Morphological Hopfield Networks - Springer Link

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Brain and Mind 4: 91–105, 2003. ° C 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Morphological Hopfield Networks LUCIANO DA FONTOURA COSTA,1 MARCONI SOARES BARBOSA,1 VINCENT COUPEZ2 and DIETRICH STAUFFER3 1 Cybernetic Vision Research Group Instituto de F´ısica de S˜ ao Carlos University of São Paulo 13560-970 São Carlos, SP, Brazil, e-mails: [email protected] and [email protected] 2 Polytechnic Institute of Orleans, ESPEO. 12, rue de Blois. BP 6744 45067 Orl´ eans Cedex 2. France, e-mail: [email protected] 3 Institute for Theoretical Physics Cologne University D-50923 K¨ oln, Euroland, e-mail: [email protected] (Received: 20 November 2002; in final form: 5 February 2003) Abstract. This paper reports on the investigation of the effects of neuronal shape, at both individual cell and network level, on the behavior of neuronal systems. More specifically, two-dimensional biologically realistic neuronal networks are obtained that take explicity into account the position and morphology of neuronal cells, with the respective behavior for associative recall being simulated through a diluted version of Hopfield’s model. While a specific probability density function is used for the placement of the cell bodies, images of real neuronal cells (namely alpha and beta ganglion cells from the cat retina) are used to obtain biologically realistic models. Several morphological measures – including fractal dimension, the excluded volume, and integral geometry functionals– are estimated for the considered cells, and their values are correlated with the potential of the network for associative recall, which is quantified in terms of the overlap between distorted version of the trained patterns and their original version. Such an approach allows the quantitative and objective characterization of the relationship between neuronal shape and function, an important issue in neuroscience. The obtained results substantiate interesting relationships between the neural morphology and function as determined by the performance of the network. Key words: neuromorphometry, neuronal networks, Hopfield models.

1. Introduction A great deal of investigations in neuroscience, from both mathematical and biological points of view, have focused on models and simulations where relatively little attention is given to morphological features such as the shape of individual neuronal cells and their spatial distribution. One of the few exceptions are Kohonen’s networks (Kohonen, 2001), where the relative positions of the cells are in great part determined by the behavior of the respective systems. While such a state of affairs is a direct consequence of the fact that much of the neuronal behavior is determined by synaptic intensity, a growing set of evidence has clearly indicated that neuronal behavior is also largely affected by the geometry at mesoand microscopic spatial scales, corresponding respectively to the spatial arrangement of cells and their intrinsic morphological features. In physics, for example, many models such as Hopfield’s (Hopfield, 1982) assume fully connected neuronal systems, where only the synaptical weights are considered. The several models developed under biophysical frameworks, such as the transmission cable theory (Rall, 1977; Levitan and Kaczmarek, 2001), also tend to overlook morphological features of the neuronal structures, concentrating on the width and extent of neuronal processes.


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LUCIANO DA FONTOURA COSTA ET AL.

One of the most convincing evidences about the importance of neuronal shape over behavior, and vice versa, stems from the fact that the mammalian cortex is permeated by topographical mappings, where the adjacencies of the input stimuli are maintained (Kandef et al., 1991). At the same time, several different types of neuronal cell morphologies have been identified in the literature (Boycott and Wassle, 1974), which further suggests a close correlation between neuronal shape and function at the individual cell level. Indeed, retinal ganglion cells have been found to have their shape so as to uniformly tessellate the visual input space with a quite regular spatial overlap (Boycott and Wassle, 1974). Topographical maps are also inherently suitable for implementing lateral inhibition mechanisms which are known to permeate the primary regions of the mammalian cortical architecture (Kandel et al., 1991). At the same time, the spatial properties of cells determine and constrain, to a large extent, the functional operation of the cell, known as its receptive field. The importance of neuronal shape on function also underlines the moth Manduca sexta sensory neurons (Tolbert et al., 2003) as well as many other structures (Costa et al., 1999; 2001; Elsevier Science, 2000). While great attention was placed on the relationship between neuronal shape and function during the first decades of modern neuroscience, noticeably through the pioneering works of Santiago Ramon-y-Cajal (Cajal, 1899) the technological advances in intra- and extracellular recording of electrochemical activity dominated much of theoretical and applied research in neuroscience during the twentieth century. Concomitantly, the motivated mathematical models tended to concentrate attention on synaptic connections, avoiding more systematic and comprehensive consideration of neuronal morphology. This situation began to change recently, with several works targetting neuromorphometry (e.g. Ascoli, 2002; Hilgetag, 2002; Shefi, O. et al., 2002; Morita et al., 2001; Karbowski, J. 2001). One particularly important aspect implied by the explicit consideration of the neuronal morphology regards the dependence between stimuli and neuronal architecture. In other words, the performance of a specific neuronal network receiving topographically organized input patterns in implementing a particular task is in great part a consequence of the congruence between the specific spatial structure and distortions of the signal and the spatial organization of the network at both meso- and microscopic spatial scales. For instance, if the input patterns are characterized by a specific natural scale (e.g. discs with fixed radius R), the connections between the cells in networks intended to recognize individual discs should be placed with separating distances smaller than R/2 and have connections longer than R. It is therefore clearly noticed that the spatial autocorrelation characteristics of both the input patterns and the neuronal organization should be compatible (Forrest, 1989; Kürter, 1990; Ji et al., 1996). Recent works providing the motivation for the currently reported investigation include the percolation approach to neuronal connectivity described in (Costa and Manoel, 2002) as well as the assessment of Hopfield models where the connections were implemented according to the Barabási–Albert scheme (Barabási and Albert, 1999; Albert and Barabási, 2002; Dorogovtsev and Mendes, 2002; Stauffer et al., 2003). That work was subsequently extended and modified to consider nonrandomly diluted architectures (Rozenfeld et al., 2002) with asymmetric connections where regions dense in cells tended to become denser, and the synaptical connections obey specific statistical density functions (Costa and Stauffer,


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2003). Both such works considered one-dimensional strings of −1 and 1 and input patterns. This paper reports the investigation of the interplay between neuronal shape and function considering both meso- and microscopic geometrical features. While uniform spatial distribution is assumed for the placement of the cell bodies, real two-dimensional neuronal cells (cat retinal ganglion cells) are considered at a more microscopic scale. The geometrical features of the cells are characterized in terms of a comprehensive set of measures. The respective behavior is obtained through a diluted version of Hopfield’s model, with the input space topographically mapped onto the neuronal ensemble and the synaptic connections being obtained as consequence of the superposition between portions of the constituent cells. The performance of the network is experimentally quantified, through several simulations, in terms of the overlap between the original stimuli and respective distorted versions. Evidence corroborating the importance of neuronal shape over behavior are obtained by estimating correlations between the overlaps and several morphological measures of the cells, such as fractal dimension (Costa et al., 2001, 2002), excluded volume, and integral geometry functionals (Santalo, 1976; Mecke, 1998; Michelsen and Raedt, 2000, 2001). The paper starts by presenting the measures adopted for quantifying the geometrical properties of the neuronal structures and proceeds by describing the considered neuronal network models and the quantification of their performance in terms of overlap indices. The relationship between neuronal shape and behavior is then quantified and discussed in terms of correlations between the performance indexes and the geometrical measures.

2. Morphological Measurements 2.1. INTEGRAL GEOMETRY FUNCTIONALS

An efficient framework for morphological studies based on integral geometry known as MIA (Morphological Image Analysis) has been recently reported in the literature (Michelsen and Raedt, 2001). The procedure consists of evaluating a set of measures known as Minkowski functionals for an image (a pattern of pixels) for each value of a parameter. This parameter can be, for example, in a black-and-white image, the radius of a parallel set dilation, leading to a multiscale analysis, or the variable threshold in a grayscale image analysis (Mecke, 1998; Michelsen and Raedt, 2001). In the Euclidean plane, the Minkowski functionals of a pattern consist of three familiar additive quantities, namely the area, the perimeter, and the Euler connectivity number. The standard definition of the Euler number from algebriac topology is the number of connected components minus the number of holes, χ(K ) = n c − n h . As our patterns of pixels represent a neuronal shape which is connected, n c = 1, the Euler number is essentially the number of holes, 1 − n h . For simple convex shapes these functionals give full (Santalo, 1976) description of how any additive functional changes as a consequence of a parallel set dilation, for example, the change in area for a convex element K such as a ball or square is given as A(K r ) = A(K ) + U (K )r + πχ (K )r 2 ,

(1)


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where A(K r ) is the area after a dilation by a structuring element of radius r and A(K ), U (K ), and χ(K ) are respectively the initial area, perimeter, and the constant connectivity which is equal to 1 for a convex body in the plane. A simple route to evaluate these functionals for a complex pattern in a pixel-wise manner relies on fundamental concepts of integral geometry of disjoint convex elements (Michelsen and Raedt, 2000, 2001). By decomposing a pattern of pixels P into the disjoint elements that compose each pixel, namely open squares, edges, and vertex, the determination of these functionals became a straightforward counting of the multiplicity of the disjoint constituents according to the formula A(P) = n 2 ,

U (P) = −4n 2 + 2n 1 ,

χ(P) = n 2 − n 1 + n 0 ,

(2)

where n 2 means the number of open squares, n 1 the number of open edges, and n 0 the number of open vertices. We then monitor the evolution of these quantities along a wide range of scales obtained by a parallel set procedure. The potential of such a framework for neuroanatomy was investigated in (Barbosa et al., 2003) assessing the ability of such multiscale representation of the neuronal form to characterize the two main classes of retinal cats cells, α and β. These cells are classified into categories by eletrophysiological measures in the same classes as done by morphological characterization, an evidence of the relationship of form and function in this case (Boycott and Wassle, 1974). Figure 1 illustrates the procedure of a parallel set dilation at two dilation radii for examples of real neuronal cells, α and β. During the dilation we store the value of the functionals, as given by Eq. 2, at each scale, producing a signature profile such as that shown in Figure 2 for the multiscale variation of Euler number. In this way we obtain a type of morphological fingerprint for the intricacies of the neuronal shape. To account for the wide scale information of the behavior of such a measure (Connectivity) shown in Figure 2, we record the integral under the interpolated curve for each cell, whereas

Figure 1. Initial and intermediary image of a neuronal cell in a parallel set procedure. The initial image at r = 0 (left) and the intermediary at r = 6 (right) for both cells (reprinted with permission from B. B. Boycott and H. Wässle, J. Physiol. (1974) 240 page 402)


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Figure 2. The multiscale Euler characteristic for both neurons in Figure 1, showing the scale range at which the cells become dissimilar.

for the fine structure we consider the difference between the interpolated curve and the actual data, thus evaluating the standard deviation of this difference for each cell. We also refer to an index (i s ) which counts essentially the periods along which the curve rises (s), falls (d), or reaches a plateau ( p) s is = . (3) s+d + p This measure expresses the spatial distribution of holes in the shape. In cases where we have just a characteristic hole size or for graded distribution of size, the index produces lower values. On the other hand, if the shape presents several typical sizes of holes appearing with gaps, the index gives high values (Barbosa et al., 2003).

2.2. FRACTAL DIMENSION

From the information at each dilation radius of the first Minkowski functional the ordinary area, we can evaluate the multiscale fractal dimension of a neuronal cell. This functional, a nonadditive shape descriptor, has performed well in characterizing complexity in neuromophology (Costa et al., 2001, 2002). Moreover the correlation of this measure with the Euler characteristic is low, suggesting a complementarity of these descriptors (Barbosa et al., 2003). Figure 3 presents the multiscale fractal dimension for the two neurons of Figure 1. It shows the different behavior of the cells at different scales and the fact that complexity is a relative property closely attached to the scale at which it is considered. For this measure, we store the maximum fractal dimension, the mean fractality, and the standard deviation, respectively max, mean and SD f for each of the cells.


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Figure 3. The fractality for both neurons of Figure 1 expresses the behavior of their complexity at different scales.

2.3. EXCLUDED VOLUME

While the previous measures concentrated on the individual shape of the neuronal contour, in this section we calculate the excluded volume, a manifestly nonadditive characteristic that gathers information along the accessible neighborhood of the cell, to indicate the promptness of a cell to connect with copies of itself. This volume is calculated in a computationally intense procedure by taking the countour of the neuron image (the cell boundary) and translating it by a vector on the lattice considering a white background. While this newly formed image does not touch the initial (fixed) image, a black stamp of the original cell (including its interior) is added onto the white background. The loop continues for all permissible translation vectors. In the end we just evaluate the number of the white pixels in the final image (the background with all black stamps). Figure 4 shows an example of the final results of this procedure to determine the excluded volume for the two cells we considered the previous figures.

Figure 4. The excluded (and accessible) volume of the two neuronal cells considered in Figure 1. On the right α3 and on the left β3 .


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3. Nonrandomly Diluted Two-Dimensional Hopfield Networks The neurons i(i = 1, 2, . . . N ) in our models are two-dimensional images of real cells with spatial reference position (x0 , y0 ). The cells are either firing (Si = 1) or silent (Si = −1) and are updated according to Ã ! X Jik Sk (4) Si → sign k

P µ µ µ with synaptic strengths Jik = µ ξi ξk (Hebb rule) if i and k are connected, where ξi = ±1, µ = 1, 2, . . . P, and P random bit-strings called input patterns, and one of them, perturbed uniformly along its extent, is supposed to be recalled by this updating rule (i.e. P associative memory). The quality of recall is measured by the overlap 9 = i Si ξi1 if the first pattern is supposed to be recovered. As can be easily inferred from the above equations, the adoption of the first pattern can be done without any loss of generality. The input space is assumed to be the two-dimensional square matrix of size M, {I(x, y), x, y = 1, 2, . . . , M}, from which χ is derived. In the example of Figure 5 (top) this space is represented by the wavy background image. To this input space a total of N identical biologically realistic neuronal cells images are added (superposed) according to a uniform density function. Two-dimensional images of alpha and beta ganglion cells from the cat retina are considered in our simulations. Such a two-dimensional ensemble of neuronal cells is superposed onto the input space in such a way as to implement topographical mapping. Thus, the input to a specific neuron is derived from the portion of the background image that falls below the cell and determined by checking if the total number of active pixels in the area under that cell exceeds a specific threshold T . The connection between two neurons i and j is established whenever at least one of the points of the cell i touches (intersects) at least one point of the cell j, defining a symmetric connection matrix, Ji j = J ji . Figure 5 illustrates one configuration, whose respective connections are given by the matrix   1000000100 0 1 0 0 0 0 0 0 0 0   0 0 1 1 0 1 0 1 0 1   0 0 1 1 0 0 0 1 0 1     0 0 0 0 1 0 1 0 0 0 J = (5)  0 0 1 0 0 1 0 0 0 0   0 0 0 0 1 0 1 1 0 0   0 0 0 0 0 0 0 1 1 0   0 0 0 0 0 0 0 0 1 0 0011000001 To enhance statistical significance, a total of six configurations, i.e. spatial realizations of the model, have been simulated with 300 cells for each of the six available α and β cells, three of each type. Every configuration is tested through a Hopfield model with 1000


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Figure 5. An example of configuration for 10 neurons laid over a background pattern (top). Their connections, illustrated by placing a dot on the center of each cell (bottom), define the connection matrix J in Eq. 5.


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MORPHOLOGICAL HOPFIELD NETWORKS Table I. Values for All Considered Measures Cell α1 α2 α3 β1 β2 β3

Stddiff

Suminterp

is

std f

Max

Mean

Exv

Sum◦

◦ R1/2

3.28 7.28 8.18 1.16 1.38 4.62

−249.27 −657.80 −887.69 −2.30 −34.79 −159.61

0.69 0.92 0.90 0.50 0.53 0.61

0.05 0.09 0.10 0.18 0.10 0.08

1.64 1.59 1.60 1.63 1.58 1.66

1.58 1.50 1.49 1.22 1.40 1.56

146794.00 144881.00 140861.00 142463.00 141180.00 143129.00

9.11 9.52 9.36 9.56 9.61 9.40

8.75 9.11 9.03 9.48 9.26 9.15

random inputs patterns. Before each training, 10% of noise (sign reversals) is added to these input patterns. Then, the overlap is calculated as a function of the number of trained patterns P and two parameters are defined: the area under the overlap curve (9 versus P) and the half integral point for the same curve.

4. Results All obtained results considered six configurations for each of the six neurons in Figure 6, tried over 1000 input patterns. Table I shows the several measures estimated for each of the six neuronal cells. Figure 7 shows the cumulative number of connections for the six considered neurons. It is clear that the varied morphology of these cells led to quite different number of connections. As expected, given the characteristic spatial scale defined by the fixed size of the cells, did not lead to scale-free behavior, except along the intermediate range of Figure 8. The number of neurons exhibiting specific number of connections is given in Figure 9. The different potential of these cells for connectivity is again evident in this figure. The average overlap for each of the six cells is presented in Figure 10 in terms of the total number of trained patterns. Figure 7 shows the phase space, considering area and half life, obtained for our simulations. Table II shows the correlations for all measures estimated in this work.

Figure 6. All the 6 considered neuronal cells. Alpha cells on the top row and beta type below (reprinted with permission from B. B. Boycott and H. Wässle, J. Physiol. (1974) 240 page 402)


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Figure 7. The clusters obtained for the six different neural cells. The features were calculated considering six network spatial configurations for each cell. Although these two features are slightly correlated, the separation of the two main morphologically classes is evident.

5. Discussion As far as the number of connections, expressed in Figure 7, impacts the network performance, we have from Table I that, for most cells, the higher the number of connections the better the performance. One of the exceptions is cell α3 , the second in number of connections and fifth in performance as measured by sum◦ . This can be explained by the fact that the network performance depends not only on the number of connections but also on the way in which these are distributed along the connection matrix. While the measures for each cell are given in Table I, the correlation coefficients in Table II provide

Figure 8. The cumulative number of connections against the number of neurons (density). Note that when the density is high all the cells behave similarly with respect to connectivity.


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Figure 9. An histogram of the number of connections. For a given number of connections, say 30, there are many β3 cells that have this number of connections while there is (virtually) no neurons of the other cell types with this number of connections.

clear indication about the interdependence of the considered features. Here we concentrate on the correlations between morphological measures and the overlap sum◦ and half-life ◦ R1/2 . It is important to observe that although these two measures resulted highly correlated (see Table II), they express distinct properties of the network. More especifically, while sum indicates the overlap over the whole number of trained patterns, providing a global ◦ captures the performance for the initial portion of the overlap curve. measure, the R1/2 ◦ therefore indicates that most of the good performance is achieved A low value of R1/2 for a small number of patterns and that the network does not work well for many trained patterns.

Figure 10. The efficiency of the Hopfield network as measured by the overlap. This figure shows for each of the six cells the decreasing performance for recognizing an increasing number of trained patterns.


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LUCIANO DA FONTOURA COSTA ET AL. Table II. Correlation Coefficients for the Set of the Measures Extracted From Connectivity, Fractality, Excluded Volume (Ex. Vol.) and Hopfield Overlap: Standard Deviations (SD f , stddiff ), Integrals (Sum◦ , ◦ ), the Monotonicity index (i ), Mean Value (Mean) and Max Value (Max) of Suminterp ), Half Radius (R1/2 s fractality Connectivity

stddiff Suminterp is std f Max Mean Exv Sum◦ ◦ R1/2

stddiff

Suminterp

1 −0.953 0.947 −0.369 −0.191 0.548 0.013 −0.231 −0.428

1 −0.964 0.284 0.368 −0.418 0.046 0.227 0.431

Fractality is

SD f

Max

Overlap Mean

1 −0.403 1 −0.342 −0.102 1 0.520 −0.959 0.219 1 0.176 −0.515 0.389 0.476 −0.272 0.661 −0.520 −0.678 −0.519 0.889 −0.154 −0.864

Ex. Vol. (Exv ) sum◦

1 −0.624 −0.611

1 0.883

◦ R1/2

1

Note. Bold face for absolute values of correlation above 0.5.

The excluded volume presents intense negative correlation with both measures of overlap. This can be explained by the fact that higher excluded volume leads to less connections and, therefore, worse performance. ◦ and the The most surprising result obtained was as the negative correlation between R1/2 mean fractal value. The first important fact to be considered is that there are situations, such

Figure 11. This illustration shows a situation where the two cells do not touch while their inscribed circle does twice. Thus if the neurons were circular they would improve connectivity and justify the high anticorrelation with the fractal dimension of the cells.


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Figure 12. An example where the fractal dimension alone fails to characterize performance. Although the area is covered efficiently inside the convex hull, the latter is alongated. This combination of factors leads to a decrease in performance.

as described in Figure 11, where low fractal shapes (e.g. the circle) present a high potential for connections. As a matter of fact, experiments performed in this work indicate that the potential for connectivity is a consequence not only of high fractal structure, but also from the shape of the convex hull of the neuron. Figure 12 illustrates the cell α1 and its convex hull. Although the cell covers intensively the space inside the hull, the later presents an elongated shape that diminishes its potencial for horizontal connections, therefore reducing the performance for associative recall. Another high correlation is obtained for the index of monotonicity i s , related to the ◦ multiscale evaluation of the Euler number, and the overlap given by R1/2 . It is important to observe from Table I that β cells have definite lower values for i s , and are at the same time among the best performers. A possible explanation for such an effect is that lower values of i s indicate that almost closed regions along the neuronal border, which tend to close at specific low spatial scale, favors connectivity. The associative recall potential of the arrangements considering each of the six types of cells is illustrated in Figures 7 and 10. Figure 10 shows the overlap in terms of number of trained patterns. As expected (see Costa and Stauffer, 2003), the overlap tends to decrease with the number of trained patterns. This is a consequence of the fact that additional patterns tend to interfere with the previously stored patterns. Interestingly, the two considered classes of cells (i.e. α and β) tend to be different, with the beta cells presenting superior potential for associative recall (the curves for this type of cell tend to populated the higher portions of the graph in Figure 10). Such a result is a direct consequence of the morphology of the six cells, which defines connection matrices with different properties. The results in Figure 7 show the phase space obtained considering two global measures extracted from the overlap curves in Figure 10, namely the total area under the overlap curves and the half-life of those curves. It is clear from this figure that the alpha cells tended to occupy the lower left


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portion of the phase space, which also indicates some positive correlation between these two measures. Generally speaking, the alpha cells tend to present smaller area and half-life and, consequently, worse performance for associative recall. Overall, the current work has addressed the relationship between neuronal shape and function considering the Hopfield model and some real neuronal cells (cat ganglion cells). A set of meaningful geometrical measures (all multiscale except the excluded volume) has been extracted from the considered cells, and the performance of neuromorphic models considering each of the cells has been quantified in terms of the overlap between the recovered and original pattern. Correlations have been obtained between the morphological and performance measures, indicating some definite trends. More importantly, the effect of diverse neuronal cell morphology over the respective behavior have clearly been characterized in terms of the overlap and global derived measures. As far as we know, this is the first time Hopfield networks have been linked to the morphology of real neuronal cells, with consistent results. Further developments of our research should include the investigation of networks explicitly differentiating between dendrites and axons, greater emphasis on the relationship between stimuli/noise structure and neuronal architecture, and varying statistical models for synaptic connections.

Acknowledgments Luciano da F. Costa is grateful to the Human Frontiers Science Program, FAPESP (99/12765-2 and 96/05497-3) and CNPq (301422/92-3) for financial help. D. Stauffer thanks H. Sompolinsky for suggesting asymmetric networks and GIF for travel support. Marconi S. Barbosa is grateful to FAPESP (02/02504-01) for financial support and to the Grace and SCILAB teams for their excellent open software.

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