence of the relationship between shape and function has been verified for the specific case of such ganglion cells. Wässle [4] verified that their receptive fields are related with the convex hull area of the dendritic X/01 $ IEEE 517

3 arborization. Related findings were verified by several researches [7 11]. They related three types of ganglion cells with respect to the function they play, (i.e. Y, X and W), and verified that these functional types corresponded to the morphological α, β and γ types of cells, respectively. The current article concentrates on α and β cells. The present work considers a group of cat retinal ganglion cells and investigates several morphological measures that can be particularly helpful to characterize each group of cells. The objective is to try to find the best combination among the presented features that classify these neurons inside each group with the smallest possible mistake. The considered features are related to the shape and influence fields of neural cells, including fractal dimension, symmetry, diameter, eccentricity and convex hull. An analysis of the correlations between the considered features is also included. 2. Neural shape characterization The features considered in the present approach are described below and listed in Table 1. These features are related to measures normally used to biologically classify these cells. Table 1. Morphological features used to characterize cells. Abrev. Morphological Feature AM Axial Moment MV Mean Value SD Standard Deviation EC Eccentricity BC Fractal Dimension by Box-Counting Method Mi Fractal Dimension by Minkowski Sausage Method CH Convex Hull Area Di Diameter IH 1-11 Influence Histograms Influence Areas by ratio IA Diameter and Convex Hull Area The diameter (Di) of each cell was obtained from the respective convex hull (CH). It was used to investigate the influence of the spatial extent of the cells in the classifications. Figure 1 illustrates the convex hull of a neural cell and the respective diameter d (the area has also been considered) Influence Area The influence area (IA) of a cell is relative to the type of interaction being investigated [5,12]. In the present case, we concentrate on the spatial coverage of dendritic arborizations, considering the area defined up to a maximum distance, Dist, from the cell, in the sense of the smallest distance to any of its points [12]. This area can be represented by Minkowski Sausage, where the influence area would be the area this sausage. Observe that the sausages give an indication of the influence area of the neurons in relation to the one specific distance Dist. It can be verified, by considering the successive area expansions, that the spatial coverage tends to saturate as the distances increase, as illustrated in Figure 2 with respect to Dist = 3, 6 and 9 respectively. Figure 1. The convex hull area obtained for a neural cell, and the respective diameter (d), which corresponds to the maximum distance between any pair of points belonging to the hull. (a) (c) (d) Figure 2. Minkowsky sausages of the artificial neuron presented in (a) with respect to Dist equal 3 (b), 6 (c) and 9 (d). Values chosen for the sake of better visualization purposes Influence Histograms Influence histograms (IH), originally introduced in Costa et al. [12], are capable of expressing, with respect to the total area affected by a specific trophic influence emanating from a neuron, the extension of regions under similar influence from the trophic field. The first important point to keep in mind is that this measure is relative to the specific profile chosen to represent the influence field under analysis. While it is possible to use profiles (or d (b) 518

4 point-spread-function) of influence varying with the position along the cell, we shall be restricted, for simplicity s sake, to homogenous profiles. The choice of a specific profile should take into account the nature of the influence factor under analysis. For instance, investigations about spatial distribution of electric fields would imply exponentially decaying profiles. The present work will assume Gaussian distributions, which are particularly relevant when analysing trophic effects such as those induced by diffusion of chemical factors around the cells. Let f(x,y,z) be a function defining the distribution of an influence factor (e.g. biochemical or ionic densities in and outside cells, or electric fields) induced by a point source, and let f(x,y,z) be a multivariate and symmetric Gaussian distribution. Assuming homogeneity, the total influence of the field around a neural cell can be easily calculated by convolving f(x,y,z) with the binary representation of the shape of neural cell, i.e. the function I(x,y,z). It should be observed that when the support of f(x,y,z) is broad, it becomes interesting to process such convolutions in terms of Hadamard products in the Fourier Domain. The resulting image, represented by T(x,y,z)=f(x,y,z) I(x,y,z), provides a faithful estimation of the spatial coverage by the trophic field defined around the cell. The process of deriving the representation T(x,y,z) is illustrated in figure Fractal Dimension The fractal dimension (FD) is a measure of complexity, which has been used in the classification of the cells since Boycott [7] showed in his work that the ganglion cells in the cat retina present different complexity. The methods that have been considered in this paper are the Box-Counting (BC) and the Minkowski Sausage (Mi) [12,13], with some alteration in order to enhance their accuracy. This alteration took into account that natural objects present limited fractality and the images are of discrete nature (spatially quantized) [13]. So, only a portion of the Log-Log curve, where the fractality is the largest, has been considered for interpolation and FD estimation. The fact that these curves present partial fractality generally is not considered in the literature, that is to say, the FD is calculated by processing linear regression about the whole Log-Log curve. In the present case, the FD by Box-Counting, considered medium values of box number to several inclinations of the grid. The grid was rotated in different angles and the medium number of boxes of all inclinations was considered [13] Axial Moments Symmetry operators have been included in vision systems to perform different visual tasks. They have been applied to represent and describe object-parts [14], to perform image segmentation [15] and to locate points of interest in a scene [16]. The definition of our symmetry operator starts from the computation of a set of first order axial moments (AM) around the center of gravity of each cell [17]. It s easy to prove that all AM s will have the same value in case of an isotropic density distribution, hence they can be used as a circular symmetry measure. For the purpose of this work we have set the number of axes to 32, according to previous experimental results. It s interesting to note that all AM s can be quickly computed in parallel as a sum of convolution filters. This approach can be also generalized for 3-D object recognition [18]. An example of the AM feature is shown in figure 4. (a) (c) (e) Figure 3. The binary representations of two neural cells exhibiting distinct complexity (a) and (b). The distributions of the trophic field around these cells, obtained by convolving a two-dimensional Gaussian with (a) and (b), respectively, are illustrated in (c) and (d). The respective influence histograms are shown in (e). (b) (d) 519

5 3. Neural Cell Classification (a) (b) Figure 4. The axial moment values of an á (a) and a â (b) cell (these neural cells figures, originally published by Dann et al., are here reproduced with permission). Three shape parameters are derived from the AM s: their mean value (MV), their standard deviation (SD), and their eccentricity (EC=AM min /AM max ), where AM min and AM max are the minimum and maximum values of the AM s. The MV and SD features, are normally distributed in first approximation. We assumed it true also in the case of the feature EC, as ratio of two normally distributed quantities. Table 2 reports the µ and σ parameters, while figure 5 shows the probability distribution. Table 2. Mean ( ì) and standard deviation (ó) of MV, SD, and EC. Feature µ α µ β σ α σ β MV SD EC Figure 5. Probability distribution for MV, SD, and EC. Since we are interested in investigating how the morphological cell classes are defined, unsupervised classification has been adopted. Two classification algorithms have been used: hierarchical grouping (Ward s method) and K-Means clustering. The first method tends to produce clusters that are easily distinguished from other clusters and which tend to be tightly packed. It reduces the number of clusters one at a time starting from one cluster per object and ending with one cluster comprising all the objects. At each cluster reduction, the method merges two or more objects. K-Means clustering involves an iterative scheme that operates over a fixed number of clusters, such that each class has a center which is the mean position of all the samples in that class and each sample is in the class whose center is closest to [20]. 4. Methodology In order to validate the potential of the adopted measures for the characterization and classification of neural cells, 50 images of cat retinal ganglion cells [7 9,19,21 24] of the types α and β, all presenting distance of the fovea smaller than 3 degrees, have been considered. A normalized version of the cells, in the sense that each cell was scaled to the same diameter, was also produced, but they were used only to generate axial moment, mean value, standard deviation and the eccentricity features since they are invariant to the scale. Influence histograms assuming a Gaussian profile with standard deviation of 4, and containing 11 uniformly distributed bins, were generated for each of images. Influence areas with Dist varying from 1 to 20 are also considered for each neuron. In order to investigate the influence of the size of the cells, the diameter of each cell was obtained from the respective convex hulls of the cells. To measure the complexity of each image, the fractal dimension was calculated by using both Box Counting and Minkowski Sausage methods. Note that in order to improve the clustering results, every feature was normalized through a statistical transformation in such a way as to present null average and unitary standard deviation. The calculation of central and axial moments has introduced new parameters with complementary information as compared to those obtained by other shape indicators. Moreover these measures satisfy the property of scaling, rotation, and translation invariance. This allowed us to improve the discrimination ratio of the two types of cells. The classifications were done using the Statistica software and involved combinations of the features using K-Means and Ward s methods. 520

6 5. Results and Discussion The linear interrelationship between neural features can be analyzed by the correlation matrix shown in Table 3. This table shows only three out of the 11 bins of influence histogram and five out of the 20 bins of influence area. It should be recalled that value 1 indicates a perfect correlation among two features, which only happens, in this case, for the same features. A negative value indicates an anti-correlation. Marked correlations (absolute values larger than 0.5) are considered strong. Classifications have been performed using all possible combinations of 2, 3, 4, 5 and 9 features, as well as by just using one and all features. Both classification methods have been applied for all such combinations. Although IA has presented a strong correlation with many other features, it resulted to be the worst feature. Alone or combined with other features, it always tends to undermine the classification. Therefore, it is excluded from the following comments. The features were grouped into the three major features groups: 1) symmetry features (AM, MV, SD, and EC); 2) complexity features (BC and Mi); and 3) size features (CH, Di, and IH), that considered the size of the neuron. Except for group (Di, IH), that presented satisfactory results (error 10% by both classification methods), any combination using only features of the same groups led to a poor result. The group (AM, Di) showed to be suitable for the classification (presented error equal to 6% both methods). This was the only group with two or three features that presented such a small error. Another interesting result was obtained by using all the features, except IA. The error presented by both methods was 6%. Most of the best classifications were obtained using groupings of 4 or 5 features together. Good results were obtained when we took at least 2 symmetry features, at least one complexity feature and IH. The IH combined with others also showed to be a good feature. Most of the worst results were obtained for groups of 1, 2 or 3 features. In such cases, the groups contained only symmetry features, or symmetries and complexity or still only a symmetry feature. The error in these cases was larger than 20% for both methods. The number of cells assigned to each class with respect to the best classification results is showed in table 4. In spite of the numbers, seemingly indicate success for all cells in specific cases, in all these case there were 2 wrongly classified cells, one in each class. It should also be observed that the relatively high obtained misclassification rates are very likely explained by the fact that the class assignment used as comparison standard was produced subjectively by diverse authors, using different criteria. 6. Conclusions A quantitative approach to neural cell classification, concentrating on cat ganglion cells, has been reported considering several geometrical features, and their combinations have been investigated with respect to two clustering algorithms. The obtained results indicated the superiority of some specific features, as well as a few incompatibilities with the original classifications originally assigned by human operators. The axial moment and diameter, in particular, resulted particularly effective. Such results indicate that the properties quantified by these geometrical features likely have special relevance to the behavior of the respective classes of cells. Moreover the features and the methodologies here introduced are general and can be easily extended to solve different kinds of object recognition problems. Acknowledgments Luciano da F. Costa is indebted to FAPESP and CNPq for financial support. References [1] Purves D., Body and Brain. A Trophic Theory of Neural Connections, Harvard University Press, United States of America, [2] Linden R., Dendritic Competition in the Developing Retina: Ganglion Cell Density Gradients and Laterally Displace Dendrites, Visual Neuroscience, Vol. 10, pp , [3] Kossel A., S. Löwel, and J. Bolz, Relationships between Dendritic Fields and Functional Architecture in Striate Cortex of Normal and Visually Deprived Cats, The Journal of Neuroscience, Vol. 15, pp , [4] Wässle H., Sampling of visual space by retina ganglion cell, In: Pettigrew J.D., Sanderson K.J., and Levick W.R., In Visual Neuroscience, Cambridge University Press, [5] Costa L.F. and T.J. Velte, Automatic Characterization and Classification of Ganglion Cells from the Salamander Retina, The Journal of Comparative Neurololy, Vol. 404, p , [6] Costa L.F. and Jr.R.M. Cesar., Shape Analysis and Classification: Theory and Practice, CRC Press, Brazil, [7] Boycott B.B. and H. 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7 [10] Stone J. and Y. Fukuda, Properties of Cat Retinal Ganglion Cells: A Comparison of W-Cells With X- and Y-Cells, Journal of Neurophisiology, Vol. 37, pp , [11] Stone J. and R. Clarke, Correlation Between Soma Size And Dendritic Morphology In Cat Retinal Cells: Evidence Of Further Variation In The γ-cell Class, The Journal of Comparative Neurology, Vol. 192, pp , [12] Costa L.F., Jr.R.M. Cesar, R.C. Coelho., and J.S. Tanaka, Analysis and Synthesis of Morphologically Realistic Neural Networks, In: Poznanski R., Modelling in the Neurosciences: From Ionic Channels to Neural Networks, Harwood Academic Publishers, India, pp , [13] Coelho R.C. and L.F. Costa, On the Application of the Bouligand-Minkowski Fractal Dimension for Shape Characterization, Applied Signal Processing, Vol. 3, pp , [14] Kelly M.F. and M.D. Levine, From Symmetry to Representation. Technical Report, TR-CIM-94-12, Center for Intelligent Machines. McGill University, Montreal, Canada, [15] Gauch J.M. and S.M. Pizer, The Intensity Axis Of Symmetry Application To Image Segmentation, IEEE Trans. PAMI, Vol. 15, No. 8, pp , [16] Reisfeld D., H. Wolfson., and Y. Yeshurun, Context Free Attentional Operators: the Generalized Symmetry Transform, Int. Journal of Computer Vision, Vol. 14, pp , [17] Di Gesù V. and C. Valenti, Symmetry Operators in Computer Vision, Vistas in Astronomy, Pergamon, Vol. 40, No. 4, pp , [18] Chella A., V. Di Gesù, I. Infantino, D. Intravaia and C. Valenti, Cooperating Strategy for Objects Recognition, in Lecture Notes in Computer Science book Shape, contour and grouping in computer vision, Springer Verlag, [19] Dann J.F., E.H. Buhl, and L. Peichl, Postnatal Dendritic Maturation of Alpha and Beta Ganglion Cells in Cat Retina, The Journal of Neuroscience, Vol. 8, No. 5, pp , [20] Andeberg M.R., Cluster Analysis for Applications, New York, Academic Press, [21] Kolb H., R. Nelson, and A. Mariani, Amacrine Cells, Bipolar Cells and Ganglion Cells of the Cat Retina: A Golgi Study, Vision Research, Vol. 21, pp , [22] Leventhal A.G. and J.D. Schall, Structural Basis of Orientation Sensitivity of Cat Retinal Ganglion Cells, The Journal of Computational Neurology, Vol. 220, pp , [23] Wässle H., L. Peichl and B.B. Boycott, Morphology and Topography of On- and Off-Alpha Cells in the Cat Retina, Proceedings of Royal Society London B, Vol. 212, pp , [24] Watanabe M., H. Sawai, and Y. Fukuda, Number, Distribution, and Morphology of Retinal Ganglion Cells with Axons Regenerated into Peripheral Nerve Graft in Adult Cats, Journal of Neuroscience, Vol. 13, No. 5, pp , Table 3. Correlation matrix among the features considered. N=50 (Casewise deletion of missing data). AM MV SD EC BC Mi CH Di IH 1 IH 6 IH 11 IA 1 IA 5 IA 10 IA 15 IA 20 AM MV SD EC BC Mi CH Di IH IH IH IA IA IA IA IA Table 4. Cells assigned to each class with respect to the best classification results. N=50 (23 á cells, 27 â cells). In all these cases, the error percentage has been 4%. Feature sets K-Means Ward α / β α / β (AM,MV,BC,Mi,IH) 83% / 100% 100% / 100% (SD,EC,BC,Mi,IH) 100% / 93% 100% / 100% (AM,SD,EC,BC,IH), (MV,EC,BC,Di,IH) 100% / 100% 91% / 100% (AM,BC,CH,Di), (AM,MV,BC,CH,Di), (AM,MV,EC,CH,Di) 100% / 93% 78% / 100% (AM,MV,EC,CH,Di) 100% / 93% 100% / 82% (AM,Di), (MV,SD,BC,Mi,IH) 100% / 89% 100% / 89% (AM,MV,CH,Di) 87% / 100% 100% / 89% (AM,MV,Mi,IH), (MV,SD,EC,Mi,IH) 96% / 100% 87% / 100% (AM,MV,SD,EC,BC,Mi,CH,Di,IH) 87% / 100% 87% / 100% 522